What is the difference between actual infinity and potential infinity?
I'm confused by the distinction actual vs potential infinity?
From wikipedia I get:
Potential infinity is a never ending process - adding 1 to successive numbers. Each addition yields a finite quantity but the process never ends.
Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity.
In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {.
Can someone explain this to me? Thanks.
From wikipedia I get:
Potential infinity is a never ending process - adding 1 to successive numbers. Each addition yields a finite quantity but the process never ends.
Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity.
In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {.
Can someone explain this to me? Thanks.
Comments (409)
The underlying dilemma is the result of different interpretations of the informal sign "..." used to denote partially elicited sets, and how these different interpretations lead to different conclusions concerning the very meaning of a set, including what sort of sets are admissible in mathematics.
Ordinarily, in statements such as {0,1,2,3,...}, the sign "..." is used to state that the "set" refers to a rule (as in this case, the rule of adding one and starting from zero), as opposed to an actually completed and existent body of entities. This is synonymous with potential infinity, that appeals to one's temporal intuitions regarding a process whose state is incremented over time.
In other cases such as "my shopping list is {Chicken,wine,orange juice,...} ", the dots might denote either
i) an abbreviation for a particular, finitely describable list that is already existent, but only partially described on paper
or
ii) An indication that a list is abstract and only partially specified, that the reader is invited to actualize for himself via substituting his own items, or rule of extension.
or
iii) a mystical sign, referring to "actual infinity" in a sense that is empirically meaningless, physically useless and logically a mere piece of syntax, but which nevertheless has psychological value in causing giddy vertigo-like sensations in true-believers when they contemplate the unfathomable.
Unfortunately, because "..." is informal notation with at least three completely distinct operational uses in addition to having private psychological uses, people continue to conflate all of these uses of the dots, causing widespread bewilderment, philosophical speculation and moral panic up to the present day.
Technically, I think that it should be #{1,2, 3,...} or card({1,2, 3,...}) or |{1,2, 3,...}| for actual infinity (cardinality symbols).
1,2, 3,... is just a sequence and not a set.
sequence: Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters.
In fact, there is another notation that is very close to set and sequence: a tuple or n-tuple: (1,2, 3).
tuple: In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.
Now, to confuse the hell out of everybody, the arguments of a function are deemed a tuple, but the typical notation for variadic functions (=with variable number of arguments) is f(a,b,c, ...), while the use of the ellipsis "..." is forbidden in tuples.
Furthermore, all these things are almost the same, with just a minute subtlety here and there ... ;-)
Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here?
Yes, I think it is. It is certainly what Wikipedia says..
The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential,[4] but not to be mistaken for physically existing. The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist physically in nature.
Of course, as it says, any representation as to whether physical infinite exists in the real, physical world is obviously out of scope in mathematics.
Actual infinity is not possible; there could always be more. Infinite is not an amount or an extent completed or capped, as extant, as that can't happen.
But you will never arrive at this number, because in order to do so you'd have to take an infinite number of steps. Which is why we call it an infinity.
If l say, l have taken 100 steps, l am using the word "steps" in a different sense, to mean a numerical quantity.
When l say infinite number of steps, the word "steps" specifies the nature of steps ( they do not end) in such a system, it does not refer to a numerical quantity.
Therefore we cannot such a definition for infinity.
It is very difficult to define infinity using any concept other than infinity itself. Hence it is often circular, self referential.
In maths, how would you interpret limits that do not exist to those that exist. For example, lim x-->o ( 1/x )does not exist but lim x--> 1 ( x-1/x^2-1) . The second one has removable singularity. Somehow we can assign useful value when using infinity but not always,so there are problems sometimes
Infinity doesn't exist as a number but a concept and l think even the concept has faults.
Without the axiom of infinity, a concept of actual infinity is not viable. That is obviously also the reason why the axiom was introduced. Otherwise, there would simply be no need for it.
The use of actual infinity is not even permitted in mathematics without axiomatizing it first. Therefore, it is perfectly ok for you to reject the axiom, but then you can also not make use of any of its consequents.
Since it is the sixth axiom in ZFC, you cannot make use of ZFC either. You will need to use an alternative set theory (of which there are actually many).
One possible problem could be that you cannot make use of any of the large number of theorems that rest on ZFC, unless they do not make use the axiom of infinity. However, it is a lot of work to weed through all of that, because it requires verifying their proofs. When rejecting ZFC, a lot of things that you would do in set-theoretical context will now be incompatible with mainstream set theory. Welcome to Hassle-land where everything that would have been simple, now becomes complicated!
Thanks all for your comments.
The axiom of infinity. So an infinite set is postulated to exist.
My statement about the infinite set of natural numbers was poorly worded.
What I should have said is that a largest natural number exists by the following argument:
Let's look at the sequence of natural numbers which I think is the "simplest" infinity we can talk about.
Natural numbers: 1, 2, 3,...
Observe how successive numbers "increase"
a) 1 to 2 the quantity has doubled (2 = 2 × 1)
b) 2 to 3 : (3 = 1.5 × 2)
c) 3 to 4 : (4 = 1.33... × 3)
d) 4 to 5 : (5 = 1.25 × 4)
.
.
.as you can see the factor (numbers in bold) is decreasing and approaching a limit which is 1. Look at larger numbers below:
e) 9999 to 10000 : (10000 = 1.001... × 9999)
f) 99999 to 100000 : (100000 = 1.0001... × 99999)
The pattern suggests that eventually there will be two very very large numbers A and B such that:
1. B = A + 1 (B is the next number we getting by adding 1 to A)
2. B = A × 1 = A (the pattern I showed you suggests that 1 is the limit of the factor by which a number increases in bold)
In other words there is a largest natural number.
The set of natural numbers does not have an upper bound, so it will always have a number that is smaller than another number. In other words, there is no largest number. If you disagree with the axiom that a set can have infinite elements, then it is possible to say that there is a certain largest number in a set but otherwise no.
The problem with axiom of infinity is that it fails to fall in one of the two categories. Intension and extension.
Some logician view that infinite extensions are meaningless as extensions must be complete in order to be well defined, so infinity cannot be defined by extensions. ( They reject Cantors proof too )
The problem with definition using intention is that they are circular.
There's a straightforward and unambiguous mathematical distinction.
The inductive axiom of the Peano axioms say that whenever n is a number, n + 1 is a number. So we have 0, and 1, and 2, and 3, ... [The fact that 0 is a number is another axiom so we can get the induction started]. However we never have a "completed" set of them. In any given application we have as many numbers as we need; but we never have all of them assembled together into a single set.
The axiom of infinity says that there is a set containing all of them.
So with the Peano axioms we may write: 0, 1, 2, 3, ...
With the axiom of infinity we may write: {0, 2, 3, ...}.
The brackets mean that there is a single completed object, the set of all natural numbers. That's Cantor's great leap. To work out the mathematical consequences of completed infinity.
I'm sure from a philosophical point of view there may be some quibbles. But this is how I think of it. the axiom of mathematical induction gives you potential infinity. The axiom of infinity gives you completed infinity.
Note that even with potential infinity, there are still infinitely many numbers. It's just that we can't corral them all into the barn. In fact in Peano arithmetic, the collection of all the natural numbers is a proper class. This is a good way to visualize what we mean when we say that a given collection is "too big" to be a set.
Quoting TheMadFool
Yes perfect sense. The axiom of infinity is a humongously ambitious claim for which there's currently no evidence in the real world. It's a bold statement. On the other hand without it, we can't get a decent theory of the real numbers off the ground. So the ultimate reason to adopt the axiom of infinity is pragmatic. It gives a much more powerful theory. Whether it's "true" in any meaningful sense is, frankly, doubtful.
There are two different domains of discussion: (1) mathematics itself and (2) philosophy of mathematics.
(1) MATHEMATICS ITSELF
(There are forms of mathematics other than classical set theoretic mathematics, but for brevity by 'mathematics' I mean ordinary classical set theoretic mathematics.)
In mathematics we don't ordinarily think in terms of a noun 'infinity' but instead of the adjective 'is infinite'. There is no object (abstract of otherwise) named by 'infinity' (setting aside in this context such things as points of infinity in the extended real system). Rather the adjective 'is infinite' holds for some sets and not for others.
Formal definitions of 'finite' and 'infinite':
A set S is finite if and only if S is in one-to-one correspondence with a natural number.
A set S is infinite if and only if S is not finite.
In mathematics itself there is not a formal set theoretic notion of 'potentially infinite'. Mathematics instead proceeds elegantly without undertaking the unnecessary complication of devising a formal definition of 'potentially infinite'.
(1) PHILOSOPHY OF MATHEMATICS
In the philosophy of mathematics, the distinction between actually infinite and potentially infinite might be described along these lines:
Actually Infinite. An actually infinite set is an object (presumably abstract) that has infinitely many members. The set of natural numbers is an actually infinite set.
Potentially Infinite. There are some philosophers or commenters on mathematics who do not accept that there are actually infinite sets. So for them there is no set whose members are all the natural numbers. Instead these commenters refer to processes that are always finite at any point in the execution of the process but that have no finite upper bound, so that for any step in the execution, there is always a next step available. For example, with counting of natural numbers, only finitely many natural numbers are counted at any given step, but there is always a next step allowed.
In constructive mathematics (not classical mathematics), perhaps, with research, one can find formal systems with a formal definition of 'potentially infinite'. But I would bet that any such system would be a lot more complicated and more difficult to work within than classical mathematics. This is the drawback of the notion of 'potentially infinite'. One can talk about it philosophically, but it takes a lot more work to devise a formal system in which 'potentially infinite' is given an exact, formal definition.
Quoting Marchesk
It is not necessary to adopt platonism to accept that there are infinite sets. One may regard infinite sets as abstract mathematically objects, while one does not claim that abstract mathematical objects exist independently of consciousness of them.
Quoting alcontali
No, that is not required. (1) There are infinite sets that are not cardinals. (2) Let w (read as 'omega') be the set of natural numbers. So w = {x | x is a natural number}. That is what is meant by {0 1 2 ...} (I drop unnecessary commas). And w itself is a cardinal, and for any cardinal x, we have card(x) = x anyway.
Here is an explication of 'set', 'tuple', 'sequence', 'multiset' in (set theoretic) mathematics:
Everything is a set, including tuples, sequences, and multisets.
A tuple is an iterated ordered pair.
Definitions:
{p q} = {x | x = p or x = q}
{p} = {p p}
= {{p} {p q}}
Then also, for example,
= <<
r> s> t>
"encodes the same information" as the sequence {<0 x> <1 y> <2 z>}.)
where f is a function whose domain is T and every member of the range of f is a cardinal. (So f "codes" how many "occurences" there are of the members of T in the multiset.)
S is a sequence if and only if S is a function whose domain is an ordinal.
S is a finite sequence if and only if the domain of S is a natural number. (There is an "isomorphism" between tuples and finite sequences. For example: The tuple
S is a denumerable sequence if and only if the domain of S is w.
S is a multiset if and only if S is of the form
Quoting Wittgenstein
There is no circularity in the set theoretic definition of 'is infinite'.
Quoting alcontali
Depends on what you mean by 'viable'. There is a set theoretic definition of 'is infinite' without the axiom of infinity. The axiom infinity implies that there exists a set that is infinite, but we don't need the axiom just to define 'is infinite'. I think you were pretty much saying that yourself, but I wish to add to it. Indeed, we agree that dropping the axiom of infinity makes an axiomatic treatment of mathematics extremely complicated.
Your claimed proof that there is no infinite set is not recognizable as a proper mathematical argument but instead proceeds by hand waving non sequitur.
Quoting Wittgenstein
No, there is no natural number smaller than the natural number 0. So maybe you meant that for any natural number n there is a natural number greater than n.
Quoting Wittgenstein
That is irrelevant because the axiom is not a definition and does not need to meet any standards of definitions. Also, we have to distinguish between two different notions of extensional/intensional. Aside from yours, there is the notion of extensionality that applies to set theory: Sets are extensional because they are determined solely by their members. That is, S = T if for all x, x is a member of S if and only if x is a member of T. And it doesn't matter whether a set is described by what you call 'intension' (such as {x | x has property P}) or, for finite sets, by finite listing in braces. For example, {x | x is a natural number less than 3} = {0 1 2}. Of course, infinite sets don't have listings such as {0 1 2}, but that does not vitiate that they exist.
Quoting Wittgenstein
Maybe there are such logicians, but even constructivists accept the proof of Cantor's Theorem and Cantor's proof of the uncountability of the reals.
And there is no circularity in the definitions of set theory. Mathematical definitions are not circular (that is, if a purported definition is circular then somewhere in the formulation of the purported definition there is a violation of the formulaic rules for mathematical definition).
I wouldn't state it that way. If we mean first order Peano arithmetic (PA), then there are not in PA definitions of 'set', 'class', and 'proper class'. Meanwhile, in set theory, the domain of the standard model of PA is a set.
[start quote of post]
This is a question from an elementary math book:
u = u + 1.
(i) Find the value of u
(ii) What is the difference between nothing and zero?
If you try and solve u = u + 1 you'll get 0 = 1 (subtracting a from both sides)
0 = 1 is a contradiction. So u is nothing. u is NOT zero. u is nothing.
Why?
Take the equation below:
e + 1 = 1
Solving the equation for e gives us e = 0. The same cannot be said of u = u + 1 our first problem.
So given the above equations ( u = u + 1 AND e + 1 = 1) we have the following:
1) u is NOTHING. u is NOT zero
2) e = zero
What's the difference between NOTHING and zero?
My "explanation" is in terms of solution sets.
The solution set for u = u + 1 is the empty set { } with no members
The solution set for e + 1 = 1 is {0} with ONE member viz. zero.
There's another mathematical entity that can be used on the equation u = u + 1 and that is INFINITY.
INFINITY + 1 = INFINITY
So we have:
a) u is NOTHING
b) u is INFINITY
Therefore,
NOTHING = INFINITY
Where did I make a mistake?
Thank you.
[end quote of post]
(1) What math book is that? What is the context? What does the variable 'u' range over? What specific operation does '+' stand for?
(2) There is no mathematical object named 'infinity' (unless it's something like a point of infinity in the extended real system - and in a context like that, the operations of addition and subtraction have special modified formulations that avoid such contradictions). And if infinite sets are meant, then operations such as cardinal addition or ordinal definitions are formulated so that they may not be confused with the operations of addition on natural numbers or on real numbers.
(3) Your "nothing = infinity" is just wordplay. As mentioned, there is not an object named 'infinity'. And 'nothing' also is not the name of a mathematical object. To say something like "nothing is not equal to itself" is not saying that there is an object named 'nothing' that has the property of not being equal to itself. Rather, it means that there is no object that has the property of not being equal to itself. So then putting an equal sign between 'nothing' and 'infinity' is nonsense.
Quoting GrandMinnow
What would be the operator in the isomorphism? Otherwise, without such operator, isn' it just a bijection? It is just a mapping between two sets, no?
Still, in my impression, the definition for morphism may be a bit ambiguous because in category theory they do not really seem to insist on the presence of such operator, while in abstract algebra they absolutely do.
By the way, I find abstract algebra much more accessible than certainly the deeper caves of category theory. It is only when they sufficiently overlap that it is clear to me ...
Thank you and can you be more specific. There are quite a number of steps I went through in my "proof".
The final steps in my proof:
1. b = a + 1 (just like 3 = 2 + 1)
2 b = a × 1
but as @Echarmion said I think a = b = infinity. Math breaks down at both ends of the whole number line: at zero and at infinity.
A very simple text. I'm quite certain there's very little ambiguity with the concepts I used.
Quoting GrandMinnow
Axiom of infinity?
Quoting GrandMinnow
How is it "wordplay"?
The solution set for a = a + 1 is the empty set { } with no members. In different words a is NOTHING, not even zero
NOTHING = NOTHING + 1
INFINITY = INFINITY + 1
Oh I see now. They may not be the same thing but just two different objects that behave in the same way. Thanks.
Sorry if this puts you off but what axioms would be necessary for the existence of natural numbers and the basic mathematical operations of + and ×? I begin from these
Oh, yes, agreed, it slipped my mind. It is indeed not just a set. Unlike in sets, the actual order of elements is also a piece of information that sequences and tuples carry. So, it is indeed more than a mapping between orderless sets.
What is the name and author of the text?
Quoting TheMadFool
The axiom of infinity is not a mathematical object named 'infinity'.
Moreover, the axiom of infinity itself is a finite mathematical object, as it is a finite string of symbols in a formal language.
Quoting TheMadFool
I explained explicitly in my post.
Quoting TheMadFool
No, you just made the same mistake I pointed out the first time.
Quoting TheMadFool
There are lots of different axiom systems for such things. For example, set theory. The existence of natural numbers is proven in set theory (even without the axiom of infinity). The existence of the set of natural numbers is proven in set theory (with the axiom of infinity). The operations of addition and multiplication are also definable and proven to exist in set theory.
Set theory proceeds from formal axioms, formal definitions, and formal rules of inference. Your argument has no apparent basis in those axioms, definitions, and rules. So I ask you what, exactly, are your axioms, definitions, and rules. Without specifying them, your argument, using such verbiage as "this pattern suggests" and then the non sequitur "in other words there is a largest natural number" is nonsensical handwaving, also known as 'waffle'.
Moreover, not just axioms, but ordinary mathematical common sense endows us with the understanding that there is no greatest natural number. Suppose there were a greatest natural number n. Then n+1 is greater than n. So n is not, after all, a greatest natural number.
I've got a question about infinite cardinalities. The following set of sets is an element of the powerset of real numbers:
{{1.2323,343.3333},{344.2,0,34343.444,6454.6444},{2323.11,834.33},{},{5 12.1,99.343433}}
So, any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers, i.e. beth2.
Now, regular languages cannot match wellformedness. So, things like matching embedded braces { } is out of the question. But I just concocted a set notation that does not use wellformedness:
[
1.2323 343.3333
344.2 0 34343.444 6454.6444
2323.11 834.33
5 12.1 99.343433
]
It is the same information as above, but in another notation. This notation is regular and can be successfully matched by a regular expression. I tried it at the test site https://regex101.com. The regex looks like this:
Since this expression successfully matches sets of sets of real numbers, can I say that it is the membership function of a set with cardinality beth2, i.e. 2^2^beth0 ?
If that makes sense, then it would be a witness to the claim that regular expressions can describe sets of which the cardinality exceeds that of the continuum, i.e. uncountable infinity.
No, that set is a member of the power set of the power set of the set of real numbers.
And I don't understand the rest of your post, starting with "any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers"
A regular expression defines a regular language. For example, a* accepts {nothing, a, aa, aaa, aaaa, ...} or (ab)* accepts { nothing, ab, abab, ababab, abababab, ... } So, that what it accepts, is a set of sequences. The question is now: What is the cardinality of the set that it accepts? If it only accepts sets of sets of real numbers, then the set that it accepts is the power set of real numbers, and that would mean that its cardinality is beth2, i.e. 2^2^beth0. Is there a flaw in what I say?
I'm very rusty in math, so let me see if I remember correctly:
Let F(x) = the set of finite sequences on x.
Let w = the set of natural numbers.
If x is countable, then card(F(x)) = w. (That's for sure.)
If x is uncountable, then card(F(x)) = x? That follows by the absorption property of infinite cardinal arithmetic? (I'm almost sure that's right. But I'd have to check that I'm not overlooking a flaw in my premises.)
And card(PPR) = beth2.
This is more a philosophical or psychological question than a purely mathematical one, but I don't have much problem understanding that the set of natural numbers and other infinite sets exist as abstract mathematical objects. And it's a pretty safe bet that mathematicians in general feel the same. Even as a child, I was introduced in a school textbook to the notions of the set of natural numbers, the set of rational numbers and the set of real numbers; and that did not seem problematic to me. And of course, as we know, whatever our feeling about the truth of real or abstract existence outside of the formal notion of existentially quantified theorems, no mathematical contradiction has been shown from ZFC.
Let's go back to the beginning.
a = a + 1
1) a is NOTHING
infinity = infinity + 1
2) a is infinity
From 1 and 2, NOTHING = infinity by the reason that if it quacks like a duck, walks like a duck, it must be a duck
I don't know if that's a mathematical reason to say NOTHING = infinity but this is very simple logic. Kindly show me where I'm wrong.
Also I mentioned in my later post that NOTHING and infinity may not be the same thing despite similar behavior. Basically, I think NOTHING and infinity are points where math breaks down. Are you trying to say that?
:brow: :chin:
Imagine what the regular expression accepts, are expressions like this:
{
{1.2323,343.3333}
,{344.2,0,34343.444,6454.6444}
,{2323.11,834.33}
,{}
,{5 12.1,99.343433}
}
So, it only accepts sets, the members of which must be sets themselves, and these member sets must only contain real numbers.
So, it only accepts elements from the power set of real numbers. (Correct?)
This regular expression seems to work like that (with members written in the alternative set notation):
So, I would like to confirm or infirm that :
There is a metaphysical problem with claiming that there are objects which do not exist independently of consciousness, and that is that these objects are imaginary. And imaginary objects are subjective, property of individual subjects. Such objects could be false, contradictory, or a logical impossibility. So if mathematical objects have this type of existence, each one needs to be justified, or else anyone could make up any imaginary thing, asserting that it exists as a mathematical object.
That is the problem with the infinite set. It is self-contradictory, an impossibility, which someone has asserted as an existing object, and other people have blindly accepted it because it is useful, without requesting justification. When we request justification, we see that "infinite set" is contradictory, as are most mathematical objects. And many which are not contradictory are irrational , like the principles of geometry.
Here's an example as to how mathematical objects are self-contradictory. Take the number 2. As an object, it is a simple unity. However, it is necessarily two distinct unities, as that's what 2 signifies, two distinct objects. So either 2 signifies two distinct things, or it signifies one unity, a mathematical object. It cannot signify both or else 2 would be 1, and that's contradictory. And so we cannot conceive of "mathematical objects" as objects, without loosing the meaning of the symbol. There is an inherent contradiction in asserting that a symbol like 2 signifies an object, because the unifying agent which makes 2 into one object has not been identified, therefore that two are one object has not been justified, and there really is no such object.
I have a left shoe, and I have a right shoe.
I have two shoes.
I have a pair of shoes.
I have a pair of shoes, which consists of a left shoe, a right shoe and a unifying principle
I have a pair of shoes, which consists of a left shoe, a right shoe and two-ness.
Unfortunately, I have two left feet. This is not as bad as having two left brains.
--------------------------------------------------------------
I have two left shoes.
I do not have a pair of shoes.
I have a left shoe, and I have a left shoe, and I have two-ness, but not pairity.
Nansen Cuts the Cat in Two
OK, you have two shoes. By what principle are these "two" things, one object? If they are a "pair" of shoes, this does not make them into an object, it is just another way of saying that they are two, a specialized form of "two". They are still not one object.
The concept of infinite set is abstract and very Platonic but not contradictory.
Furthermore, these different beth levels of infinite set sizes really kick in when you compare the set sizes of Platonic objects, which are obviously always language expressions in one way or another. Even natural languages are Platonic abstractions. For example, how many different sentences can you make in English? How does that compare to Chinese?
I tentatively guess the levels/beth numbers of infinity for English and Chinese will be the same. Still, in that case, how much is that beth level exactly, and how do natural languages compare to formal languages?
Another example. Context-free languages can in my impression express any arbitrary beth number because they can trivially handle wellformedness, while regular languages may only be able to reach level beth2 (not sure, though). That result would certainly be compatible with the Chomsky hierarchy of languages, in which regular languages are deemed substantially less powerful and expressive than context-free languages.
It is undoubtedly possible to prove a lot of these things, but I could not find any publication that deals with this matter.
Another result could be very interesting. If natural language has a particular fixed beth level, and since context-free languages can express any arbitrary beth level, there may exist context-free languages that are more powerful and more expressive than natural language. I don't know what that would mean, though.
It is contradictory, because a set is closed, complete, (as an object it is bounded, defined) whereas an infinity of anything is open, incomplete, unbounded and indefinite. I went through this in another thread recently, you weren't there.
All possible sentences you can say in English is a set. How is that closed or complete? Has anybody ever been able to list these out? I don't think so.
I already shot that fox. If I have two shoes, they may or may not be a pair. It is not the same thing. And if I cut a cat in two, there are two pieces of one cat. Also not the same thing. But what do i have to do to make them one, tie the laces together - glue the soles together - crush them into a singularity?
Most of us know well enough how to count shoes and how to count pairs of shoes and bits of cat though, and we know well enough not to count the number as another shoe. So I am happy to say that however many shoes there may be, they are all shoes and not numbers, and though there is a number of shoes, there are only shoes and no numbers, and this is perfectly clear and simple until someone points out a contradiction, at which point the explanations multiply and the clarity is lost.
So don't do it.
Yes, but you are not allowed to put physical objects inside a mathematical set.
You can only fill it up with language expressions.
So, if you cut a "cat" in two, you get {"c", "at"} or {"ca","t"}.
If you want to put a real, physical cat inside a set, you need to do that in physics or so, or in one of the other real-world subjects. Math is language-about-language only. (furthermore, real-world disciplines use a completely different way of thinking about these things ...)
So are you saying that I cannot cut a cat into two pieces? Your idealism solves the problem of contradiction but at the price of failing to account for how we actually talk about the world. I don't talk about "cat"s but about cats - and sometimes I count them. Don't tell me I'm not allowed to... damn mathematicians and philosophers stealing the language the rest of us use to talk about the world and making silly rules against talking sense.
I agree that mathematical infinity is "true" in the abstract realm of math. But that's like saying that the way the knight moves is "true" in chess. But it has no physical meaning in the world we live in. It's only true within a formal game played for entertainment. That's what I was trying to say.
I suppose so. But even in PA there are infinitely many numbers. There's just no completed set of them. So they are not formally a proper class, but we can use this idea as an analogy to what a proper class is. It's a collection that's too big to be a set. If I stated this as just a useful mental visualization or metaphor and not as a formal fact, we'd be in agreement I think.
Any language is a Platonic abstraction that is mismatched with the real, physical world; even languages that are specifically meant to describe it.
With mathematics, the situation is even worse, because it is not even meant to describe the real, physical world, but only other language expressions. Mathematics is language about language. So, the real, physical world is out of scope in mathematics.
So, but yes, agreed, talking about the real, physical world, requires another regulatory framework that tries to keep the language expressions correspondence-theory "true", hopefully without degenerating into a complete mismatch.
No it isn't. A set consists of objects, not possible objects.
Quoting unenlightened
That's quite simple, to make them one, you have to refer to them as one, and not as two. if you refer to them as two things, a pair, or any such thing, then you are talking about two distinct things. But if you refer to them as one, then you are talking about one thing. But you cannot talk about them as two things and one thing at the same time without contradicting yourself. So either "two" refers to one object, a mathematical object, in which case it does not mean two distinct things, or "two" refers to two distinct things. You can use the word either way, but you must be careful not to equivocate, so you can't use it both ways at the same time.
Quoting unenlightened
The problem is with the people who want to make mathematics do the impossible, not with the people who point out that what the mathematicians are doing when they're tying to make mathematics do the impossible, is contradictory.
You do not need to populate a set with literal values. You can simply attach a indicator/membership function that is capable of letting through literals that belong to the set and keeping out literals that do not.
S1 = { 2, 3, 4, 5 }
S2 = { x | x ? N, x >=2 and x<=5 }
S1 and S2 describe the same set. Therefore, S1 = S2.
In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset. In other contexts, such as computer science, this would more often be described as a boolean predicate function (to test set inclusion).
You can find another example for this principle in the definition for the term predicate:
[i]Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. So, for example, when P is a predicate on X, one might sometimes say P is a property of X. Similarly, the notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) is written as {x | P(x)}, and is the set of objects for which P is true.
For instance, {x | x is a natural number less than 4} is the set {1,2,3}.
If t is an element of the set {x | P(x)}, then the statement P(t) is true.[/i]
That two things are equal does not mean that they are the same. This is a known deficiency of mathematics, equality cannot replicate identity. Anyone who argues that 2+2 is the same as 4 needs to learn the law of identity, and respect the difference between equality and identity. The two sets are not "the same" in the sense of "same" used in philosophy, they are "the same" in the sense of "same used by mathematicians (i.e. equal). In philosophy, an actual thing is not the same as a possible thing, and we have a law of identity to prevent this type of sophistry, employed by mathematicians who creep into philosophical discourse without the appropriate discipline.
Well, the one expression S1 consists of literals while the other expression S2 is a comprehension formula. So, they are indeed not identical but extensional, according to ZFC's axiom of extentionality.
In the axiom of extensionality, the "=" symbol has axiomatically been assigned to express extensionality. Hence, the conclusion that S1=S2 is accordance with the ZFC axiom.
The sentence "they both describe the same set and therefore they are extensional" is therefore in accordance with the axiomatic foundation of ZFC set theory.
The use of the "=" symbol for expressing extensionality can be confusing. In any programming language that I have ever run into, the expression "S1 = S2" (or usually "S1 == S2" ) will only compare the two data structures' memory storage addresses. If they happen to be stored in different locations, even if they contain the same elements, by default, they will be considered different. Potentially comparing each element would cost computing power, and that would not be desirable as a default interpretation, when consuming resources matters (like in computing but unlike in mathematics).
Right, but as I noted, this theory is deficient. The two sets are not the same set by any rigorous standard of "same", though they are "the same set" according to the deficient standard of ZFC set theory; the law of identity being the appropriate standard for "same", not ZFC theory. ZFC theory allows that two distinct things are the same, contrary to the law of identity. Since they are not the same, your argument, which requires that they are the same, to reach its conclusion, fails. Therefore they are not even equal.
You conclude that the two sets are equal based on the assumption that the two distinct descriptions describe "the same set". They do not describe the same set, by a rigorous standard of "same", therefore you cannot even conclude that the two sets are equal.
I'm afraid I don't follow this at all. I know of no such instance.
The axiom of extensionality depends on the law of identity, which is a principle of logic and not of set theory. A thing is equal to itself. Then we define two sets to be equal if they have "the same" elements, meaning that we can pair off their respective elements using the law of identity.
It's true that in math we often identify sets as being the same type of "something" in a given context. For example the integers mod 4, the set {0, 1, 2, 3} with addition mod 4, are a very different set from the integer powers of the complex number i, {i, -1, -i, 1}. Yet the integers mod 4 (with the operation of addition mod 4) are isomorphic, as groups, to the powers of i under the operation of complex number multiplication. A group theorist will say these are "the same group" while being perfectly well aware that they're not the same set.
Another example is that if we define the natural numbers via the Peano axioms then use them to define the rationals and reals, then the natural number 1 and the real number 1 are entirely distinct sets. We regard them as the same via the "natural inclusion" of the naturals into the reals. If pressed on the details, any working mathematicians would explain this just as I have and there is never any confusion.
These issues are thoroughly discussed in a nice paper by Barry Mazur, "When is one thing equal to some other thing?"
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
I noticed that the example of 2 + 2 = 4 was given. 2 + 2 and 4 are exactly the same set. In the Peano axioms, 0 is a number and if n is a number, Sn, the successor of n, is a number. This gives an endless sequence 0, S0, SS0, SSS0, SSSS0, ... which we can use to name all the numbers. However this notation soon gets cumbersome so we adopt the definitions: S0 = 1, S2 = 2, S2 = 3, and so forth.
Now we can define addition inductively: n + 1 = Sn; and n + Sm = S(n + m). With this definition in hand, 2 + 2 = 4 is an easy theorem. If we then lift Peano to set theory via the axiom of infinity, 2 + 2 and 4 are easily seen to be the same set.
It's perfectly true that 2 + 2 and 4 are distinct strings of symbols. If we are studying strings, they're distinct. But in number theory they're the same number; and in set theory they're the same set.
ZFC was initiated by Cantor and Dedekind in the 1870s, followed by Zermelo's draft 1908 publication, followed by Fränckel's bug fixes in 1921. From day number one, there has been forceful criticism on its choice of axioms, and there still is, with lots of people proposing alternatives. Still, ZFC's dominance has only kept growing.
Whatever happens, it will be really hard to replace ZFC by any alternative, because so many theorems now rest on it. ZFC has an enormous "installed base":
Installed base (also install base, install[ed] user base or just user base) is a measure of the number of units of a product or service that are actually in use, especially software or an Internet or computing platform,[1] as opposed to market share, which only reflects sales over a particular period. Although the install base number is often created using the number of units that have been sold within a particular period, it is not necessarily restricted to just systems, as it can also be products in general. For products which are in use on some machines for many years, the installed base count will be higher than sales over a given period. Some people see it as a more reliable indicator of a platform's usage rate.
ZFC is actually also a gigantic legacy system, without necessarily being outdated, though:
In computing, a legacy system is an old method, technology, computer system, or application program, "of, relating to, or being a previous or outdated computer system,"[1] yet still in use. Often referencing a system as "legacy" means that it paved the way for the standards that would follow it. This can also imply that the system is out of date or in need of replacement.
Having broken some teeth in the past by criticizing legacy systems with a large installed base while advocating their replacement, I now instinctively refrain from doing that, because the argument will most likely fail again. One reason is the Lindy effect:
The Lindy effect is a theory that the future life expectancy of some non-perishable things like a technology or an idea is proportional to their current age, so that every additional period of survival implies a longer remaining life expectancy. Where the Lindy effect applies, mortality rate decreases with time.
Bourbaki is also known to have strongly promoted ZFC.
[i]Nicolas Bourbaki (French pronunciation: ?[nik?la bu?baki]) is the collective pseudonym of a group of (mainly French) mathematicians. Their aim is to reformulate mathematics on an extremely abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strives for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.
While there is no one person named Nicolas Bourbaki, the Bourbaki group, officially known as the "Association des collaborateurs de Nicolas Bourbaki" (Association of Collaborators of Nicolas Bourbaki), has an office at the École Normale Supérieure in Paris.[/i]
By the way, the "École Normale Supérieure" in Paris is also the school where Evariste Galois studied while working on his Galois Theory in his early twenties.
I agree with you re installed base or established mindshare. There are substantial developments in new foundations these days, category theory and homotopy type theory being the two leading candidates. In the end, foundations don't matter to the vast majority of working mathematicians. As an example if ZFC were found inconsistent tomorrow morning, it wouldn't affect group theorists or topologists or anyone else. They'd keep doing their work while the set theorists patched the problems. You'd be surprised how little attention working mathematicians pay to foundations.
It is almost literally what you will find mentioned in the page on the "Brouwer-Hilbert controversy":
In other words: the role of innate feelings and tendencies (intuition) and observational experience (empiricism) in the choice of axioms will be removed except in the global sense – the "construction" had better work when put to the test: "only the theoretical system as a whole ... can be confronted with experience".
So, what happens with "the theoretical system as a whole" ? Either it finds downstream users, or else it doesn't. In that case, what can we say about the downstream use of ZFC? Well, it is a legacy system with an enormous installed base that has been around for almost a century. Does it matter? Well, according to the formalist philosophy, that is all that matters. The status of individual axioms is simply irrelevant.
Concerning "no bearing on the topic at hand", you undoubtedly say that, because you are not aware of that famous discussion between Hilbert and Weyl in 1927, which was exactly about this. Could that have something to do with "weaker" Wiki skills? ;-)
The fact that the algebraic closures are not yoked together by a specified isomorphism is the source of some theoretical complications at times, while the fact that their automorphism groups are seen to be isomorphic via a cleanly specified isomorphism is the source of great theoretical clarity, and some profound number theory.
Yeah, the fact that automorphism groups are always isomorphic is a key ingredient in Galois theory. That is undoubtedly the "profound number theory" that he is referring to. ;-)
The Peano axiom approach calls up the full propositional apparatus of mathematics. But the details of the apparatus are kept in the shadows : you are required to “bring your own” propositional vocabulary if you wish to even begin to flesh out those axioms. The Peano category approach keeps all this in the dark: no mention whatsoever is made of propositional language.
Staying clear of the language of first-order logic may temporarily spare you from hitting the wall of Gödel's incompleteness theorems. It is the use of "?" and "?" that ransacks everything. Still, I do not see how he will manage to keep avoiding the use of language, and especially, existential quantifiers. Up till now, his Peano category approach has managed to somehow avoid their explicit use, but I am not sure that it also manages to avoid their implicit use.
The Peano axiom approach requires — at least explicitly — hardly any investment in some specific brand of set theory. At most one set is on the scene, the set of natural numbers itself. In contrast, the Peano category approach forces you to “bring your own set theory” to make sense of it.
Declaring the set theory in use to be some kind of free variable, or at least a template placeholder, is indeed interesting. However, how will he manage to not accidentally bring a particular set theory through the back door? All you need to do, is to accidentally rely on a theorem that rests on a particular set theory, in order to become beholden to it.
When we gauge the differences in various mathematical viewpoints, it is a good thing to contrast them not only by what equipment these viewpoints ultimately invoke to establish their stance, for ultimately they may very well require exactly the same things.
Ha ah! Exactly what I thought!
Representing one theory in another. If categories package entire mathematical theories, it is natural to imagine that we might find the shadow of one mathematical theory (as packaged by a category C) in another mathematical theory (as packaged by a category D). We might do this by establishing a “mapping”. We call such a “mapping” a functor from C to D.
Yes, I need a functor right now, between grammar classes of formal languages (which are always axiomatic theories). The PCRE regular language engine has custom extensions that allow it to express the grammar of context-free languages (EBNF) and match their sentences. So, now I want a functor between PCRE and EBNF; which are widely claimed to be isomorphic. So, does he know something about functors that would drastically simplify the job of producing such PCRE<-->EBNF functor? Otherwise, it may be a lot of work ... too much for me, in any case ...
Let X, X? be objects in a category C. Suppose we are given an isomorphism of their associated functors ?:FX?=FX?. Then there is a unique isomorphism of the objects themselves,
Interesting and intriguing. Unfortunately, he does not mention the proof, even though he says it is an easy proof.
An object X of a category C is determined (always only up to canonical isomorphism, the recurrent theme of this article!) by the network of relationships that the object X has with all the other objects in C.
And you usually do not even need the object's relationship to ALL other objects. A few is usually enough to know what the object must be.
A functor F: C???D from the category C to D is called an equivalence of categories if there is a functor going the other way, G:D???C such that G·F is isomorphic to the identity functor from C to C, and F·G is isomorphic to the identity functor from D to D.
If anything that you can express in ZFC, can be expressed in combinatory logic, and the other way around, then there would be a equivalence functor between both categories. Then, this equivalence functor is also an algorithm, i.e. some kind of function that accept set-theoretical expressions and translates them in combinatory-logic ones. Has anybody ever implemented anything like that?
Is 5 mod 691 to be thought of as a symbol,or a stand-infor any number that has remainder 5 when divided by 691,or should we take the tack that it(i.e.,“5 mod 691”)is the (equivalence) class of all integers that are congruent to 5 mod 691?
Well, in my own experience, "5 mod 691" is just "5" in a system that happens to have as system parameter maxint=690. We do not really care about the system parameter particularly much, because everything we do, stays inside that system anyway. In my opinion, the choice of 691 would only matter when you simultaneously deal with multiple systems that could each have different parameters. Still, I have never run into that practical situation. Another reason why it does not matter, is because this system parameter will usually be relatively large. However, it will not be too large either, because the fact that numbers wrap around that maximum boundary has a desired obfuscating effect. It nicely ransacks monotonicity. So, 232+541 = 82 (within mod 691). So, you can perfectly add up two large numbers and get a smaller one. That is not a very strongly obfuscating effect, but it still helps in cryptography.
This newer vocabulary has phrases like canonical isomorphism,“unique up to unique isomorphism”, functor, equivalence of category and has something to say about every part of mathematics, including the definition of the natural numbers.
I also believe that category theory, i.e. general abstract nonsense, is the true flagship of mathematics. Unfortunately, its theorems do not (yet) have direct applications (such as in cryptography), that I know of.
The categorical vocabulary itself, however, seems to be spreading like wildfire.
By the law of identity, two distinct sets cannot be the same. If they actually are the same, then they are necessarily one, the same set. It's contradictory to say that two things are the same. If it is the same, it is only one. Being equal and being the same are very different because "equal" refers to a multitude while "same" according to the law of identity refers to one, and only one. If ZFC states that two equal things are the same, it clearly violates the law of identity, which necessitates that the appearance of two is an illusion, there is really just one (Leibniz principle). And we cannot talk about one being equal, because there is nothing for it to be equal with.
This is a fundamental problem with the so-called "objects" of mathematics. Distinct things are allowed to be the same object, contrary to the law of identity, through the means of a principle of equivalence. Mathematicians will defend the existence of these objects, as objects, through reference to a difference which doesn't make a difference. But strict adherence to the law of identity allows no such contradictory nonsense. If there is a difference between what "2+2" refers to, and what "4" refers to, then these cannot be the same object, despite the assertion that this is a difference which doesn't make a difference.
I am concerned with the principles of the system, not any installed base, or legacy, these are irrelevant to the acceptability of the principles. I know that you believe axioms are completely arbitrary, making such things very relevant, so join the mob, if you like the "mob rules" philosophy.
There is the perennial requirement of consistency, but beyond that, anything flies, really. I am certainly very open-minded in mathematics. The more applications and users for such arbitrary concoction, the more likely that I will end up having a look at it. Arbitrary axioms are the hallmark of creativity! ;-)
Yes I do understand take this philosophical objection. If I say 2 + 2 = 4 then if they are the same object they're the same. I'm saying nothing! If I stand up and say, "I am me!" I am saying nothing. I've only affirmed the law of identity.
Yet ironically, we have a world full of people standing up and saying "I am me!" and this is of great psychological and sociological and political importance! Going off topic a little but noting the irony.
But yes I've seen this argument before. Math is meaningless because in the end it's all tautologies and saying that a thing is equal to itself.
I don't think that can really be true though. Math IS useful and meaningful because it takes human effort to determine whether two different representations of a thing are actually the same thing. Don't you agree? 2 + 2 = 4 is formally a tautology. But historically, it was a really big deal for humanity. Agree or no?
I seem to recall the old philosophical standby of the morning star and the evening star, which appear to be two different things but (upon astronomical research that took millennia) turn out to be the same thing, namely the planet Venus and not a star at all.
If you reduce everything to the law of identity, you are saying those millennia of observation and theory and hard work by humans means nothing. I don't accept that.
I'm glad I could turn you on to this paper.
I saw a little category theory back in grad school many moons go, then left math. When the Internet appeared in the 90's I was amazed to discover that category theory was being used in loop quantum gravity in theoretical physics. (This was of course from John Baez's This Week in Mathematical Physics on Usenet and later on his blog). Now category theory is in economics, biology (also via Baez) and of course computer science. Functional languages are the big thing now and they have monads, so Youtube is full of CS lectures on category theory. I find it all quite amazing to have seen this mind virus grow over the decades. Then again it takes a long time. Category theory was invented/discovered/whatever in the 1940's and was confined to math till probably the 90's. Another reason that "useless abstract math" should be valued. You never know what's going to eventually be important.
This by the way is my annoyance with the "indispensability argument" for mathematical existence. Category theory was not indispensable for computer science in 1940 but today it is. Therefore it has to be retconned as retro-indispensable; and then by analogy, everything is. Because in math, not everything is indispensable; but everything is potentially indispensable.
@Mephist and I had a monumental pages-long conversation about constructive math a while ago. You might find it interesting. All in all I learned far more about constructivism than I ever did before, and even read some technical papers on the subject. But in the end I never gained any affinity for the subject. I still regard constructive math as unnecessarily tying your hands behind your back. But of course neo-intuitionism is making a comeback via computer science and homotopy type theory. I call it Brouwer's revenge.
https://thephilosophyforum.com/discussion/5791/musings-on-infinity
Quoting alcontali
LOL. Of course I was only damning with faint praise, complementing your Wiki research but not your understanding. In that particular matter, at least. I'm perfectly well aware of the history of intuitionism and the Hilbert-Brouwer debate. FWIW I honestly did not make the connection from what you wrote, to the subject of intuitionism and constructive math. Could just be me.
Well, I really don't agree, and I think you misunderstand creativity. Art is not a product of arbitrariness, there are reasons for what the artist does, purpose, so arbitrariness is not the hallmark of creativity.
Quoting fishfry
Yes I agree, but the key is understanding the limitations of math. If some logician were to argue that '2+2' and '4' are both the very same thing, because they are equal, we'd have to correct that person, showing that these are symbols, and '2+2' clearly has a different meaning from '4'. But then we are at the position of needing to explain what it is that is signified by these symbols. If we take the Platonic route, we say that the numeral '2' represents the number 2, and we avoid the question of meaning altogether. There is now no problem of what '2' means, because '2' represents a mathematical object which is 2. But now we are totally lost, because we can have no idea what the number 2 is, it's just a mathematical object. We cannot turn to meaning, because then we might as well just go back to the symbol, the numeral '2', and ask what it means. At this point we cannot turn to Platonism and say it's a mathematical object, because we want to know what the symbol actually means, not just say that it stands for an object (the existence of which cannot be validated).
Quoting fishfry
OK, but this analogy assumes that there is a thing, an object which "morning star" refers to, and it turns out to be the same object that "evening star" refers to. We cannot do that here with mathematical objects, because as I described above, if the symbol refers to an object, then we deny that there is any real meaning. The symbol stands for a mathematical object, and this is the only meaning there is. The symbol stands for an object, and that's that. There is no meaning. This is pretty much the stance that alcontali takes, axiom s are arbitrary, so there is no such thing as the axiom's meaning, it's just an arbitrary thing.
Quoting fishfry
No, this is exactly the opposite to what I am arguing. When we adhere to the law of identity, then everything has an identity proper to itself, therefore its own meaning. This does not rob meaning from mathematics, it only establishes clear limits to the possibilities of mathematics, so that mathematicians will not believe themselves to have accomplished the impossible, like putting the infinite within a set.
That would almost amount to saying that an artist's design choices are exclusively rational, and could therefore even be expressed in formal language. My own take is that I do not believe that. I believe that artists make use of other mental faculties, that are not rationality, when making their design choices. I also do not believe that it is possible to express, even in natural language, the output of these other mental faculties.
When I look at the monad page, it says:
With a monad, a programmer can turn a complicated sequence of functions into a succinct pipeline that abstracts away auxiliary data management, control flow, or side-effects.
Now, when I think of succinct pipelines, I think of method calls that return the object itself, such as in:
var p=new Person("John Doe").age(24).height(160).weight(70);
One problem is that programmers who discuss pipelining rarely use categorical language in their discussions. So, I cannot determine if both things are related (monads versus typical pipelining practices).
I also do not really recognize the examples in the monad page. because they are mostly in Haskell, while Haskell is really a specific niche. Mainstream programming does not (yet) include Haskell.
Haskell is only number 49 in the Tiobe popularity index for 2019 with 0.174% usage. I only know one tool in widespread use that was built in Haskell: the pandoc markup format converter; which I certainly use, because in my experience, pandoc is much, much better at gracefully handle lousy input that is full of errors.
Still, I find Haskell code incomprehensible to read. It requires jumping over an enormous hurdle, without any visible payoff. It would be perfectly possible to write pandoc in a language that is more mainstream.
But then again, of the first 35 languages in the tiobe index, I only like 6:
like (c, javascript, php, sql, assembly, lua)
dislike (java, python, c++, c#, vb.net, objC, ruby, matlab, groovy, delphi, vb, go, swift, perl, R, D, sas, pl/sql, dart, abap, f#, logo, rust, scratch, t/sql, cobol, fortran, lisp)
One of the languages I like the best is Bash, ranked only number 48 (0.187%), which surprises me, because the bash shell comes pre-installed with approximately every linux system. So, I suspect that the Tiobe index drastically underestimates its use.
Same as between doing sex and an ability to do sex. That's how I'd explain this to my six-year-old.
I'ts okay. Philosophers use no categorical language; they say, "That's post-modernist regressivism" or something of the like, and they leave it at that. It's us, dilettante, who spell everything out for each other.
Yeah, but what is a philosopher?
There are obviously the grandees, and then there is everybody else who discusses the grandees. Still, you cannot become a grandee yourself merely by talking about the grandees. There are no grandees about the grandees, or grandees in grandee-hood.
In Nassim Taleb's lingo, success in philosophy has "extremistan" characteristics. It is certainly not a normal, Gaussian distribution. It is the same in music or movies. There are just a few grandees, and everybody else is pretty much a nobody. Now, young people had better be aware of the fact that having a degree in philosophy, or any subject for that matter, will not elevate them from the status of a merely nobody. If all you can bring to the table, is a degree, then you are not needed, and also irrelevant, just like millions of others.
Furthermore, none of the actors we see on a TV screen or singers on music channels have a degree in film or in music. It is the same in philosophy. Not one grandee became one by rehashing from memory what other grandees had said.
Of course, people with a degree in philosophy would want to see an intermediate level in the hierarchy, i.e. of "grandee in discussing grandees", since the whole point of getting that degree was to get some recognition. Well, no. There is no recognition whatsoever. A degree signals nothing meaningful. It does not elevate you above the populace. On the contrary, it is just a worthless piece of paper. Get over it.
On the other hand, slagging off degrees is only fun if you can bring something else to the table, but that was exactly the point I was trying to make.
Well, that would depend on how you define "arbitrary". Use of mental faculties in one's decisions negates randomness. If such decisions are arbitrary, then how do you understand "arbitrary"?
I see arbitrary in this context as "further unjustified".
For example, I do not trust what a literary critic says about why Charles Dickens wrote a particular passage. I strongly suspect that even Dickens himself did not really "know" it. It just came up to him, sourced from other mental faculties than mere rationality. So, from the outside it looks "arbitrary".
It is the same situation as with a sequence generated by a Mersenne Twister. From the outside, it looks random. From the inside, we can see that you will always get the same sequence depending on the seed that you use. Is the sequence random? For outsiders, yes. For insiders, no.
You can find an online demo for a Mersenne Twister here.
In that sense, if a true random number generator does not exist -- which is an unsettled question -- then randomness is always a subjective perception. Mutatis mutandis, if we have no clue as to how these other mental faculties work, then their output will appear as arbitrary to us.
It's not arbitrary then, it just looks arbitrary, in appearance, but it really is not. That it is arbitrary is an illusion. Would you see mathematical axioms in the same way? They look arbitrary, but they really are not. What is required to get beyond the illusion of arbitrariness is to get inside of the head of the artist. This does not mean to literally get inside, but to learn how to think in the same way as the artist. Then you will no longer be an outsider who sees mathematical axioms as arbitrary.
Agreed, but that is exactly what is not possible. These other, unknown mental faculties are out of reach of any ability to inquire them rationally. Furthermore, their assumed input could still truly be random, because there is no method available to distinguish between the output of unknown mental faculties and sheer randomness.
Yes, I see mathematical axioms in the same way. They look arbitrary but they probably aren't.
There is something uncannily recognizable to Plato's theory of forms, but the purported link between the forms and the real, physical world is unfortunately out of scope for the instrument of rationality. Still, the uncanny sensation of recognition suggests that this link is not necessarily, completely out of scope for other, unknown mental faculties.
I think we've been through this all before, you and I. I don't think that just because there is no method available, the input is out of reach. Methods come into existence, and evolve, so things which are reached by existing methods were at one time out of reach. Therefore it's reasonable to believe that a method could be developed to reach the things which presently cannot be reached. So if someone reaches a conclusion through a mental process which you consider to be by your standards, not rational, this does not mean that it is impossible to reach that conclusion through a rational process. It is possible that the required rational process could be developed.
Quoting alcontali
Yes, my point is that it requires effort to distinguish the good from the bad, but these "unknown mental faculties" may be brought into the realm of the known.
You made the statement that ZFC allows two different things to be equal. I said I know of no such example and you have not backed up your claim or put it in any context that I can understand. You must be thinking of something, I'm just curious to know what.
2 + 2 and 4 represent the exact same mathematical set. '2+ 2" and '4' are distinct strings of symbols. I don't know any mathematicians confused about this. And, as you agree, the discovery that these two strings of symbols represent the same set, is a nontrivial accomplishment of humanity and is meaningful.
I really don't understand your remark that ZFC allows distinct things to be regarded as the same. Unless you mean colloquially, as in the integer 1 and the real number 1 being identified via a natural injection.
Here is your quote.
Quoting Metaphysician Undercover
I categorically deny that claim. Please put it in context for me. As stated it's flat out false as far as I know. Of course one thing may have multiple representations; and it may have taken years, decades, or centuries to discover that fact.
I was in discussion with alcontali, referring to what was said by alcontali:
Quoting alcontali
Quoting fishfry
The point I made is that 2+2 is not the same as 4. So if set theory treats them as the same, it is in violation of the law of identity.
Quoting fishfry
'2+2' is clearly different from '4'. Each of those two expressions are composed of different symbols, having different meaning. Despite the fact that they are said to be equal, in no way are they the same. If ZFC allows that they are the same, as you say above, then ZFC allows two distinct things to be regarded as the same.
Since they are not the same according to the law of identity, by what principle of identity does ZFC claim that '2+2' is the same as '4'?
Of course 2 + 2 is the same thing as 4. I cannot imagine the contrary nor what you might mean by that claim.
But more importantly, they are the same set in ZFC. So it's not an example of your claim that ZFC allows two distinct things to be regarded as the same.
But you hold that 2 + 2 and 4 are not the same? How so? Without quotes around them they are not strings of symbols, they are the abstract concept they represent. And they represent the same abstract concept, namely the number 4. You deny this? I do confess to bafflement.
ps -- I didn't read the S1 and S2 parts of the thread, if it might help I'll go back and review them.
Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same?
Quoting fishfry
Yes, that's exactly the point. ZFC says that they are "the same" set, when they are clearly not the same by any intelligent reading of the law of identity. Therefore ZFC must employ some other principle of identity in order to say that they are the same. Can you state ZFC's law of identity?
Quoting fishfry
This is absolutely false. The symbol 2 has a meaning, the symbol 4 has a meaning, and the symbol + has a meaning. Clearly 2+2 is not the same concept as 4. Otherwise there would be no point in writing the exact same concept in two different ways, and the symbol =, which is commonly used to express the relationship between these two different concepts, would be meaningless. Do you see that 4=4 would be a meaningless equation in mathematics? Therefore it is very evident that 2+2 is not the same concept as 4, and the = sign expresses a meaningful relationship between these two distinct concepts, a relationship which is quite different from the useless expression of 4=4. The usefulness of an equation is due to the fact that something different from what is expressed on the right side, is expressed on the left side
I walked through this in detail a few posts ago. In the Peano axioms they are both the number SSSS0. In ZF they are both the set {0, 1, 2, 3}. = { ?, {?}, {?, {?}}, {?, {?}, {?, {?}}} }.
See for example
https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals
Quoting Metaphysician Undercover
We must be talking past each other in some way. I cannot conceive of anyone claiming 2 + 2 and 4 are not the same thing. I can't respond because from where I sit you're talking nonsense. If you have some subtle philosophical point it eludes me. I just can't respond. Perhaps you have a reference to support your point of view.
Here's my earlier post where I described the Peano and ZF constructions.
https://thephilosophyforum.com/discussion/comment/323461
I acknowledge that there might well be some philosophical point of view that allows you to claim that 2 + 2 and 4 are not the same thing. I've never heard of it and I don't understand what you mean, but that could just be due to my own ignorance.
But you claim that 2 + 2 and 4 are not the same object in ZFC. And THAT is an area where I am not ignorant. You're just wrong. 2 + 2 and 4 represent the same set in ZFC.
But forget set theory. You claim that 2 + 2 is not 4? The last time I heard that it was in the novel 1984 when the protagonist Winston Smith is being tortured to obtain his submission to Big Brother. He's ordered to believe that 2 + 2 = 5; and in the end, he does.
You tell me how 2 + 2 is not 4. If it's not, what is it? And have we always been at war with eastasia?
Well, first there is the understanding that the "=" symbol pretty much never means "identical". The symbol is much more permissive than that. It usually means something along the lines of "extensional" or "equivalent", depending on the axioms in use, but not necessarily "identical".
In arithmetic, "2+2=4" means that the equality is provable from number theory -- or from the larger, encompassing theory such as set theory -- by using the inference/rewrite rules of arithmetic.
From there the notion that two different, non-identical symbol streams can be "equal" to each other, with "equal" meaning that their extensionality or equivalence is provable from the main theory in which we happen to be operating.
In abstract, Platonic worlds, we almost always assume that computation does not require "effort". This view may very well fall apart in virtual worlds, generated by running computer processes. There will be some calculation effort involved for a computer process to derive that "2+2" resolves to "4". So, the more computing intensive the resolution process, the less the equality will be automatic. In the real, physical world, the problem is further exacerbated by the fact that humans tend to be slow and error prone at carrying out calculations in arithmetic.
The meaning of the "=" symbol is not only very context-sensitive, but I actually do not know of even one context in which it means "identity".
For example, in Javascript, the single "=" symbol is already taken up to express assignment. For example, "a=5" means store value "5" in variable "a". It does not want to express that "a" and 5 would be equal or so. It is rather an instruction that seeks the side effect of changing what value is stored in a.
"a === b" means "a == b" and "typeof(a)==typeof(b)".
So, '3' == 3 resolves to "true" but '3' === 3 does not, because the string '3' and the integer 3 are of a different type.
They are considered equal in the expression " '3' == 3 " because the "==" operator will carry out enough work to convert both types to a permissive common denominator, and if it can then declare a match somewhere, it will return true.
The permissiveness of the "==" operator is generally considered questionable. The practice creates dangerous corner cases. For example, 0 is considered a falsy value while "0" a truthy one. That can go surprisingly wrong in the context of permissive equality.
The following is an interesting article about the difference between === and == in javascript.
This does not show me the principle of identity. Saying that two things are the same does not make them the same. It's a hollow assertion without a principle. And I don't see any reference to sameness in your reference.
Quoting fishfry
I can't believe there is a person who does not see a difference between 2+2 and 4. The two are equivalent. And, as I explained equivalence would be meaningless, and equations useless, if there was not a difference between the left side of an equation and the right. The Wikipedia page on "equation provides an analogy. "An equation is analogous to a scale into which weights are placed." Do you see that the things on the two separate sides of a balance are not "the same"? They are said to have the same weight, but this does not make them the same thing.
So, in the case of ZFC, by what property are the two sets said to be "the same"? It's not the same weight, as in the scale analogy, nor is it the same numerical value, as is the case in the equation (what alcontali refers to as "number theory" above). What is the principle of sameness?
Quoting fishfry
If you have no idea of what equivalent means, or of how equations are used, then I don't think I can help you. If you are simply asserting 2+2 is the same as 4 without thinking about what you are saying, because it supports your metaphysics, then why don't you smarten up?
Quoting fishfry
I fully acknowledge, that in ZFC 2+2 is "the same" as 4. I am not denying this. I am saying that it is wrong, because it violates the law of identity, without any justification. If one wants to establish a principle in violation of a fundamental law of logic like the law of identity, then that person ought to provide justification for the proposed principle. Without justification, use of that principle is mere sophistry.
Quoting alcontali
Tell that to fishfry, who is arguing the opposite, that the left and right of the equation actually are the same.
Quoting fishfry
Refer to alcontali's post above. Thanks al.
I was skimming your reply looking for a point of reference, something I could understand. I came to this. I think it's a point of irreconcilable difference. I don't agree with your judgment, but I am incapable of rational response, because I cannot fathom the point of view.
This is my own personal limitation, I'm certain of that. My ignorance of philosophy is profound. I have no doubt that you have a point to make that, from your point of view, is a valid point.
I myself do not ever think I could agree with or even understand such a point of view. Within ZFC, at least, the statement 2 + 2 = 4 is a theorem that can be proven according to strict logical principles that are clearly expressed; and using assumptions that are clearly stated. Within this framework. 2 + 2 = 4 expresses an identity of sets. This is a technical fact that is beyond dispute. And set equality is defined directly in terms of the logical law of identity. I thought I explained it. The axiom of extensionality leverages the logical law of identity. So 2 + 2 = 4 is a perfect expression of the law of identity.
I acknowledge that you feel differently about this but I don't regard myself as being capable of ever understanding such a point of view. And I would not want to be able to understand such a point of view even if I could!
I'll read whatever you write on the topic in the hopes I might learn something, but I don't think it will be productive for me to engage on this.
I suppose the feeling is mutual. I really cannot believe that there is a rational human being who truly believes that 2+2 is the same thing as 4. Isn't this what we learn in basic math, first grade? You take two things, add to them another two things, and you have four things. Very good. But we can get four by adding three to one, or by subtracting two from six, and an infinite number of 'different' ways. So it is impossible that 2+2 is the same as 4, because there would be infinitely many different things which are the same as four. Does it make any sense to you, to believe that there is an infinite number of different things which are all the same? Or can you see that 2+2 is not the same as 8-4?
It is wrong to insist that our mathematical convention grounds the meaning and truth of mathematics in application. For instance, it is incorrect to claim that "the conventions of logic a priori determine that 5+7=12, whereas the physical calculation merely confirms it".
Consider, for example the addition of two summands that are so large that their summation cannot be precisely determined in any individual physical experiment, let alone by hand. Here there isn't a clear distinction between the truth of the summation according to convention, versus the confirmation of the summation bu physical demonstration.
And in visual psychology, it should not be regarded as an error if a test subject reports that he saw 5+7 as 13. It simply means that visual phenomena are not a good model of ordinary arithmetic and vice versa.
I think my eye doctor would prescribe glasses if I saw 5+7 as 13.
" {|||||,|||||||} 'equals' {||||||||||||} " isn't visually acceptable to me. The left side looks too short.
I think I understand your point but I have some counterpoints. I believe you are saying that when we say 2 + 2 = 4 we are saying two things: One, that they represent the same natural number; and two, that 2 + 2 is a legal decomposition of 4, which is not necessarily known beforehand. So 2 + 2 = 4 asserts something more than merely saying 2 + 2 or 4 by themselves. And you're right about that.
However it's not an ontological fact, it's an epistemological fact. That is, the partition of 4 into 2 + 2 is literally a matter of definition. It's an immediate consequence of the way we define the symbols. So it was true before we knew it. If you believe that math has Platonic existence, it was true even before there were humans.
You're right that before someone told us that 2 + 2 = 4, we may not have known it. So we have learned something new; but we have not made something formerly false be true. So 2 + 2 = 4 imparts knowledge of that particular partition; but it was true before we knew it. Ontologically it's an identity as I have been saying all along. But I will grant that epistemologically it is new information above and beyond the mere fact that they're the same number.
Quoting Metaphysician Undercover
Yes, we LEARN that. But it was always true. It was always an identity, even before we learned it. But I agree with you that it's new information that we have learned. It's the morning star and the evening star. They were always the same object, the planet Venus. We LEARNED that they are the same, and that was new. But it was true -- that is, it was an identity -- even before we learned it.
Quoting Metaphysician Undercover
You have actually hit on some very deep math. Subtraction isn't useful, since as you note there are infinitely many ways of expressing 4 as the difference of two integers. But if we restrict our attention to positive integers, it's a very interesting question. 4 = 1 + 3 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. so there are 5 partitions, as they're called. We say that "5 is the partition number of 4."
https://en.wikipedia.org/wiki/Partition_(number_theory)
If you happen to have seen the movie The Man Who Knew Infinity, it was this partition problem that was solved by Ramanujan. He found a formula that gives the partition number for any positive integer. It's a big deal in number theory.
But these are discoveries. God, or the Platonic universe, already knows the partition number of every integer. No new information is created by the discovery; rather, only new KNOWLEDGE is created.
So I would say that 2 + 2 = 4 is an expression of the law of identity; but we did not always KNOW that until someone discovered it and taught it to others. Is this a distinction you find meaningful?
Quoting Metaphysician Undercover
But there aren't. There are infinitely many different representations of the concept of 4, just as schnee and snow are two representations of the white stuff that falls from the sky in the winter. And you are right that it may sometimes take hundreds or thousands of years for us to discover that two representations represent the same thing. But they were always the same even before we knew that.
Quoting Metaphysician Undercover
No of course that doesn't make sense to me. What makes sense to me is that there may be infinitely many distinct representations of the same thing; and that it sometimes takes hard work to discover that fact. But when we discovered that the world was round, it wasn't flat the day before. We created new knowledge; but we did not create a new reality. The world was round and then we discovered the world was round. 2 + 2 = 4 and then we discovered that fact. Two representations of the same thing.
Quoting Metaphysician Undercover
They're two representations of the exact same identical thing. If they weren't they would not deserve the equal sign. Do you agree that schnee and snow are identical, even though one has to pick up a little German (or English as the case may be) in order to discover that?
I wouldn't even say that. '2+2' represents two distinct quantities of two, being added together. So there are two distinct units, a unit of two, and another unit of two represent here with '2' and '2'. On the other hand '4' represents one unit of four, so there is only one unit represented, a unit of four. Notice there are two symbols of 2, so two distinct things represented on the left side, and only one symbol.'4', therefore one thing represented on the right side.
Quoting fishfry
What I am talking about is ontological, because it is the objects which are represented by the symbols. We need to first clarify what is represented by the symbols before we can proceed to epistemological principles.
Quoting fishfry
It's not a partition which is represented, that would be division, four divided by two. What we have in 2+2 is two distinct units of two being unified with the symbol '+'. Conversely, we could take a unit of four, and divide it into two distinct units of two. That would be a partitioning.
So I think you have things reversed. Ontologically, 2+2 is clearly distinct from 4, but epistemologically we might say that they have the same value. They are equivalent by an epistemic principle, but distinct by ontological principles.
Quoting fishfry
It's not an identity though, it's an epistemic principle. '2' Identifies one thing, '4' identifies another thing. That two '2's has the same value as one '4' is not an identity it is a conclusion drawn from an epistemic principle, what alcontali called number theory.
Quoting fishfry
That's not an expression of the law of identity at all. That's an expression of an equation. As alcontali explained, it's a conclusion drawn from the principles of number theory. Do you know the law of identity? A thing is the same as itself.
Quoting fishfry
What I've been trying to explain to you, is that '2+2' does not represent the concept of four, '4' does. As I explained already, if both sides of an equation represented the exact same concept the equation would be useless. But equations are not useless, they are very useful for many different purposes, and that is because they express an equality between two distinct concepts. It's nonsense to say that the right side and left side of an equation each represent the exact same concept. What could an equation do for us if all it expressed was '4=4', or '2=2', or '50=50'? If both sides represented the same concept, that's all we'd have. It's only because the one side represents something different from the other, that the equation is useful.
Quoting fishfry
This is not relevant. It's a simple fact that '2+2' does not say the same thing as '4'. There is no language in which '2+2' would be translated as '4'. Each of these says something different, and they maintain their difference in all languages, so '2+2' is never translated into another language as '4'. That would be a mistaken translation. So it's very clearly a mistake on your part, to say that '2=2', and '4' "are "two representations of the exact same identical thing". It seems so basic that I can't believe you actually believe that.
They are the same according to the game of identity called as "equality theory". There is a confusion here between expressions and what they denote, "The Sun" , "The nearest star to Earth" are two DIFFERENT (i.e. not identical) expressions, yes, but they denote the same object! so when we say for example "The Sun = The nearest star to Earth", what we mean is that the object denoted by the expression "The Sun" is Identical to the object denoted by the expression "The nearest star to Earth", that's very clear, it is identity of the denoted and not of the denoting expression, of course the denoting expressions are different. That an object can be denoted by different expressions is well known, and it poses no problem whatsoever. Along this understanding the expression "2+2" is meant to denote some object x, and the expression "4" is also meant to denote some object x, however both expression (though different) denote the SAME object exactly.
We've been through this already. "Equal" does not mean "the same", or "identical". Identity is defined by the law of identity, equality is defined by mathematical principles. If you think that there is a principle of identity which makes equality into identity, then please produce this new law of identity, what you call "equality theory". I've asked fishfry for this principle of identity, to no avail.
Quoting Zuhair
You are using '=' here in a way other than how it is used in mathematics. Argument by equivocation is useless to me. Sure you can use '=' to mean identical if you want, but we are talking about the way it is used in mathematics, 2+2=4 for example, and it is not used here to mean identical.
Quoting Zuhair
This is obviously not true. There are three symbols on the left side '2', '+', and '2'. There is only one symbol on the right side '4'. That is the first indication that the right side does not denote the same object as the left side. there is a plurality of symbols on the left, so there is likely a plurality of objects symbolized on the left. Furthermore, if the three symbols on the left side are meant to signify one entity, it is not the same entity as is signified on the right side, or else it would be signified in the same way as the right side. The symbols on the left are not meant to signify the same thing as on the right, or else the same symbol would be used. If they happen to symbolize the same thing this would be by mistake, but there is no mistake here. If two distinct symbols are meant to symbolize the very same thing, this would only be intended to create the illusion of different things being represented, and there would be no reason for this except to deceive. Therefore we must conclude that they are meant to symbolize something different. If the two sides were meant to symbolize the very same thing, the equation would read '4=4', and this would be a useless equation. And if the very same thing was meant to be symbolized by different symbols this would be an act of deception. Since '2+2=4' is not a useless equation, nor an expression of deception, we must conclude that what is signified on the left side is not the same object as what is signified on the right side.
There is no new law at all. It is a schema of statements, in first order logic it would be expressed as: that x = x, i.e. everything is equal (identical) to itself, and that if phi(x) is an expression in which x occur and if phi(y) is obtained from phi(x) by merely substituting all occurrences of x in phi(x) by the symbol y, then the law is:
x=y implies [phi(x) iff phi(y)]
in a more informal manner, x is equal (identical) to y if every expression true of x is also true of y and vise verse, what we mean by true of is the truth of the denotation of that expression about objects and not the truth of its grammatical structure.
Actually equality is nothing but identity. In first order logic it boils down to substitutivity, as mentioned above.
But you need always to discriminate between what an expression is denoting and what an expression is. I already gave a simple example "The Sun" and "The nearest Star to Earth", in physics those two expressions are referring to exactly the same object but they are indeed two distinct expression! The former has two words the latter had five! So truly they are distinct expressions but they are denoting exactly the same object. In an exactly similar manner in mathematics the expression "2+2" is nothing but a functional term, it denotes an object that is exactly the same (identical) object that the constant term "4" is denoting. This is no deception, much as the different statements in the first example are no deception. The idea is about what can be called a "consequential truth" here. In the game of arithmetic the expression "2+2" is identical to "4", in the sense that they both denote the same object. i.e. this is a consequence of the axioms and rules of inference of that game, this is a theorem, and consequential fact, we need to determine exactly when two different expressions in the language of arithmetic denote the same object, just because they are different it doesn't mean that they can't denote the same object. We have a formal game here, and we want to know which of those expressions denote the same object and what are not, this is no deception, it is not even trivial, that's what we want to know.
All the rest of your account on trying to a kind of prove that "2+2" must be an expression denoting something that is different from the expression "4", is NOT correct, neither conceptually nor formally.
By the way the objection you stated that if they are indicating the same object then 2+2=4 would be equal to 4=4 and therefore would be vacant, this objection is already a property that Kant had spoken about when he defined analytic truths, i.e. its just repetition of what has been already said. Which is correct.
Consequential results are not that easy to figure out, they turn to be very tricky, that even if at the very conceptual root they are repetitions of statements, yet the recognition about which statements boils to be repetitions of which other statements is not that easy to determine and sometimes its even impossible to know relevant to a fixed set of axioms in arithmetic.
However the reality of 2+2=4 is not only linked to the above formal consequential game reasoning. One might say that all of that consequential game is just vacant, and that mathematics is not vacant as analytic reasoning is, so there must be a kind of truth to 2+2=4, something more akin to synthetic truth Kant was speaking about. The answer is that the truth of 2+2=4 is inherited from the truth of the axiomatic system in which 2+2=4 is a theorem of. The axioms themselves are not analytically derived from prior sentences, if they are consistent, then they are true of some model, and the truth of the axioms whatever it might be is inherited down to all of theorems derived in the system axiomatized by those axioms. On can say that all theorems are just repetitions of what's in the axioms, so the truth of 2+2=4 is related to the background of the axioms of the axiomatic system in which it is proven, which is CORRECT! That doesn't prevent 2+2=4 being the same as 4=4 at all, it doesn't make it trivial because it is only an aspect exposing the truth of what's in the axioms.
Again to sum it up, although "2+2" and "4" are two distinct expressions, yet they both denote the same object, much as how expressions "the Sun" and "The nearest star to Earth" are distinct and yet denote the same object.
I was going to reply to you later but just ran across this, which could not be more false.
I have repeatedly explained to you that the axiom of extensionality is directly derived from the logical law of identity. I thought that had already been mentioned by someone even before I joined this thread. So if you don't understand what it means, or want to see more detail, just ask.
If you are claiming that equality isn't identity in natural language, you might have a philosophical point.
But if you are making a mathematical claim, you're just factually wrong. Mathematical equality is identity of sets. A mathematical equality states that the sets on either side of the equation are the same set.
OK, I've said true things about '2+2' which are not true about '4'. Therefore the two are not identical. It's what I've been doing for last number of posts, explaining how '2+2' signifies something different from '4".
Quoting Zuhair
Clearly equality is not identity, because different things are true concerning what is expressed on the right side of an equation than are true concerning what is expressed on the left side. This is by your own definition of "identical", above. How can you deny this?
Quoting Zuhair
I don't dispute that distinct symbols can refer to the same object. What I dispute is that '2+2' refers to the same object as '4'. So your example proves nothing.
Quoting Zuhair
If this is true, then show me the object which both '2+2' and '4' refer to. If that object is a concept, then explain to me this concept which they both refer to. I've already explained how they each refer to a different concept, but you refuse to listen. So now it's your turn to describe to me this one concept which you believe both '2+2' and '4' refer to.
This appears to be the extent of your argument, a simple assertion that '2+2', and '4' both refer to the same object. Now see if you can justify this assertion by showing me the object which they both refer to. It will be difficult for you, because I already see that '2+2' refers to something completely different from '4', as I've explained. So you need to dispel this false belief that I have, demonstrate how the differences I described are not real, and show me how they really refer to the same object. Assertion does not justify.
Quoting fishfry
You provided no such explanation, only an assertion. The web pages you've referred me to do not support your claim. They speak of "equality", not identity. This is from the Wikipedia page on the axiom of extensionality which you referred.
[quote=Wikipedia]Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B.[/quote]
Where's the reference to the "logical law of identity" which you are asserting?
Quoting fishfry
That is not true, you are inverting what is stated in the axiom of extensionality. It is stated that if two sets have the same members, the sets are equal. It does not state that the sets are the same, it states that if the members are the same, then the sets are equal. Therefore the sets remain distinct, as two equal sets, not one and the same set.
Quoting fishfry
This is exactly the point of contention. Let's say that two things are said to be mathematically equal. By what law of identity do two equal things become the same thing?
The two expressions are of course not identical, they are indeed distinct expressions, I already said that, that's clear because the expression 2+2 contains three symbols in it, while the expression 4 contains only one symbol, of course they are not identical. But that doesn't by itself entail that what they are denoting is not identical! There is a difference between "identity of expressions" and "identity of what expressions are denoting". The expression "The sun" and the expression "Nearest star to earth" are also not identical, the first contains two words, the last contains four words, but they do denote exactly the same object. Now the identity symbol "=" between any two expressions phi, pi , i.e. the expression phi = pi , means phi and pi are denoting the same object, it doesn't mean that phi and pi are identical expressions. You are confusing identity of expressions and identity of what they denote.
Quoting Metaphysician Undercover
Your wish is my command! First one must note that expressions 2+2, 4 , 2+2=4, all of those doesn't have any innate meaning by themselves, we need to assign meaning to those symbols, otherwise they are just blind string of characters. For example the string of symbols "0 + 0 + 0 = 10" is true in Arabic language, since 0 denote number 5 in English, and 10 denote number 15 in English, while obviously it is false in English. So symbols by themselves are blind, they only acquire meaning by conventional definitions. So 2+2=4 is only true relevant to the context that assigns meaning to its symbols, for example in the system of arithmetic. Now lets take some arithmetical system, for example PA (peano arithmetic) as our background system which assigns meaning to symbols 0, 2, + , =, 4. Now in Peano arithmetic 0 is a constant symbol, it means it is an expression denoting a single object of the domain of discourse of PA. Now the expression S(x) is a functional expression it means its a term of the language of PA that denotes only one object for each particular substitution of x, similarly the expression x + y for any particular substitution of x and y, denotes a single object because + is a two place function symbol. The meaning of "phi = pi" when phi and pi are functional expressions in the language of PA means " phi denotes the same object pi denotes".
Now in PA the symbol 2 is meant to denote the object denoted by the expression S(S(0)), for simplicity let us use the notation || phi || where phi is a functional expression, to denote the OBJECT denoted by phi, so we have:
phi denotes || phi ||.
so according to that 2 is denoting the object || S(S(0)) ||.
Also 4 is denoting the object || S(S(S(S(0)))) ||
Now PA proves that the expression 2 + 2 is denoting the object || S(S(S(S(0)))) ||, which is the same object that expression 4 denotes! So by the meaning given to phi=pi in PA, PA proves that:
2+2=4
The proof of that is present in PA.
What you had in mind is an example of theoretic x meta-theoretic confusion.Which is something that almost everyone passes through!
However to veer to YOUR side, one can in some sense use a terminology that separates identity from equality, you can stress that identity is full matching, i.e. even with expressions, those would be identical only if every property associated with one of them is also to be associated with the other whether at the language level or the meta-language level, and so you'll demand that everything must match between them even the way how those expressions are written. OK, by this we can say that equality is identity of denotation, and that identity is full matching. If we adopt such terminology then of course 2+2 won't be identical to 4, but 2+2 would be equal to 4, since there is identity of denotation of those expressions. This might be plausible, but it is not often used, well as far as I know of, but it might have its virtues. not sure though.
Right, and I've explained how what is denoted by '2+2' is different from what is denoted by '4'. Principally, '2+2' denotes two units of two whereas '4' denotes one unit of four. So, there are things which we can say that are true about 2+2 which are not true about 4, and vise versa.
Quoting Zuhair
Sure, but '2+2' denotes two objects whereas '4' denotes one object. And, even if you construe '2+2' as one object, that object is divided in a vey specific way, in half. No such division is specified by '4'.
Quoting Zuhair
You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. But all I see is a claim that S(S(0)) +S(S(0)) represents the same thing as S(S(S(S(0)))). Sorry to have to inform you of this, but you haven't provided the premise required to draw your conclusion. Consequently, you have no proof.
Quoting Zuhair
You really don't seem to understand the difference between equal and identical. Here's some principles which might help.
1. Two distinct things may be equal. For example, distinct human beings are said to be equal.
2. Two distinct things cannot be identical, "the same". "Same" refers to one and only one thing, (Leibniz principle for example, if x is the same as y then x is y, there is only one thing).
3. We do say sometimes, that a thing is equal to itself, as well as being the same as itself.
Let me remedy that omission.
Before I start I hope we're agreed that there are two levels to this discussion:
1) The philosophical point that 2 + 2 is not identical to 4 because the former conveys the information that a thing, namely 2, that is manifestly different than 4, is being combined with itself to produce something entirely different, namely 4. This I take to be your viewpoint.
I might argue that point with you, but I would not be on firm footing. There are subtle philosophical issues that I'm ignorant of; but that at the very least I can see I'm ignorant of them. So I'm not entirely conceding your point; but I must depart the field. I haven't the capacity to defend my side.
2) But on the mathematical side, I claim that 2 + 2 and 4 designate identical numbers and identical sets and of that I have not got the slightest doubt. I regard this as simply a technical matter that I'm educated about and that you are about to be educated about. You may disagree but at least you know where I'm coming from.
So: I say that when in math we write [math]x = y[/math] we are asserting that x and y are identically the same. I shall now state my case. (* See note at the end).
1.1 We have the law of identity that says that for each natural number, it is equal to itself.
1) We assume we have the natural numbers as given to us by the Peano axioms. These are denoted by the symbols 0, 1, 2, 3, ... I don't know if you regard this as an objectionable premise. We have to start somewhere.
PA says:
(1) There is an undefined symbol [math]0[/math], which we call a "number."
(2) There is a function [math]S[/math], called the successor function, that inputs a number and outputs a number.
(3) If [math]n[/math], [math]Sn[/math] is a number.
There are some other axioms to make sure numbers are suitably well behaved.
https://en.wikipedia.org/wiki/Peano_axioms
With these three axioms we have an endless sequence of numbers: [math]0, S0, SS0, SSS0, SSSS0, \dots[/math]. As a matter of convention we introduce the following names: [math]S0 = 1, S2 = 2, S2 = 3, S3 = 4, \dots[/math]. I hope these are not unfamiliar.
Now we need to define the arithmetic operations. We define [math]+[/math] inductively as follows:
(*) [math]n + 0 = 0[/math]
(*) [math]n + Sm = S(n + m)[/math]
With these definitions in hand we may now evaluate [math]2 + 2[/math].
[math]2 + 2 = 2 + SS0 = S(2 + S0) = SS(2 + 0) = SS2 = SSSS0 = 4[/math]
This puts the matter to rest. The expressions [math]2 + 2[/math] and [math]4[/math] refer to the same number. It's practically a definition, following so easily from the Peano axioms and the definitions of the symbols [math]2[/math], [math]+[/math], and [math]4[/math].
If you think it means something else, you are mistaken. You may have some intuitions that [math]+[/math] means "combining two things to make some other thing," but nothing in the math supports that point of view. I can't help what they told you in first grade.
These are strings of symbols manipulated by formal rules. A computer could implement the rules. The symbols are devoid of meaning except for what we bring to them with our intuitions. And our intuitions are part of our philosophy. They are not in the math itself.
As I say that's the mathematical story. I will concede that you may have a point if you overload the symbols with your intuitions about what they mean. That's the philosophical question. If you think [math]2 + 2 = 4[/math] "means" something that the math doesn't say, then you are making a metaphysical point, not a mathematical one. If you got your intuitions in first grade, I'd ask you to update them in the light of how the math actually works.
On the math there is no question. [math]2 + 2 = 4[/math] is an identity derived directly from the law of identity, the Peano axioms, and the definitions of the numbers and of [math]+[/math]. As I say it's practically a definition.
For completeness this is only half the story. The Peano axioms aren't strong enough to develop a theory of the real numbers, for example. For that we need set theory, including the axiom of infinity. With the appropriate assignment of sets to numbers, and a definition of the successor function, the universe of sets contains a model of the Peano axioms; and the proof I gave can be lifted directly to a proof that the sets designated by [math]2 + 2[/math] and [math]4[/math] are the same set.
(*) Note -- I did not prove the starred claim that every mathematical equality is a statement of set identity. That would be part of the extension of the discussion to set theory, and I did not want to add those details to this post.
What occurs to me is that you only have rational numbers in your sets. Moreover all the subsets of R (actually of Q in this case) are finite, albeit arbitrarily large. How will you represent irrational numbers with a finite number of symbols, especially those that aren't computable? And how will you represent infinite subsets of R?
Another argument: the strings of finite length over a finite alphabet are countably infinite. It's not hard to write a program that spits them out one at a time. So any subset of those strings is at most countably infinite.
Agreed.
The positional notation always expresses rationals, and not even all of them, such as 1/3 or so. Furthermore, if the digit stream for a number is infinitely long, then it will never proceed to matching the next number. This problem looks insurmountable.
Quoting joshua
Point taken. Can't be done, indeed.
Quoting joshua
Yes, I came to realize that now.
Yes the reason is because I'm holding PA, and it shows you the rules about +, so I didn't want to go to all of that technical side. So I just mentioned that the proof is present in PA, and I didn't want to go to this technical detail. But if you follow the axioms of PA you will begin with S(S(0)) + S(S(0)) which denotes || S(S(0)) + S(S(0)) || (you won't see this explicitly written in references about PA, but that's what PA is actually saying, I'm just clarifying it), to just end up with:
S(S(0)) + S(S(0)) denoting || S(S(S(S(0)))) || .
So just go to PA to fill in the missing part, you'll see that for yourself.
Quoting Metaphysician Undercover
Thanks for this account and the two points after it. But in mathematics when we are speaking about equality we don't mean this really. Equality in mathematics which occurs between expressions, especially when it occurs between functional expressions then it meant to be identity of denotation by those expressions.
You think 2 + 2 is denoting a process that involves a combination of two units to form another unit, which is wrong. 2+2 is "describing" such a process, but NOT denoting such a process, it is denoting what results from that process, that's your error. 2+2 is a functional term in mathematics, it denotes ONE and just ONE particular object, the + is a two place function symbol, it is an assignment that sends pairs of objects to single objects per each pair, so x +y = z is meant to be an assignment that sends single substitution of x and single substitution of y to a SINGLE substitution of z. so it sends the ordered pair (x,y) to a single z for each particular substitution of x,y. so 2+2 is meant to be the object that + sends the pair (2,2) to. When we way 2+2 = 1+3 we (in mathematics) mean that the single object that 2+2 denotes is "identical" to the single object that 1+3 denotes, that's what is meant. It means identity of denotation, that's all.
I can exactly mirror you argument to say that "The Sun" and "The nearest star to Earth and Jupiter" do not denote the same object? since the first is just involving one object, while the later is involving a process of two things being near to a third object, and it involves the meaning of star, earth, and Jupiter, so it is speaking of TWO entities with a relation from them (near) towards a third entity that at the end points to that third object, so the denotation of those two expressions is distinct, which is WRONG.
We need first to agree on what constitutes a "denotation" of an expression, and then we can argue its identity.
There is a difference between the details involved in an expression and what that expression is denoting. Denotation of expressions is determined by definition of the rules of the language in which that expression is meaningful. So I agree that there is a lot of say information going on in the expression 2+2, much more than the simple reference involved in the expression 4, that's right, but that doesn't affect their denotation, because their denotation is set by the rules of arithmetic and not by these aspects, by the rules of arithmetic 2+2 is a term of the language and it can ONLY be substituted by a SINGLE object of the universe of discourse, which is as single as 2 is and as single as 4 is, it is as single as any number is (which are the singular objects of the universe of discourse of PA), this is a rule of the language of PA, that + is a function. This is a stipulation, consider it an axiom. And by rules of arithmetic (say PA) it PROVES that the single object denoted by 2+2 is exactly identical to (i.e. the same as) the single object denoted by 4. Much as physics say that the single object denoted by "The Sun" is exactly identical to the single object denoted by "The nearest star to Earth and Jupiter", even though there are particular differences in details of those expressions including differences in syntax (particular wordings, number of them, grammatical differences etc..) and difference in semantics (information involved in these sentences), still both sentences are "denoting" the same object. So definitely different expression can convey different set of information to just denote the same object, that's obvious, and 2+2=4 is just once case of that situation.
In nutshell in mathematics the denotation of 2+2 is already stipulated to be a single object that is as single as 2 is and as single as any natural number is, and the denotation of 4 is of course a single object since its a number, and that that single object denoted by 2+2 is exactly the same( is identical to) the single object denote by 4, and that's what 2 + 2 = 4 exactly means.
Good point about the rationals that would be left out. And thanks for absorbing my feedback in a good spirit. (I like that you know programming. I'm working on becoming a better programmer. )
I just wanted to add, that we can actually have a very simple system in which 2 + 2 = 4, that of first order logic and add to it primitives of identity (equality) symbolized as "=" which is a binary relation symbol, and of "+" denoting addition which is a two place function symbol, and of "1" denoting what we customarily know as one, which is a constant symbol. I'll try to coin a system in which 1 is the first number, i.e. doesn't have zero in it.
Axioms:
Equality axioms:
1. for all x (x=x)
2. if phi(x) is a formula in which x occur free, and never occur as bound, and y doesn't occur, and phi(y|x) is the formula obtained from phi(x) by merely replacing each occurrence of the symbol x in phi(x) by the symbol y, then all closures of
for all x,y (x=y -> [phi(x) <-> phi(y|x)])
are axioms
Addition axioms:
x + y =/= 1
x + y =/= x
x + y = y + x
(x + y) + z = x + (y + z)
Define: x=2 iff x=1+1
Define: x=3 iff x=2+1
Define: x=4 iff x=3+1
Theorem: 2 + 2 = 4
Proof:
By definition of 4 we have: 3+1=4
By definition of 3 we have 2+1=3, use identity axioms to replace this and get:
(2+1)+1=4
By associative law we have (2+1)+1 = 2+(1+1), use identity axioms and replace to get:
2+(1+1) = 4
then by definition of 2 we have 2= 1+1, so by identity axioms replace to get:
2 + 2 = 4
QED
Actually what is used in the above proof is only the definitions of 2,3,4, and the identity and associative laws.
I think theory of addition is complete as far as I know.
This is our point of disagreement. The law of identity does not say this, you are claiming this. If there is a law in mathematics which states that each natural number is equal to itself, it is not the law of identity. it is a law of equality. So what I am asking is, on what grounds do you say that this law of equality is a law of identity, and supplant the real law of identity with this one?
Quoting fishfry
It shows me that they are equal, it doesn't show me that they are the same. That's the disputed point, that being equal means being the same.
Quoting fishfry
Your mistake is in your citation of the 'rules'. The law of identity does not say that a number is equal to itself. I suggest you revisit the difference between identity and equality. "Identical" means the same whereas "equal" means having the same value. Do you see a difference between these two? When two things have the same value they are not necessarily the same thing. Being the same thing is what is necessary to fulfill the conditions of the law of identity.
Quoting fishfry
You keep saying this, that it is "derived directly from the law of identity", and you refer me to websites which discuss equality. Nowhere have I found the law of identity mentioned in this discussion of equality. So I really think that it is just you (and perhaps many others) who mistakenly believe that equality is derived from identity, and I am trying to point this out to you. Perhaps I am the one who is wrong, and equality is really derived from identity, but if so, where is the evidence of this?
Quoting Zuhair
Well fishfry seems to have done this already, but it doesn''t show identity, it shows equality. That is the point being discussed, you and fishfry seem to think that in PA equal things are the same thing.
Quoting Zuhair
Right, this is what I said,. By showing parts, '2+2' indicates a particular division of the object, unlike '4' which indicates no such difference. So '2+2' denotes an object divided in a particular way, in half, whereas '4' denotes no such division. Therefore '2+2' denotes a different object from '4'.
Quoting Zuhair
This is not true, '=' means equal, it does not mean identical. You are arbitrarily replacing what '=' really denotes, with "identical" and this produces a false statement. When you arbitrarily change the meaning of symbols in your interpretation, you create false statements.
Quoting Zuhair
This is totally irrelevant. What is WRONG, is to arbitrarily claim that two equal (having the same value) things are identical (the same).
Quoting Zuhair
This is what fishfry claimed to show above. But the demonstration does not show that the two are exactly identical, it shows that they are equal. Then fishfry states a misrepresentation of the law of identity, claiming that the law of identity states "that for each natural number, it is equal to itself". Where is your understanding of the law of identity?
Quoting Zuhair
What we need to agree on is definitions of "equal", and "same".
Quoting Zuhair
This is proof of your's and fishfry's mistake. You cite "equality axioms". Equality axioms are not identity axioms. You and fishfry both arbitrarily replace "equality with identity. Sophistry rules!
No! Equality rules are spoken as Identity rules by mathematicians, it just happens that equality is used more: see this site on terminology:
Glossary of First-Order Logic
Just use the find function on your browser, and search for "identity" and read all of what it says about it in that site.
The "=" symbol is used to symbolize identity, so x=y actually means that x and y are exactly the same object, i.e. they are identical, and not that they are having the same value and remain discriminate at the same time.
To be more precise, due to shortages of formal languages, it is better to call identity as indiscernibility, because under that theory in question we say that x and y are identical if the theory in question cannot have an expression phi(x) (written in the language of the theory and in which x occur) and an expression phi(y|x) [which is the expression obtained by merely replacing all occurrences of x by y in formula phi(x)] such that phi(x) is true of x and phi(y|x) is not true of y. So we say that x and y are indiscernible under the language of that theory. Of course that doesn't necessarily mean that they are in reality identical, it just means that the theory in question cannot discriminate between them and so it see them as "identical", i.e. it says that they are identical.
The indiscernibility of identicals is a famous law, and in first order logic it is the law that I wrote (and that ironically you said it is not about identity??] see:
Leibniz's law
About the first law of identity which is reflexivity law, i.e. that every thing is identical to itself, this is just a trivially true statement about identity, there is no dispute about that.
So the theory that fishfry and I are mentioning is about "identity", yes its known as equality theory, other sources name it as identity theory, but basically it is about 'identity" as indiscernibility under substitutivity, and it is certainly not about equality as common reference (which is what you think it is about), it doesn't make sense to think of it as being about common reference, why should we have a law about indiscernibility of objects that has common value under certain functions??
However, as I said you can "technically" speaking have some theories that see some objects as identical, but other theories can discern between them, yes this can happen, much as we human can see a star and think its one while in fact it is two or more stars.
On the other hand if we are to understand Equality in YOUR sense as assignment to a common object, like in having a fixed function F over a certain domain D, so we'll say that all elements of D are equal under F, to just mean they are assigned the same value (image) under F. Note here that D can have many members. This use of 'equality' is not perfect, it is mentioned in common languages like that, yes, but it is imprecise, it hides a lot of details, and certainly it is NOT what is meant by equality which is symbolized by "=" in mathematics. In mathematics when = is used it is meant to symbolize "identity", i.e. sameness of objects, and not assignment to a common value as you think.
Equality as used in PA and in ZFC, and generally in first order logic with equality, that is symbolized by "=", in those contexts it exactly means identity or sameness of objects, more precisely speaking that the theory in question cannot discriminate between x and y if it proves that x=y.
You say '2+2' denotes an object divided in half. Well I'd say: OK no problem.
You continue saying whereas '4' denotes no such division.
Yes the correct wording is that '4' doesn't denote such a division, this is clearer. However it doesn't deny it? You seem to be confusing : Not denoting phi , for denoting not phi. So you seem to be arguing that since '4' is not denoting that the object it denotes is an object that is divided in half, then it follows according to your reasoning that 4 is denoting an object that is not divided in half. This is an error. Not claiming something doesn't mean that you are claiming its negation. I'm not claiming that my son would pass the exam, it doesn't follow from this that I'm claiming that my son will not pass the exam.
So 4 not denoting that what it denotes is dividable in half, doesn't mean that 4 is denoting an object that is not divisible in half.
Absence of denotation doesn't mean denotation of absence.
Absence of denotation just signal incompleteness of information.
We are not claiming that expressions supply FULL information about what they are denoting.
2 + 2 only shows some extra-information about what it denotes more than the constant symbol 4 shows about what it denotes. That doesn't mean that what they are denoting is not the same object. I can say that Barack Obama is one of the presidents of the united states. Another time I can say that Barack Obama is one of the presidents of the united states that has a Nobel price. The first expression did NOT denote that Barack Obama had a Nobel price, yet I didn't deny it! It is only the case that the second sentence had more information, but both are speaking exactly of the same person. In a similar manner 2+2 and 4 are denoting exactly the SAME object, but 2+2 is denoting more information about that object than 4 does, but again 4 is not denying what 2+2 is denoting.
Your site provides the terminology of first order logic, not mathematics. The use of "=" is not the same in first order logic as it is in math. To equate these two is to equivocate and that is a fallacy of logic. That the symbol "=" means identical in first order logic does not demonstrate that the symbol "=" means identical in mathematics. Here's a quote from the section of the referred site on "identity" : " Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity."
In all the mathematical sites which fishfry referred me to, none of them spoke of equality as identity. So you are just continuing with your hollow assertions.
Quoting Zuhair
So you and fishfry just decided to change the name of the theory from "equality" to "identity", for some arbitrary reason. Or was that done for the purpose of deception?
Quoting Zuhair
Now I'm convinced, you've changed the name from "equality" to "identity" for the purpose of deception, and you are flat out lying here. Why do you seek to deceive?
Quoting Zuhair
Please reread this, and I hope you can see your mistake. Is '4' denoting an object? Yes. Is it denoting that the object is divided? No. Therefore '4' denotes an object not divided. Look, '4' does not denote that the object denoted is divided in half. That is clear. Therefore what is denoted is an object not divided in half. That the object, like any object, may potentially be divided in half, divided some other way, manipulated in any other way, or be converted into an infinite number of other objects, is irrelevant to what is denoted. What is denoted is an object not divided, and you are lying when you say that to interpret in this way is an error, because I can tell from the contorted way that you've written the passage that you are trying to disguise the truth. Why lie?
Quoting Zuhair
We are not talking about claims, we are talking about denotations. If '4' denotes an object, it is impossible that the object denoted by '4' is divided in half or else it would not be an object denoted, but two other objects, the halves.
Quoting Zuhair
What is denoted is what "is" denoted. If what is denoted is an object then that object is not divided in half, or else it would be two objects, regardless of whether the object is divisible. I hope you understand that it is contradictory to talk about one object which is two objects.
Quoting Zuhair
This is utter nonsense. What is denoted is what is denoted. It's completely nonsensical and illogical to say that the denotation could possibly include an infinity of other things which are not actually included in the denotation. Can't you see that you're just blabbering nonsense in an attempt to cover up your lies? The denotation is of something specific and what is not included in the denotation is not denoted. It's nonsense to argue that something else could have been denoted, therefore we should allow that what could have been denoted is part of what was actually was denoted
Quoting Zuhair
But '2+2' denotes two objects, each with a value of two. What do you think the '+' sign is there for, decoration?
Quoting Zuhair
You're not even talking about identity here. Do you know what predication is? What you call "more information" is predication. "Barack Obama" is one subject. What you predicate of that subject is something different from what the name denotes. Let's say our subject is "Barrack Obama". We can say many things about this subject, many different predications, one of which might be "is one of the presidents of the United States that has a Nobel Prize." How could this lead to the conclusion that "one of the presidents of the United States that has a Nobel Prize" is the very same identity as "Barack Obama"? Likewise, let's assume '4' as our subject. We can say many things about 4, many different predications, one of which might be "is equal to '2+2'". How does this lead to the conclusion that 2+2 is the very same identity as 4? That's nonsense. "More information", or predication, is something completely different from identity.
I can't respond to this. You're factually wrong. There's only one set {0,1,2, pi}. There isn't "another" set that happens to have the same elements and is therefore equal. Any set with the same elements is identical to this set.
If you don't get that or you don't want to get that or you think I'm completely wrong, that's your privilege. You're making mathematical claims that are false. It's ok. A lot of people do that and I can't fix them all. I've said my piece here. All the best.
Yes thanks for making that point. In fact Russell and Whitehead famously took 400 or whatever pages to prove that 1 + 1 = 2 directly from logic; and presumably they could do 2 + 2 = 4. The only reason I didn't mention it is that I'm not familiar with the development of numbers as in R&W. My knowledge is mostly in the math domain which means I need to start with the Peano axioms. But Russell and Whitehead is probably the right answer to how you show that 2 + 2 = 4 is a logical identity.
Perhaps you can clarify this point for me then. The law of identity is that a thing is equal to itself. Why wouldn't this apply to numbers? A rock is identical to itself, the number 3 is identical to itself. You are claiming the former and denying the latter? Perhaps this is a clue to why we disagree. How can a number not be identical with itself by virtue of the law of identity?
Here is your error, you think that '2 + 2' denotes TWO objects. This is wrong. You are not understanding the operator "+", this is a two place FUNCTION symbol, you need to read some logic related to mathematics, i.e. foundational work on mathematics. "+" is a two place FUNCTION, it means that it is a ternary relation that sends a pair of objects to ONE object for that particular pair, so suppose you are summing 9 and 8 here the addition "+" function would send the pair {9,8} to ONE number that is 17, in other words view addition as some process that at each time it has TWO INPUTS and ONE output such that whenever you input the same values again you get the same output again. Now it is important to understand what the expression "2 + 2" means, it means the OUTPUT of summing 2 with 2. In other words the expression "2+2" denotes the object that the operator + would send the pair {2,2} to. I hope this is clear. So '2 + 2' by definition of functionality of "+" cannot denote two objects. The appearance of two symbols in it, i.e. the symbol "2" appearing twice, doesn't mean that "2 + 2" is denoting two objects at all, "2 + 2" is the VALUE of the function + for the pair {2,2}, and it is ONE object. You are not discriminating between "denotation" and 'information "predication" accompanied with that denotation', '2+2' denotes ONE object and only ONE object which is the value of the + operator on the pair {2,2}, but '2 + 2' carries information [this is not denotation] related to that denotation, that the single object denoted by '2+2' can be split in half, i.e. it is the value of a pair having identical projections, that doesn't mean that it is denoting two objects at all. On the other side the expression '4' is a constant symbol, it also denotes a single object, but a constant symbol is a zero function symbol, so it does NOT carry with it any additional information about what it denotes, but at the same time it doesn't denote absence of any kind of information about what it denotes, so it doesn't denote an object that is not divisible in half, i.e. cannot be the value of + function from a pair with identical projections, it cannot assert that negative information about what its denoting because it is a ZERO place function symbol.
Your main error is that you think that "2 + 2" is denoting two objects.
possibly I don't agree with R&W on that. + is not a logical operator, it is a mathematical operator, but as you know we can speak logically about extra-logical concepts, we can add them to any logical system, but of course the result is not a purely logical system, but a logically compatible system you may say, usually refereed to as logically extended system. The trivial complete system that I've depicted is not a pure logical system, it is a logico-mathematical system. I think it's complete. i.e. not subject to Godel's incompleteness theorems, but I'm not sure really.
One object but two digits.
You can erase one digit and be left with an object here, but not with 4. Clearly there's a subtle difference, just as between soup and its components.
Well PA is a mathematical system. Most formal mathematical systems nowadays are stipulated as extensions of logical systems, in particular first order logic with identity. And it is about those mathematical systems that I was speaking. Even older mathematical systems like ordinary math, all those can be recaptured more effectively as systems extending first order logic with identity. I've shown you the axioms of first order logic with equality and you replied that the equality sign in them is not about identity, when I showed you that this is just a terminology preference, and that it is also named as first order logic with identity and I showed you the rationale behind those axioms and its relationship to the informal notion of identity, you replied that this is not mathematics. In reality all older mathematical systems that you know of can be formalized as extensions of first order logic with identity, and in those systems the symbol = is taken to represent identity.
Now the question is what about older systems that are not formalized as extensions of first order logic with identity, can we understand the = in them as something other than identity, the answer is yes of course, it can be taken to be an equivalence relation, and I agree you'd better name them as equality, since it is not necessarily the identity relation. But formal recapturing of them as extensions of first order logic with identity with the = sign taken to represent identity, is by far a much sharper and more well defined an rigorous approach.
But anyway your argument that the expression '2 + 2' is taken to represent two objects is outright false, even in ordinary math the expression '2 + 2' is taken to denote a single natural number that is sent to by the + operator from the pair {2,2} [more precisely one must write it as (2,2) since it is an ordered pair], it doesn't denote two natural numbers as you think, because + is a FUNCTION.
Yes! On the informal level I would agree, but formally NO. It only symbolizes a number that the operator + is sending the pair (2,2) to. It doesn't speak of anything of that number having two or more digits. In reality the best representation of four is as four strokes, but this is besides the formal system of arithmetic actually.
But four ones is not one four, due to the obvious notion that it's a single stroke.
I think the point trying to be made here is that a shared value does not denote that two things are identical, as obviously they have at least one differing quality.
Take a stick.
Snap it in half.
Is it the same?
Only specific mathematical systems are based in first order logic, perhaps ZFC is one of them. Now, the question is where does ZFC derive its meaning of "=", from traditional mathematics, or from the law of identity. As we seem to agree, though you are intent on making an impossible reduction, the two meanings of "=" are distinct. Since the axioms of ZFC do not mention the law of identity, but mention a theory of equality, I think it is quite obvious that ZFC derives its meaning of "=" from traditional mathematics, and not the law of identity.
Quoting Zuhair
It is not a terminology preference. In traditional mathematics "equal" means having the same value (and I'm sure you are fully aware of this), implying that two distinct things may have the same value. In the law of identity "same" means one and only one thing.
It is very clear that ZFC derives its meaning of "equal" from the traditional meaning of "equal", and not from the law of identity, because ZFC does not cite the law of identity, and as we've seen, it allows that two distinct things are "equal". Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity.
Quoting Zuhair
There you go, continuing with your lies. You are fully aware that this is not true, being the well-educated individual that you are. Yet you assert it anyway! Why lie? What's the purpose?
Quoting Zuhair
Didn't you just say "all" older mathematical systems can be formalized as systems where "=" represents identity? And now you ask about those which cannot. Oh what a tangled web we weave when first we practise to deceive.
Quoting Zuhair
Exactly, an "ordered pair". And an ordered pair is two objects. Why say that this is false? Your propensity for lying never stops amazing me.
I didn't ask about those which CANNOT, I asked about those which are not. I mean are not presented in a formal manner as an extension of first order logic with identity. Of course they can be formalized as an extension of first order logic with identity, but I'm asking about when we don't do that and leave it un-formalized. Then in this case how are we to understand "=" symbol in them. I'd say that it is not necessarily the identity symbol. Yes that is correct of course.
Quoting Metaphysician Undercover
if ZFC is presented as an extension of first order logic with identity, then of course "=" would stand for identity. IF we don't do that, then of course it would not necessarily stand for identity.
Quoting Metaphysician Undercover
Not only ZFC, you have PA (peano arithmetic) nowadays presented as an extension of first order logic with identity. And there are many other systems also so presented, you can read about reverse mathematics. Anyhow almost all of traditional mathematics before the era of set theory and modern mathematical logic, nearly all of it can be re-formalized as extensions of first order logic with identity systems, and of course the "=" in them would be understood to represent identity.
Quoting Metaphysician Undercover
As I said above, this depends on how you formalize ZFC, if you formalize it as extension of first order logic with identity then the = symbol in it would stand for identity. If not then it can stand for some other equivalence relation.
Quoting Metaphysician Undercover
I don't know why you keep assuming that I'm lying? Anyhow. The fact that nearly all of traditional mathematics can be formalized as extension of first order logic with identity is well known, you can see reverse mathematics for that. And you can serf the web for Harvey Friedman's grand conjecture, etc..
Quoting Metaphysician Undercover
The ordered pair of the two objects, here in your example (2,2) is not what is denoted by "2 + 2", I'm trying to tell you that but you keep refusing to listen, the expression "2 + 2" is the object that the + operator send the ordered pair (2,2) to. To clarify this: the + operator is sending the pair (2,2) to some object call this object k, to represent that for you by an informal sketch:
(2,2) ---+---> k
Now "2 + 2" is that object k, in other words "2 + 2" is not denoting the ordered pair (2,2), No! '2 + 2' is denoting the object that the operator + send the pair (2,2) to, and that object, i.e., k is exactly the natural number denoted by the symbol 4. In other words "2 + 2" is denoting exactly the same object that 4 is denoting. That's the easiest way to understand it.
You may say No. not necessarily, 2 + 2 is denoting an object k, and 4 is denoting an object L, where L is not identical to k, but L is equal to K, i.e. L is possessing some relation R to k where R is some equivalence relation that can occur between distinct (non-identical) objects. So according to this 2 + 2 is denoting an object that have the relation R to the object denoted by 4 where R is some equivalence relation that is not necessarily the identity relation, of course the intention is that the = sign stand for that equivalence relation R. OK this is a possible case of course, but this is more complicated! It is much easier to stipulate that R is the identity relation itself.
Anyhow as I said before if you present arithmetic or any mathematical theory that contain the symbol = as an extension of first order logic with identity, then = would be taken to symbolize identity itself. if not then it can stand for some other equivalence relation.
I hope that settles matters.
Obviously this cannot be done validly, because "=" does not always mean "same", something you refuse to acknowledge, for some strange reason.
Quoting Zuhair
Sure, you can interpret "=" as identical, but that would only produce a false presentation based in the fallacy of equivocation, as the website which you referred me to clearly states: "Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity."
Quoting Zuhair
The evidence shows that you know what you are asserting to be false. This is lying.
Quoting Zuhair
There is no "k" though. What is symbolized is "2+2", two objects and an operator, not one object "k". So this object represented by "k" is not represented by "2+2", it has been wrongly created by you mind, false imagination, nothing here represents it.
That this is the case is evident from the fact that you proceed to say "that object, i.e., k is exactly the natural number denoted by the symbol 4". There is no symbol "4" in the expression "2+2". The object "k" is only on the right side, where the operator sends the ordered pair. So I can ask you, what does "k" really represent? Does it represent what "4" represents, as you say here, or does it represent what "2+2" represents, as you say above? You are only contradicting yourself.
Or perhaps that little arrow represents the same thing as "=", and all you are doing is stating "2+2=4".
However you look at it, there is no operator signified by "4", so it is very clear that "2+2" does not represent the same thing as "4". You've just stated my case for me. It is your repeated demonstration that you clearly understand that "2+2" signifies something different from "4", though you assert the opposite, which makes me say you are lying.
Quoting fishfry
The law of identity doesn't say that a thing is equal to itself, it says that a thing is the same as itself. In formal logic, "the same as" is represented by "=". So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. I am arguing that no mathematical axioms can be interpreted in this way because it is fundamental to mathematics that the two sides of the equation represent distinct things, while the law of identity indicates that "the same" refers to one and only one thing.
OK, that's fine. OF course just to make it more precise. I said almost all of mathematics before the era of set theory can be formalized as an extension of first order logic with identity where the symbol "=" is taken to mean "identity" i.e. "being the same as". Actually this is a well known result, actually most of that kind of mathematics can be formalized in second order arithmetic, you can read about it in reverse mathematics which also can be re-formalized as an extension of first order logic with identity. Actually ZFC itself can be formalized as an extension of first order logic with identity, and ZFC is way stronger than almost all of mathematics before the era of set theory. This is a very well known result.
You say that it is fundamental to mathematics that the two sides of 2 + 2 = 4 must represent distinct objects.
I say that if = stands for identity, then it would mean that 2 + 2 denotes (represents) exactly the same object that 4 represents (denotes). Obviously you object to that, you say that there is something fundamental against this.
what is that fundamental aspect that enforce us to interpret = sign as some equality relation other than identity. Notice that identity relation is a kind of equality relation, but the converse is not true, you can have an equality relation that is not identity. OK. But why = as used in mathematics, for example in arithmetic, why it is not reducible to identity in your understanding?
Notice that Peano arithmetic which is a very famous theory of arithmetic, is indeed formalized nowadays as an extension of first order logic with identity, of course with the understanding that "=" is taken to represented identity relation and not any other kind of equality.
If there is something fundamental to mathematics against the use of = symbol in it to represent identity, then how PA is formalized as such??? How ZFC is formalized as such and it is generally regarded by many as the official foundation of mathematics? Both are indeed formalized with = in them understood as identity.
What you are saying is that the current foundational systems of mathematics are committing a fundamental error? (notice that most of those are coined as extensions of first order logic with identity) According to your account they must instead represent the = as an equivalence relation that can hold between distinct objects, and that the object denoted by 2 + 2 must be considered as a distinct object from that denoted by 4. This is strange? why?
I introduced the object k as an intermediate clarification step, of course it is not mentioned by 2+2.
I just want you to answer this question
does the expression "2 + 2" denotes two objects or one object?
I know that it contains in it the symbol 2 twice, that is clear, but do you think just because of this containment, then it ought to "denote two objects"
Take the following example: "The planet between planets Venus and Mars"
Obviously this expression contains expressions "Venus", "Mars" and each is denoting an object. So it does contain denotations of two objects in it. BUT it itself denotes ONE object that is the object denoted by expression Earth. What we mean by "denote" here is the object that is the subject of speech of that expression, which is obviously the object denoted by expression Earth in English. This is an example of an expression that contains denotation of more than one object within it, but it itself only denotes ONE object.
In a similar manner 2 + 2 is denoting ONE object.
2 + 2 is equivalent to the expression "The result of summation of 2 and 2"
Or sometimes we express it as "The sum of 2 and 2"
2 + 2 means "the natural number that results from adding 2 to 2"
So '2 + 2' is denoting a single object, although it does contain inside it two occurrences of a denotation, yet it is denoting a single object, similarly 2+3 it contains two distinct denotations, but it itself is denoting one object which is the object that results from adding 2 to 3.
Now whether the SINGLE object denoted by '2 + 2' is itself the same (identical to the) object that is denoted by '4', is something that I personally think it to be the easier and simpler way to formalize. If we say No, the object that 2 + 2 is denoting is different from the object 4 is denoting but it is "equal" to that object, and here equality can be understood as a kind of equivalence relation (a relation that is reflexive, symmetric and transitive), I think this is a more complicated way of looking at it.
Of course the interpretation of = as equality relation is weaker (logically speaking) than interpreting it as identity relation. Many times people prefer or feel more safe with holding weaker assumptions. And so it indeed can be justified as a kind of cautious philosophical approach to the matter. However, I still think that identity, albeit being a stronger interpretation, yet it is much nicer and sharper, and actually much easier formally speaking than the more general equality notion.
Try to formalize PA yourself using "=" as an equality relation. You'll see how cumbersome it would be. Interpreting "=" as identity simplify formalization to a great extent.
Agreed. And I mentioned this to MU. I said that one can indeed interpret the '=' sign as some equivalence relation, no doubt, like that of synonymy, or actually any equivalence relation, of course this can formally work. But the formalization would be more cumbersome, because you are holding to a weaker concept than identity, you'll loose all the merits of identity, which shortens formalization to a great extent. Philosophically speaking one might prefer to hold to the weaker interpretation, but formally speaking, it is not the preferable one. For example how would you DEFINE 4. Using identity I don't need to introduce 4 as a primitive symbol, since I can define it, since it is the unique object that 2 + 2 is denoting. However you cannot define it as such when "=" is just an equivalence relation, you'll need to introduce 4 as a primitive notion, that said you'll need to introduce all naturals as primitive constants of the language, which is in some sense cumbersome.
Quoting Zuhair
And, of course, this re-formalizing is a logical fallacy, equivocation, as I've demonstrated.
Quoting Zuhair
What is on the right side of the equation is not the same as what is on the left. For example, in "2+2", there is as you've described, an "operator" signified, which is not signified by "4". Therefore it is very clear that what is on the right side is not the same as what is on the left side and so to interpret the "=" as signifying "the same" is wrong.
Quoting Zuhair
I don't believe that PA is actually formulated as such. None of the websites which fishfry referred me to, to support this claim supported that notion. Those websites described PA as based in equality theory, not identity. I think that those people such as yourself who insist on this notion are practising sophistry. I said that way back.
Quoting Zuhair
No, I think those who interpret, or "coin" these systems as based in identity are committing a fundamental error. We discussed the axiom of extensionality, it defines equality, not identity. It is incorrect to say that this axiom of equality is an axiom of identity.
Quoting Zuhair
It is clear that "2+2" must be interpreted as denoting two distinct objects or else the "+" symbol is left meaningless. This is obviously a problem for the ontology of mathematical objects. We commonly think that every instance of the symbol "2" would denote the very same object. This would be a Platonic object, a number, symbolized by the numeral "2". But if this were the case, then we'd have a problem interpreting the "+" symbol. You'd have two instances of the same sign, representing the same thing, "2" with the "+" symbol representing a relationship between an object and itself. You cannot add an object to itself, so what could this "+" sign possibly represent as a relationship between an object and itself, other than identity, "2" is the same as "2"? But this is not what "+" represents. It represents a relation between what is represent by the first "2", and what is represented by the second "2", and if there is a relationship between these two which is not a relationship of identity, they must be distinct objects. Therefore what is represented by the first "2" in "2+2" is necessarily a distinct object from what is represented by the second "2".
Quoting Zuhair
But "the result of summation of 2 and 2", refers to two objects. There cannot be a summation without a multitude. You just want to say that "2+2" means "4", but it doesn't, it means something different than "4" and you don't seem to be capable of respecting this.
Quoting Zuhair
You're so ridiculous that it's becoming funny. "The natural number that results from adding 2 to 2" is "4". If someone wanted to say "the natural number that results from adding 2 to 2" they would just say "4". Instead, they say "2+2", because this means something different from "4".
Quoting Zuhair
I think I now see why you lie about this matter. Identity is seen by you as a stronger position. You want your position to appear stronger. So you will continue to lie in the claim that you are in the stronger position, attempting to deceive people into believing that your mathematics is stronger than it actually is.
I take no responsibility for and neither endorse nor necessarily agree with anything written by anyone on this site but myself; nor do I necessarily disagree. I have no idea why you are quoting some other poster's thoughts to me on this subject. I've written plenty to you already that you haven't engaged with, including a proof directly from first principles that 2 + 2 and 4 are identical.
You claimed the other day that the law of identity does not apply to the number 3, or numbers in general. At that point I assumed you've simply given up rational debate and/or recognized the impossibility of your own position. Talk me down please.
all of this is wrong. + is a binary function symbol which means it is a ternary relation symbol, it is a relation between three occurrences of symbols, it relates the first two symbols to a third occurrence of a symbol, here it relates the two occurrences of 2 to a third occurrence of a symbol which is 4, this is explicit when you write it in relational terms as +(2,2,4), but when it is written in functional terms here the confusion would raise since you don't see the third occurring symbol (which is 4) you only see two occurrences of 2 linked by + sign in between, here it means that + is relating the two occurrences of symbol 2 to the symbol '2 + 2', you see here the expression '2 + 2' is acting as a symbol denoting an object of the language.
in your views '2 + 2' represent two distinct objects operated upon by the + operator. While the common view is that '2 + 2' denotes the natural number that results from running the + operator on two occurrences of 2. It is like the expression "The planet between planets Venus and Mars", it does mention two distinct denotations those are the planets Venus and Mars, and it does mention an operator running on them which is the "between" operator. However what it denotes is non of those, what it denotes is the planet Earth, which is ONE object. Notice that there is no symbol or word inside that phrase that symbolize what the total phrase is denoting, however the total phrase itself does denote planet Earth. Similarly '2 + 2' is an expression that mentions denotations of objects by two occurrences of the symbol 2 and an operator running on them, yet the total expression (i.e. all three symbols in 2 + 2 in that sequence) is denoting non of those, what is denoted by the total expression '2 + 2' is a single object that can be what is denoted by '4' if you interpret '=' as identity, or it can be another object that is related by some equivalence relation to the object denoted by 4, anyway the whole expression of "2 + 2" is not denoting multiple objects, no , it is denoting a single object, because + is a FUNCTION.
Just because an expression contains (mentions) inside it different denotations, operators, relations, etc.. doesn't mean that it is denoting those, or that it denotes multiple objects, no it can be using those to denote a single object that is non of them (as it is the case of 2 + 2).
Quoting Metaphysician Undercover
Yes I agree, many of them do it that way. But definitely there are formalizations of PA as an extension of first order logic with identity, but they often don't mention the axioms for identity since they consider it as part of the underlying logic, which in this case it is usually taken to be "first order logic with identity".
For ZFC, it is usually formalized in first order logic with identity, but sometimes formalized using one primitive that is the membership symbol. However most formulations of ZFC are extensions of first order logic with identity. And that suits set theory, since if = doesn't represent identity why should we define 'singleton' sets after the equality relation then?
We went through this. You provided no such proof, it was just an assertion.
Quoting Zuhair
Why the change? We were talking about objects, now you switch to symbols. Let's maintain consistency. We were talking about what the symbols denote. Tell me what "+" denotes, not how it relates one symbol to another. If it denotes a relation, then what is it a relation between, distinct objects, or one and the same object?
Quoting Zuhair
If we have two occurrences of 2, then 2 here is a symbol. That symbol represents one object, the number two. If we have two occurrence of the same natural number, 2, then the only natural number represented is 2. What does your operator do, show that 2 is the same as 2?
Quoting Zuhair
This all depends on what your operator "+" represents. You cannot jump to conclusions without explaining what the symbol represents. We have two occurrences of the same object, 2. We have an operator which expresses a relationship between them. As I told you already, the only reasonable relationship between the same object is that of identity. So is the operator "+" expressing a relationship of identity between the two occurrences of the same object? If not, then I think that the two occurrences of "2" are not symbolizing the same object. If there is an expressed relationship between two objects, which is not an expression of identity, then the two objects must be distinct objects.
I provided a direct proof from first principles. You might have questions or specific objections. But to claim I did not provide a proof means you happened to not see my lengthy post, or ... well I just don't know. It would be helpful if you'd specifically engage with the proof I gave; not claim I didn't give one.
No the operator + doesn't express a relationship between the objects those symbols are denoting, for example lets take the expression "3 + 5" you seem think that "+" here is representing a binary relation between the object denoted by 3 and the object denoted by 5, which is wrong. The reason is because + is NOT a binary relation, it is a "TERNARY relation". Every binary FUNCTION is in reality a ternary RELATION, and + is a binary function. The operator + here is a relation between three objects, one expressed by 3 and the other by 5 and the third by the expression "3 + 5". Let me try to give a helpful analogy that of the relation "son of", when we say for example "Issac is the son of Abraham and Sarah" now the relation "Son of" is a ternary relation, it links an individual to his two parents, so three people are involved in this relation, so for the example above Son is linking Sara, Abraham to the son of Sarah and Abraham. Similarly when we see "3 + 5" we are seeing THREE terms of the language, those are "3", "5" , "3+5", you know that the symbol + is not a term of the language. So the + sign here is understood to be a ternary relation that links objects denoted by the terms of the language which are "3","5","3+5". Each of those terms is denoting a single object, so 3 is denoting one object, 5 is denoting one object, and '3+5' is also taken to denote ONE object (because 3 + 5 is a binary function symbol and so it is a term of the language, so it denotes one object (despite having parts of it that denote other objects)).
With the case of 2 + 2 matters becomes more confusing, here + sign is relation that links the object denoted by 2 to itself and to the object denoted by "2 + 2", so it doesn't just link the former object to itself, No! it links it to itself and to the object denoted by 2+2 which is not equal to 2. This is a little bit confusing. To give an analogy is a little bit more complex. Suppose a country X only allow adoption of a child to an adult if one adult has a job (earns an income) and an adult that know how to work at house (cook, clear, wash,etc..), so in this case it allows it between maximally two adults and one child, "adopted son in country X" is a ternary relation, but it can also occur between two objects sometimes, if a single adult has a job and also is capable of doing house work, so you can have an adopted son of Mrs J and Mrs J, it means he is the son of Mrs. J that earns a job and of Mrs.J that can do house work.
So to be more precise the operator + in "x + y" means a ternary relation between the object denoted by x in the first role, and the object denoted by y in the second role and the object denoted by "x + y"
so + sign in "2 + 2" means a ternary relation that links the object denoted by 2 in the first role and the same object in the second role and the object denoted by "2 + 2".
That's why I was saying that '2 + 2' is a FUNCTIONAL expression of the language, it denotes a single object (because it is a functional expression) even though parts of it (which are 2 in first role and 2 in second role) are denoting other kind of an object, still what "2 + 2" is denoting is something else other than what any of its two terms shown in the expression are denoting.
I think you tend to think that denotation of an expression is the sum total of all denotations of its parts, for example the denotation of expression "The planet between Venus and Mars" in your sense is the total denotations made by all parts of that sentence, now Venus and Mars are parts of that sentence and each is denoting a separate object. Now the total expression (all the six word words in that sequence) is definitely denoting a SINGLE object which is of course planet Earth. However in your sense you take the denotation of the above phrase to mean the set of objects denoted by expressions Venus, Mars, "The Planet between Venus and Mars", so in your sense denotation of an expression is the total denotations made by all denoting parts of that expression. While in my sense I take the denotation of the sentence to mean what the total expression is denoting, which in this case it would be Planet Earth, which is a single object. According to that line of terminology the total denotation of 2 + 2 is of course not the same as the total denotation of 4, that's obvious, because the first 2 is denoting an object but in the first role, the second 2 is denoting another situation which is the same object but in a second role, and the whole expression "2 + 2" is denoting a third object. While 4 is only making one denotation, i.e. of a single object, because it is an atomic expression, it has no denoting proper parts, it has only itself as a denoting part. But here when we say 2 + 2 = 4, we are not speaking about the total denotations involved in 2 + 2 , no we mean what is denoted by '2 + 2', and here it means a single object that is related by the = relation to the object denoted by 4.
In nutshell the + operator in the functional term 'x + y' is a ternary relation between the object denoted by x in the first role and the object denoted by y in the second role and the object denoted by 'x + y'.
The problem was that you were treating "+" as a binary symbol, so you thought that the expression "x + y" doesn't denote an object that might differ from the object denoted by x and that denoted by y. No we have THREE objects in play, and not two.
There is another way of interpreting the + sign which is as a kind of a relation between ordered pairs of the input objects to an output object, so + in the expression x + y only means a relation that sends the ordered pair of the object denoted by x and the object denoted by y, lets symbolize that pair by (x,y), this ordered pair is an object of course, now + sends (x,y) to the object represented by 'x + y'. Now we come to what does the ordered pair means, I'll use the originally posed set theoretic ordered pair of Wiener
(x,y) = {{{x},0}, {{y}}}
So for the case of 2 + 2 = 4, the + operator is the relationship that sends the object
{ {{2},0}, {{2}} } to the object denoted by '2 + 2'.
You see here the + operator is interpreted as a binary relation between an ordered pair of two objects and some output object. But even here it doesn't mean that it is a binary relation between the two input objects, so it is not the binary relation between objects denoted by 3 and 5 in the expression 3 + 5. But it can be interpreted as the binary relation between (3,5) and the single object denoted by 3+5.
Is it possible you missed this?
https://thephilosophyforum.com/discussion/comment/328116
Also you claimed that the law of identity does not apply to numbers; that for example 3 is not the same as 3. Please clarify or retract. Thank you.
ps -- Originally you claimed that in ZFC there are equalities that are not identities. I know of no such instance nor have you presented a single such example. Clarify or retract please. The only examples I know are natural injections, such as identifying the integers with the copy of the integers contained in the real numbers; and in casual contexts where we call two isomorphic groups "the same" when we know that we mean isomorphic. Other than those two contexts, I know of no instance in which mathematical equality is anything other than set identity and logical identity. I have challenged you on this point and found your responses lacking in specificity.
No, I didn't miss it, we went through it already, your first premise is false:
Quoting fishfry
The law of identity states that a thing is the same as itself, not that a natural number is equal to itself. This is the problem, you keep asserting that the law of identity says something about equality, when it does not. It says something about identity. So you wrongly proceed to claim that mathematical principles which say something about equality are base in the law of identity. That is why I kept asking you to back up this claim, that there is a law of identity which states something about equality.
Quoting Zuhair
So it's now becoming clear to you that "3" and "5" in "3+5" denote separate objects, just like "2" and "2" denote separate objects in "2+2".
Quoting Zuhair
Right, we have a multiplicity of objects denoted.
Quoting Zuhair
You are claiming that a "binary function" is an object. I see no justification in this. A function is an activity, or a relation. Each of these may be a property of an object, or a relation between objects, but is not an object itself.
Quoting Zuhair
You haven't justified your claim the "2+2" is an object, nor your claim "+" represents a ternary relation. I think you have fallen back into your habit of lying.
It is a language expression. In principle, it has nothing to do with real-world "objects".
There are rewrite rules that allow for carefully reducing this language expression, "2 + 2" to other language expressions e.g. "1 + 1 + 2". The complete sequence of permissible rewrite operations that demonstrates that the symbol stream "2 + 2" is extensional with "1 + 1 + 2" is called a "proof".
The expression "2 + 2 = 1 + 1 + 2" does not necessarily mean anything outside the system of basic rules that define that system. Seriously, it is completely self-contained with no reference to the real, physical world.
The main problem here, mostly at the term infinite, which is nmpossible for me. It is not unlimited, but it's not limited by. "Not limited by" is much more make sense than "unlimited".
God is infinite? Unlimited? Can do anything? God is unlimited, therefore God can be less/more than God's self? It's impossible. It against axiom: "thing can't exceed beyond thing itself (without additional from outside) - from a glass of water can't be poured into a gallon of water.
I prefer say, God is not limited by.
Agreed. But for those who request some kind of referential interpretation for the symbols, i.e. semantics, it would be nice to try figure that out as I was doing with MU. But essentially you are right. The matter is that arithmetic is nothing but a game played with symbols.
Those are not my claims. Please read about the syntax of first order logic which is the background logic used in foundational systems of mathematics. Please read what it means to be "terms" of the language, and also read about "functional terms" in particular and how to differentiate it from relational expressions. You'll see that there is a difference between a relational expression for example x R y, which means that x bears the the relation R to y, for example 1 precede 2, this is a relational sentence you see two "terms" linked by a relation symbol, here that expression is not a term of the language. The usual interpretation is that terms of the language range over OBJECTS (i.e. elements) of the universe of discourse, while relational symbols do not range over elements of the universe of discourse. That said the symbol 2 is taken to denote a single object in the universe of discourse because 2 is a constant symbol, while the expression "x" is a term that ranges over many objects of the universe of discourse, this means that x can be substituted by many objects of the universe of discourse. On the other hand the relation symbol = is not substituted by any object in the universe of discourse, because it is a relational (predicate) symbol and it is not a term of the language. Now the expression "2 + 2" is by definition of the syntax of first order logic, is considered to be a "term" of the language, because the + sign denotes a FUNCTION, and the rule is that when you have a function symbol F and you have a string of terms (x_1,x_2,...,x_n), then the expression F(x_1,x_2,..,x_n) is considered as a "TERM" of the language, which means that it denotes OBJECTs in the universe of discourse, and moreover if each of the variables x_1,x_2,..,x_n is replaced by a "constant" symbol like for example c_1,c_2,..,c_n , then F(c_1,c_2,..,c_n) is taken to denote a SINGLE object in the universe of discourse. Now addition "+" is considered as a binary FUNCTION symbol, so +(2,2) is considered as a term of the language that denotes only ONE object in the universe of discourse. Now we often write +(2,2) using the infix notation 2 + 2. so 2 + 2 is a term of the language, and so it denotes an object in the universe of discourse. Those are the rules of first order logic.
Any binary function is a ternary relation, please read the syntax and rules of first order logic.
I'm speaking about matters that are standard definitions of syntax of first order logic, its non of my manufacture.
Now if you have two terms x,y, and you have a binary relation symbol R, then the expression "x R y" does NOT denote an object of the universe of discourse, now if you substitue x by some constant a and y by some constant b, then you have the expression a R b, now this expression is something that can either be true or false, i.e. its a proposition, so a R b doesn't denote an object because truth or falsehood is of propositions and not of objects.
But if you have two terms x,y, and you have a binary function symbol F, then the expression "x F y" does indeed denote an object of the universe of discourse, and it denotes ONE object for each substitution of x,y by constant symbols, now substitute x by a constant a and y by a constant b, then the expression a F b (which is the infix form of the prefix form F(a,b)) will denote an object and it is not a proposition, i.e. the expression a F b is not something that can be true or false, notice that for example 2 + 2 is not a proposition since it is not something that we'd say about it being true or false, while for example 1 < 2 is a proposition because it is a relational expression of two constants liked by one relation symbol. Also notice that 2 + 2 = 4 is also a proposition because it contains a binary relation symbol "=" that links TERMs of the language, so 2 + 2 is the term on the left side of = and 4 is the term on the right side of =. IF 2 + 2 was not a term, suppose for example it was a binary relation expression and + is expressing a binary relation between 2 and 2, then 2 + 2 = 4 won't be a proposition because the left side is not a term and any proposition involving a binary relation symbol must have the left and right side of that symbol being "terms" of the language, because relation symbols are symbols that symbolize links between OBJECTs of the universe of discourse, and those objects can only be denoted by TERMs of the language.
To demonstrate this with an example: Take the expression "Mary is the Mother of Jesus and James" this sentence itself is not denoting an object, its denoting the relationship of Mary to Jesus and James, so it's denoting a ternary relation between a mother and two of her sons, and this relationship is itself not an object, and it is indeed a proposition that can either be true or false, so this sentence is an example of a relational expression, it doesn't by itself denote an object. That's very clear. But on the other hand take the sentence "The mother of Jesus and James", here you are seeing a functional expression, now this expressing is DENOTING an object which is Mary, here "the mother of" is a function symbol, and at the same time it is a ternary relation from "the mother of Jesus and James" to Jesus and James. Notice that "The mother of Jesus and James" is not an expression that can be false or true? No, it is a functional expression that is denoting an object (which is Mary) and not a relation between the two objects Jesus and James order for it to be true or false.
I feel that your problem is that you were thinking of the "+" sign as a binary relation symbol linking two terms of the language and so the expression 2 + 3 would NOT denote an object. Which is wrong!
By convention the "+" sign is a binary function symbol linking two terms of the language, and so the expression 2 + 3 would BE denoting an object.
Hope that helps!
Sorry but you've lost my interest, too much lying and too much nonsense..
Quoting Zuhair
I explained the problems with this position. If the numeral 2 always denotes the same object then the only relation between "2" and "2" is identity. So when I write "2+2" both 2s denote the same object and the expression is meaningless unless "+" symbolizes identity. Then "2+2" would say 2 is identical to 2. But "+" does not denote identity, so either the 2's denote distinct objects or the "+" is nonsense.
Can you explain what the "+" denotes in a way which is not nonsensical, adhering to your stated principle that the symbol 2 always denotes the very same object? You now have claimed that "+" is a function which denotes an object. So with "2+2" you have one object, another object, then the first object again. That's meaningless nonsense. Or, "+(2,2), which denotes one object and two instances of a different object, still meaningless nonsense. how could there be two instances of the same object?
Quoting Zuhair
I hate to have to bring this to your attention, but you are getting further and further from showing that "2+2" is the same as "4", because you are making "2+2" more and more complex, while "4" is simple. Clearly they are not the same object, and your demonstrations are simply proving this.
Quoting Zuhair
The problem is that you have repeated stated that "2" denotes an object. Unless the "+" annihilates the existence of the object denoted by "2", to create a new object, then "2+2" cannot denote an object as well as "2" denoting an object at the same time, without contradiction. So if "2+2" denotes an object, by what means is the object denoted by "2" annihilated in favour of this new object denoted by "2+2"? And, if "2" no longer denotes an object its meaning is lost, such that "2+2' can no longer be equal to "4".
That's really strange. Just see the example of 'The mother of Jesus and James", this sentence is denoting a single object that is Mary, also Jesus in it is denoting an object and James too and those objects are different from Mary. Just because the whole sentence is denoting a different object from what some of its parts are denoting, it doesn't mean that it annihilates the existence of the objects denoted by its part. This is like saying if the above sentence denotes Mary the it annihilates the existence of an object denoted by "Jesus", and an object denoted by "James".
I can't see how the analogy is relevant. Jesus and James denote distinct objects. The two 2's denote one object. Even if we take "3+5" which clearly denotes two distinct objects, and say that "+" makes this into one object, this is completely different from "Jesus and James", because "Mary" does not make Jesus and James into one object. "Mary" refers to a completely different object with a relation to both Jesus and James. So if "+" makes "3+5" refer to a single object with a relationship to the objects "3" and "5", like the relationship which "Mary" has to "Jesus and James" how could this object be the same as "8"? Mary is a completely different object from Jesus and James, and not at all equivalent or the same as "Jesus and James". So the analogy really fails.
Do you see what I mean? It's very clear that if "+" makes "3+5" into a single object with the same relation to the two mentioned objects "3",and "5", that "Mary" has with "Jesus and James", this newly created object cannot be said to be the same as "8", because "8" has a completely different relationship to "3+5" from the relationship which "Mary" has with "Jesus and James"..
But the situation here is much more difficult, with a deeper, more fundamental layer of complexity. We were talking about "2+2", which you claim indicates two instances of the same object. This contradicts the law of identity already. So, before there is any point to discussing how "+" makes two objects into one, you need to demonstrate how it is consistent with your principles to treat two occurrences of "2" as denoting two different objects. According to the law of identity, there must be two distinct objects denoted here, but you claim that mathematical principles deny this. So, before you can talk about the "+" making two objects into one, we need to deal with whether "2+2" denotes two objects or one.
Finally you are nearly getting what I mean. Yes exactly I'll re-iterate what you wrote because it captures what I said in a very good manner, but I'll add my words in betwen in two brackets
Quoting Metaphysician Undercover
That's exactly what I mean.
Now you made the quesiton Quoting Metaphysician Undercover
But 8 also refers to a completely different object from objects referred to by "3" and "5", and "8" is also not at all equivalent or the same as "3 and 5". I see the analogy is perfect! Why you say it fails?
(notice that 3 and 5 is not the same as 3 + 5, 3 and 5 is the totality of the objects referred to by 3 and 5, it is not what the + operator sends 3 and 5 to. The totality of the object referred to by 3 and the object referred to by 5 is NOT equivalent to the object referred to by 8, those are different objects, the latter is refers to an individual object, the former refers to a totality of two separate objects, so they are not the same nor are they equal).
You seem to confuse the pair of 3 and 5 which is usually written as (3,5), with (3 + 5), No! these are of course two distinct objects, much as the pair (Jesus, James) is different from (the mother of Jesus and James) are different. The analogy about this point is perfect really!
To complete the analogy:
The object referred to by "The mother of Jesus and James" is Equivalent (or the same as) the object referred to by "Mary".
Permit me to write it using = as:
The mother of Jesus and James = Mary
Now, the object referred to by "3 + 5" is equivalent (or the same as) the object referred to by "8".
so we have 3 + 5 = 8.
I see a perfect analogy, where do you see it fail?
Yes + sends objects denoted by the symbols it occurs between, to some object. The objects denoted by the symbols the symbol of + is written in between (in infix notations) would be sent by the + operator to an object as specified by the rules of arithmetic. Just because + occur between two symbols doesn't mean that the objects those two symbols are referring to are distinct objects No. For example "2 + 2" here the first and second "2" which are linked by + symbol, both of those do refer to exactly the same object, why? because 2 is a "constant" term of the language, so it can only refer to a single object in the universe of discourse. Now the + operator would refer that single object (symbolized by "2") into the object referred to by the symbol "4", that's it. So it referred one object (even though it had double reference, i.e. referred to by two symbols) to an outcome object. Like in saying that "Mary is the mother of Jesus and Jesus", its only saying that Mary is the mother of Jesus, the double presence of Jesus didn't change anything, it doesn't mean that there are two sons of Mary by the name Jesus.
But of course for the analogy of Mary being the mother of Jesus and James, with 3 + 5 = 8, here the analogy in some sense breaks because the mother of Jesus and Jesus is still Mary, while 3 + 3 doesn't equal 8. So regarding this point the analogy fails. Well we don't expect analogies to agree on all points anyway because we already know that the relation mother is not identical with the relation addition.
Hope that helps!
If I use the word same instead of equal does that satisfy you?
The fact that you say you read my post and this is your complaint means we're done. You have no substantive reply? I showed a proof from first principles that 2 + 2 and 4 are identical. You ignore it?
You're not debating in good faith.
You could have said, "Oh I categorically reject the work of Giuseppe Peano and everything he stands for." Or, "I didn't understand the chain of equalities." Or SOMETHING. Anything. But you simply will not engage substantively.
You're asserting a falsehood. There is no case in math of an equality meaning anything other than identity; whether of abstract objects (logical identity) or sets (set identity or equality). Set identity is the same as set equality.
If you don't agree that's your right, even if you haven't and can't show a single example to support your claim. But in all this time you have not presented an argument. And you have never engaged substantively. And from me to you, you're factually wrong. All the best.
ps -- I apologize if this comes off rude or confrontational. I'm genuinely frustrated that you won't engage on the substantive technical points I made. I presented a proof from the Peano axioms that 2 + 2 and 4 are the same thing. Same as in same. Same as in equal. Equal is the same as same. You simply chose not to engage. I find that too frustrating to continue the convo. It's not you, it's me.
It fails because you are arguing that the "+" makes "3" and "5" into one object, an object which is the very same as "8", but there is no way to make "Jesus" and "James" into one object which is the very same object as "Mary". Do you see what I mean?
Quoting Zuhair
I know that, but it's not the issue. The issue is how does the "+" turn two distinct objects into one? And further, how is that created object the very same as "8"? Your analogy does not turn Jesus and James into one object. And even if we could somehow understand Jesus and James unified as one object, this object would in no way be the same as Mary.
Quoting Zuhair
There is nothing in "3+5" to take the place of "mother", there is just Jesus and James. If you remove the "and" from "Jesus and James", and replace it with "mother", you have "Jesus mother James". You have nothing to show whether the mother is of Jesus, James, both, or neither. So it is really the "and" in "Jesus and James" which acts as the function, to show that one object "mother" is related to both Jesus "and" James. So let's replace the "+" with "and", then we have "three and five", just like "Jesus and James", and the analogy is good. What unites Jesus and James into one object, like you claim three and five are united as one object? Clearly it is not "mother". It is some logical principle which allows us to speak of multiple things as one object.
By applying this logical principle we can create objects through some sort of synthetic unity. Consider "2" for example. It denotes an object, a number, but within the meaning of "2", there is two distinct things signified, united within one object, 2. This is a synthetic unity, the object 2, is created by a principle of union which unites two other objects..
Quoting Zuhair
You don't see the problem here? Let's say each "2" in "2+2" denotes the same object. Also, "+" sends objects denoted to some object. We have only one object denoted, "2". There are no other objects denoted, just "2" and the operator "+". If "+" sends "2" to any object other than "2", it could be any object, randomly selected. There are no rules of arithmetic which would allow that the object "2", and the operator "+", could give us an object other than "2".
Quoting Zuhair
This is what I mean, that selection of "4" is a random choice. Why not "5", or "8", or any other of an infinity of possible objects? Why does that + operator send the single object "2" into the object "4", and not some other object?
Quoting fishfry
No, that's the point you cannot validly substitute "same" for "equal". It will produce equivocation. You don't seem to understand this.
Quoting fishfry
You showed no such proof. You showed that they are equal, according to equality theory, and you claimed that equality is based in the law of identity which it is not. Therefore you have no proof. Do you understand? Your proof requires that the law of identity says something about equality, which it does not. Therefore you have no proof.
Quoting fishfry
We're just rehashing the same thing. It may be the case, that in math equality means identity as you claim, but as I explained to you, this is not "identity" as defined by the law of identity. So you (and all other mathematicians who say this) are using "identity" in a way which is inconsistent with the law of identity. Do you understand that the law of identity states that a thing is identical to itself? So to use "identity" as equality and then claim that equality is supported by the law of identity in that way, is equivocation, plain and simple.
Quoting fishfry
My example is the law of identity. It is very simple, and very explicit. A thing is the same as itself.
So for example, let's define equal to, as identical to, as you are wont to do. Then we'll adhere to the law of identity for our definition of identical. Clearly "2+2" is not identical "4" according to the law of identity, therefore "2+2" is not equal to "4" by your own definition of equality. That is a substantive argument if I've ever seen it. Now, Zuhair has wasted pages trying, to no avail, to demonstrate that "2+2" actually is identical to "4". But Zuhair is just digging a deeper hole, recognizing the truth that "2+2" is not identical to "4", neither is what "2+2" signifies identical to what "4" signifies, so Zuhair tends to lie in an attempt to get out of the hole.
Quoting fishfry
How can you say this? All human beings are said to be equal, but no two are the same. Many things are said to be equal which are not the same. Do you recognize this? If so, how can you say that "equal is the same as same"?
Because the rules of arithmetic and and the arbitrary definitions dictates that! I showed you how formally this can run in a prior comment on a system that is by far much easier than PA. I'll re-present it here:
2 is Defined as the object that the + operator would send the single object 1 to. (this is: 1 + 1 = 2)
4 is Defined as the object that the + operator would send objects 3 and 1 to. (this is: 3 + 1 = 4)
3 is Define as the object that the + operator would send objects 2 and 1 to. (this is: 2 + 1 = 3)
Notice that the choice of symbols in those definitions is really arbitrary, there is no control whatsoever on choosing 2 as the symbol for what the + operator would send the object 1 to, nor there is any control on choosing the symbol 4 to represent the object the + operator would send the objects 3 and 1 to, and same arbitrariness apply to choosing the symbol 3 to represent what the operator + is sending the object 3 and 1 to. Yes these are arbitrary. But once made, then we cannot change them, since + is a function, it permits only one outputs per specific input objects.
Now from those definitions and from the rules of arithmetic, it is here where arbitrariness would stop, because we'll be enforced here to say that 2 + 2 = 4, in other words the object the operator + is sending the object 2 to is 4.
Proof:
by definition of 4 we have: 3 + 1 = 4 [this is an arbitrary definition as you said]
by definition of 3 we have: 2 + 1 = 3 [this is also arbitrary definition]
by rule of identity (the substitution schema) we can substitute identicals! so we have:
(2 + 1) + 1 = 4
By associative law we have
(2 + 1) + 1 = 2 + (1 + 1)
by identity (substitution schema) we substitute identicals to have:
2 + (1 + 1) = 4
but by definition of 2 we have 1 + 1 = 2 [arbitrary definition of 2]
by identity (substitution schema) we substitute identicals to have:
2 + 2 = 4
QED
So arbitrariness ends after we have made the choice of the definitions and the choice of the axioms and the choice of the inference rules, after making those a machinery would set in and it dictate how the arbitrarily chose symbols would related to each other.
So YES, definitely part of it is indeed arbitrary, the starting part! Quoting Metaphysician Undercover
Yes, there is a big debate whether this is to be called as a "logical" principle. Nowadays it is generally held to be a mathematical principle. This is the principle of "Set", a set is what turns multiple objects into one entity, it turns Jesus and James into one entity which is the pair {Jesus, James}. But still you are confusing the + operator for "and", you think that the + operator is that joining logical principle that turns multiple objects into a single entity, you are confusing the + operator for the set operator, which is not correct. View the + operator just and a "sending" rule, a rule that sends objects to objects that's all. So for example we can view "Mother" in the above sentence as a "sending rule" it sends the pair {Jesus, James} to another object which here happens to be Mary. EXACTLY a similar thing is happening here the "+" operator is sending the pair {3,5} to another object which by rules of arithmetic and arbitrary definitions this other object is enforced to be 8. The rules of arithmetic (after making the arbitrary definitions of each number) would "control" this assignment (sending) of objects, it would control which object the + operator would send the pair {3,5} to.
Think of the + operator as a sending scheme, that's all.
I didn't say that + would make 3 and 5 into one object, I said it will send them to one object, if I did say that it makes them into one object, then I only meant that it would send them to one object. + is not a merging process, it is an assignment scheme.
Well, that is not completely right. I understand very well that "=" can be understood as equivalence relation, I concede to that. I personally would prefer it to be understood as "identity" relation, since this "at least to me" would make matters easier to control from the formal workup side, identity is a the sharpest kind of equivalence relation. But anyhow it's not necessary to interpret "=" as how it is used in mathematics as the strong notion of "identity". But some theories I think would fare far better if they do that, for example Set Theory, here to say that the set X defined for example as: for all y ( y in X if and only if y=empty set ), this is usually symbolized as {{}} or as {0}. Now to say that this set is a "SINGLETON" set is to say that it has only ONE member, but if we just stipulate that "=" is an equivalence relation that can occur between distinct (non identical) objects, then it would be absurd to label it as singleton since there is no guarantee for it having just one element, it can indeed have MANY elements all being "equal" to the empty set, I would have called it "EQUALTON". So I think in the context of set theory, the equality sign is better to be understood as identity relation. By the way I need to conceded that this is also not necessary 'technically speaking' since indeed we can take equality to be just an equivalence relation, but to me that would make matters murky on both the informal intuitive and the formal technical workup accounts, i.e. I mean as far as set theory is concerned.
For the more mathematical looking foundational theories like PA, of course it suffices to interpret equality as just being an equivalence relation, and that it can hold between distinct objects, that might give us some room of freedom in making some interpretations, actually PA is presented officially with its equality symbol just stipulated as an 'equivalence relation' plus some simple closure principle on naturals, so indeed it is officially treated just as an equivalence relation closed on naturals.
There is (implicitly) "sum of". (Not that the analogy follows through completely, as @Zuhair points out.)
But anyway, instead of trying to clear up his actual or perceived misunderstandings about out-and-proudly platonic math concepts, shouldn't the mathematicians offer the finitist (especially since he objects to the identity of the 2's in 2+2) cardinal arithmetic and see if he is satisfied with that?
No you did not show me that. You showed me "ternary relation". You did not show me how one object and the "+" function makes a completely different object.
Quoting Zuhair
See, you have inconsistency here. First you are talking about sending one object to a completely different object, then you are talking about turning two objects into one object.
Quoting Zuhair
OK, but this does not satisfy what is required in order to say that "3+5" denotes the same object as "8", which is what you are arguing, that these two are identical objects. All it shows is that the two objects "3" and "5" are related to a third object "8", like Jesus and James are related to Mary. It does not make Jesus and James identical to Mary. Your example introduces a fourth object, " the mother of". Mother is not a sending rule at all, it is a named object.
Quoting Zuhair
The debate is whether "2+2" denotes the same object as "4". If the sending rule only "sends" "3+5" to the third object "8", and does not make those two objects into the object denote by "8", it does not fulfil the requirement of saying "3+5" denotes the same object as "8".
Quoting bongo fury
The analogy is ill, and unacceptable. What acts as the sending operator in "Mother of Jesus and James", is the :"and". It sends both "Jesus" and "James" to the same object, "the mother of". Zuhair wants to make "mother" the sending operator, sending "Jesus" and "James" to "Mary". In reality though, "Mary" is just another name for the object called "the mother of" and the analogy is way off base.
Quoting Zuhair
What are you saying, that an empty set is the same as a set having only one member, "0"? You know that 0 is a mathematical object, just like any other integer don't you? therefore the set is not empty.
As I say, though, at what point does arithmetic become a philosophical puzzle for you, or your child?
Have addition as union of disjoint sets. Is that ok?
It was a typo, a Freudian one if you like since I agree that this particular error makes you totally right and me totally wrong. I get that.
From my end it was a totally honest error. As a math person I'm so used to talking about equality that I never think about logical identity. I'm learning a lot from this thread. I typed "equal" because I've done it a million times in math; and the only time I talk about an identity is something like [math]e^{i \pi} + 1 = 0[/math]. a specific numeric and/or algebraic identity meaning it's true by virtue of syntax. It's a logical truth. But as I think of it, that's logical identity too.
More than ever I see that mathematical equality is the same thing as logical identity. The same morally and the same technically in any mathematical framework you like.
By the way if called on to do so, I could drill that symbology down to an identity of sets. The thing on the left and the thing on the right are the same thing. It was true even before Euler discovered that fact in 1740. At that moment it became new knowledge; but it did not suddently become a new truth about the world. It's a mathematical truth. It was always true.
And I do acknowledge that my fingers typed equality when in the context of the discussion I should have used the word identity; and that this purely random or perhaps Freudianly determined; either way, it was one hell of a bad typo. I see that it generated a lot of confusion. Please just substitute "identity" or "logical identity" in my argument. My apologies.
To sum up my view I would say that
* On the philosophical aspect, I suspect I have much to learn. I'd be surprised to find out that logical identity is somehow different than mathematical equality; but I would not be surprised to be surprised.
* On the math aspect, you're just wrong. But neither of us has said anything new for quite some time, and I have nothing to add. Only that I'm disappointed at a personal level that I took the trouble to work out an immaculate technical proof; and you are just totally disinterested in actually following and engaging with the argument. It's your privilege not to engage, but that is definitely a disappointment at my end. I'd say to myself "Pearls before swine" but I'd never say such an uncharitable thing in public. But the phrase did pop into my head, and that does about sum up how I feel about the mathematical aspect.
I actually just popped in tonight to mention that someone, possibly a member of this forum, posted the following to philosophy.SE today:
https://philosophy.stackexchange.com/questions/67227/is-there-a-difference-between-equality-and-identity
Some interesting thoughts there.
Really? With a child, discussing how the set of 2 pens here plus the set of 2 pens there makes a set of 4?
Wouldn't you want to be ready to climb down from platonist notions or foundations ("2 on the number line", or "the class of all pairs" etc.) and agree that the two separate concrete pairs of objects were being compared and found "equal" in cardinality or size, just as two pens might be found equal in weight, or in length? In other words, equivalent, and in the same equivalence class by this or that mode of comparison (in this case cardinality)? But obviously not identical?
Or would you want to get them with the platonist program straight away, and make sure they understood that 2 on the number line "sends" with itself in a two argument function returning at 4?
Notice they will soon learn to equivocate anyway between identity and equivalence, like any good mathematician not presently embroiled in philosophical or foundational quandary.
Not that @Metaphysician Undercover will be happy with any cavalier embrace of equivocation.
Quoting fishfry
Yes, the irony... that competence in maths should not only involve easy equivocation imputing (with the equals sign) absolute identity here and mere equivalence (identity merely in some respect) there, but then also involve an "identity" (e.g. site menu) sign meaning only a batch-load (for all values of a variable) of cases of "equals", the latter still (in each case) ambiguous between identity and mere equivalence! (The ambiguity removed only by a probably unnecessary commitment to a particular interpretation.)
I think we're making some progress, and I think this is the heart of the disagreement between us. So maybe we could start with this statement about "symbology". Do you agree with a distinction between the symbol and what is signified by the symbol? The symbols themselves are objects which we read, and these objects, the symbols, have differences between the right and the left, such that they are not the same symbols.
Furthermore, according to the law of identity, two instances of what we call 'the same symbol", are not actually the same. So this "S" is not the same as this "S" (by the law of identity), considering them both as objects. So when we use "same" in this way, to say that they are the same symbol, we are using "same" in a way which is not consistent with the law of identity. This sense of "same" is based in some principle of similarity, not in the principle of identity. As a consequence of this, we have instances where the same word (as "the same symbol") has a different meaning, depending on context. And equivocation can result. This is clear evidence that the phrase "the same symbol", uses "same" in a way which violates the law of identity. Do you agree that this sense of "same", by which we say that this "S" is the same symbol as this "S", is inconsistent with the law of identity?
Quoting fishfry
I would have to find out what you mean by "logical identity". In general, the sense of "identity" employed by logicians is not consistent with the sense of "identity" described by the law of identity. If you read Stanford on "identity", you'll see two distinct senses. One they call qualitative identity, and the other they call numerical identity. Numerical identity is what is described by the law of identity, identity is specific to each and every thing, a thing is the same as itself. "The same" means one and the same thing, and this is identity. Qualitative identity identifies through qualities, a description. If two things have the same description, they are treated as the same. This sense, qualitative, is the sense generally employed in logic, because logic proceeds with descriptive terms, and predication. But a careful logician will recognize the difference between subject and object, and that predication deals with subjects rather than objects. The law of identity deals with objects rather than subjects, while the logician deals with subject. So when the logician employs "logical identity" it is an identity based in qualitative identity, (the subject is identified through some sort of description), rather than numerical identity which is what the law of identity deals with.
Quoting fishfry
I told you the problem with your proof. The very first premise is false. So it doesn't matter how immaculate your proof is, it is unsound by that false premise. There is absolutely no sense in me wasting my time following all the points of your argument when you start with a premise which is clearly false. Until you fix that premise, or show me an argument which does not require it, there is no point.
Quoting bongo fury
That pretty much sums it up.
I thought we got over that point. I agreed with you that "=" is NOT necessarily the identity function, so why you are returning the discussion backwards. I agreed with you that if you interpret "=" just as an equivalence relation (as it is officially formalized in PA for example), then of course the object that the + operator send objects denoted by 3 and 5 to, is NOT necessarily identical with the object denoted by 8. We already passed this point. The debate now is not about that. The debate is about what is the operator +. To me it is nothing but an assignment scheme, i.e. a sending rule, nothing more nothing less, it sends maximally two objects to a third object. Actually although I don't want to go there, one of the intended interpretation of arithmetic is as a closed syntactical system, i.e. non of its expression denotes anything external to it, so for example under that line of interpretation the symbol 2 means exactly that symbol itself, and so for example 2 + 2 has "distinct" symbols on the left and right of the + sign, and although they are "similar" in shape, yet they are two different objects since they occupy different locations on the page, each 2 is denoting itself only. Now also 4 denotes itself only, also to further agree with you 2+2 is denoting nothing but itself (the totality of the three symbols) and so it is NOT the same as 4, not only that every individually written 2 is not the same (identical) to the other, and the equality in 2+2=4 doesn't entail at all identity of what is on the left of it with what's on the right of it, its only an equivalence syntactical rule, and can be upgraded to a substitution syntactical rule without invoking any kind of identity argument at all, and the whole game of arithmetic can be understood as a closed symbolic game nothing more nothing less. This is the extreme that one can go with interpreting equality as just an equivalence relation and not being identity, we'll need to revise our definition of "constants", "functions" to accommodate that. But still we need to maintain that expressions like 2 + 2 denotes an object while expressions like 2 > 1 denotes relations (linkages) between objects and such that expressions like 2 + 2 cannot be labeled as true or false since they are by the rules of the game not propositions, while expressions like 2 > 1 are propositions and they are to be spoken about of being true or false.
I know you agreed to that, but I could trust you because of propensity toward lying. And, you've continued to argue that "2+2" is the same (in the sense described by the law of identity) as "4". So your agreement appeared to be worthless in that matter.
Quoting Zuhair
OK, so I made my point, you now agree with me, and perhaps we don't have anything more to debate.
Quoting Zuhair
I really do not know what you mean by "sending" "two objects to a third object". You seem to think that sending is not a relationship, so what is it? How are Jesus and James "sent" to the Mother, when their true relationship to the Mother is that they are from the Mother?
Quoting Zuhair
OK, I can agree with this.
Quoting Zuhair
See why I accused you of lying when you were arguing something opposed to this? I knew you were intelligent enough not to actually believe what you were saying.
Quoting Zuhair
Why must "2+2" denote an object? Each symbol, "2" is itself an object. The two 2s are distinct symbols, distinct objects. Now the question is what is denoted by each of these symbols. In common language, a word has different meanings depending on the context, and this is how we know, without a doubt, that the different instances of what we call 'the same symbol", are actually different symbols, having different things denoted by each of them. Perhaps, the first "2" denotes something different from the second "2", and the "+" denotes a relation between these.
I agree that "2+2" on its own is not a proposition, but "2+2=4" is. So what is "2+2" on its own? It is just a part of a proposition, terms which need defining. If we add "=4" we complete the proposition and give some meaning to "2+2". Now, consider that as human beings, we require a proposition to identify an object. Yes, it is true that the object has an identity distinct from that given by us, and that is what is stated by the law of identity, (that the identity of the object is proper to itself), but we as human beings also want to give the object an identity, for our sake, and this cannot be done without a proposition. Therefore "2+2" cannot identify an object, because it is not a proposition, but "2+2=4" may be a proposition which identifies "4" as an object.
Now each of the 2s need to be defined, or identified, So we can define one 2 with "1+1=2", and the other 2 with "1+1=2". What defines "1" though? We have four different 1s here. In general, "1" signifies the fundamental unity, an entity, or an object, and each application of the symbol "1" is use to signify a different object. That's how we count, continually adding a new and different "1". Each instance of "1" is itself a different symbol, a different object, also denoting a different object, allowing us to count a multitude of different objects. And this is clear evidence that each instance of a symbol like "1", or "2", or "3", signifies a different thing, otherwise when we count, by continually adding "1", we would just be counting the same object over and over again and the count, the total, or the summation would be invalid because it requires that there are actually that many different things, for the total count to be valid.
No problem with two 2's in 2 + 2 being denoting different objects, since they can be interpreted as denoting themselves and they are of course distinct. Now the + sign is just a symbol here that we are seeing it between these two symbols combining all three to get 2 + 2 Now this is by itself as you agreed with me is not a proposition, clearly it is not something that we'd label as true or false. Now generally speaking when we are in a mathematical language we must specify which symbols are taken to refer to objects (even if to themselves) which we call as "terms" and which symbols are taken to refer to "relations between objects" we call them "predicate" or "relation" symbols. Now not every string of symbols constitute a statement of the language, and what we mean by statement of language is actually a proposition, something that can be said of as being true or false. The smallest kind of propositions c are constituted of a relation symbol and all terms that it relates, so if for example you have R being stated as an n-ary relation symbol, so the string of symbols expressing that R would be to concatenate it with n many term symbols in some specific (prefix, infix, etc..) so if R is a binary relation then we concatenate it with two symbols, and so on... which only makes sense because we think of relations as relations between "objects" and objects are represented by "terms" so for a binary relation symbol R we have propositions of the general syntax of:
term R term
In nutshell relation symbols link terms. So for example = is a binary relation symbol, so it must occur between two term expressions, i.e. expressions taken to represent objects. Lets take (2 + 2 = 4)
Now for = to be a relation symbol it must occur between terms, so the totality of whats on the left of it must be a term and so is what's on the right of it, 4 is clearly a term, so 2 + 2 must be a term, otherwise if 2 + 2 doesn't signify a term (i.e. a symbol referring to object) then what = is relating to 4? either 2 + 2 is a relational expression (similar to 1<2) but those are not put next to relation symbols, image the string
1< 2 = 4, it doesn't have a meaning, it is not a proposition, or 2 + 2 might be neither a proposition nor a relation symbol, but this is like for exame 2+ = 4 here "2+" is an example of a string that is neither a term nor a proposition, it even cannot be completed with =4 to produce a proposition.
In order for "2+2" to be completed with "=4" to produce a proposition, then 2 + 2 must be a term of the langauge, and thus denoting an object, even if that object is the string of the three symbols itself!, otherwise we cannot complete it by adding to it a relation symbol and a term after it.
Notice that not every string of symbols in a language are taken as well formed formulas of that language for example 2 + 2 = is a string of symbols, it is also incomplete, it doesn't represent a term nor a relation, even though it is composed of two terms (the "2") and another term (2+2) and a relation symbol =, but here it doesn't constitute a proposition and it is not itself denoting a term. When you add 4 to it of course it becomes a proposition. So not every part of a proposition is a term or a proposition, examples are 2+, +2, 2=, =4, etc.. all are neither proposition nor terms
2 + 2 is definitely not expressing the occurrence of a binary relation between the two 2's, otherwise it would have been a proposition and we know that 2 + 2 is not a proposition, we know that 2 + 2 = 4 is a proposition and we know that = is a binary relation sign, and we know that it only links terms, so 4 must be a term and 2 + 2 must be a term, and so 2 + 2 must denote an object, now the first 2 denote itself, the second 2 denote itself (those are distinct objects occupying different places in a written expression), and the string 2 + 2 is itself also denoting itself, so the total denotations involved in 2 + 2 is three kinds of denotations each of the 2's denoting themselves and the total expression 2 + 2 also denoting itself, while the expression 2+, +2 , do not denote neither a term nor a relation they are incomplete expressions.
The equality sign = only says here that the object 2 + 2 is equal (i.e. related equivalently) to the object 4, of course they are indeed distinct since 4 is clearly distinct from the symbol 2 + 2, yet they are equal.
2 is referring to an object (which is itself here), but to identify it in relation other symbols by using the particulars of a certain language (for example in arithmetic those mount to +,x,=,< etc.. symbols) then we'll need propositions, but those can only occur by relating it by a relation symbol to other term symbols so 2= 1+1 won't have any meaning if 1 + 1 was itself not a term of the language denoting some object (which can be taken here to be the string 1 + 1 itself), otherwise if 1 + 1 is not an expression denoting an object (i.e. a term) then how can we related 2 to it via the equality symbol = which is a binary relation symbol (sometimes called two place relation symbol), the whole string of symbols would be meaninging much like writing 2= 1<3 i.e. 2 is equal to (1 being smaller than 3), this is meaningless, it is not a proposition, same if we say 2 = 1 + 1 and envision 1 + 1 as a relational expression expressing a binary relation + occurring between 1 and 1, then we be saying ( 2 is equal to (1 having + relation to 1)) which is meaningless because an object is equal to an object and not to a relation. While if we treat 1 + 1 as a term of the language, lets say it denotes itself, so here 2=1+1 would be (2 is equal to the string 1+1), and this makes sense, since this equality is just a syntactical equivalence.
I think, if the "2" is denoting itself then it is not denoting anything, it is simply a 2. If we say that it is a symbol, and therefore denoting something, then to say that it denotes itself is nothing other than to say that it denotes nothing. A symbol which denotes nothing is not a symbol, so the "2" which is supposed to denote itself is simply an object, denoting nothing.
If we bring this object into a logical operation, it is now a subject. It is a subject because we can move it around at will, use it as we please, it is subject to the will of the logician who uses it. What the subject "denotes", is dependent on how the logician uses it. and this is determined by definitions. As denoting something, the subject is a symbol, and it may denote anything, object, relation, etc., but in logical proceedings it need not represent anything..
Quoting Zuhair
A "term" which refers to itself, is not acceptable to me. It is not really a term as per the definition, but a subject employed for trickery, deception. "Term" as per the definition requires that the symbol represent something, and to represent itself is to represent nothing. So the use of a "term" which represents itself is a ploy to avoid the restrictions of the definition, which dictates that the term must represent something.
Quoting Zuhair
I see a problem here, the possibility of category error due to confusion between the object which is the symbol (term), and the object which the symbol denotes. You say that relation symbols link terms. Therefore, what is related are the terms, the subjects. And, we must remember that it is not necessary that the subjects represent objects, because the trickery employed which allows that a term represent itself, such that the represented object is really just the subject representing nothing.. This allows that the mathematician may be just playing with subjects, establishing relations between terms which do not represent anything, just like logicians do, but it is in defiance of the definition of "term". The definition of "term" disallows this, but the crafty mathematician has found a loophole.
Quoting Zuhair
I agree with this, except that the term must denote an object, as I described above, the mathematician has found a way to employ terms which do not denote objects. Let's just say that the term is a subject, which is a special sort of object, one which is subject to the will of the logician, and it need not denote anything. I'm sure you've seen examples of formal logic, expressed in 'symbols' which do not denote anything. These examples are used in teaching, to demonstrate the logical process. The process is shown using terms which do not denote anything. Mathematics is supposed to be more rigorous, requiring that an object be represented. It is intrinsic to mathematics that objects be represented because if no objects are represented the distinction between numbers is meaningless.
Quoting Zuhair
So, lets start from the beginning, and enforce the rule of definition, in a rigorous manner. If "2" is a term, it must denote something. What it denotes must be something other than itself. This means that we must assign meaning to "2". That requires a proposition. We cannot just assume that "2" denotes itself, in order that it's properly a term, we must say what it denotes. I suggested "1+1=2" as a proposition which defines what "2" denotes.
Quoting Zuhair
Do you realize that this is all one sentence? Maybe you could express it in a more comprehensible way? As I explained in the last post, if "4" is defined by the proposition "4=2+2", and "2" is defined by "2=1+1", then we must turn to the definition of "1+1", as a term, to make "2" intelligible. And, it is necessary that each time the symbol "1" appears it denotes a different object, or else "1+1" is unintelligible.
What you are saying is definitely intelligible and sane, there is no problem with it. Although I might disagree with you about self referring terms, but no problem really. We can take terms to be "denoting" objects OTHER than them, no problem at all. So we must have a universe of discourse those expressions are speaking about. Anyhow. That won't change matters so much, since it is agreed that a relation symbol symbolizes a relation between the objects denoted by term symbols that this relation symbol syntactical is coupled with. To give an example of that, lets take the relation symbol "=" denoting equality, here = is a binary relation symbol, so it symbolizes a relation occurring between the objects denoted by the symbols that the = symbol links. Which symbols the = links, the answer is that it links the expression 1 + 1 to the expression 2, so the = sign here represented an equality relation occurring between the objects denoted by these expressions. It is always the case that relations are between objects, and so relation symbols must link terms, because terms are the symbols that denote objects, this is because the symbolization must copy what is symbolized. Since equality is a binary relation between objects, then the symbol for equality, which is "=", must be written as linking symbols that denote objects, since = links 1+1 to 2 then 1+1 must denote an object, and 2 must denote an object. That's why 1+1 must be an expression that denote an object.
In reality 1+1 is a tricky expression, it has many denotations, let me present those
The first 1 denotes an object
The second 1 denotes an object
The string 1+1 denotes an object
[All these three objects denoted can be distinct, since equality is not necessarily identity]
The + sign is denoting a ternary relation that is occurring between the above three objects.
[Imagine that like the the expression "the mother of Jesus and James" here Jesus and James are denoting persons, the whole expression is denoting another person "Mary", "the mother of" is denoting a relation between objects denoted by Jesus, James and by the total expression above.
Now lets take the expression 1 + 1 = 2
Here we have all of the above four denotations, and to it there are the following denotations
2 is denoting an object [which can be distinct from all of the above denoted three objects]
= is denoting an equality relation between the object denoted by 1+1 and the object denoted by 2.
That's a fantastic explanation of Formalism. I know that you don't like it, well, but by the way its really a nice account explaining my intentions. Yes the whole of arithmetic can be interpreted as just an empty symbol game, and saying that a symbol represent itself is next to saying that it is not representing anything, I agree. You may say an empty symbol is not a symbol, well its a character and that's all what we want, we may call it as "empty symbol", its a concrete object in space and time (even if imaginary) and it serves its purpose of being an "obedient subject" to the wimps of logicians and mathematicians. I really like it.
This is indeed a plausible stance! But I think formalists won't agree. I still think that we can have distinctions between numbers even if they are meaningless. There are still meaningful matters that the formalist would hold to, that is the deductions carried in the system, those are non-trivial pieces of knowledge. But again it would be difficult for the formalist to account for the success of some mathematical disciplines in science and various applications, that's where it hurts when it comes to formalism.
OK, now we're back to the same problem. In "1+1" each "1" is a term denoting a distinct object. Therefore there are two objects denoted. How does it come about that "1+1" denotes a single object, as a term in itself?
Quoting Zuhair
I don't see that this is "always the case". Why can't a group of terms be related to another term through the same relation, like in your analogy, a group (Jesus and James) are related to Mary?
Quoting Zuhair
So the third is the one which needs to be justified. How does "1+1" denote a single object?
Quoting Zuhair
I explained why this analogy doesn't work. "Jesus and James" is analogous to "1+1". "The mother of" is a term, denoting an object, a person "the mother", who has an implied relation dictated by the definition of "mother". It does not denote a relation, it denotes a person (object) who has a specific relation.
Quoting Zuhair
Hey thanks, I'm glad you liked it. Here's the problem though. In logic, we can learn the logical procedures, by playing the "empty symbol game". It is useful for that purpose of teaching procedure, but beyond that mode of practise, it's useless, like an activity not being applied, what Wittgenstein calls language which is idol, or on vacation. To be useful there needs to be substance, subject matter, the symbols must be applied (represent something), to have meaning, and allow the logical proceedings to actually do something.
Arithmetic is different though. It is based in symbols and relations, rather than procedures. So the foundations of arithmetic involve symbols (1,2,3, etc,) which represent objects and the relations between these objects, relations which are determined and inherent within what the symbol represents (its object). This is somewhat different from the foundations of formal logic, which involve what we can and cannot say about an object.
Because of this difference the notion of an "empty symbol game" in mathematics is illogical. Arithmetic and mathematics are structures of coherency, every symbol has its place in the structure according to what it represents, so that an empty symbol would not fit into the structure at all, having no place. Imagine if there really was a symbol in arithmetic, like "2" for example, which represented nothing other than itself. It could have no relationship with any other symbols like "1", "3", "4" etc. because if it did have such relations with other symbols, they would have to be inherent within what is represented by "2". Then '2" would not just represent itself, it would also represent these relations, as "2" actually does.
Think of the object represented by "the mother of...". Not only is a particular object represented, but inherent within that way of representation (how the term is defined) is also its relation to other objects. This is the way it is with the terms of arithmetic, inherent within the object represent by "2" is all the relationships with other arithmetical objects by the coherency of the definitions. This is very important in geometry because some fundamental definitions or axioms, (like the circle has 360 degrees for example) are very arbitrary. If all the definitions which follow, and build up the structure based on this axiom, are not consistent and coherent, then the conceptual structure is useless. The object represented by the symbol "circle" might be completely arbitrary, but it is necessary that there is an object which is consistent with the objects represented by the other terms, thus making the symbol useful, or else everything is incoherent and meaningless.
I read the whole of your message about why symbols must denote other objects etc.. Its nice and very DEEEEEEEEEEP, indeed; and I won't differ with you about those points (nor necessarily fully agree with them). I'm concentrating on the conventional terminology specially the syntactical ones.
To many mathematicians (well at the least those who stick to formalism) they are willing to understand mathematics as just a game of empty symbols because that would figure out the deductive streaming in those syntactical games we call as mathematical systems. So we have games with specified rules, we cannot go against them because these rules are fixed, and all of what's in the game is in reality "subject" to these rules.
Now one of the rules is to regard an expression like 1 + 1 to be denoting a single object. There is no justification for that at all, it is a rule of the game, you might object to it being unsubstantial, etc.., yes you may, but that won't change anything, the rules of arithmetic stipulate that 1 + 1 IS a functional expression, and a functional expression always denote an object, whether it does it in a substantial manner (i.e. denote an object other than itself) or whether it does it in vacant manner (like it being denoting itself), it doesn't matter, in both scenarios its a fixed rule of the game that 1 + 1 refers to some object.
The reason is because the "+" operator is stipulated before-hand to be a primitive "binary FUNCTION symbol" And by fixed rules of the game of logic and arithmetic when an n-ary function symbol is coupled with its n many arguments in a formula (which must be terms of course) then the *whole* expression is taken to denote some object (that is besides the objects denoted by its arguments which are shown in the formula). So binary functional expressions for example have 3 object denotations and not two as it appears, in general any n-ary functional expression has n+1 object denotations, although you will only see two objects (arguments) written in the expression. For example lets take the successor function S of peano arithmetic, here S(2) means "The successor of 2" which is a functional expression, you have 2 denoting an object and you have the whole expression i.e. "S(2)" also denoting an object, which is equal to 3. But you don't see it in the expression. Actually this is the main difference between "functional" and "relational" expressions. In a relational expression you see some n-ary relation symbol with n-many terms and this would be a proposition, and there is only n many object denotations, it would be a proposition because its speaking about a relation between those n many objects. While with functional expressions you'll also see an n-ary function symbol with n-many terms, BUT this won't be a proposition, because its not speaking about a relation between n many objects, no, actually its referring to an (n+1)_th object linked to the n many objects symbolized by the n many term symbols you see in the expression, so the total number of denotations is n+1 many object denotations and not n many object denotations as it is the case with relational symbols.
If 2 + 3 was denoting a relation between 2 and 3 and that's it, then it would be a proposition, because either 2 has the relation + to 3 or it doesn't have it, one of these two situations must be true, so it would denote a proposition, but clearly this is NOT the case, we don't deal with 2 + 3 as a proposition at all, we don't say it's true or false, so 2 + 3 must not be something that denotes a relation occurring between two objects, so what it is then? by rules of the game 2 + 3 is short for "the result of addition of 2 to 3" that's what it means exactly, and so 2 + 3 is referring to an object resulting from some "process" applied on 2 and 3 and that process is addition, that's why we call it as a functional expression, because its there to denote something based on a process acting on its arguments, and not to depict a relation between the two objects denoted by its arguments.
I think that's the best I can do in explaining common usage of these expressions.
I'm only explaining the rules of the syntax and its relation to denotations, which is something controlled by the rules of the game. See: rules of syntax of first order logic
You need to review these rule for yourself and see whether I'm telling the truth or not, since I know that you don't trust me!
Good news. I'm working on a reply in case it takes a while. I do think you're failing to distinguish between:
* The philosophical question; and
* The mathematical question.
When you send me to SEP and make subtle (and interesting!) points about the nature of identity, that is part of the philosophical problem. About which I have already stipulated that I'm ignorant and open to learning.
But on the mathematical point, you still won't engage and that's still frustrating to me. You reiterate that my first premise is false but I thought I clarified that. Could you please repeat exactly what I said that you think is false? I'd like to respond but I am actually not clear on what you're referring to. I already acknowledged using the word equal when I meant to write identical at some point.
I'll spend some more time reading your interesting post. You make a lot of good points about symbology. I'll give it all some thought.
I'm a Platonist when it suits me, and a formalist when that suits me. My derivation of the theorem 2 + 2 = 4 in Peano arithmetic was purely a formal exercise. When doing math it's helpful to think like a Platonist. When doing mathematical philosophy it's often helpful to think like a formalist. I'm a conceptual pluralist in that way.
I'll surely grant your point that two oranges are different than two fish; and that each pair is an instantiation of the abstract concept of two. But I am being careful to not talk about the world at all. Frankly I am not trying to convince anyone that two pens plus two pens is four pens. I take no interest in such mundane applications!
My only hard claim in this thread at this moment is that 2 + 2 = 4 is a theorem of Peano arithmetic. I can base the rest of my argument on that. But in the end its a formalist argument. I have no idea how many pens is two pens plus two pens. Maybe they're four for the price of three. I have no knowledge of such things.
Quoting bongo fury
We don't burden young minds with higher mathematical abstractions. In fact that was the great failure of the educational fad of "new math," which was coming into vogue around the time I was too old to be scarred by it. After that failed they tried "new new" math, and some other things, and now they've got Common Core about which I hear awful things. I simply am not discussing early math pedagogy. I'm not discussing the subject of what we should teach children about numbers.
Quoting bongo fury
No I don't agree at all. Most people will never care one way or another. And you have snuck in a word NOBODY is talking about, equivalence. In math there can be equivalences between very different things. So that's a red herring, a distortion of the argument. No mathematician obfuscates identity versus equivalence. On the contrary, mathematicians are very precise about the distinction.
Quoting bongo fury
I am not equivocating anything. In set theory, 2 + 2 and 4 are strings of symbols that represent or point to the exact same abstract mathematical object. That's a fact.
Quoting bongo fury
You're making strawman attacks, assuming facts not in evidence and imputing error to others when in fact your own thinking, or at least your writing, is muddled. I could not track that last paragraph.
Some mathematicians are incredibly careful and thoughtful about these issues. See Barry Mazur's famous essay, When is One Thing Equal to Another Thing?
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
You know that scene with the cinema queue in Annie Hall, where Woody is lucky enough to (ahem) marshal Marshall McLuhan against the bloke quoting him? I'm seriously tempted to ask Professor Mazur to see if he can read my previous post, which has so annoyed you, without recognising it as a passable expression of ideas to be found (to my delight and no surprise at all) in the first section of that pdf (for which I'm grateful).
I feel badly misunderstood, but hey, this is the internet...
Another reason to forego point by point corrections to your post here is that I wanted in the first place to shorten the thread, not lengthen it. As I dared to remark right away, between the vast magisterial tracts talking straight past each other,
Quoting bongo fury
IOW, why not be...
Quoting fishfry
And, as @Terrapin Station deserves credit for often saying (and on this at least we should take notice), "one thing at a time, please".
Well you'll need to justify this claim. I've never seen it stipulated that the "+" is a binary function symbol. Nor have I seen it stated that when a binary function symbol is used with two terms, that the whole expression must be taken to represent one object. That such and such convention interprets things in this way does not mean that this is a fixed rule of mathematics. It is just one of many possible interpretations, and interpretations are often illogical or incoherent.
Regardless, we seem to agree that the two 1s in "1+1" represent distinct objects, so what needs to be explained is how this function "+" makes these two objects into one. We cannot just stipulate that these two objects are one, because that would be contradiction, so the function must do something to avoid such contradiction. The function must represent a process which makes them into one.
Quoting Zuhair
I agree that "+" cannot denote a relation. It must denote a process, or function, as you call it. But I disagree that it signifies "the result of addition", it signifies the process of addition itself, not the result of the process. So in your example of "2+3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+". There is no result of this process denoted, and therefore no third object denoted, just the process. Perhaps the third object you had in mind is "5"? Then wouldn't you say that "2", and "3", along with the process of addition results in "5"? But you really have no result of the process of addition until you state the sum. The process is something carried out by the thinking mind, not the symbol itself.
Quoting fishfry
You know this is a philosophy forum don't you? So it's likely that you should expect that we are discussing a philosophical issue. If you want to discuss a mathematical issue, maybe a different forum would be better.
Quoting fishfry
This is the false premise you stated:
Quoting fishfry
That is not the law of identity. The law of identity is the philosophical principle which states that a thing is the same as itself. In mathematics there are theories of equality, and perhaps axioms of equality, but these are not laws of identity. So what I was asking for was if you know of a law of identity which states that things which are equal have the very same identity.
Quoting fishfry
It's one thing to make these sorts of assertions, but another thing to justify such claims. This would require showing how this string of symbols "2+2" denotes the exact same abstract mathematical concept as this symbol "4".
Those are present in Peano arithmetic in a very clear manner. You can review a full treatment of them. That they are not fixed rule of mathematics, might be, but they are fixed rules of first order logic that function symbols represent an object and these can take complex form and not just the constant or the unary form.
Quoting Metaphysician Undercover
Of course + denote a process, more precisely a "function", of course it doesn't denote the result of a process, I never said that. What I'm saying is the whole expression of "1 + 1" is what is denoting the result of a process, and for that particular string it denotes the result of adding 1 to 1. I'm not sure if you are understanding what I'm saying. I'll try to break it down for you, take the above expression, i.e. "1+1, lets take its parts and see their denotation abilities:
The first symbol 1 : This is a denoting constant (i.e. a zero place function symbol), it denotes an object.
The "+" sign: this doesn't denote an object, but it denotes a process, more specifically a "function"
The second symbol 1: This is a denoting constant, it denotes an object
The string 1 + : This is an "incomplete expression" it doesn't denote an object
The string +1: This is an incomplete expression also, not denoting an object.
The string 1 + 1: This is a functional expression: IT DENOTES AN OBJECT.
So '+' denotes the process of addition itself, but "1 + 1" denotes the object that results from applying the process of addition on two "1" symbols.
It is definitely a rule of the game in logic that the total expression of 1 + 1 (i.e. the three symbols in that sequence) is denoting an object, that's definite, because it represents the result of a functional process. You cannot change this. This is NOT an interpretation of the symbols, to say that they are illogical, equivocal, erroneous, NO! It IS a rule of the game of arithmetic and the underlying logic.
And it makes full sense, because equality is a "binary relation" symbol between OBJECTS, so the symbol for equality, i.e. "=", must LINK two term symbols, i.e. two symbols that denote objects, otherwise the syntactical expression won't copy the semantic content. so you have the structure
"term - binary relation symbol - term"
Here we have 1 + 1 = 2
so = is linking the expression '1+1' to the expression '2', so '1+1' must denote an object. Otherwise the whole expression would be meaningless, it would be equality between what and what?
I tend to think (I'm not sure) that you think that 1 + 1 is an "instance of the process of addition" i.e. the process of addition is itself a big process it doesn't only work on 1's but also on any two naturals, now "1 +1" is just one instance of this addition process. It appears to me that what you have in mind is the following interpretation:
"instance of a process - binary relation symbol - term"
so here possibly you are thinking that 1 + 1 = 2 means that
(the process of adding 1 to 1) = 2
And that this would be a proposition that defines 2 by it being equal to an specific instance of the process of addition.
If I got you right, I think this interpretation can possibly work, I'm not sure, but definitely it is against known rules of the game, because relation symbols link terms and not expressions denoting process instances to terms, unless the "process instances" are understood as some kind of objects? and so expressions denoting them would be "terms"! Well that would be another issue.
In nutshell, the conventional way of understanding 1 + 1, well lets say in first order logic extended with the symbols of arithmetic, is for it to be the object that results from the function + operating on the two 1 symbols.
That + is a function, see this
No! There is! Please see my message that just precedes this, where I've pin pointed where is that object and I showed you all denotations involved. But to just reply to this here. I'll say:
In "2 + 3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+", and also we have an object denoted by the total string "2 + 3" itself. I didn't mean 5 at all, since 5 is not shown in the expression "2 + 3". The reason is because "+" is stipulated by the rules of arithmetic and underlying logic to be a FUNCTION, and by rules of the game any function symbol if written with its argument 'terms', then the whole expression of that function symbol and its argument would be denoting of an object. We don't have any mentioning of 5, yet, it is the rules of arithmetic that later would prove to you that the object denoted by 5 is equal to the object denoted by "2 + 3". Remeber equality is a relation between OBJECTs.
In set theory, yes that is the case. But of course you can reject set theoretic interpretation of arithmetic. But fishfry is saying the truth about set based implementations of arithmetical concepts. In set theory the symbol 2 denotes the von Neumann ordinal { {}, {{}} }, also the expression "2 + 2" is a functional expression that denotes the von Neumann ordinal { {}, {{}}, { {}, {{}}}, {{{}}} }, and also the symbol 4 is taken to represent exactly the set { {}, {{}}, { {}, {{}}}, {{{}}} }. This follows from the definitions given to those symbols and ZFC, and of course the inference rules and axioms of ZFC. Those are the usual interpretations held in ZFC.
hmmm...., let me think about that, I'm really not sure if "identity" really arise in mathematical system per se. But if you consider first order logic as a kind of mathematical system, since it is a part of "mathematical logic", then of course there is a theory about identity. I'll speak about formal difference between an axiomatic theory of identity and an axiomatic theory of equality if that helps.
An axiomatic theory of equality basically presents equality as an equivalence binary relation, it basically contain the following three axioms:
1. Reflexive: For all x (x = x)
2. Symmetric: Forall x,y (x=y implies y=x)
3. Transitive: For all x,y,z (x = y and y=z implies x=z)
That's all.
Now an axiomatic theory of "identity" stipulate identity as a substitutive binary relation, most of the times it uses the symbol "=" to signify "identity" and not just equality, it basically contain the following axioms:
1. Reflexive: For all x (x = x) [in English: everything is identical to itself]
2. Substitution axiom schema: if phi(x) is a formula in which the symbol x is free and never occur as bound and in which the symbol y never occurs, and if phi(y|x) is the formula obtained by merely replacing some or all occurrences of the symbol x in phi(x) by the symbol y, then all closures of:
for all x,y (x=y implies [phi(x) \iff phi(y|x)]), are axioms.
That's all.
So the axiomatic systems of these two notions are clearly different!
Now identity theory proves all axioms of equality theory, but the converse is not true, i.e. every two identical objects are also equal to each other, since identity relation is an equivalence relation, but not every equality relation is an identity relation. For example the relation "bijection" which means that two sets can have their members linked to each other in a one to one manner, this relation is clearly an example of an equivalence relation (i.e. equality) because it fulfills the three axioms of equality theory, yet bijection is not the identity relation, since we can have two distinct sets that are bijective to each other, and so it violates the substitution scheme.
Identity relation simply states sameness of objects, and it does that by fulfilling the above axioms of identity theory especially the substitution axiom, which mean that any description of the object x also holds if we replace the symbol x and the symbol y, i.e. it is also a description of the object y, and vice verse, so the identical objects are said to be indiscernible from each other using our language, i.e. relative to our language they appear identical.
However in mathematical systems and even in logico-mathematical systems like PA, I see = axiomatized after equality theory! However in more deep formal systems like set theory and Mereology the = symbol is usually taken to represent "identity" and not just equality, and usually ZFC and Mereology are formalized as extensions of first order logic with "identity" rather than with just equality, although most of the time these terms are used interchangeably in set theory and Mereology but vastly to mean "identity" and not just equality, since the axioms about them are those of identity theory and not just of equality theory.
Quoting Zuhair
I looked through the Wikipedia page on the syntax of first order logic, which you referred me to above. This is what I think. In the expression "2+2=4", the "2", and "4" symbols are predicate symbols. A predicate symbol represents an "element" with a relational definition. Because of this, expressions like "2+2=4" cannot be considered as terms, and therefore there is no basis to the claim that any objects are represented here.
Quoting Zuhair
That's clearly not the case. A function, or process is indicated by "+". There is nothing to indicate "the result" of the function. Consider for example cause and effect. The cause may be represented or described without representing the effect. When only the cause is represented, there is nothing to indicate what the effect is. Though it is true that "effect" is implied by "cause", unless one already knows that x cause has y effect, y would not be indicated by stating "the cause is x". "1+1" represents a process, it does not represent the effect or result of that process. This is evident from the fact that one might state "1+1" without knowing that the result is "2". And of course this is the natural process of summation, we write down the numbers to be summed before we know the result of the summation.
Quoting Zuhair
No, absolutely not. It may be true that "+" denotes the process, but there must be something which is active in the process, or else you just have a type of process indicated, with no specifics. So the two "1" symbols denote the elements involved in the specified process, and there is nothing to indicate what is caused by the process, "the result" of the process.
Quoting Zuhair
It is an interpretation, a faulty one. There is nothing in the rules to say that the expression represents "the result" of a process. You are making that up, back to your old habit of bullshitting again.
Quoting Zuhair
According to the definitions on the referred page, these numerals are predicate symbols, therefore they denote elements. The expression is not meaningless though, it demonstrates a relation.
Quoting Zuhair
Until you demonstrate how numerals like "2" and "3" denote objects rather than elements, as predicate symbols, which is clearly explained in the rules, I think we ought to stop saying that these denote objects.
Quoting Zuhair
That's the point. Fishfry keeps trying to switch out "equality" for "identity", as if the two have the same meaning. But "identity" has a very specific, well defined meaning in philosophy, and no such definition in mathematics. So Fishfry's use is either an attempt at equivocation, to smuggle the philosophical meaning into mathematics as if "equality" means the same as philosophical "identity", Or else he just brings a non-defined word into a mathematical usage which would leave it meaningless.
Quoting Zuhair
Perhaps you can do for me what I've asked of fishfry. Show me an axiomatic theory of identity which is proper to mathematics.
Quoting Zuhair
I haven't yet been shown these identity axioms of ZFC. The one which fishfry steered me to, the axiom of extensionality is clearly stated as an equality axiom. So if it is taken to represent identity, I think that's a faulty interpretation. It presupposes some sort of identity with the use of "same", but it doesn't stipulate what "same" means.
I wrote to you the identity relation in ZFC. I already wrote that explicitly it is the reflexive and substitution axiom schema, those are the identity theory of first order logic, and ZFC is *usually* formulated as extension of the rules of first order logic and those axioms of identity theory. However *sometimes* ZFC defines "identity" in the following manner "it also uses the symbol = to mean "identity" and not equality"
Define (=): x = y \iff \forall z (x \in z \iff y \in z)
However, this approach is not favored a lot, the majority would define axioms of ZFC as an extension of identity theory (i.e. the reflexive and substitution scheme).
Quoting Metaphysician Undercover
No this is wrong! It seems you didn't read it well, the expression 2, 4 are called zero placed function symbol, or simply constants, and those are TERMS of the language and they denote objects. That's the usual presentation in Peano arithmetic and most mathematical system. However, we CAN formalized 2 and 4 as predicates that's not a problem at all, this can be done. But it is not the usual thing.
All the information I've seen shows that the reflexive axiom and the substitution axiom are equality axioms. Why do you think they that are identity axioms?
Quoting Zuhair
I saw no such rule, to dictate that "2" and "4" are "zero placed function symbols", on the page you referred. I think you're making this up. Any way "zero" would indicate an absence of objects.
Quoting Zuhair
We agree then, that there are no objects denoted by "2+2=4"? On what basis would you claim that "2+2" is identical to "4" then?
No of course we don't agree. The usual formulation is for 2 to be a constant (zero place function symbol), that's the usual convention. Now there are some second order logic theories that can interpret arithmetic in a manner that 2, 1, etc.. are predicate symbols, but those do have equality of predicates axioms in them. I'm not willing here to discuss these versions because they are un-important. The usual ones especially for peano arithmetic is for 1,2,.. etc to be constants and so they are terms of the language denoting an object in the universe of discourse. You need to read PA and first order logic very well. from my discussion with you, it is clear that you are so ignorant of even very well known conventions. You are simply discussing matters that you don't know much about, for example you don't know that constants are zero place function symbol, which tell a lot about how much you know of commonly held syntax. You need to read some foundation of mathematics book, then you can come a speak about it.
Are you still unwilling to accept a difference between equality and identity? I thought we agreed to that difference a long time ago.
Quoting Zuhair
Why would I want to waste my time doing that, when I find inconsistencies and contradictions in the conventional interpretations of the very first principles?
What? that's really strange. You need to first read it and then know about it then you should decide whether its worthy or not. You need to get a good book on mathematical logic like Mendelson's, or Shoenfield's, or Suppes's logic, then you can read Peano arithemtic, and then Set theory. You need to read them carefully, solve the exercises , etc.. it is not something that you'll manage to know on one glance or so.
After you manage to learn about mathematical logic, Peano arithmetic and Set theory, then you can start discussing matters about them, or matters that they are used to be a foundation of, which is most of known mathematics. Otherwise the discussion would be really very poor.
https://www.math.uni-bielefeld.de/~frettloe/papers/wikibuch.pdf
Equality is in the least sense an equivalence relation and that's it, it doesn't necessarily satisfy the substitution schema. However when logicians are speaking about equality in the sense of satisfying the substitution schema, then in reality they mean "identity", so for example if you formalize peano arithmetic on top of axioms for = that are the reflexive axiom and the substitution scheme, then in here what you mean by = is exactly "identity" and 2 + 2= 4 exactly means the OBJECT denoted by expression '2 + 2' is the object denoted by expression 4. If you don't formalize Peano arithmeetic on top of substitution scheme but you keep the reflexive axiom and add to it symmetry and transitivity axioms, then here you are using a weak notion of equality which is just some non-specific equivalence relation, here the substitution scheme doesn't work because its not there, and you can have the object denoted by 1+1 not being identical to the object denoted by 4.
Most mathematical logicians would like equality to be interpreted as identity because the substitution scheme makes matters easier.
And again and again symbols like 0,1,2,3,... all these symbols are CONSTANTs of the language, they are defined constants (except 0 which is a primitive constant symbol) they are TERMS, they are in reality zero place function symbols (i.e., function symbols without arguments), each of those denote ONE object in the universe of discourse. = is a relation symbol, and + is a two place function symbol, which is in turn a process that sends object(s) referred to by two occurrences of symbols to an object referred to by a third expression. And it is always the case that functions and relations are associating objects denoted by terms of the language.
That's the convention, it is simple and crystal clear, it is not involved in any contradiction at all.
Arithmetic is in the LEAST just an game played with empty symbols, and so symbols are just subjects to the rules of the game that's it; in the MOST they are about something other than those symbols, like about some platonic world of mathematical objects.
Mathematics as a whole range between strict formalism to outright Platonism.
I personality hold that the formalist side is the purest form of mathematics, when mathematical systems are trying to capture some particular subject matter, here is it becomes what I call as "applied mathematics" in the most general sense, and most mathematics is coined as applied mathematics to a platonic realm, however usually those are considered as pure forms of mathematics and what is considered as applied mathematics is reserved to their application in empirical sciences.
Good Luck!
Then why did you waste so much of your time discussing this with me?
Quoting Zuhair
I know enough but the substitution axiom to know that it deals with equality not identity. The difference is that two distinct things may be equal, but they cannot have the same identity. The substitution axiom allows that one thing may be substituted by another equal thing, so it clearly accepts that these are two distinct things, not one and the same thing, with one identity. The substitution axiom allows that two distinct things, with differences between them, which don't make a difference to the purpose of the logician, may be substituted as equals. But clearly, that there are differences between them means that they are not one and the same thing, as required by the law of identity.
No! Unless these differences are indescribable by formulation of the language. Once you are in a logical theory then what decides identity of something in it should be in relation to what the theory can describe. Indescribable difference are immaterial inside the theory, and the two objects would be considered identical by the theory because it cannot discriminate between them by its language, so it considers them "IDENTICAL", it sees them as identical (not just equal). The substitution scheme says that if we have x = y then whatever is true about x is true about y and whatever is true about y is true about x, which mean that "all equals are identical"! More precisely speaking all equals are indiscriminable. "Equality" in the sense of being just an equivalence relation, has nothing to do with this principle, why should it demand something like substitutivity? for example bijection (i.e. equality of sizes of sets) is an equivalence relation and a logical theory do discriminate between bijective objects. Clearly there is no need to demand something like the full substituition principle if we are just thinking of the equality relation as some equivalence relation. But if we are thinking of equality as indiscernibility and thus "identity" from the inner perspective of the theory, then we'd add such a strong principle. Of course you can raise the point that this is just indiscernitibility of identicals, while the other direction which is identity of indiscernibles is not granted by the substitution principle, which is true of course, but that is only because of the weakness of first order logic. So the intention from the first order logic point of view is that first order identity theory (reflexsive + substitution axioms) that its trying its best to capture the *identity* principle.
I just wanted to add, that first order identity theory does not allow adding to it objects that can obey the substitution principle and yet be non-identical. So it in some sense does imply identity of indiscernibles. For suppose for a proof by contradiction that we can add to identity theory two primitive constants x,y, and suppose we add the schema
x =/= y and [phi(x) \iff phi(x|y)]
in other words x is distinct from y and yet they are fully substitutive.
This cannot be because: let phi(x) be the formula x=x, now let phi(x|y) be the formula x=y, i.e. substitute only ONE of the occurrence of x in x=x by y, then we'll have
x =/= y and [x=x \iff x=y]
by modus Ponens we have
x=x
x=x implies x=y
-------------------
x=y
then we'll be having: x=/=y and x=y, A contradiction!
This assures that first order identity theory does speak of = as identity and not just some equivalence relation.
That's the key point, the limitations of the language. The law of identity puts identity of the thing within the thing itself, such that even if the human being cannot discern the differences (due to deficiencies of sense, language, whatever), but can still recognize two things as distinct, we can say that the two are distinct. Therefore the law of identity represents a recognition of the limitations of the language system, the inability of the human being to adequately identify certain objects.
When we approach mathematical axioms with the recognition that equality is not identity we uphold this principle which represents the limitations of the language. If we ignore this principle, and insist that equal things are identical we become ignorant of the limitations of the language, and we will start to believe that mathematics is capable of doing what it is not capable of doing. Belief that a tool is capable of doing what it is not capable of doing is a dangerous belief.
Quoting Zuhair
Sure, within the theory there is no difference between the two objects. But in application, and theories are useless unless applied, there is a difference between the objects which the theory is applied to. Because of this, within the theory the two objects are said to be "equal". Therefore the rules of the theory recognize that the two objects are not actually the same, and express this recognition by using the word "equal" and not "identical". But within the theory, the objects are treated as if they are identical and this is a deficiency of the theory. If we ignore, or even deny this deficiency, we are in a world of self-deception.
Quoting Zuhair
Whichever things that are said by the premises to be true about x are also true about y. But that does not mean that the two are identical, it just means that they are treated equally by the theory.
Quoting Zuhair
Of course you add a strong principle, but a strong principle which is false (as yours clearly is) is a deceptive and dangerous principle.
Quoting Zuhair
You see why I claim there is contradiction in the very first principles? What point is there in making exceptions to the first principle, because you know it is wrong? Why not just admit that the principle is not a principle of identity, but a principle of equality, it doesn't have the strength which you desire it to have, and get on with the use of the system, understanding that it has its weaknesses, instead of trying to hide its weaknesses and disguise them to create the illusion of strength?
OK, what you are saying in this last posting is understandable, I in some sense agree with most of it.
There is something nice in your conception about 'equality', you view the substitution schema to mean 'equal' treatment given by the theory to the related objects, and not as indiscriminability which is the synonym of identity. So equality is not just an equivalence relation, but also a substitutive equivalence relation. That's nice. But I'd say that this is very near to identity, since WITHIN the theory they are seen "identical" [and not just being equivalent], but outside the theory that can be discriminated, so you want to give a term that describes what's going inside but at the same time alerts us that this is not necessarily what's going outside! That's fine. No problem. But again I would consider such a kind of "equality" relation far stronger than just being an "equivalence" relation, i'd consider it as some kind of quasi-identity relation, i.e. some equivalence relation that is the nearest possible relation to identity that the theory in question can describe. I agree that to be on the safe side, it's better to term it as "equality", although I still maintain that the primary intention was to capture "identity" that failed for first order theories "externally" [but not internally]. I'll officially use "equality" but I'll metaphorically use "identity", because sometimes "equality" can be understood as merely an equivalence relation, which is way weaker than what you are describing here.
I should add, that I'm really amazed by the last posting. Really strange. In this posting you appear to know exactly the official stuff about identity, and in-depth really! So your account was excellent. And the correspondence was indeed fruitful (at least for me). Unfortunately your account on other aspects of the syntax of first order logic and of arithmetic like not knowing that 0,1,2,.. are CONSTANTs, and that those are terms of the language, i.e. symbols denoting objects, and that the expression 1 + 1 is the value of the function + on arguments 1 and 1, and thus 1 + 1 being a TERM of the language also is denoting an object (besides the objects denoted by the two 1's in it), etc.. Also you not discriminating between a predicate (relation) symbol and a constant symbol, so you thought that 0,1,2,.. are held conventionally as PREDICATE symbols (although one can indeed make a formalization that can interpret them as such, but this is not desirable, and definitely not the convention), those aspects of your response were really very poor, and reflects great shortage of knowledge regarding the common conventions held by foundational mathematics regarding the main logical language which is first order logic and one of the most formal languages that are directly connected to mathematics, that is the first order language of arithmetic. Anyhow your account on equality was very good, I hope your knowledge increase one day about the syntax of first order logic, and of Peano arithmetic and set theory, etc.. so that we can have correspondence would be by far more fruitful and productive.
OK, let's say equality is a "qualified" identity. This means that it is a relative identity. In relation to the specified theory, the two objects are identical when they are said to be equal. But we all know that they are not really identical, that's an artificial simplification which is theoretical only.
Quoting Zuhair
What would be the point of this though, really? Let's say that this "quasi-identity" relation is "the nearest possible relation to identity that the theory in question can describe". How near this equivalence relation actually is to true "identity", would be completely dependent on the theory's capacity to describe. And unless we had some way of determining true identity, and comparing the identity produced by the theory, we would never know the theory's capacity, or how close the quasi-identity is to true identity. And if we had a way of determining true identity why would we be using the theory which employs quasi-identity.
The real issue I think, is what I explained to fishfry earlier. The purpose of equations in mathematics is to compare similar things in an attempt to determine the differences between them. So we find all the ways in which they are the same, "equal", and we are left with the differences. If the right and left side of the equation both represented the very same thing, then there would be no difference between the things represented, and the equation would be useless.
So I don't think that equality even aims for identity. If the two equal things were really identical, then we wouldn't be employing equality to determine this. We employ equality when we know that the things are different and we want to understand the differences between them. That's why the principle of identity is actually completely different from the principle of equality. But if we had some way to quantify the difference between "identical" and "equal", then we'd have the basis for accurately determining the difference between equal things.
Quoting Zuhair
Let me refresh your memory. I didn't say that apprehending those symbols as predicate symbols is conventional, I said it's what I think, meaning it's the way that I see them. That's my interpretation, not the conventional interpretation. As you may have noticed, I don't see things in the conventional way.
Quoting Zuhair
Thanks for the encouragement Zuhair, but following common conventions is really not what I enjoy, I find that rather boring. So I like to look for those bits of meaning which are omitted by the conventions. Generally, they are omitted because they are what's taken for granted. But what's taken for granted, is left as an unknown, like when the religious take God for granted. So for instance, Newton's laws of motion take inertia for granted, so what inertia is, its nature, is left without an approach, and it remains in the realm of the unknown. Here, in set theory, identity is taken for granted, so what it means to be "the same" is left in the realm of the unknown.
Yes, I agree. I generally agree with ALL of that posting really. And sorry for confusion about your stance from the conventions, I see know what are you trying to do, but I honestly see that for one to decipher those hard subjects, then one must read at least some of the conventional work done by foundational mathematicians on that. But again that's fine. Also I agree that concepts like 'identity', 'set membership' and even 'natural number' , and 'part-whole', are all very hard concepts when one try to dig down into their basis. Not easy at all. Nice correspondence!
Yes!!!
OK, I've reconsidered. I recognize that making stupid comments about conventions which one is totally ignorant of is not good for a person's integrity, so I think I will take some time to educate myself on some of those basic conventions you've referred to. Thanks Zuhair.
I misunderstand many things. My apologies if that is the case in this instance. LOL at the Annie Hall reference, one of my faves.
You deliberately re-quoted exactly the line that I apologized for, explained as a typo, and corrected in my previous post. Why? You do know you're strenuously arguing against a typo for which a correction has already been issued, don't you?
Quoting Metaphysician Undercover
Weren't you the one who originally made a mathematical claim, which I am refuting?
You claimed that in ZFC they misuse the law of identity in some way. I challenged you on that and you have not produced evidence.
YOU are the one who made a strictly mathematical claim about ZFC. And who can't defend it with facts.
Beyond your factual incorrectness, I found this a very patronizing and hostile remark. Did I misread it?
You made a specific claim about ZFC, an abstract mathematical system. I challenged you on your mathematical claim. You then say I have no right to talk mathematics? What kind of low-end game is that?
It couldn't have been a typo because you continued afterward, to make the same mistake. Look:
Quoting fishfry
Quoting fishfry
No, I said that equality in ZFC is not based in the identity of the law of identity. I explained why this is the case.. You insisted that the equality of ZFC is based in identity, so I asked you for a citation of a law of identity which ZFC is based in. I'm still waiting.
I see no good faith on your side (claiming my typo isn't a typo??) but I'll stipulate that you disagree. I'm done with this thread. You should, in the fullness of time, go back to the detailed proof from the Peano axioms that 1 + 1 and 2 are identical. You would learn something. The fact that you refuse to engage with that proof makes this thread irrelevant to my life. You asked for a proof, I gave you a proof and now you want to quibble that you have no obligation to read it? For weeks on end? Is that your idea of honest discussion?
Regarding the proof: That's how you show that 1 + 1 = 2 and that moreover, IF you believe that = means something other than identity, that 1 + 1 is identical to 2. I deny that mathematical equality differs from identity in set theory, except in a handful of casual conventions that can easily be rigorized on demand. You CLAIM they have different meanings but have not even attempted to defend or explain your claim but only seem to be avoiding the question. I use set theory because YOU are the one who invoked ZFC, claiming, and after all this time without evidence, that in ZFC two things that are not identical are asserted to be equal. I categorically deny that (except for as usual a handful of conventions such as embedding the integers in the rationals in the reals etc.) You made a claim about ZFC. I tell you that you are factually incorrect. You have failed to produce an example of your claim. You choose not to engage on the Peano proof, which contradicts your belief about 1 +1 and 2. I see no logical continuation of this dialog.
Again I do understand that you don't see things this way. That's what makes horse races.
Have a nice evening.
This is ridiculous. I've explained numerous times in this thread how equality differs from identity. "Identity" applies to one thing, the same thing, its identity. So "identity" relates to what makes one specified thing other than everything else. "Equality" applies to two distinct things which are judged to be "the same" in a specific way. You might consider that "identity means "the same" in an absolute way, whereas "equal" means "the same" in a qualified, relative way. The example I gave is that two human beings are equal because they have the same rights, but they are not the same, because they each have a distinct identity.
Your task, as it has been since we first engaged in this thread, is to demonstrate that in mathematics "equality" is "identity".
Quoting fishfry
That is the point you keep asserting, without justifying. All the information you referred me to speaks of "equality", and there is no axiom which indicates that equality is identity. Your so-called proof relied on the premise, that "=" means identical, but the axioms don't bare this out. The information you referred me to spoke of "equality" not identity, so that premise is taken as false unless you can justify it. So unless you can justify your claim, produce this information whereby it is dictated that equality is to be interpreted as identity, I will continue to conclude that you wrongly interpret these axioms.
And if you insist that this is "the conventional" interpretation, that is not a justification. All this means is that "the conventional" interpretation is wrong, as I've demonstrated.
I understand what you're saying. You're wrong about mathematical objects. The number 2 is identical to the number 2. The number 2 + 2 is identical to the number 4. Identical as in your definition. There is only one thing, the Platonic number 4, which we may denote as 4 or 2 + 2 or the positive square root of 16 or 3.999... and many other representations.
I understand what you are saying and I deny it.
I do of course recognize contexts in which what you say is right. For example in group theory, two distinct groups that are isomorphic are often taken to be the same with respect to isomorphism. And in univalent foundations, we take as an axiom that isomorphism is equality (I'm paraphrasing greatly here but that's the essence as I understand it).
https://en.wikipedia.org/wiki/Univalent_foundations
So ok for that aspect of things.
But in ZFC, the domain of discourse in which you originally claimed that identity differs from equality, I tell you that you are incorrect. But I have said nothing new, I've written the same things over and over.
I hear you saying that in math the axioms speak of equality but not identity. I take that as profoundly insignificant, you take it as profound. This I believe is where we differ.
I went through this with Zuhair already. It is impossible that the numeral, the symbol "2" represents the same object every time it occurs. If it did, then both the 2's in "2+2" would refer to the same thing, and 2+2 would not equal four because there would just be the same two.
This is fundamental to the nature of counting. Each thing referred to by "1" must be a different thing, or else there would not be a multiplicity. "1+1" must represent two distinct things, or else it would not equal to two things. And "1+1+1" must represent three distinct things or else there would not be three. And so on, and so forth, each occurrence of the same numeral "1", must represent a different thing when we count, or else there is no multiplicity, only the same thing over and over and over again; and the sum of the count would be invalid because there would only be one thing being counted,.over and over again.
Quoting fishfry
That's what I've been saying, you keep asserting the same thing over and over and over again, without justifying your claim. I've demonstrated how equality is different from identity. So unless you can demonstrate how it is that equality is identity, in set theory, all you are doing is demonstrating that you misinterpret.
You've just swapped in the term numeral for number. That's a particularly low form of false argument. It's like saying that two isn't two because some people call it zwei or dos or deux.
If you deny that the number 4 is the same as the number 4 you are entitled to your opinion, but that kind of sophistry is of no interest to me.
I did want to add that earlier when you said that mathematical equality is not identity but rather only equality in a certain context, you are thinking of equivalence or isomorphism. Mathematical equality is identity, not mere equivalence or isomorphism. You're simply wrong about that.
The idea that equality means that two "different" things are "the same" is nonsense. Equality means that two distinct expressions or representations of a thing refer to the same thing. 2 + 2 = 4 is an identity. I can't help what your grade school teacher put in your young and uncomprehending head. It's tragic that by your own admission your mind is stuck in the third grade.
Quoting Metaphysician Undercover
You have never done so, If you had we could talk about it. You have indeed expressed belief in the false claim that mathematical equality does not express identity. Repeating a false claim does not constitute a demonstration. On the contrary. Mathematical equality DOES express identity.
If you proposed an argument rather than just a repeated false claim, we could talk about it.
But in the end you have now said, and not for the first time, that you don't believe the number 4 is the same as the number 4. There is no conversation to be had (at least on this topic) with someone who professes such an obvious falsehood.
https://en.wikipedia.org/wiki/Law_of_identity
I just explained this. When the symbol "4" is used twice in "4+4=8", it must signify a different thing in each of the two instances, or else 4+4 would not equal 8. If the two 4's both signified the same group of four, there would not be eight, by putting together the two things represented by the two 4's, there would only be the same four. Therefore the two 4's in 4+4 must signify different things or else 4+4 could not equal 8.
If you have anything of relevance to say, address my post, show me how it is possible that when you count, and you add 1+1+1+1 etc., each instance of "1" signifies the same thing. If you cannot address this issue you are just blowing smoke, saying that "1", or "2", or "4", always represents the same thing.
Quoting fishfry
To be clear, I do not believe there is any such thing as the number four. Aristotle decisively disproved this Pythagorean idealism (currently known as Platonic realism) many centuries ago. What is the case, is that we use this symbol, written as "4", and each time it is used it signifies something, usually something different from the last time. When it is used, it may or may not signify the very same thing as in another instance of use, but in the vast majority of instances it signifies something different each time. Therefore the symbol, or numeral "4", does not represent the number four. This is a false assumption.
Quoting fishfry
I was expecting this sort of reply, and I've already addressed it:
Quoting Metaphysician Undercover
I can not relate this sentence to anything that I know nor to anything that makes any rational sense. I hope you'll forgive me, I cannot continue this conversation. You are factually wrong about this and you are not making me understand your reasoning. In other words if I thought you were wrong but I could say, "Ok I see where he's coming from," that would be fine. But I can't even do that.
The '4' in s 4 + 4 = 8 signifies two different things else 4 + 4 would not equal 8?
I find that absurd beyond the point of my being able or willing to respond. Of course the '4' signifies the exact same thing every time it is used, namely the number 4. If you are not willing to stipulate that then we have no common basis for conversation.
I get that you are sincere in your beliefs. From my viewpoint you give me nothing rational to respond to.
But perhaps you could give me a reference that supports your view. Earlier I noted that Wikipedia supports my view that logical identity and mathematical equality are the same thing. I'm willing to grant that Wikipedia is often wrong. But at least I have one reference. Give me something to work with, else I can't respond.
ps -- I should add this so you understand why you are wrong. It's a basic principle of math that the same symbol means exactly the same thing each time it's used in an argument or equation. For example when we say that for all even natural numbers n, 2 divides n, then even though n ranges over all possible even numbers, in each particular instance n means the same thing each of the two times it's used.
Likewise when we say 4 + 4 = 8, it's basic to all rational enterprise that the symbol '4' refers to the exact same thing each time it's used. Without that, there could be no rational communication at all. Natural language is symbolic. If I say that today it's raining and today it's Thursday, and you claim I can't assume that "today" refers to the same day each time I use it, then we'd all still be in caves. You couldn't say "pass the salt" without someone saying, "What do you mean pass, what do you mean salt, what do you mean "the"? You are denying the foundation of all symbolic systems from natural language to computer programming to math.
What exactly do you mean that '4' refers to two different things in 4 + 4 = 8? The burden is on you to justify denying the entirety of scientific and indeed rational discourse.
It's not a matter of giving you a reference, it's just a matter of whether you understand the reason or not. Do you know how to count? Say you have "1", and that 1 signifies something. And, you have another "1", and that 1 signifies something. In order that these two 1s, when they are put together (1+1), can add up to two, they must each signify something distinct from the other. If each of the two 1s signified the very same thing, there would not be two things, only one. Do you understand the reasoning here?
When we count objects, each object is counted as one (1), and so each object is represented by the symbol "1". So we count them, 1 plus another 1 makes 2, plus another 1 makes 3, plus another 1 makes 4, etc.. Each "1", must necessarily represent a distinct and separate object from every other "1", or else we would not have the multiplicity implied by the count, "2" "3" "4", etc.. It's not the case that the fourth object counted, when we point to it and count it as "4", is represented as 4, each distinct object is represented as 1. And, that each 1 represents a distinct object is absolutely necessary, or else the count would be invalid.
Quoting fishfry
You are clearly wrong, and have given this absolutely no thought, or else you would see how wrong you are. When I say I have 2 chairs at the table, and I need another 2 chairs at the table to have 4, so that I can accommodate my guests, it is very obvious that each instance of "2" must represent a distinct pair of chairs. If the two 2s represented the same pair, I could not get four chairs out of them. I would be stuck with only one pair of chairs.
Quoting fishfry
Again, you are very obviously wrong here, and you have clearly given this no thought or else you would see immediately how wrong you are. In the equation "4+4=8", each "4" must represent a distinct group of four things, or else they could not produce the sum of eight things. If both the 4s represented the same group of four things, there is no way to get a group of eight things, which is what is signified by "8".
Try looking at it this way fishfry. There is a difference between what a symbol "means" (as said in your fist paragraph above), and what a symbol "refers to", (as said in your second paragraph above). So we can say that a symbol must always have "the same meaning" in order that it be useful, but the symbol doesn't necessarily refer to the same thing each time it's used. I use the word "house", for instance, and we say that it has the same meaning each time I use it, but I use it to refer to many different things which are all houses, so it doesn't always refer to the same thing.
What is important to understand here is that the phrase "the same meaning" does not use "same" in a way which is consistent with the law of identity. "Meaning" is the type of thing which varies according to circumstances, matters of context and interpretation. So in reality, even though we think "that the same symbol means exactly the same thing each time it's used", and this is necessary for a symbolic system to work, the very opposite of this is what is actually true. There must be nuanced differences in the meaning of "house", each time that I use it to refer to a different house, or else people would always think that I am referring to the exact same house each time I use the word. So "same" here really means "similar", and this is a qualitative identity, which is not what is described by the law of identity.
Qualitative identity is used to say that two things are equal, or "the same" according to some principle, or inferred criteria of judgement, but it does not mean that they are "the same" in the sense dictated by the law of identity, which would require that they are not two distinct things, but one and the same thing.
This is a good reference: Fundamentals of Model Theory.
Thanks for the reference Zuhair, but I really can't read the symbols used. It's like learning a new language for me, and it's a type of language which is even more difficult than a normal language, which I can't learn a new one anyway because that itself is very difficult for me. I have enough trouble with English.
Having said that, I see that the paper takes the premise of Platonic realism, assuming that symbols like "2" represent a thing called a number. This is the premise which I've been explaining to fishfry is incorrect. I believe that to adequately understand the use of mathematics it is necessary to apprehend that each time these symbols are used, in different circumstances, they represent different things, dependent on the circumstances of the application. What the symbol represents is not "a number", but a specific and unique object determined by the application of the mathematics.
So for example, when we count something, there is necessarily something which is being counted. One might just count, and claim to be counting "the numbers", having no tangible objects being counted, but as I explained, this is not a valid count. If nothing is being counted except "the numbers", then the start and finish are arbitrarily chosen, and the conclusion of "how many", which is what is determined by a count, is also arbitrary. Therefore any such count (how many), cannot be properly justified, it is just a function determined by the rules of the count, which are arbitrarily chosen. This is just an exercise, a practise, to demonstrate an understanding of the rules, like practising logic (as we discussed), where the symbols do not represent anything. If one were really going to count "the numbers", the count would never be finished. Therefore a count of "the numbers" can never be a valid count.
In reality then, the symbols in such a practise do not represent anything. Logicians recognize this when they practise the laws of logic using symbols which do not stand for anything. They know that using such symbols is just an exercise to help them learn the laws of the system. But for some reason, mathematicians like to say that such symbols actually stand for objects (Platonic), things that they call numbers, and such. But we all know that such objects are just imaginary, and have no real existence whatsoever. So we ought to recognize that these mathematicians are just fooling themselves, claiming the real existence of non-existent imaginary objects, immersing themselves into this fantasy world which the paper calls "model theory".
Well, you definitely have some point of view here. But model theory is not altogether useless. It's easier to understand mathematical theories as speaking about abstract models, since those models do not contain properties that the mathematical rules do not entail by themselves, so its prudent to say that 1 + 1 = 2 is speaking about some process working on some abstract objects, since there is nothing in 1+1=2 to confer additional properties to what 1 and 2 represent. Yes one can certainly use the above rule in applications like in adding an apple to another to get two apples, but the properties of Apple like it having a seed for example, a DNA, etc.. all those are particulars that are not inferred from 1+1=2, so we need to abstract away those properties. Moreover if we speak in the strict formal sense then 1+1=2 can stand by itself as a syntactical game prior to any application, and so the abstract model of it would indeed provide nearer semantics to the formal essence of 1+1=2. Platonism is the easiest way to go about mathematics. But of course that does't necessarily entail that its true. Indeed as you suggest mathematical statements has their value and probably "justification" in being applied to something external to them, something that is not arbitrarily chosen. So their semantics might break down to their multiple applications, each at a time. From the philosophical point of view this applicative reduction might look more prudent, but from the pure mathematical point of view, definitely platonic models would be preferable, since they are more direct engagements of what those mathematical statements are saying.
OK, now the question here is why does "2" represent one object, and not two objects. Intuitively I would say that the first "1" in "1+1=2" represents an object, and the other "1" represents an object, and "2" represents two objects. By what principle do mathematicians assume that "2" represents a single object, which might be called the number two?
Quoting Zuhair
Perhaps Platonism is the "easiest way", but it is really nothing more than a cheat. Instead of recognizing, and understanding that a phrase like "1+1=2" is completely useless, and therefore meaningless, unless it is applied towards some real objects, in a real situation, the mathematician wants to say that it is implied within the phrase itself, that real objects are referred to. But this is contrary to the nature of language itself. In no instance of language use, is it inherent within the particular instance of usage, that there is necessarily objects being referred to. That this is the case, that no word necessarily refers to an object, is what allows for the existence of deception. So, claiming, or asserting that there is necessarily objects referred to, with a phrase like "1+1=2", is itself an act of deception, because there is really no language which can necessitate that if the word is spoken there is necessarily a corresponding object.
Now, the key to understanding, I believe, is to recognize that using "2" is an act which makes two objects into one object. We refer to the pair as if they are one object, using the numeral "2", but we have to remember that what is really referred to is two distinct objects, which are only made into one object through this artificial process, this synthesis, which is accomplished by someone uniting them, putting them together as one object, simply by calling them "2". So if we assert that "2" stands for one object, the true essence of this object which it stands for, is that it is really two objects which is only one object because we say that it is, and we have made it thus (one object), simply by saying that it is.
Quoting Zuhair
This is what I dispute though. The platonic model does not really engage with what the mathematical statements are truly saying. It is simply a cheat, an easier way for the mathematician, a way to avoid analyzing and understanding what the statements are really saying. Look, "2" really says two objects, and the mathematician just says consider those two objects as one object. It doesn't matter to the mathematician that there are no real principles whereby the two are considered as one, we'll just take it for granted that the two are one, and this will allow me to make all sorts of neat axioms. So the mathematician might assert that "2" says one object, and this is "what those mathematical statements are saying" but in reality we all know that the meaning of "2" is two distinct objects. So what the mathematical statement is really saying is that there are two objects here. But what the mathematician is saying is just bear with me, and consider that these two are one, so that I can perform my magic.
That is very funny. As in ironic When I wrote those two paragraphs I used "means" and "refers to" as synonyms. You take me to be saying two different things. I take meaning and reference to be the same in this context. But even "same" is a loaded word for you so I hope you're not going to go down another rabbit hole here.
I read the rest of your post and some of the interaction between you and @Zuhair. I see that you're sincere and knowledgeable about ... something. I can't figure out what because you won't supply a reference. You said that in "4 + 4 = 8" the two occurrences of the symbol "4' do not refer to (or mean?) the same thing. From my point of view there is simply no further conversation to be had. You're clearly serious, you're not trolling me. But when I try to take you seriously, I can't understand what you're saying. There is only one referent (or meaning) of the symbol "4" in the context of elementary arithmetic. [There could of course be other contexts, such as modular arithmetic in which "4' means some equivalence class mod some other integer]. But in the context of elementary arithmetic, "4" means the number 4.
I am simply not prepared, either by philosophical erudition or even the slightest interest, in debating this proposition. If you didn't seem so learned I'd honestly think you're trolling me.
What would help would be a simply clear example of WTF you are talking about. If the two occurrences of "4" in "4 + 4 = 8" refer to (or mean) something different, TELL ME WHAT THEY MEAN. Don't just toss out more paragraphs of obfuscation. Show me what you are talking about.
And -- secondly -- why won't you engage on the specific disagreement we're having about mathematical equality? I claim it is logical identity. You claim it's what's normally called equivalence, congruence, or isomorphism. This is a point we could engage on but you won't engage.
You said a while back that a grade school teacher once told you that an equality is between two different things. I'm afraid that experience imprinted an incorrect idea in your mind. You must let it go. With a handful of well-understood exceptions, mathematical equality is logical identity. You have not presented any specific examples to the contrary.
You should give me a reference. If this is from some branch of philosophy or some philosopher's idea, let me know what that is. As it stands I think you were just warped by your grade school teacher.
One thing I see on discussion forums is that sometimes someone is arguing a point of view but not being up front about it. Someone claims uncountable sets are incoherent and twenty posts later it turns out they're a diehard ultrafinitist of the crank variety. I don't care if they're a crank but if they'd just start by saying, "From an ultrafinitist point of view ..." then I could engage with them. But without that information, their point of view is not comprehensible.
If your ideas are original, say that. If they follow from someone's work, say that. Give me something to hang on to. Because as it stands you're just saying flat out incorrect things, and waving your hands instead of giving hard facts, evidence, and examples to support your point.
Quoting Metaphysician Undercover
To the extent I do, you're wrong. And to the extent I don't, I really wish you'd give me a reference so that I have some idea where you're coming from.
I want specifics.
You made the outrageous and on its face absurd claim that "4" means something different each time it occurs in "4 + 4 = 8". Tell me what different things they mean. If you can't articulate the difference, perhaps your ideas aren't as clear as you think they are. Surely that's fair.
I understand the general difficulty in having both 4 symbols in "4+4" representing the same object, its indeed not that easy to fathom. I'll try to give here a situation were this can be understood. Its a hypothetical scenario to clarify that this can be the case.
Let's say we live in a country were people live in tribes, now each tribe exactly has 50 men and 50 women, and the progeny of each tribe are separated from their fathers to constitute another tribe, the law dictates that marriages must be fixedly arranged between "tribes" that is if a man of say tribe A marries a woman from tribe B, then all men from tribe A must marry woman from tribe B only, and the same applies for woman, i.e. if a man from tribe A marry a woman from tribe B, then every woman from tribe A must marry a man from tribe B. Now lets fix that when a tribe is married to another tribe, then the result is also 50 girls and 50 boys, and that those would be separated from the parent tribes and so constitute another tribe.
Now the country sets two kinds of descriptions, one is Predicative description, and the other is Functional description.
The predicative description given by the country is the predicate "||" to signify "is married to" and this occurs between TRIBES. While the predicate "m" is used to signify marriage between persons. So the general statement in a laws of that country is:
RULE: For every tribe A for every tribe B (A || B if and only if for every male a of A there is one female b of B such that: a m b, and for every woman a of A there is one male b of B such that: a m b).
Now we have the situation: A || B to mean tribe A is married to tribe B (according to rules above).
Now this is a predicative formulation, why, because A||B is a "proposition", it something that can be true or false, and the symbol || is denoting a "binary relation", so it is a "predicate" symbol.
Notice that we can have the situation were tribe S can marry itself!!! so we can have S || S
Notice that S occurred twice in the proposition "S || S" but still it denotes ONE object, although this object is a totality of many individuals, however that whole of many individuals is considered here as one object. So repeated occurrence of the symbol symbol in an expression doesn't denote different denotation, no here S repeatedly occurred in "S || S" but it still carries the same denotation, namely tribe S.
This also shows that we can have a binary relation between something and itself OTHER than identity, for the expression S || S could have been false? while S is S is always true! Of course this is understood, for example we can have " \not [Sarah hate Sarah]" this is an expression having two occurrence of a symbol that is "Sarah" and yet it refers to the same object, and the binary relation between them that is "hate" can be negated (i.e. its negation is the true statement). And clearly the relation "hate" is not the same as the relation "identity", although it can occur between objects and themselves.
Now the country further uses the following notations to express "is the progeny of tribes", that is:
"P (A || B)", this is read as "The progeny tribe resulting from marriage of A to B"
Notice that expression "P(A||B)" is a "denoting" expression, it denotes a TRIBE. So the expression "P(A||B)" is NOT a predicative expression, since it clearly does NOT constitute a proposition, it is not something that can we can say of being true or false. "P(A||B)" is denotative and not declarative.
But we need a declarative statement "i.e. a proposition, or a predicative expression" about what that denotative expression "P(A||B)" is about? Here were "=" will trip in, to complete the picture and turn it into a proposition. Here the country stipulates:
P(A||B) = C
Now this is a proposition, it is say that the progeny tribe of tribe A married to tribe B , is , tribe C.
Notice here that in that country tribe S is married to itself, and it resulted in tribe Q, so we'll write that as:
P(S||S) = Q
Now we have two distinct occurrence of the symbol S on the left, but still it has the SAME denotational coverage! Both symbols of S denote the same object that is " TRIBE "S" ".
So we can have the same object undergoing some process with ITSELF to resent in other thing, like what happened with S.
The problem with expression 4 + 4 = 8 , is that it in some sense "hides" information, it should have been written as: R(4+4)=8, to mean "the result of adding 4 to 4, is, 8", that would have been more informative. Anyhow mathematicians and logicians shorten that to just 4+4, but what is actually meant is R(4+4). The expression 4 + 4 is deceptive, it gives the impression that "+" is a binary relation occurring between what's denoted by symbol "4" on either side of it, as if it is declaring that "4 is added to 4", which is not what's intended, the foundational mathematicians stipulate "+" as a two place function symbol, and they mention it in the rules of the language, which are often not written explicitly in many contexts, and so it would be considered understood that when they write 4+4 then they mean a denotative expression and not a declarative one, and that 4 + 4 actually means "the result of adding 4 to 4". Anyhow.
Of course you can object to the notion that A,B,S,etc.. here when used to denote "tribes", then they are not actually denoting "individuals", and of course that is correct, they are denoting "multiplicities", but still when B is used it always denote the SAME multiplicity. Whenever we hear B the specific 50 men and 50 woman in that country that were recorded under name "B" would come up into our minds. So all occurrences of B have the same denotational value! or lets say "coverage". IF we accept a totality of multiple individuals as ONE object that is the sum object of all of those, then B would be said to denote ONE object along that understanding.
Let me explain what I mean by the difference between what "4" means and what it refers to. What "4" means to me is that there is four objects signified, which are classed together in a group. And "4" always has the same, or at least a very similar meaning to me every time I use it or see it used. It means four things grouped together. However, many different times when "4" is used, it is used to reference different groups of four. So for instance, someone says get me the four books off my desk, or the four bases in baseball, or get me four winter tires, etc., the "4" refers to different things in these different instances of use. Despite having a very similar (what we call the same) meaning every time it is used, it refers to different things.
Quoting fishfry
So imagine there are four chairs, and we represent those four chairs with the symbol "4". Now we want to add four more chairs, so that we might have eight chairs, so we represent these four more with the symbol "4". We can express this as "4+4=8". You must understand that the first four chairs, represented by the first 4, are not the same chairs as the second four chairs represented by the second 4, or else there would not be eight chairs. Do you understand this? Whatever group of four objects which is referenced by the first 4 in "4+4=8" cannot be the same group of four objects which is referenced by the second 4, or else there would not be eight objects.
Let's apply this to the most fundamental level of arithmetic. Let's say that the symbol "1" represents an object. If we add another object, and represent this with "1", so that we can say "1+1", don't you see that each of these 1s must represent distinct objects in order that we could get two from this? If each of the 1s represented the same object, how could there be two?
Quoting fishfry
When they taught you elementary arithmetic, back in primary school, didn't they show you a group of four objects, represented by the symbol "4", and then another, distinct group of four objects represented also by the symbol "4", and if you added these two groups together, there would be one group of eight, represented by "8". If there was only one group of four referred to by both instances of "4", how could you ever get eight? Clearly these two instances of "4" must refer to distinct groups of four, just like I was taught in primary school.
How did you ever get the idea that each instance of usage of the symbol "4" must refer to the same thing? Surely they did not teach you this in school, in elementary arithmetic. Where did you get that idea from?
Quoting fishfry
Sorry, but I am not an elementary arithmetic teacher. But weren't you already shown this in primary school? This 4 references this group of four objects, and that 4 references that group of four objects. Obviously they must have shown you that each 4 necessarily represents a different group of four objects, or else it would be impossible to add them together and get eight. When this is what we were taught in primary school, where does your notion, that each 4 must reference the same group of four objects come from? Surely you must recognize that arithmetic would not work if this were the case.
Quoting fishfry
What reference do you need? It's so obvious, that if you cannot see it, I don't know what else to say. If the two 1s in "1+1" both represented the very same object, then there is only one object represented. Do you understand this? And if there is only one object represented, then it is impossible that there is two (or any other multitude) of objects here. Do you understand this?
Quoting fishfry
I already explained to you the difference between identity and equality, more than once. You keep asserting that in mathematics the two are the same, providing absolutely no evidence to back this up. I've already demonstrated that you simply interpret the use of "equality" in mathematics as meaning "identity". And despite me asking for them over and over and over again, you have provided no principles to support this interpretation, just the same assertion, this is what "equality" means in mathematics. So I conclude that your interpretation is a misinterpretation.
Quoting fishfry
How many more examples do you need? Put an object on the table, represent it with "1". Represent the same object with another "1". Say this "1" added to that"1" gives me 2, and voila, see if you have two objects on the table? No you still have only one, the same object which was represented by both 1s. Try it with "2". Put two objects on the table, and represent them with the symbol "2". Represent the very same two objects with another 2. Put those two 2s together in "2+2", and say voila! 2+2=4, so I now have four . There seems to be a problem, you still only have two, the same two represented by both 2s.. Try it with "3", and the problem will just be getting bigger. I could go on with example after example, and watch the problem get bigger and bigger.
Now try it my way. Put one object on the table and represent it with "1". Put another, completely distinct object on the table, and represent it with another "1". Now put 1 and 1 together, say "1+1=2" and voila! you actually do have 2, for real this time. Problem solved! Each "1" must represent a distinct object, if 1+1 actually equals 2.
I'm sorry Zuhair, but I really can't follow your example. It's quite complex, and as I said I'm not good with symbols, so I just get lost trying to figure out what you're saying. Here's something to think about though, which might be relevant to the case of tribe S. There is nothing to prevent one from using "4" to signify the very same object (group of four) in multiple instances of use. The problem is that in many instances, like in "4+4=8", it cannot signify the same object. But in some cases, like "4=4", or 2+4=4+2, it can signify the very same thing. This is why we can say that a thing is equal to itself (identity is an equality), but we cannot say that two equal things are necessarily identical (equality is an identity). So we find that identity is a very special sort of equality. Perhaps it's an absolute, perfect, or ideal form of equality.
Yes, it can! If you followed by tribe example, you'll see that you can do that! consider 4 to be the name of the set of all four member sets, now you join any member of that set with a member of it that is disjoint form it, and the result of all such unions would be 8 member sets that would be collected in a different set named as 8. It is very similar to the tribe condition. If you have the patience (which I agree is difficult) to follow the whole example I gave, you'll see the analogy. It would be solved in a very nice manner through conceiving numbers as denoting "sets", well actually sets of sets.
Of course the concept of having sets of sets is not a nice concept and not easy at all, but it can be interpreted in hierarchical labeling of collections, but that's another story.
"Lost in symbols, hey!", me too really, I wonder if one can can get rid of that symbolic approach to mathematics and use instead of them understandable words within some rigorous language.
I don't think we can talk of sets here, because set theory already premises that "4" in one set refers to the same object as "4" in another set, and this is the false premise which I am trying to expose. So to be talking of sets is to already assume what you are trying to prove (begging the question).
Anyway, I'll try again to decipher your example. I think that the problem with the example is that once the symbol represents a group (tribe), then the individual members of the group loose their identity. So you say "a tribe is married to another tribe", when in reality each member of one tribe is married is to a member of another. This allows you, as a mathemagician, to set up a sort shell trick, where the actual thing under the shell, meaning the persons being referred to by the tribe name, is hidden.
Consider this:
Quoting Zuhair
In this case, what it is which makes "tribe A is married to tribe B", a proposition which is either true or false, is the individual marital status of the individual people. Without this there is no truth or falsity to the proposition.
So you proceed to hide the status of the individuals by saying that the proposition is either true or false. Now it appears like the status of the individuals is irrelevant, because the only relevant thing is the truth or falsity of the proposition. But in reality the truth or falsity of the proposition is still dependent on the status of the individuals. Therefore we must consider the status of the individuals.
Quoting Zuhair
So this situation, in which tribe S is married to tribe S, if we consider the status of the individuals, must be analyzed. Remember the conditions, a male must be married to a female, so a person could not marry oneself. And, if the 50 girls of tribe S married the 50 boys of tribe S, there would only be 50 marriages, but in the case of tribe A being married to tribe B, there would be 100 marriages. This is exactly the quantitative difference I am talking about. If both the 4s in "4+4=8" represented the same object, then there would not be eight here, there would only be four. This is exactly what when S is married to S, there is not 100 marriages, only 50. But when A marries B there is 100 marriages. That's a substantial difference in the quantity of marriages.
Quoting Zuhair
So this is not really true. S is married to S really denotes that the females of tribe S are married to the males of tribe S, which is substantially different from what is denoted by A is married to B.. So one S represents the males, while the other S represents the females. You have divided S into the subgroups MS and FS, and you ought to say that MS is married to FS. And now the mathemagician's shell trick of equivocation has been exposed. You claim that the same thing lies under each S, but in reality half of tribe S is under one S, and the other half of tribe S is under the other S. This is the only way to speak of S being married to S. This is verified from the fact that Q, the progeny of this union, is only half of C, the progeny of the union of A and B.
No, the laws of the country doesn't specify a tribe of one gender, tribes can only be named if they have 50 women and 50 men. Notice the definition of marriage between tribes doesn't say what's the total number of marriages, so although you have 50 marriages between tribe S and itself, and 100 marriages between tribes A and B when they are different, still both cases are concealed by the laws, and both receive the same description of being "married tribes". The other point is that for the case of S and S each couple would given birth to 2 children one male and one female, and that would make the progeny tribe made of 100 people 50 men and 50 women and so would constitute another tribe according to the rules of the country, which is tribe Q. While each married couple of tribes A and B only give birth to ONE child, but totally they'll have equal amount of girls and boys. That's how the country breeds! Those are fixed game rules. So yes there are differences even in how the resulting tribe came into existence upon marriage of the tribes, but still the rules of the country are insensitive to those difference and thus collects them under the same parcel. So the nutshell is that we'll have the same treatment of S married to S as of when A is married to B, despite the inner differences.
Of course there would be some hidden details no doubt, but the point is that there are indeed hidden difference, but since the definitions involved are blind to those differences they would pass the same. Like when we say for example "MAN" this denotes a lot of grown up males, but there are still many differences but all fall under the same SHELL.
So definitely sets, tribes, etc.. do conceal differences, that's the point of them really.
You're missing the point of the criticism. What is "concealed", is the fact that half of S is married to the other half of S. In the case of A and B, all of A is married to all of B. So "S is married to S" does not mean the same thing as "A is married to B", because each "S" in "S is married to S" only represents half the entire original tribe of "S", whereas each of "A" and "B" remain consistent in representing the entirety of the tribes. Do you get the hint of equivocation in what "S" represents?
There is inconsistency in the application of the rules for what the symbols stand for. "S" is used to stand for the entire tribe of 50 men and 50 women, as stipulated at the beginning. But in "S is married to S", according to the example "A is married to B", the two "S"s must stand for different groups (tribes) which are married to each other. So, one S represents the women of the original S, and the other S represents the men of the original S. In other words, there is equivocation in the meaning of "S". Do you see this? "S is married to S" doesn't mean the whole tribe is married to itself, as consistency with "A is married to B" would imply, it actually means that half the tribe is married to the other half. Therefore each S in this case signifies half the tribe, whereas "S" was originally used to represent the whole tribe.
Quoting Zuhair
The hidden difference is the difference in what "S" signifies. But that difference qualifies as equivocation, so the example is invalid.
We're not talking about chairs. Four chairs over here are different than the four chairs over there.
Once again you are avoiding the question. We are talking about 4 + 4 = 8. You claim the two instances of '4' represent or stand for or refer to or mean two different things. I categorically deny that. I have repeatedly challenged you to explain that remark and you deflect by talking about chairs. You have no argument. You got confused by your grade school teacher and you can't get out of that psychological box.
You have claimed that in ZFC things are claimed to be equal that are not identical. I have categorically denied that and challenged you to provide an example. You have repeatedly failed to do so.
You have claimed that mathematicians use the word equality when they really mean congruence, equivalence, or isomorphism. I have categorically denied that (with certain well-understood casual figures of speech in particular contexts) and challenged you to provide a specific example. You have repeatedly failed to do so.
You have no argument but you have your misunderstandings and a lot of words and handwaving.
I ask you to introspect on the point that if you can't come up with specific examples, perhaps you don't understand your own ideas as well as you think you do.
You asked for an example, so I gave it. What objects do the two 4s in "4+4=8" refer to in the example, if not the group of chairs here, and the other group of chairs over there?
Quoting fishfry
No, you're not paying attention fishfry. I very specifically made a distinction between what the two 4s refer to, or stand for, and what they mean. They each mean a very similar thing, that there is a group of four objects representedby the symbol, but they refer to, or stand for, distinct things, like the two distinct groups of four chairs.
I implore you, please try to understand this. It's as if you refuse to distinguish between what a word means and what a word refers to. Here's an example. I might talk about my "computer". Do you know what "computer" means? Do you understand, that what "computer" means is something completely different from the object referred to when I talk about my computer? What it means is something completely different from what it stands for or refers to. These two are categorically different and to conflate them is a category mistake. That's the distinction I'm trying to make when I say there is a difference between what "4" means, and what the two 4's each refer to in the expression "4+4=8".
Quoting fishfry
That's ridiculous. I've repeated over and over again, that mathematician use "equality" to mean equality as defined by the axiom employed. And, the axioms do not define "equality" as identity. It is you who keeps making the incorrect assertion that mathematicians use "equality" to mean identity.
Quoting fishfry
I gave you so many specific examples, like using 4+4=8 to recognize that putting two groups of four chairs together makes eight chairs. and also the very the act of counting things. Are you unable to read or something?
I came here tonight to append a note to my previous post, which I composed in my mind before I saw this ... ahem ... remark of yours. It changes nothing. Here is what I wanted to say:
* I apologize if I sound strident. And I don't want to be strident. But in truth I've been trying to leave this conversation for a while. I stated my intention twice already and weakened.
On my part I'd be so happy to simply agree to disagree. Over time I'll go over your posts with an open mind. I would like to understand your point of view and in the process I might well learn something. I'm much more of an open-minded fellow than I sometimes appear. Make an argument I can understand, and I may well agree with you.
But we're talking past each other. I hereby agree to disagree with your point of view as expressed in this thread. We're talking past each other and no productive dialog is occuring.
* Ok that's pretty much what I wanted to write. But then I saw the quoted text ... and whether it's true or not, it comes off as having a bit of an edge to it. So neither of us is attaining our highest selves here. Let's let this go and meet again perhaps in some other thread.
But you are right in fact. I am not paying the slightest attention to your argument. That's another sign I should depart the thread. I'm making my arguments and you are making yours but nobody has gotten any more enlightened in many a post. So I'm out, unilaterally. I do not concede the point but I'm all outta ammo.
And yeah when I show up to apologize for sounding strident, I click on your post and I see your little snark and of course I don't read another thing. Becoming a self-fulfilling prophecy.
I do promise and commit to reading through your posts in much more detail at my leisure. I will also pick up the conversation between you and @Zuhair, which I haven't read but can see that it's an exploration of your ideas by someone who seems to at least know what you are talking about.
Bottom line I have no idea what you're talking about. But it's cool. Peace.
Honestly I failed to see the "equivocation" you are referring to. "S" represents the WHOLE tribe, it represent all 50 woman and 50 men, i.e. it represents the collection of 100 persons, 50 of which are women and 50 of which are men, and this meaning remained consistently throughout the application, it NEVER changed at all. So I don't see any equivocation at all.
"50 men of tribe S are married to 50 women of tribe S,
AND
50 women of tribe S are married to 50 men of tribe S."
this completes all the required conditions for fulling "marriage" between tribes per the rules of that country. So accordingly the proposition S || S (i.e. S is married to S) is true.
Quoting fishfry
Quoting fishfry
I assume then, that you still do not understand the distinction I made between what a symbol means, and what it refers to, or stands for. Perhaps if you read up on the kind/token distinction, that will help you.
When you say "S is married to S", it is quite clear that one S represents the fifty men and the other the fifty women. But you claim S represents the whole tribe. Hence the charge of equivocation.
Quoting Zuhair
Your use of "AND" as a conjunction between the two expressions above provides the necessary ambiguity for your equivocation. "S is married to S" can refer to one situation only. Yet you use two distinct expressions. Since you allow that "S is married to S" represents the two distinct situations expressed above, the charge of equivocation is justified.
Not it is NOT justified! Because we are using the "AND" in the GENERAL case of definition of marriage between any tribes A,B (whether A, and B are the same tribe or not), the general rule is:
IF
[50 men of tribe A are married to 50 women of tribe B
AND
50 women of tribe A are married to 50 men of tribe B]
THEN
A || B
I only applied that rule to the case of tribe S, where 50 men of them are married to 50 women of them and 50 women of them are married to 50 men of them as well. Just substitute S instead of A and S instead of B, and you get the conclusion S || S. No equivocation at all.
You are refusing to acknowledge the equivocation in your use of "AND" in the rule. In the case of "A is married to B", quoted above, "AND" is used as a conjunction between two phrases which refer to two distinct sets of circumstances. In the case of S is married to S, "And" would refer to two distinct descriptions of the same set of circumstances.
So, in case (1), of "A is married to B", you have situation Y ( "50 men of tribe A are married to 50 women of tribe B") "AND", situation Z ("50 women of tribe A are married to 50 men of tribe B").
But in the case (2) of "S is married to S", you have the situation X, with two different descriptions of X ('50 men are married to 50 women', "AND" '50 women are married to 50 men').
See, in case (1) you are saying there is situation Y, "AND" situation Z. In case (2) the conjunction "AND" joins two descriptions, saying of the situation X, this description "AND" that description are true. Therefore there is equivocation in the your use of "AND", which is unacceptable for a "rule", one says 'there is situation Y "AND" situation Z', while the other says 'this "AND" that are true of situation X'.
Quoting Zuhair
Making such a substitution alters the meaning of "AND" Therefore the example employs equivocation.
You can assume what you like. What's true is that I've given up interacting with you. Your persistent rudeness reflects badly on you. I'm fully aware of the subject matter that you claim I'm ignorant of. You persistently avoid engaging with anything I write. You're a nasty piece of work. Unpleasant.
Thus '4' is a reference set to match one to one to an unknown set to determine its 'size' or quantity.
OK, now what happens if we remove the "unknown set" which is matched to the reference set, so that we can just deal with the reference set itself? Is the reference set an object itself, does it contain objects, or what does it mean to be a "quantifier"?
So, in relation to the subject of this thread, is a mental construct, "the set of integers" for example, properly called "an object"? If so, then we have an infinite object, which seems incoherent because objects are known as objects by understanding their boundaries. If a mental construct is not an "object", then what exactly is "the reference set"? If we use the reference set for comparison, making a one to one relation in the act of counting for example, and if the reference set doesn't exist as an object, how does it exist?
A more meaningful adjective for the set of integers is extendible. Using Peano type axioms of formation, we can always make a larger integer, but never a largest integer.
This suggests it is the process, not the object that is without limit. Since the symbol ‘1’ for the unit represents an immaterial abstraction, we can use as many as needed from an inexhaustible supply. The set of integers will manifest itself as a finite set for as long as the forming process continues. As you said, human thought is only familiar with things having boundaries. We can only measure things with boundaries.
Our world is based on abstractions of the mind, since we can’t comprehend the reality of it.
Yes, I'm refusing this. "AND" here is "logical conjunction", it specifically means a function from the truth value of each statement linked by "AND" to the truth value of the whole statement in such a manner that the truth value of the whole statement (i.e. the two statements linked by "AND", and "AND" itself) is positive (i.e. is true) if and only if both statements linked by "and" have positive truth values. So "AND" here has a specific role assigned to it, and that role serves as its meaning. It has nothing to do with the imagined mixed roles you are speaking about. So back again, all of what our rule is saying is that the following:
IF the statement (50 men of tribe A are married to 50 women of tribe B) is TRUE
AND the statement (50 women of tribe A are married to 50 men of tribe B) is TRUE
THEN the statement "tribe A is married to tribe B" is TRUE.
AND is specifically the logical conjunctive article, nothing more nothing less. Now let apply this to tribe S were 50 men of tribe S are married to 50 women of tribe S, we have the antecedent:
The statement (50 men of tribe S are married to 50 women of tribe S) is TRUE
AND
The statement (50 women of tribe S are married to 50 men of tribe S) is TRUE.
Then by the above rule it follows that
tribe S is married to tribe S.
There is no equivocation whatsoever!
What I'm speaking about is simple blind following of the rules, nothing more nothing less. "AND"
is given a constant meaning throughout applications of it whether A is B or whether A is not B.
No equivocation!
Actually, it suggests that the "set of integers" is not an object. And, the fact that the interval between two integers can be divided indefinitely indicates that numbers are not objects.
Quoting Zuhair
You've just inverted the equivocation by insisting that "AND" means the same thing in each case. If we are to conclude, by following the example as a "rule", that tribe S is married to tribe S, there will be equivocation in the use of "married". The phrase "is married to" will mean something different in "tribe A is married to tribe B", from what it means in "tribe S is married to tribe S".
What this demonstrates is that the "rule" is faulty. If the rule is followed, there is produced a conclusion which is consistent with the "rule". But there is inconsistency between what "married" refers to in the rule, and what "married" refers to in the conclusion. Equivocation is created by following the rule. Therefore the so-called rule ought not be taken as a rule, rather it should be taken as something which could lead one into deception.
No! you are confusing matters. Notice my original statement:
Quoting Zuhair
Notice the "if and only if", the above statement is a DEFINITION of "||". Notice that it was symbolized by another symbol from "m" which was given to marriage between individual.
Marriage between tribes (symbolized by ||) has NO meaning by itself, it is just a string of letters, the country gave it a meaning by the statement after the "if and only if" above. So you cannot say it leads to equivocation of meaning or anything like that, because its meaning is understood to be fully traceable to the specifications building it posed by the rule, in other ways that rule is a DEFINITIONAL RULE. Without it you have no meaning of tribal marriage at all.
In those rigid kinds of definitions, there is no room for equivocation or the alike. These are strict rule following machinery. Equivocation is out of question here.
The point is that the symbol "||" refers to a different situation in S||S than it does in A||B. Therefore the rule produces ambiguity in the use of that symbol, and the possibility of equivocation. If we assume that "||" has the same meaning in each case, we are deceived by equivocation. Therefore the rule is a faulty rule, and ought not be accepted.
Quoting Zuhair
Very clearly this is not true. Your "DEFINITION of '||" allows that two very distinct and different types of situations are referred to by "||". Therefore ambiguity is inherent within the definition itself. Your "DEFINITIONAL RULE" is ambiguous, and equivocation in the conclusion follows from that ambiguity.
Yes, you are proceeding with "strict rule following machinery", but there is ambiguity in the rule itself, the definition of the term. And this ambiguity produces equivocation in your professed conclusion, that "S||S" describes a situation similar to "A||B". That the two different situations are similar is an illusion created by the ambiguity inherent within the definition of "||".
What is the 'closure' of the concept of an integer? I'm sympathetic to intuition-ism and constructivism, but I don't agree about this familiarity only with boundaries. The concept of 'for each' is quite natural to us. A proof by induction does help us intuit a truth about an infinite set of numbers.
This isn't to deny that certain problems can crop up. But finitism, for instance, has its own problems.
S||S is a particular case of A||B; also C||D when C, D are disjoint tribes is also a particular case of A||B.
To complicate the situation we may even allow for asymmetric gender partial Overlaps between tribes, like in saying there are tribes K, L where 20 men are shared between tribes K,L and say 12 woman are shared between tribes L,K, still we can get the same rule applicable to them! So there is a spectrum of possible overlaps, all those would be cases of A||B. Of course in each specific case there will be additional features that discriminate this case from others, for example the particularities of the tribes themselves, also the particularity of the number of actual marriages between the tribes, etc.. all of these doesn't matter, since they all meet the definition of ||. This is like variation in particularities of objects fulfilling a predicate, for example the predicate "is a circle", now not all circles are really a like, they might vary in their size for example, in their colors, etc.., that doesn't affect them all being circles. No equivocation at all. Similarly the relationship || between tribes has strict definition, and whenever that definition is met, then the relationship holds between the respective tribes, variations in particularities of individual actualization of that relationship are immaterial as immaterial is the size of the circle in meeting the definition of a circle. A circle is a circle whether its big or small in size, similarly A||B holds whenever tribes A,B fulfill the definition of ||, whether the actual marriages between the two tribes is 100, or 50, (or any other number in case of partially overlapping tribes). No equivocation at all. Equivocation might arose only when || is APPLIED in a manner that doesn't depend on the mere definition of it, that confuses different applications of || and attributes the same consequence to these as if they were the same, but that's something that has to do with APPLICATION of ||, and actually with a kind of non-careful application, i.e. an erroneous application of the relationship ||, it has nothing to do with the mere definition of || at all.
The whole matter began when I wanted to coin a relation that can exist between something and itself other than the identity relation! So the relation || as I defined in the example can occur between a tribe and itself, and also can occur between distinct tribes, so its not the identity relation. As far as the "application" of relation ||, there is no equivocation at all.
So identity is not the ONLY relation that can occur between something and itself.
But
Identity is the ONLY relation that can ONLY occurs between something and itself.
I'll grant you that as true. But the point is that there is ambiguity as to what "||" signifies. So, we must be careful not to equivocate.
Quoting Zuhair
That's right, there need not be any equivocation if we respect the differences. The equivocation would only occur if we say that one circle is "the same" (as specified by the law of identity) as the other, on account of them each being a circle.
Quoting Zuhair
That definition of "||" is not so strict as you seem to think. You describe it as "the relationship || between tribes", as if it is necessarily a relationship between a plurality of "tribes". Yet in the case of S||S it does not represent a relationship between tribes, but the internal relations of one tribe. Do you apprehend this difference? This is the difference between what is internal to an object (part of the object), and external to the object (not part of the object).
We name an object, "X" for example, and we can describe relationships between this object and other objects. Or, we can describe relationships internal to that object, as relations of the parts, but these relations are not relations between the object X and other objects. The former is properly relations of the object, the latter is relations of the parts (as objects themselves).
Now, you conflate these two with your definition. You say that if the parts of tribe A have a specified relation with the parts of tribe B, we can call that a relation between the objects named by A and B. This is not true though, there is no such relation between two objects "A" and "B" being described. There is only a group of parts (as objects themselves), with specified relations between these elements. This is very evident in the case of S||S. There is not two objects "S", with relations between those two objects, there are relations between elements and a rule which produces the claim of two objects named S with a relation between those two objects. It appears to me, like you may have some system for naming a particular group of parts by names such as "A", "B", and "S", but this assignment might be completely random, so we cannot say that these names refer to objects. They refer to groupings of elements, the elements themselves being objects
Therefore you have some random, or principled groupings of objects, and these groups of objects have received names, like "S". These groups are not objects, but artificial groups. You also have the members of the groups, which are the real objects that exist in certain relationships with each other. Your system for naming the groups as objects is ambiguous because it doesn't distinguish between internal relationships (relations between members of the group) and external relationships (relationships between a member of one group and a member of another group). Because of this failure, to distinguish the internal from the external, which is an essential aspect of an object, your groupings cannot be understood as objects. Very clearly in the case of S||S the two S's cannot be understood as signifying the same object with a relationship to itself. It really signifies that the objects which are the members of group S have the necessary relations with each other, to apply the rule, and say S||S.
The rule is ambiguous because it allows that internal relations between the members of a group are treated in the same way as relations between members of one group and members of another group. This annihilates, negates, or renders insignificant, the boundaries of the group, leaving the groupings as meaningless
Quoting Zuhair
Let's be specific with the terms here. I said that in the case of "4+4=8", if "4" is to represent an object, each of the two 4's must represent a distinct object. You gave S||S as an example of a case where each of the two S's represents the same object. Now I've demonstrated that "S" does not represent an object at all, but an artificial grouping, which cannot be apprehended as an object due to an inability to distinguish which elements are internal to (part of) the group, and which elements are external to (not part of) the group. The artificial grouping does not follow the rules of having a meaningful boundary, which is necessary for an object.
Now, it appears like an artificial grouping can be created which does not follow the rules of what "an object" is. The law of identity, what you call here "the identity relation", applies specifically to objects. If you can create an artificial group, which is not an object, it's quite clear that this artificial group might not follow the rules of what an object is. So your example really shows nothing, because your artificial group is not consistent with "object", therefore the identity relation is not applicable to your artificial group.
Hmmm.. I see the confusion here, OK, when I said a relationship between tribe(s), I only meant that each of its arguments is a tribe. That's all. It doesn't indicate plurality. It doesn't indicate that those arguments must be distinct from each other. For example identity is a relationship between object(s), it doesn't mean that the arguments of identity are distinct objects, of course not.
As regards false grouping argument of yours, and that tribes are not objects etc.... I object to this argument. A tribe is a well specified entity, it refers to the totality of specified individuals. And in the example I've outlined that each group of 50 men and 50 women that go and register themselves in the registry of the country as a tribe, then those would be called a tribe, so a tribe in this case is the totality of all those individual objects so registered. This is a well specified entity. Now whether || is not sensitive to "internal" relationships within a tribe versus "external" relationships, and that this would blur up the boundaries of tribes, etc.. argument of yours, this is not correct. Yes definitely || is not sensitive to internal x external relations between tribes, however this doesn't entail that the tribes are not well defined entity, it only means that the relation || is not sensitive to boundaries of tribes, that doesn't mean that the tribes don't have clear cut boundaries. Clearly each tribe is a well defined entity and what is external to it is very well demarcated, it is what is not a registered member in it, and what is a member of it and what is not, is well defined in the registry of that country. So each tribe is SHARPLY demarcated, and in that sense it is indeed an object, although a plural kind of object rather than a singular kind. It is this insensitivity of || to boundaries of tribes that cause it to be able to occur inside a tribe and for other tribes to be in-between them (outside each of them to the other), yes that's what cause it to be a relationship that can be between something and itself for some objects and also at the same time can occur between something and other things for some objects.
So we do have S||S, each of S represents the SAME tribe (which is indeed an object), and yet || is not a relationship of identity! So the occurrence of a same symbol on either side of a relation symbol doesn't entail that each occurrence must stand for distinct object.
Can't elaborate on my response. There is no human experience with 'infinite' unbounbded/without limit entities. Cantor was an illusionist, who fooled many people. That's it.
Well the though police aren't going to kick down your door for thinking so. Nor will they harass you for denying the theory of special relativity.
Personally I think you are caught up in a conspiracy theory here. And conspiracy theory (one of the great if disorganized religions of our time) seduces by presenting itself (deceptively) as the opposite of credulity. You imply that basically all mathematicians are fooled. This is a belief in something highly unlikely that you are unwilling to justify except in terms of an amateur's hunch. What I think you're missing is that all of this metaphysical jazz is what you are bring to the situation. Math is dry and technical. It is philosophically agnostic. Individual mathematicians may have metaphysical beliefs, but those beliefs don't play a role in proofs.
Cantor didn't prove something 'magical' about the 'real world,' even if he himself thought so. Or at least mathematicians are not at all bound to experience Cantor's work like that.
OK, since a "tribe" is meant to be an object, you are defining "object" with these principles. Accordingly, an object is a whole, composed of the totality of specified parts. And, there may be an overlap between the parts of one object and another, such that the parts of one object may also be the parts of another object.
Now, I suggest that you are missing a very important (essential) aspect of an "object" in your definition. For the existence of an object, it is necessary that the parts exist in specific relationships to one another. This is why your definition, and consequent "rule" is ambiguous, as I've said. An "object" is not a random collection of parts, in any random relations, but the parts must exist in a very specific way in relation to each other, in order to constitute an entity. Those relationships are essential to the existence of any object, such that when the specified relations cease to exist, the object ceases to exist, despite the fact that the totality of parts may continue to exist. In other words, "totality of specified individuals" is insufficient for "well specified entity", and your claim that "a tribe is a well specified entity" is absolutely false. It is this ambiguity (lack of definition) in the relationships between parts of your so-called "entity" which allows for your mathemagistic sophistry.
Quoting Zuhair
As explained, this is absolutely false. Your entire argument above, relies on the truth of this false assertion, which you seem to think that if you repeat it enough it will magically become true.
Quoting sandman
Very true, that's why I like to call people like Cantor mathemagicians.
Quoting Eee
Yes, those metaphysical beliefs clearly do play a role in mathematical proofs because they are entrenched in the axioms, as foundational support for those axioms. And Cantor is a good example. What is at issue here is how we conceive of an "object".
When JFK was assassinated, the general population could not accept that an ordinary individual could remove a popular public figure, so some thought it must be a conspiracy. I was never an advocate for that. Tragedies don't discriminate.
I said 'many', not 'all'. SR is a great theory, and has so much experimental support, why is it still questioned.
Here is a paper that questions the 'diagonal argument'.
https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4
Well you seem to refuse tribe as an object, well this is a deep point. Anyhow to me a tribe, a herd, a bunch, etc.. all of those are objects, and they are well specified objects as long as each individual member of them is a well specified entity. Anyhow I don't think I can discuss refusal of such clear kinds of objects.
Cranky.
Wittgenstein also criticized the diagonal argument. He was wrong but his objections were at least coherent and interesting.
It's not a good paper. And Cantor was definitely sophisticated enough to see what he is supposed to have missed. The author is instead failing to see.
Quoting sandman
What you still refuse to see is the proof is solid in the same way that a game of chess is legal. All of the moves are according to the rules. It does not prove something metaphysical about reality. Or at least the rules are agnostic about their 'real world' meaning.
Speaking more personally (indulging in a real-world interpretation), I experience its intuitive content this way. If someone claimed to have a way/algorithm to list all infinite sequences of bits, I'd know they were wrong. I would just flip bits along the diagonal and have a sequence they didn't include on their list.
Correct, under your definitions, I refuse a 'tribe" as an object. If we are to consider a "tribe" as an object, the relations between the members are essential to the existence of that object. Remove those relations and you have no object.
You seem to think that you can randomly point to a bunch of objects, say "abracadabra", and suddenly there is another object, which exists as the unity of those objects you have pointed to. Sorry, but that's not reality.
I agree that a community's embrace of a set of axioms manifests among other things something like metaphysical preferences or basic intuitions. But this is obvious. And where are the mathematicians that deny it? You and @sandman might as well complain about the rules of chess for not conforming to your metaphysical preferences.
Anti-Cantor cranks are fencing with their own shadows. To be a mathematician it suffices to prove things using 'the rules.' One can think of it as a game with symbols. One can also, to be sure, think that one is doing the True Metaphysics. One can, as I do, think of it as working within a system that strives imperfectly to articulate and accord with intuitions of space, quantity, and algorithm. Imperfectly! I like non-mainstream versions of mathematics. They are fascinating. No need for dogmatism or a fixed position. And that's also how I enjoy philosophy.
So to me the idea that mathematicians are true believers is in general ridiculous, and, in my experience, most online anti-Cantorism is purveyed by those who seemingly can't even play the game agnostically. That would take work, serious interest, and not just self-inflating online conspiracy theory.
I don't see how that's an adept analogy. Metaphysical principles are based in how one apprehends the nature of reality. Chess is a game which we can choose not to play if we don't like the rules, how does that relate to reality? Not even by killing oneself can one choose not to partake in reality.
Quoting Eee
If the rules do not conform to reality, then I'd have to ask you what are these proofs sufficient for? I'd say they're sufficient to produce unsound conclusions. What exactly would the mathemagician be proving, if one uses poorly formed rules? I'd say that the mathemagician would be proving that confusion follows from the use of poorly formed rules.
Quoting Eee
I don't like playing games, I'd rather be engaged in something meaningful.
Quoting Eee
That's the point, why put serious effort, work, toward something which is just a game? I know that athletes do it. Sorry, but I'm not interested.
So in your sense if I bought two applies today, then I only have two objects, that is the apples themselves, there is no other object that is the totality of these two applies, i.e. the sum material of these two apples, i.e. an object such that each of these two apples is a part of it, and that doesn't have a part of it that is disjoint of these two apples. To my naive understanding, I see it obvious that there is that object.
I see each apple as an individual object. The claim that an apple is an object is justified by sense perception of its existence separate from other things. If you show me two apples and assert that the two are one, then you need to supply a principle to justify this claim.
Let's say that the two apples are "the same", in the sense that both are apples. So we place them both in that category, the set of apples. Notice that "the same" here is not being used in a way which is consistent with the law of identity. The apples are not really "the same" in that sense, as they remain two distinct objects. Now, what is one is the category, or set we have created called "apples". The apples are not really unified to be one, they are judged as being members of one set, according to the principle (the Idea) whereby we class them both as apples. This is explained by Plato in his famous theory of participation. The distinct objects partake in the Idea. It is well explained in The Symposium.
But it is important to note the deficiency of the theory of participation which is developed in Plato's Parmenides, which some people argue leads to the refutation of Pythagorean/Platonic idealism by Aristotle. Let's say that the Idea, which is "the set of apples", provides the unity whereby the two apples are judged as one. It is the Idea itself, which is one object, not the two apples. The problem is that no matter how many apples partake in this Idea (the set of apples), the Idea as one object does not change. Apples may come and go from the set, as time passes, but despite this activity the set itself, as an object never changes. We see that the sense of "the same" here means that the Idea is unchanging over time, whereas "the same" in the law of identity allows that the same object may be changing as time passes.
This makes the "Idea", or "set", as an object, passive and unchanging, and therefore independent from, separate and distinct from, the objects identified through the law of identity, which are the members of the set. Through generation and corruption, identified objects which are members of a set, like apples, come into being and cease being, while the idea, or set itself is supposed to be unchanging.
So when you show me two apples and say that they are one object, you might as well show me three or four apples, a thousand, or a million apples, and each time you are showing me the exact same object, the set, or Idea of apples. That object is the Idea, the principle whereby you unify them as one. But that seems like nonsense, that showing me two apples, or four apples, or a million apples, is showing me the exact same object. Therefore it appears like we cannot properly refer to these "Ideas", or "sets", as objects, because we do not allow them to change when change is warranted by what is exhibited. Showing me two apples cannot be showing me the same object as showing me a hundred apples. Therefore either the two apples do not properly make an object, or we do not understand the way that this object, being the Idea or set, changes when members are added or subtracted from it.
No that is not correct. If you show me three or more apples, the totality object would be some OTHER object. I just showed you two particular apples (those that I've bought today), and I asked you simply if by today when I bought them, do I have an object that is the whole of both of them, and I explained this object in terms of Part-whole relationship, an object such that each apple (that I bought today) is a part of it, and such that it doesn't have a part of it that is disjoint (doesn't share a common part) of both these apples. This object is the smallest object that has both of these apples as parts of. It is simply this object that I've asked you to tell me whether it exists or not. I didn't speak about a Naming category (like the one you've spoken about) nor did I pose the problem of changing material of a set. I said at the moment when I bought both of these apples is there the object that I've defined or not? To me this 'whole' or 'totality' object of these two particular apples, to me I say, is as concrete as the existence of each apple, and it is an object as each apple is an object. Of course this object can be ruined with time, as each apple can be, and actually only when one of the apples constituting it would start to ruin. But this is another question. My simple question is whether such an object exists in the first place.
By showing me the apples you are not showing me the "totality object". This you would show to me in your explanation of "part-whole relationship", etc.. That is why number as an object, if it is an object, is something other than the objects which you use to demonstrate its rationality. The proper demonstration is an explanation, and the apples are just props.
Quoting Zuhair
So by the time you are talking about a number of apples, "number" is completely abstract, and you are applying that abstract idea back, onto the apples. The claimed object which is signified by "2" does not have any apples as a part of it, it is abstract. And you cannot demonstrate that object to me by showing me two apples, because such a demonstration can only be done through an explanation of what it means to be 2.
Quoting Zuhair
That claimed object, the "totality object", is what I see no reason to believe exists. You made some claim about "part-whole" relationship, but I see no reason why two apples, ten apples, or a million apples is a "whole" anything. Therefore there is no "whole" to this object which you are describing, and your descriptive terms are misleading. You are assuming that two apples is a whole, without any reasons why two apples may be a whole, and why it doesn't take three, four, five, or the proper totality of all apples, to make a whole. Surely there are more apples than two, so by what principle do you apprehend two as a whole?
This is what Plato's theory of participation demonstrates to us. The "whole" is an Idea, the parts participate in the Idea. Now the existence of the Idea, as the whole which unifies the participants producing the object, must be supported or else the whole theory of participation falls apart. "One" is fundamentally a whole, that is its essence, a unified entity, but "one" is fundamentally different from "two" by the difference between a singularity and a multiplicity. So what principle will you introduce to support your assertion that "two" is a whole, just like "one"?
All of those have their wholes, For any predicate that hold of apples there is a totality of all apples fulfilling that predicate. And those totalities would be different totalities if the apples constituting them are different. But I've just presented to you a particular case. There is nothing special about two here or three or any number.
I like to present matters in a Mereo-topological manner. Now a unit is an object that is not a whole of two separate (not in contact) parts, and at the same time it is separate from any other object. A totality of units is a collection. The smallest collection is a unit. An element of a collection is a unit part of that collection. So the unit collection is the sole element of itself. Multipleton collections are those that are constituted of many units. So they are not the elements of themselves. Now for the sake of simplicity let's assume the ideal condition of all units being unbreakable and actually in-changeable over time, and they won't be in contact with other objects at other moments of time. So no unit object can be split into two separate objects at some other moment of time, nor it would be a part of another unit object at other moment of time. Of course this is an ideal condition. Under that assumption we can have stable totalities and thus I can extend any predicate in the object world as far as that predicate only hold of unchangeable unit objects. If the units are breakable (as it is the case with the real object world) or can come in contact with other units to form bigger units (as it is the case with the real object world) then this method fail, or at least becomes very extremely complex.
Set theory can be explained as an imaginary try to REPRESENT stable collections of units, by stable units. So any two stable collections (i.e. their units are unchangeable over time) would have distinct representative units (whether those representative units are part of those collections or external to them) as long as they are not the same, and each collection is only represented by one unit. This theory of representation of collections by units, is the essence of Set theory. Of course the representative units are ideal, i.e. unchangeable over time. Now while element-hood of collections are being unit parts of those collections, yet "membership" in a set is another matter. Membership in sets can be defined in two ways, l personally like the definition of them being elements of collections represented by the unit, i.e. every set is actually a unit object that represents a collection of units, now those units of the represented collection are the members of that set. Let me put it formally:
x is member of y if and only if there exists a collection z such that y is the representative of z and x is an element of z.
We start with the non representative unit, i.e. a unit object that do not represent any collection of units, this would stand for the empty set. then go upwards in a hierarchy. What I call as the representational hierarchy, where collections are represented by sets (units) and sets themselves are collected into collections, which are represented by sets, etc..... This step-wise hierarchical approach enables a gradual transition from the less complex to the next more complex to the next, and so on... So a nice way would be to start with the empty object (the non representing unit), then to the collection of all sets (i.e. units) representing parts of that empty object, then to the collection of all sets (i.e. units) representing parts of the resulting objects, etc... According to this view a set is always a unit, and that unit act to represent a collection of units.
We can extend the representational hierarchy as long as we don't have a clear inconsistency with it. This way we can encode almost all of mathematical objects in that hierarchy.
Sets not only can represent finished collections, it can also represent unfinished collection, as long as the process of producing the elements of that collection is well defined, like the process of making the naturals by succession from prior naturals and so on.. we can have a set that would represent the process of that natural production. And those are the infinite sets of naturals.
So set theory of mathematics like in ZFC are just a theory about representation of actually finished collections and of potentially non-finishing collections.
I just wanted to put you in the picture, that sets (as used in mathematics) are different from the collections I've spoken about. While the genre of collections is the same as the genre of their elements, sets on the other hand can be totally external to the collections they represent and can indeed be of a different nature. There is a lot of confusion between collections and sets, even in standard text-books of mathematics, and especially there is the confusion between element-hood of collections and membership in sets, that many mathematical textbooks on set theory introduce sets in terms of collections and set membership in terms of element-hood of collections, and this is a great confusion. Sets do not function as collections, no they function as unit representatives of collections, thereby enabling us to speak of a hierarchy of multiplicities within multiplicities and so on... So the set concept is a stronger concept than the collective concept. The former is representational and latter is mereological.
Right, that's why I say you're a magician. You claim that you can point to any random objects, and say those objects make another object, a whole. Your claim here amounts to an assertion that you can state "the predicate" which pertains to these two apples, or three apples, or any other number of apples, but no other apples. In reality there is no such predicate, it is a hollow claim. Therefore you have no principles for what constitutes an object, only "if I say it's a whole, then it's a whole". A principled "whole" has a defined completion, not the hollow (and sometimes impossible) claim that it's possible to define the completion.
Quoting Zuhair
See, you are using "totality" as if any collection of things is a totality of things. Now "totality" is meaningless and redundant because any collection of things is, by your usage, automatically a totality of things. But to have a proper totality, which gives "totality" meaning, it is required that we name a type of thing, and sum the complete number of those named things. To say that any random number of things is a "totality" is nonsense, because it's not the total of anything, it's just a random collection.
Now you claim that any random collection of elements is a "unit", but you have no principle of unity to substantiate that assertion, only the hollow claim that there is such a predicate which unifies them while excluding others. In reality, the elements cannot be a unit unless they are united by something which produces a unity. To say that any random collection of elements is a unity is to utter nonsense. By what means are those elements united?
Do you not respect the fact, that "a collection" must be defined? And, to complete the specified collection, to produce a totality, or "whole", it is required that the entire collection be summed. To insist that my collection is 'definable', and demonstrably the totality or whole of that 'definable' collection, does not justify the claim that it is a totality or whole. The definition must be produced, and it must be demonstrated that there are no other elements which would fulfill the conditions of that definition in order to justify the claim that the collection is whole, a totality, complete.
Quoting Zuhair
If I correctly understand what you are saying here, a "set" is a collection. As a "unit" the set is complete, a totality, or whole. The completion is determined, fixed, perhaps even "caused", by the definition, the predicate. The definition makes the unit definite, and that is complete.
Here is the issue which I see. In reality, it is only possible that the definition may produce a complete collection, in some cases it may be the case that a complete collection is impossible according to some definitions. Completion of the collection is really dependent on the quality of the definition. A good definition may produce a complete set, while a bad definition will produce a set which is impossible to complete. Notice your opening sentence in the paragraph above: "Set theory can be explained as an imaginary try to REPRESENT stable collections...". The operative word here being "try". So if we take a poorly defined set, and try to produce a completion, or whole, or if we assert that a poorly defined set has produced a complete whole, this is a mistake.
Quoting Zuhair
I see this as a self-refuting, contradictory start. The empty set would be complete, whole, with no members. It requires a definition which nothing could fulfill, a definition of nothing. But it is the definition itself which produces the entity, the unit, so the empty set is at the same time a unit, and also nothing. In common terms, to try and produce the empty set is to try to produce something out of nothing. There is no such thing as the empty set, it is an impossibility, by way of contradiction.
Therefore I propose as the true start, the definition of a unit, what it means to be an object, complete, whole, a totality. We cannot start with an empty symbol because that creates infinite random possibilities for nonsense. We need to start with what the symbol is really supposed to represent, a set, which is an object, a unit, or whole. In this way we may restrict the use of the symbol, to eliminate vain attempts to produce an impossible whole.
Remember, the possibility of completion is directly dependent on the quality of the definition. Therefore we ought to restrict poor quality definitions which are not conducive to the possibility of completion. Such definitions may produce the illusion of completion when no such thing is possible, allowing for misleading, or deception. This can be done with a proper definition of what it means to be a unity, a whole, complete or totality.
So the empty set, as a starting point ought to be replaced with the set of one, a whole, an object, complete, whole and total in itself, by its very nature; so that the principal or primary set is consistent with itself, and not self-contradictory as the empty set is.
Quoting Zuhair
One problem, the empty set cannot be a unit, as described above, that's self-contradictory. It's an object composed of nothing.
Quoting Zuhair
I would say that this confusion is actually produced by the convention of allowing for empty sets. By allowing for a set with no members we produce a separation between the set, and the members of the set. But no such separation is warranted. The set itself is the object, the elements comprise that object, and the object is created by the definition. The quality of the definition determines the definiteness or definitiveness of the object. A vague definition produces a vague object. So the set, and the elements or members, must be one and the same, in order that the set be the object composed of those elements, while the definition is what has separate existence. Therefore judgement lies in conformity between the definition and the elements, such that a poorly defined set makes a vague object, or in some (impossible or contradictory) cases not an object at all.
I never said that, nor did I claim it. Actually what I said refutes that!
Quoting Metaphysician Undercover
You didn't correctly understand what I was saying!
Quoting Metaphysician Undercover
I cannot agree more! Of course, and that's what I was saying. But you totally misread what I was writing. I think because of you "apparently" not having experience with the topic of Mereo-topology.
What I'm saying is a little bit complicated. Seeing your comments, I realize that you completely mis-understood me. But I do concede that what I wrote was too compact.
Lets come to what I meant by "UNIT", I mean by that an individual. For example an apple is a unit, while the collection of two separate apples is not a unit. Now I envision a unit as an object that is not the whole of two separate objects, that is at the same time separate form other objects. This has something to do with separateness and contact. So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material. While the collection of some two separate apples is not like that, you have one apple being a part of that collection and the other apple also being a part of that collection but you have a breach of material between them, i.e. the two apples are separate, i.e. not in contact with each other and no part of that collection is in contact with these two parts, such collections are NOT units, they are collections of separate units. The only collection that is a unit is a collection that have one individual, like the collection of one apple, like the collection of one bird, etc.. those are unit collections. You need some experience with Mereology (Part-whole formal study) and connectedness (Separate-contact) formal study, joining both fields you have what is known as Mereo-topology. You need to be familiar with the axiomatization of Mereo-topology, in order to get the grasp of what I'm writing here. These are particular concepts, they are not that philosophical, but of course they can be realized on philosophical grounds.
I define "collection" as a totality of units, of course that totality itself may be a unit (in the case the collection has only one unit part of it), or might not be a unit (like a collection of multiple units: like of two apples, 10 cats, etc...). I need to stress here that "being a unit" or not, has nothing to do with the collection being definable or not, even if it is definable after some predicate still the collection if it contains many units, still it is NOT a unit. Being a unit depends on the continuity of the material in the collection, and not on definability issues or the alike. The only collection that is at the same time a unit, is the singular collection, i.e. the collection having one element, i.e. has one unit part. Otherwise collections having multiple elements whether definable or not, are always not units.
A set (as that term is used in set theory) is a unit object that represent a collection of units, like in how a lawyer represent a collection of many accused persons. Each accused person is a unit object (because its material is in continuity, and it itself is separate form other material) and the lawyer is also a unit object, so here you have an example of some Representation relation where a collection of unit objects (that is itself (i.e. the collection) not a unit since there are many accused person in that collection of our example) that is represented by a unit object (the lawyer). That was an example of EXTERNAL REPRESENTATION. On the other hand there is INTERNAL REPRESENTATION where a single unit in the collection can stand to represent the total collection, like for example when the HEAD of some tribe represents the whole of its tribe in some meeting of head of tribes. The head of a tribe is a unit part of that tribe, and yet it can represent the whole tribe. Any group (collection) of people can always chose one among them that can stand to represent the whole group. This is internal representation.
The usual set theory with well founded sets is a theory of external representation of collections of representatives of collections of representatives of..... It is about tiers of representation of collections.
The empty set can be ANY non-representing individual object. For example take any particular apple. This can serve as the empty set, since apples are not representatives of collections of representatives..
Now take some unit object that serves to represent the chosen apple above (the one we called the empty set). This must be different form that apple, because the apple is not a representative of anything, while that object is representing that apple itself. This latter object would act as the singleton set of the empty set, denoted by {{}}. Now you can take a third object that act as a representative of the collection of the apple (the empty set) and the object that represents that apple (the singleton of the empty set), now this representative object would be the set of the empty set and the singleton of the empty set, denoted by { {}, {{}} }. And so on....
So one need to discriminate sets (which are unit collections that act as representatives of collections) from collections (which are totalities of unit objects). If one manage to fathom that discrimination, then one can of course understand the difference between being an element of a collection, which is being an individual (i.e. a unit) part of that collection, and between being a "member" of a set, here a set is a representative of a collection, and with well founded sets, they are always external representatives of collections (like in the lawyer, accused example), now being a member of a set is actually to be an element (i.e. a unit part) of the collection represented by that set. Membership of sets is a representational act, it is a kind of a singular representational act. Discrimination between the concepts of Collections and their elements, from Sets and their members, is vital for a proper understanding of the subject of sets and classes, and it is something often misunderstood, and misrepresented even at official text-books unfortunately.
Actually If I was to rename matters, I'd call collections as sets, and what is termed as "sets" in set theory I'll call as representatives, and epsilon membership, I'll re-name as "representation step". Anyhow
One needs to be careful! Not every collection has a representative! Even some well definable collections might not have representatives. Although this largely depends on what is meant by "well definable".
I hope you can re-read my prior posting with this clarification.
As about the question of random collections and defined ones, this is another matter, I didn't allude to those yet. I want to define the basic terms, and then if we have some agreement over those, we can go to those issues. But basically I do agree with the sentiment that ALL collections are aught to be definable!
Quoting fishfry
The author giving just one reference and that being the Wikipedia page of the diagonal argument is telling by itself.
And seems like the author is simply confused about infinite sets. And one really has to understand how different the reals are.
Sorry for the misunderstanding, I'll try to stay on track.
Quoting Zuhair
I tend to think that this is not a very good representation of what an object, or "unit" really is. It isn't based on an accurate description of the relationship between parts and wholes. A "unit" for you is something with existence separate from other units, yet its parts do not have such separate existence. In reality though, there is vagueness in what constitutes "separate". Due to this vagueness, there may be discrepancy in judgements as to what is the unit, and what is a part of a unit, depending on one's perspective. For example, the apple is really a part of the tree. Its generation, existence and subsistence is dependent on the tree, such that as soon as it gains "separate" existence it starts to degenerate. Also, consider a "unit" like the earth. You might think of it as "separate" from the sun, but really it only exists as a part of the solar system. Then the solar system only exists as a part of the galaxy, and so on. And if we look the other way, we are faced with the question of why this composition of molecules which is "the apple" is properly "the unit", and not the molecules themselves. After the apple separates from the tree, the molecules of the apple separate from each other in the process of degeneration. That is why I think your determination of what constitutes an object or "unit" is rather arbitrary, and dependent on one's perspective.
Quoting Zuhair
Now your description of a "collection" doesn't seem to provide principles to distinguish between an artificial collection and a natural collection. So for example you do not distinguish between a collection of apples in a bag, placed there artificially, and a collection of apples hanging on a tree. The tree might be an object, a unit, and the collection of apples exist as parts of that unit, but they could also be rearranged as parts of an artificial collection.
Quoting Zuhair
So this appears to be the critical question, what constitutes a "breach to its material"? The apples hanging on the tree clearly have no breach of material and are therefore part of the tree, but that's a simple example. Is there a breach of material between the earth and the sun, when the two are connected by things like gravity and light?
On the other hand, an artificial collection might very well be connected by something. Apples in a plastic bag are "connected". Apples of the same variety are "connected". Furthermore, when we manufacture things like cars for example, we connect parts together to produce a unit. So the distinction between artificial and natural, though it serves as an example, is not even a good distinction itself.
The issue here seems to be what constitutes a "material" connection. You would say that having a material connection to something else negates the status of being a "unit", making the thing a "part" of a unit instead. If we switch to Aristotelian terms we'd replace "material" with "substance". In his "Categories", "substance" in the truest and primary sense, is defined as that which is neither predicable of, nor present in, a subject. Notice that this produces a more rigorous restriction than your "material" connection. Not only do we have "present in" as a restriction, but also "predicable of". So for example, if X is predicable of Y, X cannot be given the status of substance, and cannot therefore be a unit or object. This would extend your category of material connection to include predication as representing a material connection.
I believe that the goal here is not to produce the artificial/natural distinction mentioned above, but to distinguish between substantial and non-substantial, or material and non-material collections. Consider my criticism of your last post accusing you of a "random collection of elements" which you flatly denied, accusing me of misunderstanding. If a collection is truly random, the so-called "parts" of that collection are actually units, there is nothing substantial connecting them, and the collection itself cannot be an object or unit. Therefore the "parts" of that collection are not properly "parts". But if the parts are connected for any valid reason, this must qualify as a substantial, or a material connection. Then the whole of the collection is a valid object or unit, and the parts cannot be understood as independent objects.
Quoting Zuhair
According to what I've explained above, I dismiss your criteria of "the continuity of the material", as being too vague, and replace it with the Aristotelian concept of substance. Therefore any valid "collection" is itself an object or unit, the parts having a substantial relation to one another, demonstrating the existence of a "whole". The parts are therefore not independent units. Having something in common for example, cannot be taken as merely coincidental, and must be understood as indicating that the parts are not independent objects, but parts of a whole.
Quoting Zuhair
I foresee an issue with this concept of a "singular collection". I'm afraid it might be somewhat contradictory like the empty set, or simply purposeless. Let's say that every object is unique, as per the law of identity. Any individual thing which we come across could therefore be a singular collection. However, it is pointless to make such a collection, because the reason for making a collection is to acknowledge relationships between things. So we could only place a thing in the category of "singular collection" if and only if there could be no relations between that thing and anything else. Having a relation would make it a part of a collection negating the status of "singular collection". Perhaps we could keep the category of "singular collection", but it would most likely remain an empty category. It's not an empty set though, but an empty category, because under my categorization a valid set (reasonable relations) constitutes an object.
Quoting Zuhair
So this is where your system gets very confused, and mine becomes much more practical. In your example, why not simply say that the collection of people represented by the lawyer is itself an object? They all have X in common, so they have that valid relationship to one another, and therefore exist together as that mentioned object. There is no need to assign to a member of the group the task of representation, such as the person who represents the group, and hand the group real existence through that representation, the group already has real existence through the real experience which they share, which constitutes a real relationship. Such real relations make real objects. And, in the other example, the tribe has real existence as an object, due to the relations between members, it does not require a "head" of the tribe, or representative of the tribe, to give it real existence as a collection or object. Requiring that the collection has a representative creates all sorts of problems, beginning with the representative's real capacity to adequately represent the collection. See, under this system, the collection, as an object, can only be apprehended as an object, to the extent provided by the representative. But the representative is not the true object, and we are better off to look directly at the object to understand its true existence.
Quoting Zuhair
Yes, this complexity is exactly why your system is bound to failure. As I described above, any individual object may be used to create a "singular collection". But that is to assume that the individual has no relations to anything else, and this produces an empty set. To make the set meaningful, another object must represent the singular collection, but that negates the status of singular collection. So your whole set system is based in something meaningless, or even contradictory, the empty set, which is represented by the singular collection. It's like you're building your sets bottom up, when they need to be produced top down to have any substance. The set must be principled on the real existence of parts to a whole, as an object, and not based on a part which is meant to represent a whole.
Quoting Zuhair
This is evidence that your system is faulty. We need to recognize a collection as an entity itself, and not rely on a representative. A representative is often incapable of representing to us, the "thing" which is responsible for the real and valid existence of the collection. And this is proven by the fact that some valid collections have no representative.
No this is wrong. A collection can exist and be apprehended without having any representative, or even if it has a representative, the apprehension of the collection need not depend on it. Having representatives is and ADDITIONAL feature. It enables that collection to be a member of higher collections through the representation relation.In other words, representative singulars are only essential for having a hierarchical development of collections of collections of collections... etc.. It is not essential for our apprehension of the collection itself, which could be described in fairly specific manner without reliance on having a representative whatsoever.
What I'm trying to achieve is a hierarchical buildup like bringing separate bricks, define a collection of them, assign some brick (external to them) to act as a representative of them, actually just a label of the collection of those bricks, now there are other representative bricks representing other collections of bricks, now put those representative bricks into collections and also assign other bricks as representative of those collections, and so on... going up. Each brick is a unit, but a collection of separate bricks is not a unit. It is something like this envisioning that I want to construct.
OK, let me simplify this method. Lets use the concept of Mereological atom. An atom is an object that doesn't have parts other than themselves. Now a totality of atoms, is a collection. and an element of a collection is being an atom part of that collection.
Now the buildup I want to speak about is to have a Representation relation that assigns in a unique manner to each collection of atoms, some atom that act as a label (name) for that collection, I call this uniquely distinguishing label as a representative. Of course a collection of atoms might have a representative or might not have any one.
The buildup is to have collections of atoms, each of these collections is represented uniquely by a unique atom, now the next tier is to have collections of those representing atoms, and those collections would in turn also have representative atoms, and so on....
Set theory is about such a hierarchical build up.
Is this artificial. Yes it is! Not only that even the representation relation can be a fixed one. Much as naming symbols are arbitrary in nature.
You can call this hierarchy of names, or Naming Hierarchy. Each name can be understood as a mereological atom. Now we have collections of names, those collections themselves have names that names them, then we have collection of names of collections of names, then we have names of those, then collection of those, then names of those..etc... I'm claiming that Set theory of mathematics thrives in such a naming hierarchy which I happen to call the representative hierarchy.
One need to completely disentangle the concept of representation (unique naming) from that of collection, that's the point that I'm trying to insist on here. A collection of mereological atoms satisfying a predicate \phi, lets denote it as C^phi, is the totality of all of those atoms, i.e. C^phi is an object that has each of those atoms (satisfying phi) as a part of, and such that any object that has each of those atoms as a part of, would have C^phi as a part of! This is substantially different from the *SET* of all \phi atoms, lets denote it by S^phi, here S^phi is the atom that names the collection C^\phi. Now being an *element* of the collection C^phi means being an atom that satisfy phi, that is a part of C^phi, while being a *member* of set S^phi means being an atom that is part of the collection named by S^phi, so it doesn't necessarily mean being an atom that is a part of S^phi. In some sense a set is one step higher than its members, while a collection is not higher than its elements.
My outcast is descriptive, while your's is largely etiologic!
Collections can be fairly described and recognized up to identity without resorting to any representative of them, yes I do agree with that. Representatives are neither essential for the existence nor for the characterization of a collection. However representatives of collections are essential for developing a hierarchical account about collections, i.e. when we want to speak about collections of collections of collections, etc...
I don't believe in random collections, yet I don't refute them. However the concept of "random" seem to be different to me than to what you mean by it. It seems from your accounts that you call a totality of unconnected parts as a random totality, because there is NO etiologic like connection between its parts in your sense, so you call such collections as arbitrary, random, etc.. While to me the concept of random only raise versus definable. To me a definable collection of separate unit objects, is itself an object, and it is not a random object because there is a strict "descriptive" rule that joins its separate unit parts. However that descriptive joining of its unit parts should NOT be understood as a kind of "connection" between its unit parts that renders them inseparable, otherwise those would seize to be units, the unit parts still remain "separated" since there is no material (or if you like call it substance) that joins them together, so they remain separate apart, even though they are descriptively linked in some manner. Now as long as there is a descriptive characterization of the collection in a unique manner, i.e. the collection is a definable entity in terms of its unit parts, i.e. like in saying it is the totality of all unit objects satisfying predicate \phi for example, then this definability is (to me) against saying that this collection is a random collection. To me randomness only arise if there is no such a definition, so we have an object that is the totality of unit objects and yet there is no description of those unit objects in our language. Those so called as indefinable collections are really what can be said to be random collections. So for example there is no clear etiologic connection between some particular tear drop and some particular orange, but we can descriptively define a collection of both of those individual objects. That collection is an artificial hybrid, a chimeras, still it is not random, since it has a unique description. Now if a collection is definable in terms of being a totality of unit objects satisfying some particular predicate, then we can assign a representative unit object that can serve to label it (i.e., represent it). However can we assign a label to an undefinable collection? My guess is NO, we CAN'T. Because intuitively we can only label what we can describe. That's why in my philosophic line of thought all sets in set theory are ought to be definable! That is they are names for definable collections!
However my account is different totality from your account. You refuse to admit a collection of "unconnected parts" being an object, to you there should be a kind of necessary relationship between the parts of an entity for it to be an object. That's why you call any try to describe a collection of unrelated objects, as an object, as being magical, since it brings to existence something out of nothing, to you it is some kind of fuzzy entity that doesn't qualify of being an object. While to me it qualifies as being an object no matter how much fuzzy it is, actually even if it is indefinable, still it is an object, so in principle I admit the possibility of the existence of fuzzy collections as well as indefinable collections. Yet I don't see such collections as useful, and I would be sympathetic with a line of argument rejecting their existence. But I don't admit fuzzy sets, since sets are labels, representatives, and it would be against the nature of naming to have them name fuzzy collections, this would be very confusional. And to say that we can name indefinable collections is even contradictory, it is like naming the unnameable. You see the difference of how I use the words "collection" and "set" above.
Are you saying I was excessively judgmental? Perhaps.
I don't see how this could be true. There is nothing to validate the collection as a "collection" without a representative. In both your examples, the lawyer and the tribe, the collection cannot be apprehended as a true collection without the representative. The representative is the thing which tells you things about the collection such that you might apprehend it as a collection. Without the representative, you might apprehend any group of objects as a collection, or not as a collection, with nothing to tell you whether it truly is a collection of not. If you move to predicates to validate the existence of the collection, rather than a representative, then you move to my system. But then we must work top down instead of bottom up. And, what would be the point to having a representative if you could apprehend the collection without a representative?
Quoting Zuhair
But this is not how we understand collections ontologically. We proceed toward understanding them by determining what they have in common, not by looking at a representative. So it seems to me like this idea of representatives is a step in the wrong direction.
Quoting Zuhair
It appears to me, like creating a hierarchical account through representatives would be prone to arbitrariness and error. Any hierarchy needs to be created through reference to real relationships between objects, not reference to representatives.
Quoting Zuhair
I would say that this is not even a totality. A totality must be the completion of a defined collection. if the parts are "unconnected' then they cannot be a defined collection.
Quoting Zuhair
So how could "unconnected parts" make a collection, or a totality?
Quoting Zuhair
Here's the issue which you do not quite seem to grasp. If there is a "descriptive rule" which joins parts, then those parts have a real commonality, which joins them and renders them inseparable in reality. For example, two apples have a single descriptive rule, and they also have a common background, they came from a tree, and they have a common origin. This makes them part of that whole, which is understood through the common origin. We can also say the same thing about two oxygen atoms they have a common source. Being descriptively linked implies a material connection. So I don't think it's correct to make the separation which you want to.
Quoting Zuhair
Actually, I'm looking for compromise, by allowing that a descriptive rule implies a material connection.
Quoting Zuhair
The fuzziness is now due to the inability to determine the material connections through the descriptive rules. The descriptive rule indicates that there is a material connection. But we cannot get to the material connection through the descriptive rule, and that's why the representative idea is faulty. We cannot get to the real connections which constitute the real existence of the object, through the description. We need to determine the real relationships.
Well I do agree that having a common description imply some material connection, but that connection is not the connection that imply inseparability. You can call these connections "loose" connections, as opposed to "tight" connection which is what causes continuity (inseparability), so if object K has tight connection to object L then they are in continuity, i.e. they are not separate, ie. they are in contact; while if object K has loose connection to object L then they are separate. Now what I call as an "individual" or a "unit" or sometimes I call as a "singular", this is an object that posses tight connections between parts of it in such a manner that it is not the totality of two parts that loosely connected to each other, and at the same manner the object itself must not possess tight connections to external objects. Now if we have many individuals such that there is a description that isolates them from others, i.e. there is a description common to all of them but not to other objects, then those individuals would be said to be LOOSELY connected by this descriptive joining, so they are still separate form each other. Now this would be an object! I call it a collection of objects satisfying this property, so it is a collection and its elements are the individuals that are loosely connected in it. So far for collections.
But this is not enough. You need representatives, or actually NAMEs, you can also call them tokens, or labels, those would be singular objects (units) that we arbitrarily assign to each collection, but provided that the assignment works along unique lines, I mean each collection is assigned only one name, and each name only names one collection. So although the choice of which object would name a collection is arbitrary, but once done naming of other collections cannot use that name, so the naming function is not totally arbitrary. Of course this is not Ontologically innocent, it involves adding unrelated material into the picture!
But why names? why should we assign an external object that is singular to act as a name to a collection that may have multiple elements, so why represent a multiplicity by a singular object? With external naming, there is no clear intimacy between the name and what is named, the assignment is arbitrary for that particular aspect. And this is what actually happens with naming generally, its artificial, for example the names used in language are all arbitrary, there is no special connection between the string of letters "horse" and the animal group it is used to represent. So that's the question: why we should bring an external object that doesn't bear a necessary relationship to a collection and make it act as a name, actually a "representative" for that collection?
The answer is to develop a hierarchical account about collections! This cannot be done in an efficient manner without the use of singular names. The idea is that through this artificially made unique naming process, we can define a new relation, called "membership", that act to copy the relation of element-hood in collections but raises this relation to the name of the collection, and since names are singular objects so they can be elements of collections (while collections when they are non-singular objects cannot be elements of collections, so we can't have a hierarchy of collections in collections using directly the "element-hood" relation!!!), so all elements of a collection wold be "members" of the name of that collection. The "name" of a collection, is what we call as "set" in set theory. So for example the set of objects k,l, denoted by {k,l}, is actually the name given to the collection whose only elements are k,l. so k,l would be "members" of that set, i.e. they bear the membership relation to the NAME of that collection, which is the set itself. Through this copying process of elements to members, one can speak of a hierarchy of sets that are members of sets and so on.... And so indirectly speak of collections of collection of...This would give the powerful mainframe needed to interpret almost all of mathematics.
Now you might be suspicious, and actually object, to such a buildup. Since its pivotal rule is built up through an intermediary that involves some arbitrariness, which is the choice of a name per particular collection of course. So its like building a big building that involves multiple big junks of tightly connected material put on top each other using light joining material, so the the whole buildup is bound to fail!
I'd say it would be extremely difficult to make that hierarchical buildup without using names, I tried myself to figure out these possibilities. For example you can use various grades of tightness of connections, like in saying we have: degree 0 loose connection (which are the tightest connections), degree 1 loose, degree 2 loose, etc.... where for each i the i+1 loose connection is looser than the i loose connection. We can do that and define elements of a collection by those bearing the loosest kind of connection between each other and internally of course they use harder degree of connections, and so on...
This can be done but largely on disjoint collections. When there are overlaps, for example like with the case of power-sets, then here it would become very bleak. And even worst in trying to capture non-well founded sets, that it becomes even impossible to use this method for that sake.
So we needs NAMES, to do the intermediary role in developing a hierarchy of sets of sets of..,etc.. It is the simplest way to do it! And this proves to be very powerful logically speaking, that almost all of mathematics can be encoded in it.
This build-up is not due to me. It is largely David Lewis's idea. But here I used 'sets' as names, while in his approach he use them as collection of names. The approaches are equivalent, but mine is more extensional than his. Also related work can be found in point-set topology although taking different technical paths.
According to this line of thought of mine, to me, sets being names, then they out to be assigned only to "definable" collections. Because with naming procedure, you need name something that you can describe first. And so to me all sets must be parameter free definable, i.e. they must name collections that are parameter free definable collections. And naming must proceed in a hierarchical build up from the simplest to the most complicated in a step-wise manner. So definable collections that are not the result of that buildup cannot have names, and this include the collection of all singular labeling objects (names) , and of all singular names that do are not part of the collections they label, etc.. those collections usually called as "big classes", are not reach-able by a hierarchical naming build-up from below, so they cannot be named, even though they are definable!
The nice corollary to this line of thought is that it proves the axiom of choice! and actually of much stronger form of choice, of a definable global choice!
So having a pertinent line of thought about what sets really are, can solve some technical problems, like with the famous problem of axiom of choice here.
We agree to an extent, but you seem to have some ambiguity. Material connection is what makes parts into an object, and when its an object, you say that the parts cannot exist as separate objects. Yet now you say that material connection does not imply "inseparability". I would agree with this if we clarify by distinguishing between actually separate, and separable.
I find your distinction between loose and tight connections to be very fuzzy, and I think we can replace it with the more substantial distinction between temporal continuity and spatial continuity. Suppose for example, that the apple exists as a collection of molecules. There is a tight connection between the molecules, and an even tighter connection between the atoms, and this spatial continuity indicates that these parts are not actually separate. Yet the molecules, and atoms are in principle separable, and this is a function of the temporal continuity of the object. The parts are not separate spatially, yet they are separable temporally.
So we need to distinguish temporal continuity from spatial continuity. If the parts are inseparable in the sense of temporal continuity, then the object is eternal, and it would appear contradictory to even talk about the object as being composed of parts. Such an object is the fundamental, or base "unit". It can have no parts because that would imply that the unit is separable in time. The spatial extension of such an object is dubious.
From experience, it appears like spatial extension implies parts which are separable. It's difficult if not impossible to conceive of a object that occupies space, which is not divisible into parts. If you think about this, you will probably be able to conclude that the actual unity, which is an object with spatial continuity, is really an illusory "unity" due to temporal separability. The spatial continuity which we observe really hides the underlying temporal separability, The parts of the spatial object which are connected through a spatial continuity, may appear to have a loose or tight spatial connection, but the true strength of that continuity will only be understood through an understanding of the temporal separability. Since the temporal aspect is what hands the parts of the object separability, the nature of time is better understood through the terms of discontinuity. Therefore we end up with spatial continuity which is an illusion, and temporal discontinuity which describes the reality of the object in terms of separability. The reality of the eternal object (true temporal continuity) is dubious, as synonymous with the object that has no spatial extension.
Quoting Zuhair
OK, I'm with you here. I apprehend a name as an object, which is a representative of something else, as described. And I wonder why there is a need to bring names into the picture. The type of object which a name is, I think is a tool, and this tool has the purpose of understanding. It's the tool we use for understanding.
Quoting Zuhair
This I find to be very confused and I am unable to follow. I agree that we may use names to develop a hierarchical account, that is part of understanding, but where I find confusion is in your failure to recognize the distinction between naming an object, and naming a relation. These are two distinct uses of a name, like the difference between noun and verb.
So if we name "membership", what that name actually refers to is the relationship implied by "membership". Now we must guard against deception. We can name membership when no reasonable relationship has been identified. Therefore there is no point to naming "membership", unless to deceive. The relationship ought to be named directly, without the medium "membership". Furthermore, what follows from this, is that this "name", which is the name of the collection, but actually represents membership in the collection, which in itself represents a relation, is an even further layer of representation. So we have three levels of representation now, the name represents the collection, the collection represents membership, and membership represents a relationship. Plato warned us against such multi levels of representation, calling them "narrative". Any hierarchy produced in this manner would be extremely unreliable, as we ought to refer directly to "the good" to produce a hierarchy. Names are tools used for understanding, so the good here is understanding. Multi levels of representation are conducive to confusion rather than understanding.
Quoting Zuhair
Exactly! A name is an object with extremely unreliable temporal continuity so we ought not construct structures using names. And, since it is required to produce a multi level representation to make the name into a building material, this makes the structure even more unreliable.
Quoting Zuhair
But there is no need for a hierarchy of sets, that is a faulty premise. What is needed is to understand the relationships between things, that is the good which we might put the tools (names) to use toward. The hierarch is produced naturally by understanding the nature of continuity, and the strength of relationships. So we begin with some fundamental determinations concerning the nature of continuity; for example what is prescribed above concerning spatial and temporal existence, and we proceed to name relationships according to their strength, producing a hierarchy of relations. There is no need for a hierarchy of sets.
Actually from experience with mathematics. We do need a hierarchy of sets. The other alternatives are not so promising.
I find temporal versus spatial separability not easy to fathom. I'll have a better try to fathom your notes about it.
I do admit that there is some ambiguity with my characterization of "loose" versus "tight" connection.
I view the connection between adjacent parts of an apple as being tight connections, while I view the descriptive joining of many apples satisfying some predicate as being a kind of loose connection. "Separability" in my sense means absence of tight connections, so loosely connected objects are separate, while tightly connected objects are not separate. I define a "unit" or a "singular" as an object K such that for any part x of K, the part of K that is the complementary part of x, denoted by x'^K, is in contact with x, i.e. x'^K is in contact with x. So when an object is a unit, then it is not the totality of non-tightly connected objects, and the other condition is that a unit must be not be in tight connection with an external object, i.e. it must be separate from external objects, so it can bear loose connections to external objects, but it cannot possess tight connections with them.
So a collection is a totality of one singular object, or a totality of many singular objects loosely connected to each other through descriptive joining of its singular parts through having a common description that isolates them from other singulars not fulfilling that description.
A name is a singular (a unit) that as you called it a "tool" that helps us understanding, here it helps us direct our attention to a specific collection, in such a manner that we can speak of multiplicity of collections in a hierarchical manner. Each collection has only one name, and each name only names one collection. Sets are singular names of collections. They are not names of relations. Naming of relations is a different subject, and I've never attempted to speak about it in any of my prior comments. I've been always speaking about naming collections, and so speaking about naming objects, and not relations.
When a set say set x names some collection C, then we call each "element" of C (i.e. each singular part of C) as a "member" of x. In some sense membership would copy element-hood but transfer it to an object external to the collection, that is to the name of the collection. But you need not confuse "membership" as a name for "element-hood", No! That is not the case. Membership is not a name, it is a relation, so it is not an object.
Now through membership relation and sets (i.e., names of collections), one can easily define a hierarchy of sets. And that build-up proves to be an extremely useful tool in our understanding of many mathematical entities. And the witness to that is SET THEORY. In particular ZFC set theory (Zermelo-Frankel set theory with Choice), which proves to be very powerful in understanding mathematical entities and rules, through the iterative buildup of a hierarchy of sets.
Of course for the development of set theory, all of our units are un-breakable over time, and they don't change their tight connections with time, so they are remotely different from natural objects which rut over time or combine with other objects to build bigger units, etc... Here in the platonic mathematical imaginary world, all individuals (units) have non-changeable tight connections over time. So they are as you said "eternal". Then we can freely form collections of them using the descriptive tool, and with the help of the naming relation, we can speak of a hierarchy of them, which helps us encode almost all of mathematical entities in it. Thus serving as a FOUNDATION for MATHEMATICs.
I'll try to keep track with your notions of temporal versus spatial continuity, and come back with comments about it.
I'm speaking within the confines of a mathematical realm, some platonic realm in which time doesn't cause any change to connection relations. So what is actually separate is always separate, so separable is separate, and so temporal x spatial connection is immaterial in this realm. We only have spatial connection and separation. That said we need to revert again to loose versus tight connections.
My account entails that the existence of connections between parts of an entity is what qualifies that entity to be an object. So having loose connection is fairly enough for that quest. You don't need tight connections between parts of an entity to qualify it for being an object! NO! loose connection can do the job, so an entity in which loose connections between its singular parts exist, is perfectly qualified of being an "object. However, you need tight connections to form units (singulars) but units are just special kinds of "objects", so an entity that has tight connections over its parts and it itself doesn't have that kind of connection to external objects, that would qualify it to be a unit object. But objects need not be units. They can be totalities of loosely connected units, or what I call as "collections". So as such collections qualifies for being "objects". I hope this resolves the confusion.
Now the loosest kind of a connection is the one made through descriptions. And so an entity of separate singular parts that are commonly describable in a manner that isolates them from objects not fulfilling that description, is the least needed condition to qualify it as an object. And we have this situation with collections as I defined.
The above coupled with a naming function that name collections by singular names, is enough to build up the required hierarchy in which almost all of mathematical objects can be carved!
As attractive as it sounds, this proves to be extremely difficult. Experience along such lines are moot. Its hopeless. Without a hierarchy of sets, or similar structure, there is almost no hope to encode most of mathematics. You will only have the sketchy picture of prior to the twentieth century mathematics. But again this is one of the most useful kinds of mathematics.
Ok, I agree it would be eternal since its not actually breakable. But why it can have no parts? Any object is itself a part of itself. Perhaps you mean it doesn't have "proper parts" [parts of an object other than itself]. I'm under the impression that you think that an object must be breakable to into parts in time in order for us to say that it has parts. But this is not correct. Even if we have an object that is eternally not breakable, still it can have many parts connected by tight connection in a manner that renders it a unit, it doesn't mean it doesn't have proper parts, it only means its no breakable to them, but it can have them always as parts of it. In real life having eternal objects is itself faulty. So the idea of eternal objects is hypothetical or actually imaginary, it suits framing mathematical objects, because usually we work with mathematical objects in some Platonic realm, and that realm appears to be time free, sometimes even disrupt spatial reasoning as well, anyhow.
Now if we work in an imaginary space in which time has no effect, i.e. doesn't change connection relations of objects to each other, still it is imaginable for those objects to have parts, so having parts is not a function of temporal separability as you hold. Not only that, still without time we can fathom of having objects that are composed of units that are loosely connected to each other. So we can have collections having many elements.
However, we may reach into a definition of "loose" and "tight" connections using this spatial x temporal distinction. For example you can say that:
x tightly connected to y if and only if at all times x is connected to y;
while
x is loosely connected to y if and only if sometimes x is connected to y and sometimes x is not connected to y.
According to this definition a "true unit" would be an object that is never breakable nor is continually in connection with an external object.
Of course in a mathematical realm in which time is not operable, like "most" of mathematical contexts, then all unit objects in that realm are true units.
Of course in an imaginary context one can incorporate time into the mathematical world and actually use that time-dependent definition of "true unit" and of loose and tight connection. So a collection would be either a true unit or a pseudo unit (a temporally appearing continuous object that is temporally separable from external objects), and being an element of a collection would be being a true unit part of that collection. Then we introduce naming of these collections with true units, and everything would run as I intended.
The problem is that this would add additional features to the picture, namely temporarily, which is not all that desirable in a mathematical realm. For the purpose of defining sets, we can simply hold the dichotomy of loose and tight connection as primitive concepts without relation to time. Our aim is largely descriptive. Since set theory serves as a foundation for mathematics then the particularities of what decides the "units" of a certain mathematical discipline is stuff related to the particularities of that discipline itself, so in Geometry units would be "points", in arithmetic units would be "numbers", in set theory units would be "sets", etc.... Here we are only concerned in introduced a general descriptive framework that can be applicable to diverse mathematical disciplines, and possibility even non-mathematical spheres of knowledge as well! For example the idea of having a "true unit" in time, might be useful in understanding the ontology of time and space?
I think that "collections" and "sets" are pervasive, so they can be used in any field of knowledge. That's why I'm dealing with that matter in a merely descriptive manner. So consider relations of "connection", "tight connection" to be primitive relations. Then using logic, we can define the terms "singular" (i.e. unit), collection, element, set, member. And that's all what we need. Then build the hierarchy in a gradual manner, and you are allowed to go as high as consistency permits. That set you get all the extensions of ZFC set theory, and thus encode any area of knowledge really, but specially mathematics.
Quoting Metaphysician Undercover
I couldn't manage to follow that really. But in my usage when I used names, I used them to name "collections" which are "objects" and not relations. When I write "member" or "membership" this is a symbol to denote the membership relation, I don't mean by those symbols to be "names" those are not names, those are definable relation symbols. Names only are assigned to name collections which are objects. I don't think that a hierarchical build-up would be confusional, why? The definitions I gave were very strict. So I'm using "names" in a particular context, that is to name collections. symbols used to "name" relations are not called as "names" here.
Quoting Metaphysician Undercover
You said I'm not discriminating between naming of objects and naming of relations. I need to see where exactly I made this confusion. I introduced names for the specific context of naming collections, and I defined collections as totalities of loosely connected singulars (units) and I justified my claim that such entities are totalities and such totalities are objects. So I was all the way speaking about naming objects. Where do you see me introducing names to name relations and confusing those relations as objects? I never said we can name for example the relation 'membership' and I've never introduced a name for it (in the particular context of name that I've used) where do you see me speaking about introducing a name for the relation "element-hood" and speaking about such naming? The relations that I've used are "connection" tight and loose, part, membership, element-hood, where do you see me speaking of attaching names to those? I never said that! I only spoke specifically about introducing singular names for collections, and I specifically defined what "collection" mean, and I justified that being an "object".
The point is that your description of the distinction between "loose" and "tight" does not provide us with an indication as to what these terms really mean. The reality is that the parts of a collection are either loose or tight depending on what type of relation they have with each other. Therefore there are all sorts of different types of sets, which may be named dependent on the relations between the parts. We cannot just classify loose and tight sets, just like we cannot just class soft and hard physical objects, there are all sorts of different type of objects which we name, like 'cars' and "houses". The naming of a set, as it is supposed to be the naming of an object ought not be any different. The name ought to indicate to us what type of an object that particular set is, by indicating something about the relations of its parts, or its use, or something like that. W#hen someone says "house", or "car", it brings to mind a specified type of object, the same ought to be the case when sets are named.
Quoting Zuhair
So this is a basic type of set. The set "C" has "members", "elements", that is what is specified of this type of set. Notice that there is no specification as to the relationship between members, the relations might be either loose or tight. Would you agree that the so-called "empty set", and the set with only one member, are different types of sets from the set which has members. We could have a set "A", which is empty, and a set "B" which has one member, and then the names A, B, C, would each specify a different type of set, therefore the name would be useful. We could then proceed to look at the different types of relations between members and have all sorts of subcategories of type C, but A and B would be distinguished as completely different.
Quoting Zuhair
Unless your hierarchy is built on real principles of distinguishing real difference between types of sets, that hierarchy will be arbitrary and misleading. Therefore the hierarchy is not built on the names themselves, but the type of set which the name identifies. We cannot make up relations between named sets, without first having a complete grasp of the relations between the members of the sets themselves, which gives us an understanding of the type of set, because such relations would be completely imaginary. Understanding the relations between the members, gives an understanding of the type of set, and understanding the type of set will allow us to proceed toward establishing relations between the types of sets. ZFC proceeds without a proper understanding of the types of sets, to produce imaginary relations between imaginary types.
Quoting Zuhair
As I said, the Platonic eternal, unchangeable object must be considered as imaginary, impossible, and must be excluded as not real, until its reality may be demonstrated. Therefore to base a set theory in this assumption is to start from a false, unjustified premise.
Quoting Zuhair
But this is unreal, impossible, therefore I deny an such "speaking" as irrelevant, because you are speaking illogically, of the impossible as if it were possible.
Quoting Zuhair
But this is total nonsense. In the "platonic realm", all objects are eternal, so the distinction of loose and tight means nothing because one cannot be more susceptible to corruption than the other. And "loose and "tight" cannot refer to spatial relations because eternal objects are non-spatial. Therefore your specified relations are completely illogical, there is nothing to justify "tight" or "loose".
Quoting Zuhair
Again, this is illogical nonsense which has been accepted into set theory, the idea that a thing can be a part of itself. The adoption of that idea into set theory is representative of the breakdown in logic of set theory. "Part" means 'some but not all of', and "whole" means 'all of'. If we redefine "part" such that 'all of' may be 'part of', then we loose the essence of the distinction between part and whole, rendering a corrupted ontology.
The eternal thing cannot change, that's what defines its being as eternal. But if a thing is composed of parts, then the parts must exist in relations to each other. Sense observations demonstrate to us that all relations are changing (relativity). Therefore it is impossible that an eternal thing is composed of parts.
Quoting Zuhair
As I've shown, your distinction of "tight" and "loose" amounts to meaningless nonsense. If the relations between an object's parts are so tight that these relations cannot change with an eternity of passing time, the claim that there are parts in relation to each other, rather than one changeless entity without parts, is unjustified. If the relationship between two so-called "parts" of an object cannot change, then the claim that they are "parts" is not justified. That is one object, a whole without parts.
Quoting Zuhair
Exactly, therefore we must establish proper principles to distinguish between imaginary eternal objects, and real objects. An eternal object, by nature of what it means to be eternal, cannot consist of parts. This presents a problem to set theory. A set, by its nature consists of parts. Therefore those who work with "set theory" must resist the contradictory notion of treating sets as if they are eternal objects.
Quoting Zuhair
Now, don't you see this as illogical? There is no such thing as a space in which time has no effect. that's like assuming absolute rest, which relativity theory has ruled out as impossible. So if we adopt this premise as a basis for "set theory", we are working from a premise which inherently contradicts relativity theory. If one were to apply this "set theory" in physics, all sorts of confused conclusions would follow from that application of contradictory premises.
Quoting Zuhair
However, as I've shown, it is illogical to speak of "relations" in a realm where time is not operable. Relations must be justified. If the relation is spatial, it cannot be free from time. If you propose a type of relation which is not spatial, such a proposition needs to be substantiated, justified.
Quoting Zuhair
The glaring problem here is that mathematics is fundamentally a tool which is used to understand the relations between objects, from the most fundamental, "order". In the mathematical realm, change to relations is as you say, not "desirable". However, change to relations is reality. So if mathematics assumes changeless relations, because this is desirable, but changeless relations are not real, then there will be a problem in the application of mathematics.
So we need to look directly at this fundamental idea of changeless relations, and determine whether we can make sense of it. I suggest that there is a type of relation, which is demonstrated by the existence of time itself, which may be changeless. This is "order". We can assume an "order" which is given by time itself, a changeless order which is the passing of time. However, we need to make this consistent with relativity theory.
You Can! if your aim was to FOUND (i.e., lay the basis for) matters with. We leave "tight" and "loose" like blanks to be filled with the relevant application. So tight an loose are left as "primitive" concepts, those would take different meanings according to the working application. Of course collections would have different meaning across all applications, but they will have consistent meaning within the same application. Like how number 1 can have different meaning across applications.
I also want to note, actually an apology, that what I'm saying here about Mereological, well actually Mereo-topological, understanding of set theory is not the conventional line. It has been loosely suggested by David Lewis, but not with all such detail. So set theory is not dependent on those views I'm posting here. But to a great extend those views can make one understand what's going on with set theory as far as applying the rules and the logical flow within these set theories is concerned.
I find your idea that an object cannot have parts unless its subject to temporal separability as un-supported. Especially under imaginary grounds. We can fathom the imaginative line of having an eternal unicorn. Now this entity (which do not exist in the real world) does have parts, for instance the corn is a part of the unicorn, its head, tail, legs, etc.. all of these are parts of a unicorn, so the unicorn is not a mereological atom, even if it was an eternal being. So descriptive imaginative wise we can fathom what it does mean to have eternal objects with parts. But in the real world if one thinks that an object if composed of parts then those parts must have been in existence "before" the whole, i.e. every object must be "formed" after parts by some force connecting those parts that occurs in some moment of time of course after the existence of the parts, then according to this synthetic hypothesis (which holds mostly in the real physical world), then of course one would be bond to reject eternal objects being synthesized from parts, since there should have been a moment where those parts were separate and then after that another moment came where they'll possess a relationship to each other that caused the unit of that object.
In the mathematical realm, we don't adhere to such observations. The mathematical realm is changeless with time (unless time itself is adopted in some mathematical models where change is studied) and objects in that realm can be dealt with as having parts (proper parts I mean) without having to have a formalization (synthetic) moment. That applies to classes (i.e. collections) with many elements, where their singular parts can be understood as their elements, now those multipleton collections do have parts and they are supposed to be eternal in the platonic realm. Now that is obviously false in the physical realm. But this doesn't mean it is false of every realm! Platonists holds that the platonic realm exists, so it is a true realm, while fictionists think its false.
I'd say even if that platonic realm is FALSE (i.e.doesn't exist), still, the logical-mathematical rules displayed in them are not necessarily false. And they can hold of some real scenarios, and so can possibly find applications, and that what really matters!
Then it would be impossible to create a reasonable hierarchy like you were talking about, if the meaning of tight and loose could vary.
Quoting Zuhair
Of course we can imagine things which would violate that proposed law, but the whole point is to exclude from our principles, things which are physically impossible. If we allow mathematical principles to include imaginary things which are physically impossible, and we apply mathematics within physics which employs inductive conclusions that exclude such things as impossible, then we will be employing contradictory premises in the very same application, as I described.
Relativity theory denies the possibility of eternal unchanging relations between parts (absolute rest). But if mathematical principles allow for eternal unchanging relations, then we have contradictory premises. To resolve this problem we cannot change our description of the physical world without loosing accuracy. So we must change these mathematical principles to provide consistency. That it would be difficult to make such changes, or that the existing principles are supported by simplicity, is no excuse.
Quoting Zuhair
Sure, such mathematical rules would be applicable, and in the vast majority of cases they would give us very accurate conclusions. That is because the vast majority of cases don't deal with things like eternal objects, and infinity is never approached. And, when eternal objects are approached (fundamental particles for example), we can be fully aware of the faults within the principles, and take the conclusions with a grain of salt. However, as these faulty principles become more accepted, and work their way deeper and deeper into the hierarchical structure of the mathematical axioms, their application becomes more commonplace. At this time, they are subsumed by other principles, and we could loose track of when they are actually being applied, and not notice the mistakes which they produce.
You can! The hierarchy would be more of a mold, a frame, that suites a generality purposes. Of course in the particular application the hierarchies would differ.
By the way do relativity theory speak about rules about the mathematical objects used to write its laws with? Aren't those mathematical objects a part of the theory? I don't think relativity theory assumes that numbers for example have a mass, or that they move with a speed less than light, etc.. Those mathematical objects are fixed, eternal, unchangeable. It's the physical objects that the rules of relativity theory applies to. I don't think that the mathematical objects and rules that it uses has anything to do with relativity theory. Imagine that number 1 for example will rut with time? That's crazy! Isn't it.
I agree that mathematical objects are ideal. And when using them in applications one must be cautious about hidden mathematical assumptions that might causing blurring or even faulty theories.
We use possibly fictional objects to display the mathematical rules with, because this is the most evident way in which it can be presented. Most of these rules, as well as the objects manipulated are non-spatio-temporal. But I think we can have pseudo-spatio-temporal objects representing mathematical worlds, thus in some sense approximating the real world. But I think also that nothing of the rule physical world law about physical objects would be applicable to these realms either.
In nutshell I think that set theory involving "collections" "elements of collections" "sets" "members of sets" etc.. all of these can be well understood in terms of Mereo-topology in a fairly easy manner. The hierarchy of sets is I think very essential to understand higher kinds of mathematics. For who's to say even those can possibly find some application in the real world?!
Objects existing in relationship to each other are objects existing in relationship to each other. If physics uses contradictory premises concerning objects existing in relationship to each other (the premises of relativity, and the axioms of mathematics being contradictory) then there is a problem.
It is irrational for you to claim that the "fixed, eternal, unchangeable" objects of mathematics are not subject to the laws of physics, unless you were to produce principles to support a dualist ontology. In that case, we'd have two distinct types of objects, and we'd have to start all over with our discussion of what constitutes an "object", starting with two distinct "objects". If we do not adhere to true principles, derived from the real existence of objects, we might as well allow that the construction of mathematical objects (being imaginary) does not need to adhere to any principles at all. What's the point in even assuming parts, and loose or tight relations at all, when it would be much easier to have eternal objects which have no parts whatsoever? Then the collections of such objects (sets) are not objects at all, but imaginary collections.
It seems to me like you want some half ass sort of compromised system for the existence of "mathematical objects" where you adhere to the principles of physical objects (loose and tight connections) to an extent, but when the principles of physical objects contradict the principles of eternal objects, which you desire to assume for the sake of simplicity, you are ready to throw these principles out the window in order to cling to the false facility of Platonism..
Quoting Zuhair
The problem is, that mathematicians are manufacturing, creating, objects. These objects might be completely fictional, imaginary, and not intended to represent the real world at all. Or, in application, these objects might be intended to produce a representation of the real world. We need to decide which is the case. Are we using mathematics to model the real world, or are we using mathematics to create fictional, imaginary worlds? What good is the wishy washy position of saying that these objects are "in some sense approximating the real world"? Then those who want to use mathematics to model the real world will be dissatisfied, and those who want a fantastic, purely fictional mathematics will also be dissatisfied.
Yes, I think there is an intermediate position. Mathematics is producing rule following obedient fictional objects and scenarios. However, those happen to have applications in the real world. I suspect that the matter is not accidental. There is seemingly some common grounds between imagination and the real world. Some rules about arithmetic works fine when applied to real objects, and it really succeeded in increasing our understanding of the real world around us. On the other hand obviously there are rules that are not applicable to the real world like having infinitely many numbers, etc... I think logically obedient rule following imaginative scenarios do have some common grounds with reality.
This is TRUE of many mathematical disciplines. For example a lot of set theory stuff is so imaginary that it might not even find any application at all. However, no one can really tell. Even imaginary numbers turned to have applications, even non-Euclidean Geometry turned to have applications. The problem is that we don't know really what our reality adheres to, or even what discourse about obviously imaginary objects could be useful in applications about the real world. There must be some shared realm for those applications to exist. The problem is that if we take Quines-Putnam indispensability argument, then even those non-spatio-temporal features of mathematical object might need to be accepted as part of reality, even though not a physical concrete kind of reality, but some kind of reality there!? The mathematician usually do not bother with these philosophical ground. All of what he cares for is the analytic consequences of his assumption, which for clarity and simplicity they are usually stipulated outside of the confines of space or time or both, or within the confines of some imaginary world that has its own space and time characteristics, as well as its own part-whole relationship with respect to eternity issues in it. Most mathematicians work primarily in a Platonic world! Philosophy comes later!
I agree that correspondence with the real world is not accidental, and these principles are adopted for usefulness. But I don't think that any non-useful rules would be accepted. The reason why there is infinitely many numbers is so that we can count anything. It doesn't matter what the world might consist of, we will still, in principle be able to count it because we have infinitely many numbers.
I think we ought to consider a difference between corresponding with the real world, and being useful in the world. The two are clearly not the same. Mathematical rules I believe, are produced to be useful. This means that they do not necessarily correspond with reality, nor do they even have "some common grounds" with reality, they simply interact with reality by means of us using them. Perhaps this process, the activity of interaction, may be called a common ground, but we have to be careful to recognize that although it "grounds" the mathematical rules, it doesn't ground the real world. So for example some people say that the laws of physics describe the foundation for existence in the universe, but this is not really the case. The rules of physics are how we apprehend existence, in the universe, but we may be missing a whole lot, and therefore the rules of physics don't really describe the foundation for existence in the universe.
Quoting Zuhair
The problem with usefulness, and pragmatism in general, is that many things can be useful, in many situations. If you need to pound in a nail, you can pick up a rock and hit the nail, instead of using a hammer. So speculators may think up wildly imaginary theories, and people applying the mathematics will pick up what is available, and put it to use. Therefore we need some standards of efficiency, or something like that, by which to judge usefulness. This is complex. We need clearly defined goals, which in itself is difficult because our own goals are often not clear to us. Then we need the means for judging whether the goals are adequately being achieved.
Quoting Zuhair
I agree with this, and I am completely on board with you here. Maybe, as philosophers, we can analyze this separation between the Platonic world of mathematicians, and the real world which we live in. The difficult thing here is to understand how there can be such a separation in the first place. Let's say that the separation was created, it was manufactured, produced by dualist principles. Like the example above, with infinitely many numbers, the goal was to enable us. To measure the world, we need a measurement system which transcends the world, it must be capable of measuring anything possible in the world. As you say, we don't know what's in the world before we measure it, so we must have a system capable of measuring anything. Therefore the measurement system is based in the assumption of infinite possibility, whereas the real world consists of limited possibilities.
Do you agree that this is the basis of that dualist separation between the real world and the Platonic world? The human mind apprehends the world as consisting of numerous possibilities. In order for it to understand each, every, and any possibility, the mind assigns to itself, the capacity to understand infinite possibilities. But that assignment is wrong, because the human mind is restricted by the real world, being a part of the human body, and so its capacity to understand is really restricted. So the human mind has created this dualist premise, and all these dualist principles, in an attempt to give itself the capacity to understand anything, and everything, when in reality it doesn't have that capacity. That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it.
Yes. Actually I find myself in total agreement with what you said in this particular last comment. Of course we are speaking about mathematics judging it according to its possible usefulness. Some mathematicians might reject to look at mathematics from this perspective, seeing it external to what really constitute its cor content.Seeing it like judging chess by how much profit it brings to the chess player? They might insistent that mathematics is the discipline devoted to some imaginary world were there is no impediment to carrying out strict rule following scenarios, so giving itself the maximal freedom in doing so, and enjoy that practice all for itself without caring about whether it would be applicable or not. Something like the position of "Art for Art" and "Art for people". Your views here suite "Mathematics for science", while some mathematicians might insist on "Mathematics for Mathematics". But I agree with you that the importance of mathematics is related to its role in furthering our understanding of the world, and if it didn't have that rule, it wouldn't have had all that fame and seriousness in studying it, it would have been something like chess, a mental game, a kind of sport, or even art.
I agree with the duality policy. The real issue is how to judge when a mathematician is going a stray? I mean as far as possible contribution to knowledge is concerned (i.e. application). I think a real foundation of mathematics must help direct mathematicians towards producing more beneficial mathematical theories. But how to judge this? I think this is a very important question? We need a foundation for applicable mathematics! But I'm almost very sure that a lot of mathematicians, possibly the most, wouldn't care the hell for that, they'll view it as too restrictive, and favor diving deep into the world of logically obedient rule following scenarios, no matter how wildly far their imaginative worlds are from reality. Sometimes I think this is like the dualism of religion and state in secular states. Let the mathematicians dive deep into the imaginary platonic world they like, and let science work with its strict observance to reality moto. The important matter is not to confuse both. We only need to coordinate both at applications!
Quoting Metaphysician Undercover
The real problem is even if it is false, still the logically obedient strict rule-following themes it negotiates can prove to be extremely useful, even if in part. The real problem is that we'll never know at which stage it will "run out its course"? Possibly one day foundations for 'applicable' mathematics would issue, having clear cut edge between what is beneficial and what is not?! Perhaps by then this platonic dream would vanish! perhaps?! but I don't really know where such a thing would start? or even if it could start really? Until such alternative is found, we'd better keep the current dualist stance.
I find that hard to imagine, Mathematics for Mathematics. What would this consist of, people studying and producing mathematical principles just for the sake of doing that, and none of them actually doing anything with the mathematics? So people create mathematics, they study to understand mathematics, and they never apply the mathematics. That's a very odd thought. But in the university I went to, Mathematics was in the Arts department. I think it was studied as a useful art though.
Quoting Zuhair
Well you're right, mathematics as an art provides a freedom which is appealing to many. One can demonstrate all sorts of very beautiful things just by applying mathematics to mathematics. That is actually the beauty of mathematics, its very nature is incredibly beautiful. But suppose we can separate math for math's sake, creating beauty in mathematics, from applied math, which is math for some other purpose. The pure art of mathematical beauty would just be there to look at and think about, and the artists would have to warn people against trying to apply that math (some mathemagicians could put some real freaky tricks in there which are thought to be incredibly beautiful). The other mathematicians, creating mathematics for a purpose, would have to be disciplined so as to reign in that freedom, and keep things directed toward the proper goal. I think the two have already been confused as eloquence is becoming more and more of an important part of theories.
Quoting Zuhair
It may not be that difficult. To begin with, any application in which the infinite is approached, in any way, is an application where the false premise of Platonism is causing a problem. The mathematician can devise all sorts of different ways to deal with the occurrence of the infinite, but these just disguise the problem. The very nature of infinite, and the nature of application (being practise), makes it impossible that the infinite could be encountered in any application. The mathematician might say 'we have to be able to apply the infinite, it's part of mathematics', but really all that the infinite is, is a thing of beauty, a beauty which is negated by any misguided attempt to apply it.
I'm not really sure of that? But as part of history of mathematics, mathematics prior to the 19th century were very cautious when speaking about the infinite. They actually almost avoided it, and only used finite kind of numbers and entities (albeit infinitely many). Not only that! Most of hard core mathematics can be encoded in very weak systems of set theory, however those systems need the infinite most of the time, but they are very weak infinite systems like first and second order arithmetic.
Can you describe this "need" for me? If mathematics prior to the 19th century got along fine without speaking about the infinite, where does this need to apply the infinite, in set theory, come from?
Suppose the purpose of the infinite is, as I described. It is assumed so that we can measure anything. No matter how large the magnitude which we've already encountered, we can always measure something larger. That is the principle of the infinite, it's an open ended scale so that we can always go bigger in our comparisons (measurements) Perhaps you don't agree that this is the reason why we assume infinity, but let me start with that assumption anyway.
The problem I apprehend in set theory is that there appears to be a perceived need to measure the infinite itself. Why would we want to measure the infinite? Suppose you're applying mathematics in measurement, and infinity rears its ugly head. (Consider that the infinite is the most beautiful mathematical principle, as a principle devised for unbounded usefulness, but when it occurs in practise it is the most ugly situation). What is indicated, by the appearance of infinity in the practise of applying mathematics, is that we've encountered something which we cannot measure. That's what I think. What I also think is that the proper way of dealing with this situation is to take a very good, analytical look at the thing we are trying to measure, in relation to our principles of measurement, and determine why infinity appears. The principles being applied are not properly related to the infinite to allow the thing to be measured. The purpose of infinity is to enable us to measure anything, so if it appears in the measurement, there must be an incompatibility between the principles applied, and the thing being measured. We must therefore determine this incompatibility, and devise the proper principles suited to measure the object we are measuring.
However, it appears to me, like set theory takes the wrong approach when the infinite rears its ugly head. Instead, set theory proceeds toward an irrational resolution of this problem, by devising a way to measure the infinite. It's irrational because the infinite, as a principle which gives us the capacity to measure anything, can only be effective to this end, if it is separate (transcendent) from the things being measured: it is necessarily an Ideal. If we allow it to be placed in the category of things which we can measure, we negate the transcendent nature of the infinite The principle, the infinite, can no longer give us the capacity to measure anything by transcending everything, because we've allowed the principle to become diluted, by designating it as one of the things which we are trying to measure.
This is indeed a very big subject. And what you've said is reputable. Indeed some imminent mathematicians took that stance, and even a much more aggressive stance of even not allowing for existence of large finites (see finitism and ultrafinitism) let alone infinities. One can indeed write whole volumes on that subject. However, I'll present a possible counter-view:
In agreement with your assumption of having the infinite as a capacity for unlimited measurement in applications about finite objects. I'd say that It is not just infinitude of natural numbers that we need for the sake of such unlimited measurement, we need to stipulate useful relations and functions (operators) on them, so relations like "equal", "smaller than", "greater than", and functions like "summation", "multiplication", "exponentiation" etc.. all of these are needed. So we need SENTENCES in a language that uses those functions and relations between natural numbers, those sentences would be the axioms, and can be the theorems deduced in the systems having logical and mathematical inference rules starting from those axioms. So I insist that we need "sentences" in the language of arithmetic, which doesn't include only natural numbers, but also includes relations and functions on those natural numbers. Now due to Godel's incompleteness theorems, there is no effective system that can capture all true sentences of arithmetic! Now notice here that I'm speaking about sentences of arithmetic and their terms only range over natural numbers, i.e. there is no infinite object whatsoever symbolized in that language, so it is totally about finite objects or descriptions about finite objects, so it is the kind of language that we think it can possibly have applications in our world, viewing all objects in our world being finite. Now we come to the role of theories about the infinite, i.e. theories that speak about infinite objects like actual infinite sets for example, which as you said, and I think it is generally agreed that our physical world seems to be incompatible with their existence in it. However, despite this incompatibility those infinite theories can prove some true arithmetic sentences (those that only range over natural numbers using finitely long formulas) to be true that the theories restricted to the finite objects fail to prove! Now we want the infinite to give us the capacity of measurement as you said, but you need the tools for those measures, and the tools for those are not just the existence of infinitely many naturals, but we need sentences about some relations and operations on them, and the main problem is that we don't have a theory restricted to the finite realm that can effectively give us all of those sentences, which are infinite in number by the way. Or even if those useful arithmetical sentences are finite in number, still we don't have a theory about finite objects that can capture all of those finite sentences, or even if we can have, we don't know which theory is that. So theories speaking about infinite objects can indeed prove some of those arithmetical sentences about the finite realm of them, and those can be useful sentences. So that's why we go to the infinite. Actually Hilbert was under the impression that all of infinite mathematics is just a conservative extension of the finitary one, and that all of what we need is what those infinite theories speak about the finite realm of them. However, he turned to be wrong. And this makes it even more necessary to go to theories speaking about infinite objects, since those can be indispensable for that sake, since those are stronger theories (deductively speaking) and so can prove sentences about the finite realm of them, that weaker theories limited to finite objects cannot do. So this would indirectly support the measure-ability view that you've raised! We go to theories about the infinite, in order to have more and more useful SENTENCES about the finite world! And so use these sentences necessary to produce the additional tools for measurements in our world, and so furthering knowledge about our finite world.
If the theories about the infinite realm proves to be indispensable for harvesting some useful arithmetical sentences about the naturals, i.e.that no theory limited to the finite realm can effectively prove them, then this would make the case for mathematical investigation of the infinite!
Not only that, sometimes those infinite theories might not be indispensable for the above sake, but being more powerful they make proofs about such sentences much easier to have, and so one can contemplate many versions of equivalent systems about the finite world, much easier than when working with theories restricted to just finite objects. It is this ease that is also important, since it would have heuristic value to discoveries in the finite realm.
Of course there is still the objection that such additional sentences about the finite world that theories encountering infinite objects prove, these would prove to be FALSE arithmetical sentences, and so be misleading rather than useful, and would be deemed as waste of time and efforts. That's of course possible, but until such an argument is provable, the window is still open for mathematical investigation of the infinite, under the hope that it might play the above-mentioned role, and so we need to give it the benefit of doubt!
Of course there are other more radical objections to your line of view, like the mathematics for mathematics viewpoint, and like the other direction objection that is our physical world itself being of ACTUAL INFINITE reality and that our current physical theories and observations being erroneous about that aspect, etc... I didn't want to go to those, because I honestly think that the bulk of evidence supports a finite (or at most potentially infinite) outcast of our universe, and that mathematics ought to be useful in understanding that universe, and therefore I approached it from that perspective as given above.
Finite provides boundaries that allow measurement.
Mathematics tries to manipulate the imaginary (mental) concepts using methods designed for the finite.
Am still waiting for anyone to measure a one ended stick.
Quoting Zuhair
I take it that you are saying here, that we might need to allow for infinity in any scale of measurement. If that's what you're saying, I don't think it's true. I think that each parameter, "greater than", "heavier than", "denser than", etc., has its own definition. This creates a sort of category, and the descritpiton of the category provides the limits to what is measured within that category, therefore infinity is excluded by these descriptions or definitions. We might say that each of these categories is a specified "quality" and the scale is produced to enable measurement of that quality. The determining features of that quality exclude the possibility of infinity within the scale. So it doesn't make any sense to say "infinitely great", "infinitely dense", "infinitely heavy", "infinitely hot" or any such thing, because these are defined qualities, and to fulfill the criteria of any quality requires that the thing being measured can be related to the thing which forms the scale for measurement of that quality, and this excludes infinity..
Quoting Zuhair
The "SENTENCES" act to describe the various qualities, and they may be set up as rules for application of the numbers. So a scale consists of sentences which are rules for the application of numbers. The most fundamental sentences are the axioms which are the most general rules for application. This is where it gets tricky (watch out for mathemagicians). The question is, are there any true general rules (axioms), which are applicable to all mathematical applications, or, is each set of axioms tailored to a particular type of application (measuring a particular quality). In terms of "sentences" then, are there any sentences which may act as rules for all mathematical applications, or is every sentence designed for a particular type of application. I suggest that we allow the possibility of a sentence which allows the use of mathematics for any scale, to measure any quality, as a fundamental axiom, and this would be a sentence describing the infinitude of numbers. Some qualities would require one type of scale, others another type of scale, and the numbers must be infinitely pliable to adapt to all scales. The fundamental axiom therefore, would be an axiom of order, order being required for any form of measurement, also allowing for infinity so long as the infinity is ordered. Allowing for infinite disorder is nonsense.
Quoting Zuhair
You ought to be able to see that Godel's approach is backward. A sentence is useful for describing a quality. Once the quality is described, we can take the mathematical principle of infinite applicability and apply it to the described quality. But it doesn't make sense to try and turn things around, making the infinite applicability of mathematics into a quality which can be described by a sentence. So naturally, Godel cannot find that sentence. The infinite applicability of mathematics must be inherent within mathematics itself, and therefore quantitative, and not qualitative. It cannot be described. Even my above description, using "order" does not do justice to "quantity", because it attempts to hand quantity a quality, which is to assign that impossible sentence. So the meaning of "quantity" and "infinite" must remain independent from any descriptive sentence which would assign to these a quality. Such an assignment would be a restriction to the thing which has been designated as unrestricted.
So we cannot approach in that backward manner of attempting to assign a restriction to the unrestricted. We start with the unrestricted, "quantity", and proceed toward measurement by using sentences of restriction which are derived from the thing to be measure. We observe the thing to be measured, and we produce sentences of restriction which are applied to the mathematics, restrictions which are designed to enable measurement of that particular type of thing. So there is no random or arbitrary restrictions placed on the natural numbers, each restriction is placed for a particular reason, dependent on the apprehended quality.
Quoting Zuhair
According to what I've said above, we cannot come to theories about the infinite in this way. We accept the infinite as a starting point. We make theories concerning the things we observe, and restrict the infinite for application accordingly. To turn around, and face the infinite, with the intent of restricting it for no particular purpose is an irrational move. That's what theories about the infinite do, they restrict it with descriptive sentences. And if this is done for no purpose other than to describe the infinite, it's crippling.
I would think that if we want to proceed toward understanding the infinite, we must approach from a different direction, other than mathematical axioms, which by their nature are composed to restrict the infinite for various purposes. We must therefore approach from the premises which assign to "quantity" its infinite capacity. This means that we must understand "infinite" in terms other than descriptive terms; descriptive terms being applicable to quality only, and used in mathematics for the sake of restricting quantity. Are there sentences which give to "quantity", "infinity", without resorting to description?
Quoting Zuhair
The infinite gives us the appropriate capacity for measurement simply by assumption. We assume that we have that capacity, and so long as we do not restrict it, it persists, as the fundamental premise. We make the sentences concerning relations and operations according to our observations, and the qualities which we desire to measure. We never need "all of those sentences" we produce them as required, dependent on our observations.
At this point, I think we ought to distinguish between the object itself, and the observed qualities of the object. The restrictive sentences are always produced for the sake of measuring particular qualities. We do not assume the capacity to measure the object itself. This seems like it would be a little nonsensical as all of our observations are of particular qualities. What would we be trying to measure, as the object itself? Therefore the observed finitude of the object is a function of its qualities. Observation of the object's qualities, and the conclusion that qualities are real, a fact of the object, produces the conclusion that the object is itself finite. So to say 'an object is finite' is to say nothing more than 'an object has finite qualities'.
In all of our observing and measuring of qualities, we really do not ever get to what it means to be an object, and this is what it means to be something which "has" qualities. So here, to understand the existence of the object itself, we must turn to something other than mathematical principles. From these other principles we can begin to understand "the object" in a different way, as a fundamental unity (perhaps as various qualities unified). And unity in relation to multiplicity is the fundamental principle of mathematics. So when we assume a unity of a multiplicity of qualities, as an object, we have one thing which is at the same time many things, and potentially an infinite number of things, so long as we maintain the distinction between the thing (one) and the qualities (many).
Quoting Zuhair
According to what I stated above, the idea of "infinite objects" is a misguided one. The object, as the thing, is always one, a simple unity. The multiplicity as one, is what is unified, under the named and identified "infinite objects". And this multiplicity is a property of the identified thing, we might call it the qualities of the named thing, it consists of numerous things. To speak of a multiplicity of objects is to class those together as one unity. Then "the numerous objects" is a quality of that multiplicity which is referred to as one unity. So we have an object which is described as a multiplicity. We allow that the multiplicity which composes that one identified object, may be infinite. But it is incorrect to refer to that proposed infinity as an infinity of objects, because it is really an infinity of parts, the qualities of that mentioned object, which is the named collection. The thing identified as "infinite objects" is really the object itself, so "objects" ought not be used here and it is really nonsense to speak of "infinite objects".
Quoting Zuhair
I think it is important to recognize that a sentence about something will describe a quality. As a quality, the thing referred to is finite. We might allow that the object itself, with that quality is infinite in the sense of potentially having an infinity of qualities,, but this is a self-defeating assumption because it assumes an object which is immeasurable, and the purpose of assuming the infinite is to make all things measurable. And that is also why it is irrational to allow that the infinite itself is an object. We allow for the possibility of infinite qualities to account for the unknown qualities which we have not observed. But this assumption of "'the possibility of infinite qualities" is only made because we know that our knowledge will never be complete. It doesn't indicate that we assume that there actually is an infinity of qualities to any object because this would be assuming the object as fundamentally unknowable.
Hi Metaphysician Undercover! I also cannot understand what you've wrote. I think we are departing a part.
My argument was a very simple argument. I was simply speaking about Peano arithmetic "PA". Peano arithmetic only speaks about natural numbers, which you can in some sense imagine them as indices of the quantity of members in finite sets, so we are speaking here about measurements related to FINITE objects, and so the language of PA doesn't encounter any mention of the infinite, although it allows for the potential of having infinitely many naturals, but it doesn't speak of that infinitude of naturals as an object on its own right. The only objects that PA speaks about are naturals which are in some sense measures of finite objects. So generally speaking PA would be the kind of a theory that is expected to have applications about objects in our finite (or potentially infinite) universe. So all sentences written in the language of PA are statements about finite objects, so they all speak about the state of affairs related to finite objects, as we regard them to be potentially applicable!
The problem is that MOST of sentences written in the language of PA are not provable in PA. So we are missing a lot of sentences that might have useful application in our real world, because PA cannot prove them. However those arithmetical sentences can be proven from theories that encounter speech about existence of infinite objects, like set theory for example, so ZFC can prove arithmetical sentences which cannot be proven in PA. Notice that I'm speaking about arithmetical sentences, i.e. sentences about natural numbers, i.e. statements about measurement of the FINITE, so those are statements that can have applications in our real world, and some of those sentences are provable in ZFC while PA cannot prove them!
Not only that there are sentences that PA happen to prove,i.e. theorems of PA, but yet the proofs of those sentences are too long to be first discovered depending on PA alone, that it is much easier to prove in ZFC, because the proofs are much much shorter. In this way ZFC can help us discover theorems of PA itself.
You see what I'm trying to say here. That's why we revert to ZFC, because it is more powerful and more expressive than PA, that we can even prove theorems of PA itself using ZFC in a much easier way! And also using ZFC we can in addition prove sentences written in the very language of PA itself, that PA itself cannot prove. And those sentences can express facts about measurements of finite objects, and so might have applications.
I read all of the above account of yours carefully. It is really nice. I should say that it added a lot to my knowledge. So Thank you for that.
There is a lot of depth to what you are saying. I won't here make my final stance with or against it, since I need myself more time to examine it in a deep sense. For now I'll outline the potential difference between what you've presented and the commonly held views in foundations of mathematics.
What you said runs against standard set theory. For example you adopted the strategy of assuming the infinitude of observations, of possible qualities (largely its a stance made because of our incompleteness rather than being a truthful statement about the real world, as you maintain). However you refused to put the infinite among those qualities, and you emphasize that its not, it is just a pure quantity. The problem is that the common modern understanding in set theory of the infinite runs to the opposite of that. It views a lot of grades of infinitude, and the absolute allowance of all observations is something that NO formula in set theory or logic can describe. We may paraphrase it as saying that there no limit to measurability made inside theories, but this here applies to various degrees of the infinite as well as to finities. So you are not differentiating between the 'absolute capacity' of measurement, which is sometimes ironically called by some set theories as the absolute infinite, [which you call the "infinite" by the way], and the various grades of the infinite, the latter ones are using your terms qualities, and they can be measured in an effective manner, while the former one (the absolute infinite) is what you cannot measure nor can formalize it as an axiom, and using your terms I would describe it as not really a quality, its a pure quantity (using your terms), this absolute infinite is something that no set theorist tries to capture by its axioms or theories, its an unlimited tendency of measurements. But the scope of measurement in your case is limited to the finites, while for the traditional set theorist it doesn't stop their, it can encounter various grades of the infinite, he will answer you that you are confusing $\omega$ [the first infinite] with the absolute infinite, and this is wrong. The scope of allowable measurements with the set theorist is vastly larger than the narrow one you are allowing. That's the difference.
However, my argument above (the one you've answered to) is not that deep. It only says that theories that have capture SOME infinite objects, are vastly stronger (deductively speaking) than theories that only capture the finite, that's why technically speaking, those stronger theories can help even prove some theorems of strictly weaker theories that only speak about the finite, not only that it can prove theorems spoken about in their language that those weaker theories cannot prove, and those sentences are of the kind speaking about infinite objects and relations between them and properties of them, so they are (generally speaking) the kind of sentences expected to have application in our finite world. This mean that theories speaking about infinite (as well as finite) objects can aid in measurements of the actual world via proving those sentences of them that are concerned with the finite realm of them. It supply us (technically speaking) with more and more sentences about finite objects, and so enrich our knowledge base and potential to make descriptions in our finite world. Its a pure technical issue. So they are useful and can make contributions to our finite world, although they are theories that have the capability of speaking of infinite objects (pure unities with infinitely many qualities).
Just being a pure unity with infinitely many qualities, it doesn't mean that it is unknowable, or that it is not measurable! That's the basic message that set theory is telling us. The earlier philosophers who thought that, like Aristotle, were mistaken!!! Cantor clearly showed how to make descriptions of various grades of the infinite, and he successfully did so, without being encountered with any inconsistency so far (it has been more than a century since then!).
Since contemplating SOME of the infinite as objects, or as qualities in your terms, did enrich our parcel with sentences about finite sets and numbers, which can possibly have applications, then it enriches our potential for application, so we are justified in doing such contemplation. Its a technical utility point of view.
"What is the difference between actual infinity and potential infinity?"
All matter can potentially be part of a black hole.
Here's the issue I have with this position. Let's assume that numbers are measures applied to finite objects. Prior to application, we need a rule, or rules of application, and these are in the form of sentences. Naturally, the sentences, rules, must be based in some actual understanding of what a finite object is, in order that the application be useful. For example, in geometry the goal is to measure the spatial limitations of objects. The objects are finite because they have spatial limitations, and the goal is to measure these limitations. So we have developed some understanding of these spatial limitations through observations, and produced axioms of geometry from this understanding.
Now let's go deeper, and relate this principle to the axioms for applying numbers. The goal here is clearly to measure objects, but we must be careful in the assumption that the objects are finite. We could probably say that natural numbers are intended to count finite objects, but if we consider rational numbers, which are infinitely divisible, the object loses its finitude in that way. Now we would have two distinct rules applicable to the use of numbers for measuring objects, rules for measuring finite objects and rules for measuring infinite objects. Because of this, we would need further rules to distinguish whether the objects to be measured are finite or infinite. Therefore we need some principles for understanding the infinity of an object. In the example of geometrical axioms, above, we have a clear understanding of the finitude of the objects to be measured, they have spatial limitations.
Now we want to say that these spatial limitations do not provide a complete, or absolute form of finitude, because the object with spatial boundaries, may be infinitely divisible. The rational numbers allow us to make such divisions, but then the irrationals pop up, and we see that our ways of dividing finite objects are somewhat deficient. So I think we can say that there are natural limitations to dividing objects, which makes infinite division of an object irrational, but we really do not understand these limitations.
The dilemma now is how do we restrict our rules for the application of measuring objects, such that objects are necessarily finite, and therefore measurable, when we do not know the real principles whereby an object is restricted to being finite. We have rules of geometry which restrict the spatial extension of an object, but we do not understand, and therefore cannot produce the rules, to restrict the "intension" (idiosyncratic use) of the object. Until we properly understand the intension, we do not have adequate rules to restrict the object's intension, therefore we cannot truthfully say that the object being measured is finite. This is a problem inherent within division, that we have irrational ratios, (a type of contradiction), which becomes very evident in harmonies and wave problems. We do not understand the real spatial constraints which restrict the division of objects, so the rules which we apply are simple modifications to infinite divisibility. Therefore our rules are rules for measuring infinite objects which we pretend, by tweaking the rules with terms like Infinitesimal, are actually rules for measuring the finite object.
Quoting Zuhair
Now, with set theory you jump to the assumption of "infinite objects". What grounds this assumption? We use numbers to measure objects, and the rules for measuring are based in the natural restrictions of the object. Having restrictions is what makes them measurable, but also what makes them finite. In the last post, I called these restrictions "qualities". The qualities of an object are what we measure.
Let's say that because an object has restrictions, boundaries, making it finite, this allows that there is a multitude of objects, more than one. Now we want to count that multitude and this is different from measuring an object (its restrictions, qualities), it is measuring a quantity. This is completely distinct from the act of measuring objects, it is an act of measuring a quantity. Therefore we can make a conclusion here. As explained above, we need principles, statements, rules based in the understanding of what an object is, prior to making rules about measuring objects, now we need an understanding of what a quantity is, prior to making rules about measuring quantities. Again, we are confronted with the very same problem described above, which is the consequence of our inability to understand the nature of divisibility. The plurality, multiplicity, "quantity", is completely dependent on how we divide things up into individuals. The rules which we have for dividing our environment into individual things are the rules which govern the "quantity" of things.
Quoting Zuhair
I view this distinction between the absolute infinite, and various grades of infinite as unjustified, and actually a category mistake. The absolute, and the relative are categorically distinct. To place "the infinite" in both categories, is actually impossible, and all it does is give two quite distinct meanings to the term, inviting equivocation. For example, it's like the difference between absolute rest and relative rest.in physics. If we accept the principle that all rest is relative then absolute rest is meaningless to us. But if we accept absolute rest as a meaningful proposition, then relative rest can no longer be called "rest" because "rest" is just assigned as a reference point, while the thing designated to be at rest is still in movement compared to absolute rest.
The same is true for absolute infinite and various grades of infinite. If infinity is relative, various grades, then "absolute infinite" is meaningless because if there was an absolute infinite it would mean that the other various infinities are just not true infinities and ought not be called such. So all this does is give "infinite" two very distinct meanings, inviting equivocation.
Quoting Zuhair
This could only be true if we could develop an understanding, and therefore applicable rules concerning what it means to be an infinite object. To me, as I've outlined above, the idea of an infinite object only arises because of our inability to understand the nature of divisibility. We do not understand the true constraints and restrictions on divisibility in our world, so we posit infinite divisibility. This creates the infinite divisibility of an object, and the idea of an infinite quantity of objects, both. An infinite quantity of objects is derived from the infinite divisibility of the universe. Therefore I believe that "infinite object", or "infinite objects" represents a misunderstanding of the natural boundaries which objects have. Unless "infinite object", or "infinite objects" can be given some real meaning, as referring to something real in the world, supported by real principles, it's simply nonsensical talk..
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It doesn't matter whether its makes sense in relation to the real physical world or not! The point is that it can serve as a strong reducer of proof lengths of theories that only speak about objects related to our real world, thereby it can aid our analytic deductive reasoning about our real world, it makes it technically easier to analytically follow up truthful assumptions about our own world, because it "shorten" proofs. That rule of making it easier to follow up the analytic consequences of truthful assumptions about our real world, is a very useful tool, as as far as this property is concerned then it makes full sense to introduce them as Tools to help understand our reality in a much faster and easier way than without them.
If it doesn't make sense in relation to the real world, then it cannot be a true premise. Therefore the proofs which are derived are unsound. This is the matter of "eloquence". Eloquent proofs are not necessarily sound proofs. You are persuaded by the ease of the proofs, not by the soundness of the proofs.
Unfortunately you are not following what I'm saying. I'm speaking of two roles that a stronger theory (deductively speaking) can play. The FIRST role is in proving sentences in a language about finite objects that are NOT provable from some known finite theories having SOUND principles. Here you can make your objection above. But your objection above doesn't extend to the SECOND role. The second role is that stronger theories do facilitate provability within SOUND theories themselves!!! I need you to concentrate here, suppose you have a sound theory, by that I mean a theory whose axioms are coined in relation to the reality, i.e. they are realistic rules, and so they are sound, i.e. they meet reality, now supposing that this theory is consistent, then provability of a consistent theory whose axioms are sound, is always sound, i.e. its theorems always conform to reality, since provability (in consistent systems) preserves soundness of axioms. Now I'm speaking about those kinds of theories which we think that they'll have many applications in the real world. Now even for those theories, the stronger theories which are not reality based (like those speaking about the infinite), still do possess much expressive analytic power that can enable them to prove theorems INSIDE the realistic theories in a much shorter manner, here with this situation the proofs are sound of course, because they are only shortening proofs of a sound theory. This technical assisting role is very important. You and others in philosophy might underestimate it, because this second role is in principle dispensable! But there is a great difference between "in principle" and "in practice", I'd agree that they are in principle dispensable, but in practice they are not, because we are humans, so theorems of sound axiom systems that are provable from very long proofs will not be discovered by the human mind, while the assisting stronger systems would enable discovering those theorems because they can prove them in shorter steps, and then afterwards we can go back to the original sound theory and find the long proof of those theorems. So the stronger imaginary (potentially unrealistic) theories can play the role of a technical guide of the weaker more realistic ones. Truly the FINAL acceptance of the theorems would be by establishing their provability in the sound systems by finding those very long proofs, but finding these long proofs can be assisted by provability in the stronger theories, even if the stronger ones are not realistic. The reason is because the stronger ones have more expressive analytic power. Its a pure technical matter.
I don't understand why you call these theories, which are not based in sound premises "stronger theories". They are clearly weaker.
Quoting Zuhair
The problem though, is that the so-called stronger theory is unsound, and therefore the conclusions produced are unsound. It may be that some of the conclusions will later be proven to be true, but some might later be proven to be false. So there is really no point in using the so-called stronger theories, because they cannot give us any certainty in the conclusions.
You refer to such a theory as a "technical guide", and say that they are aimed at practise. So lets say that they are like hypotheses. We apply them in the attempt to prove whether they are true or false. So we must be willing to reject them when they are proven to be false.
*stronger* is a logical term. Theory A is stronger than theory B if and only if every statement provable in B is provable in A, but not every statement provable in A is provable in B.
Quoting Metaphysician Undercover
Yes! Of course. If the stronger theory proves to mislead us about provability in the weaker theory most of the time, then we reject it. On the other side if most of the times it proves to be helpful, i.e. assist us in proving theorems of the weaker theory, then we would say that it is playing a conservative extension role over the weaker theory most of the time, then we adopt it as a guide only.
You take theory B, and add some unsound premises and voila, theory A. Everything in B is provable in A, and even more is provable in A due to the additional (unsound) premises. I don't see how that's helpful. Sure, it gets you to the conclusion you want quicker, but that conclusion is unsound, fabricated by adding the premises require to produce the desired conclusion.
It is sound! because it is the pure conclusion of the original sound theory. I'm just using theory A as a facilitator and then I'm checking it again in the pure theory B, so there is no harm. For example you have a certain argument for a proof in theory B (the reliable theory), but there are some missing steps, you translate the argument to theory A, sometime theory A manage to fill the gaps, then you go back to theory B itself and check if that filling is correct. There is no problem with that approach as long as we are not depending solely on what theory A is doing. To facilitate matters is an important practical tool. Its no magic, we know why it works, because it increase the expressive power the logical deductive system, it supply you with more tools to solve analytic problems, many times the difficulty in proving a theorem (in the original reliable theory) is not due to you being in the wrong direction, no, its due to poor analytic tools, those are supplied by the more powerful theory that extends A (even if the additional premises of A are unreliable), and that's why it manage to fill the gaps, but we always go back to the original theory and check the complete proof which is solely this time coming from theory B (the reliable theory). Theory B is the final arbiter. We start from B and we end by B; theory A is just an intermediate step quickening provability in B, just a facilitator. That should be right!
I'll put this as a separate reply. Because it speaks about role 1 (see prior comments). Now role 1 is the real issue. Because role 2 is not fundamental, but it is a practical tool, that is to build technically powerful conservative extensions over theories in order to shorten proofs thereby easing provability in the original reliable systems, that was role 2 and there is no problem with it, it is just a technical facilitator. But I'll present here role 1, which is what really matters! With role 1 we take a reliable theory B and add to it axioms about *infinite* objects, under the assumption that those axioms are SOUND. The real take here is that we don't really know if our universe admits existence of a parallel realm to it that admits infinite objects, such that this parallel realm is indispensable for understanding our own universe with. Obviously the parallel realm here is meant to be the Platonic mathematical realm. So lets put this as a HYPOTHESIS. And then add axioms about such a realm that admit infinite objects, and so strengthen the original reliable theory with those additional axioms, to give this idea the benefit of doubt of being true.
Now how to decide if this is true or not. We simply try it and check. Here the stronger system would not be just a facilitator as it was the case in role 2, no, here it would prove additional theorems about the reliable sector of our world, sentences that the original reliable system cannot prove. Now we go and check if those additional proved sentences have applications, if so, then this would mean that the added theory is increasing our knowledge about our real world, and thus the axioms are true! This is challenging! If it fails and proves misleading, then we REJECT the extended system from being a part of useful mathematics, and only keep it as a piece of beautiful analytic school of art (Mathematics for Mathematics).
Lets see how this fair in practice. Now we know that every *finite* fragment of ZFC is arithmetically sound!!! That is: it doesn't prove false arithmetical statements! Now true arithmetical statements are always considered as "possibly applicable" because they are statements about finite objects, they are the kind of statements you've desired as giving unlimited possibility of measuring!!! Now in applications if we are going to apply a fragment of ZFC then this would be of course FINITE, i.e. well take finitely many theorems of ZFC and work within their deductive closure, now ALL arithmetical sentences proved in such a fragment are guaranteed to be TRUE, and so potentially applicable!!! That's why ZFC practically speaking works as a foundational system for useful mathematics.
The problem is that these axioms are saying things about a Platonic realm of infinite objects, so it's not easy to determine whether they are misleading or not. This is why we need metaphysics, to make that determination. An axiom can be very useful yet still misleading. That's the problem with "use" as a principle, and pragmaticism in general. The success which is derived from use is itself misleading. When we have success we are uninspired to look for a better way. The tool might be the most primitive, awkward tool, but if it brings us success in what we are doing, then we are not inclined to look for a better one, That success misleads us because it hides the fact that we really need a better tool, by making it appear like we have the tool we need
Yes, I would agree with that of course. But it need not be so really. The issue here is that if we have something useful, even if it is not the best, we ought not to reject it, we initially accept it, on pragmatic basis, but at the same time we look for a better system. But until this *better* system is in our hands, we keep working with what we have, despite is metaphysical (or other) weaknesses. I think this is a correct policy.
I agree in principle, but I think resolving that type of issue is far more complicated in practise. The way that we do something, including the way that we think, is essentially a habit, so we need to look at this as a matter of breaking a habit. The first issue, is as I said in the last post, if "the way" is observed as bringing success, it will not be seen as something which needs to be changed, and the motivation to find a "better way" will not exist. So the first step is to point out all the problems with "the way" existing, to inspire innovation and invention of a new, better way.
So the second issue is the way that we break a habit, and this is a complex issue. First, it takes very strong will power, and having someone demonstrate a better way (a different course of action toward the same end), I believe is insufficient. And this has to do with the ends involved, "the way" being a means to an end. The end is suited to, conformed to, or particularized by the means, over time through repetition, just as much or more than the means is designed, or conformed to meet the end in the first place. In other words, what we want in general, our goals, is shaped by our practises. This is how the habit sets in, we get accustomed to, and grow to like, and be comfortable with, the luxury that a particular action brings us. The required luxury (goal) will conform to the practise because no particular luxury is necessary. This allows the forming of a relationship of necessity between that particular luxury and the behaviour, the relation of cause and effect. To break the habit requires that we apprehend this particular luxury, which the action brings us, as not good.
Therefore we cannot assume that there is a "better way" of achieving any particular goal which is already being achieved by the existing way, because the causal relation of necessity between the action and that particular goal has already been established. The goal and the way exist in a causal relationship of necessity. We need to dismiss that particular goal as an incorrect goal, by replacing it with a "better goal". So it's not a matter of demonstrating a better way, it's a matter of demonstrating a better goal. Having a different goal will necessitate finding a different way.
There might be numerous distinct strategies for this. One would be to cease the practise cold turkey, leaving a hole where the luxury provided by it once existed, then finding another goal to fill that hole with. Another might be to find that other goal first, then replace the old practise with the new practise as the one necessary for the new goal. However, this latter method is sketchy because goals are not well defined, and they tend to shape themselves to the practise, as is evident in "the habit".
It still requires will power. It may take a long time to convince people that a particular habit is bad, but once it is recognized as bad, without the will power to stop they will continue to do it.
It is not enough to point it as bad, with mathematics you must demonstrate an alternative system with superior utility, something that is better. Once a system with superior utility is at hand, the exchange would be immediate, you won't need any will power. People will readily exchange older cars for new more efficient ones if they can afford to. Its a pragmatic argument.
There is a problem with pragmatics though, which I explained in the prior post. Utility is judge in relation to the end, as the means to that end. And the particular end, in its particular nature, is particularized, itself conformed, by the means. This leaves the means and the end in a necessary relationship of cause and effect. This particular end can only be produced by this particular means. There is really no such thing as demonstrating a better way, because a different way will bring about a different end. Therefore to judge the utility of mathematical systems it is required to judge the goals, or ends of the mathematicians.
This means that we must judge what the mathematician is trying to do with the system, rather than what the mathematician is actually doing with the system, and relate what is attempted to what is actually produced. Now, "the good" exists in potential, as the thing striven for, the desired end. When we look directly at "the good", it looses its nature as something particular, and is apprehended as a generality. For example, you might have a craving for a particular type of junk food, but when you look at that desired thing as "the good", you see that many different types of food will fulfil your hunger. As "the potential" to fulfill the need, there is always numerous options as to the particular thing to fulfill the need. So this is the first step to overcoming the deficiencies of pragmatism. We need to de-particularize the nature of 'the good" (as the end, what is sought), apprehending its true nature as something general. In understanding "the good" in this way,(i.e. the truth about "the good"), we see that pragmaticism has no bearing, it has no grounding, because utility can only be judged in relation to a particular end, and there is no such thing as the one and only end which will fulfill the need. Therefore pragmatism may be dismissed as insufficient because it provides no "system" for relating one particular end to another.
This is the way to judge a habit. We apprehend, and list all effects brought about by that action, these are "the goods", the ends produced by the action. And, we also apprehend and list the true goods, what we understand as the truth concerning the goodness or badness of these effects. The habit can be judged as bad, if there are bad effects of the action. As I stated earlier, every time that "the infinite", "infinity", or the mitigated "infinitesimal", occurs in a mathematical application, this can be judged as a bad effect of the mathematical habit.
Quoting Zuhair
I must say, I don't really know what constitutes a mathematical "proof", so this example is lost on me. However, I would say that any proof which utilizes "infinite", or "infinity", is not a sound proof. Infinity, by its very nature is unresolved, so assuming it as a premise of a mathematical proof, for the purpose of resolving an unresolved issue, only creates an illusion of resolving the issue by premising that the unresolved (infinity) has already been resolved. In other words, the unresolved is inherent within the premise, so the conclusion doesn't really provide a resolution.
I will remind you, that Pythagoras demonstrated the irrational nature of the square. The relation between two perpendicular sides of a square produces the infinite, which as I argued above is bad. This makes the square a truly impossible, or irrational figure. And, all "powers" are fundamentally derived from the square. Therefore any exponentiation is fundamentally unsound in relation to a spatial representation..
No! not always, if the proof is carried in a FINITE fragment of ZFC, and the proved statement is an arithmetical statement, then this is already known to be SOUND, i.e. any finite fragment of ZFC (even though it speaks about infinite sets) if it proves an arithmetical statement, then that arithmetical statement is part of TRUE arithmetic, i.e. it conforms to a proof that only relies on finite objects.
Not only that! It is expected after knowing Wiley's proof of Fermat's Last Theorem (which he actually did it in a theory even stronger than ZFC! Even though mildly so) that it can even be carried in Peano arithmetic, which is of course part of TRUE arithmetic, that's what experts on the proof say, so suppose for the sake of discussion that this happens, then that would be a clear example of a theorem of PA (a theory solely about the finite world which is HIGHLY applicable, actually the most applicable theory ever) had came to be proved first via ZFC, and that knowing that proof in that higher system served as a guide to proving it the lower reliable system. So a theory basically about the infinite did help us understand provability within a theory about the finite, a kind of a detour though it to simplify matters!
From my perspective, ZFC has unsound axioms concerning the nature of objects, as we discussed earlier. Therefore any proof using ZFC is unsound.
Quoting Zuhair
But your use of "stronger theory", as you explained, really means a theory with less rigorous criteria for the soundness of its premises, and is therefore actually a weaker theory, less sound.
Quoting Zuhair
That a conclusion from a theory with unsound premises happens to be consistent, or "the same" as a conclusion from a theory with sound premises, might be completely coincidental. You seem to be forgetting about all the wasted time spent using that theory with unsound premises to create conclusions which are inconsistent with the sound theory, to focus on one conclusion which coincidentally happens to be consistent, in an attempt to justify use of the unsound theory.
If you are working within a FINITE fragment of ZFC, then the result is always arithmetically SOUND (that if ZFC is consistent). It's a matter of technicality.
OK, might want to jettison calculus then . . .and all the technology we use as a result.
(This thread demonstrates why one of my profs sixty years ago advised his class of grad math students not to take a course in mathematical logic.)
But it looks like everyone is having fun!
The point is that "usefulness" as a principle to base judgement on, is misleading. The existence of "bad habits" demonstrates this fact.
Name me ONE conclusion that ZFC proved about arithmetic that is not sound?
Context means everything:
(1) a=a+1 (no finite solution. In complex analysis a would be the point at infinity - corresponding to an actual point at the north pole of the Riemann sphere)
(2) a=0
for k=1 to 100
a=a+1
next
(now what is a?)
I'm still mulling over "bad habits" in math. Sloppiness; jumping over points in a proof assuming they are true; assuming a hypothesis and then proving it; muddling a proof so badly other mathematicians can't verify it; etc. Using infinity or infinitesimals are the least of our concerns. :nerd:
I have no idea what you're talking about. Don't worry, it's me, not you. :grin:
Quoting Zuhair
I don't need to name any, they are all unsound. We've discussed the fact that the axioms lack truth, in how they describe objects. The axioms are the premises, and soundness requires true premises. The premises are not true, therefore the conclusions are not sound. Are you using a different meaning for "sound"?
I would say that having axioms (premises) which instead of being based in the reality of the objects we are familiar with, are based in some imaginary assumptions about imaginary objects, is far worse than sloppiness. Untrue axioms have an effect reaching into many applications, whereas sloppiness is specific to the particular application. And, sloppiness can be caught and corrected through various means of verification, whereas it requires good metaphysics to determine the truth about axioms.
[math]\lim_{n \to 0} 1/n = \infty[/math]
Actual infinity is then:
[math] 1/0 = UNDEFINED [/math]
Quoting Metaphysician Undercover
I never admitted that they are not sound. They are indeed sound of what they are describing in the platonic sense. And if platonic sense proves to be indispensable for discovering our reality, by then this would prove it to be sound. So the question of soundness of those axioms and its relation to application is still unsettled. But if they were unsound as you claim, then they must bear wrong theorems, i.e. we need to see MANY arithmetical consequences of those theories that violate true arithmetic. Why we are not seeing any? What's available in practice witness to the contrary direction, i.e. the arithmetic sentences proved in them are true, and actually it is provable that any consistent finite fragment of ZFC is arithmetically sound. You gave a metaphysical argument against set theory which I don't totally agree with.
I'm not convinced that metaphysics can determine anything and I'm not sure what is meant by the "truth" of axioms. This is an interesting discussion and a few of you are probably analytic philosophers who know much more set theory than me. All I can say is that practicing mathematicians usually avoid these discussions unless they are in these sub-disciplines. This commonly employed mathematics seems to describe most of the physical world. And, yes, the point at infinity exists, as I've described it.
But one can certainly entertain opposing ideas. That's metaphysics. :cool:
I thought we agreed that Platonic objects are not true objects. We assume these objects for some purpose or utility, but they do not have any real existence as objects. So if we create premises which describe Platonic objects as objects, when they really are not objects, but fictional objects, then these premises are false and therefore unsound.
Quoting Zuhair
This is not the case. "True arithmetic" is arithmetic as defined by the accepted axioms. So if unsound axioms are accepted into "true arithmetic", then we might see no such consequences if there is consistency between the unsound axioms. And of course, there is consistency in the unsound premises of Platonic objects. The undesirable consequences only become apparent in application, because the premises concerning the nature of an object are inconsistent with what an object really is. You can see these undesirable consequences in the particles of particle physics.
Quoting John Gill
Zuhair actually appears to be quite knowledgeable about mathematics and its axioms.
Don't yout think it is a sort of problem, that mathematicians would avoid discussions concerning the truth or falsity of their axioms? Think about other disciplines, a physician for example. Do you think a physician would be comfortable applying principles of medicine without any concern for whether the principles are true or not.
EXAMPLES?
Not at all. There are mathematicians whose expertise lie in set theory and foundations. Let them do their job. Non-standard analysis lives within a mathematical model that, to the best of my knowledge, is consistent. It assumes (axioms) the existence of infinitesimals and infinity with symbols representing them and rules for manipulating these symbols. Can you determine the "truth" or "falsity" of these axioms? (no fair resorting to "manifest")
You want examples of why conclusions drawn from false premises are unsound? Come on. Try this. The full moon is ten miles away. I can walk ten miles in four hours. Therefore I can walk to the full moon in four hours.
Quoting John Gill
Consistency does not mean that the premises are true, that's the problem. A system with complete consistency, applying false premises will still give unsound conclusions.
Quoting John Gill
Yes, we can make those judgements. We have to look at what the symbols represent, and make judgements on the reality of that If the symbols are supposed to represent objects, we can apply the law of identity, as proposed by Aristotle in his battle against sophism. We discussed this to considerable extent already in the thread. For instance, in the case of "2+2=4", I argued that if each 2 represents an object, then each 2 must represent an object which is distinct from the object represented by the other 2, or else there would be no equality with the object represented by 4.
This is the fundamental principle of counting. 1 represents an object. We add to that another object, represented by 1 (1+1), and we now have counted 2 objects. In order for there to be 2 objects in that count, each 1 must represent a different object. And, we can proceed indefinitely, to count numerous objects in this way, so long as we recognize that each time we add 1 to the count, it must represent an object distinct, and different from the objects already counted. If we count the same object over again, the count is invalidated, like in the case of the person who counts money folded over in one's fist, so that the same bills are counted twice..
I want examples of those, I mean of the undesirable consequences in the particles of particle physics. Which known examples you are referring to?
It's called the uncertainty principle.
Oh! but that's cornerstone in Quantum mechanics, isn't it? I always hear about a lot of strange conclusions in quantum theory like all possible worlds being actuated, and all intermediate states being there, even between life and death, etc.. I don't know if these are actually of any importance. But do you think that all of those strange results are due to the nature of the platonic realm in which the mathematics of that mechanics is coined? I thought that uncertainty principle had nothing to do with the mathematics involved, it has something to do with inability of have complete form of measurement which is due to the nature of the objects studied and not to the mathematics involved in them. Not sure, really. Can you clarify the picture to me?
Mathematics is our means for measurement. I already proposed, and you somewhat accepted, that the reason we have "infinite", or "infinity", as a feature of our measurement scale, is that this assumption gives us the capacity to measure anything. If an object appears to extend beyond our capacity for measurement, (i.e. beyond the infinite), this implies that "infinity" is not being properly applied. We cannot ever blame the nature of the object for our inability to measure it, because this is self-defeating, killing the inspiration required to devise the means for measuring it.
That's not what I've asked about in my last comment. I wanted to know how the "uncertainty principle" is the error of applying an unsound mathematical system to particle physics? I wanted to know what are your objections to the uncertainty principle? and why you think it is the mathematics involved in it that are the source of the problem? I thought the source of the problem is our "physical" means of measurement not the mathematical side of it. Can you elaborate on this specific issue, I mean exactly that related to the uncertainty principle.
Without a doubt, there is a physical limit to the human capacity to observe our world. We observe with our senses. Molecules are at the limit of what we observe with our senses. It may be argued that we taste and smell them. But any particles smaller than this are beyond our capacity to directly observe. However, we devise instruments to extend this capacity of observation. Mathematics is used in this extension, The information from the instruments is interpreted with mathematics, and assumed to be made commensurable with the things observed by the senses through the principles of application, axioms and physical theories.
Quoting Zuhair
I have no "objections" to the uncertainty principle, uncertainty is the natural product when we arrive at the limits of our capacity to understand. But the issue is, why does the limit appear here, why can't we extend our understanding further. And the answer is that we do not have the principles required to enable us to go beyond this point.
It is not an issue of the human capacity to observe, because we already extend that capacity with instruments. Nor is it an issue of the "physical means of measurement", because we create and produce these, the instruments for measuring, as required. Therefore we ought to consider that the problem, which is causing this limit to appear before us, is a manifestation of the principles by which we interpret the information.
Thanks, John. Also from that wiki page, I'll add "...the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value."
To give Zuhair some credit though, the problem is derived from the Fourier transform which deals with our physical capacity to measure frequencies. The problem I see, is that instead of working to find the principles and axioms which will get us beyond this apparent limit, the physicists and mathematicians, accept this limit, and base their principles on assuming this limit as the real limit (special relativity), so that any real things beyond this limit will remain in the realm of uncertainty. Mathematics is inherently unlimited (infinite), but when we put a limit on physical existence, as SR does, then there is no need for mathematics to extend beyond that limit, so we accept principles which allow infinity to be reached at that limit. There is no longer any need to allow for mathematical principles to extend beyond this apparent limit, because it is assumed that the limit is real, therefore there is nothing beyond it to be measured. The result is that anything real beyond that limit cannot be measured because the mathematics has been shaped so as to disallow this, an all that is there is left as unknown.
Its nice to be informed of that. But my knowledge about those issues is damn sketchy. And so I have no say in such subjects. Thanks for your informative reply.
I was always under the impression that mathematics can supply us with descriptive arsenal that help us discover matters easily. Due to human nature people often don't see (i.e. overlook) what they don't seek. I think that without having descriptive account on "orbits" like those of Ellipses, Parabolas, and hyperbolas that mathematics beforehand supplied us with, it could have been very difficult to observe how the planets moves, and it would be very difficult to predict their movements. Possibly similar things might apply with the uncertainty principle. I don't know really.
Wiki: "The term Planck scale refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 1.22×1019 GeV (the Planck energy), time intervals around 5.39×10?44 s (the Planck time) and lengths around 1.62×10?35 m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists have no scientific model to suggest how the physical universe behaves. The best known example is represented by the conditions in the first 10?43 seconds of our universe after the Big Bang, approximately 13.8 billion years ago. "
This is a very interesting subject which you bring up here, but my opinion is somewhat opposite to what you say. I think that mathematics allows us to make many very accurate predictions based on statistics and probabilities, without having any accurate description of the mechanisms involved. So for example, Thales apparently predicted a solar eclipse in 585 BC. I think it's common that we observe things, take note of the patterns of specific occurrences, thereby becoming capable of predicting those occurrences, without understanding at all, the motions which lead to those occurrences.
So the ancient people observed the motions of the sun, moon, planets, and stars, and described these motions relative to their point of observation, and could make predictions based on those descriptions. But the motions they described were completely different from the motions we describe of the very same bodies, today. And we say that they were wrong. However, we still insist that motion is relative so we don't even really have the right to say that they were wrong.
Quoting John Gill
That's ironic, the numbers approach infinity (limitless), as the condition approach the limit. What this indicates is that the limit is created by the principles which govern how the numbers are applied. The limit is created by the numbers approaching infinity, and the principles of application dictate when the numbers will approach infinity.
Not so for Planck Time. You'll need a real, live physicist to discuss this properly. It used to be that this limit was variable according to some physical features.
I don't see where you differ with me. Mathematics can also speak of patterns that had not been yet observed! Because it tackle all possible structures in an unlimited manner. That's what I meant when I said *before-hand*, if we had good mathematics about ellipses, parabola, hyperbola, etc.., even before we observed the movements of planets, that knowledge would make it easier for the astronomer to discover the pattern of movement of those planets, because as I said: many times humans don't see what they don't look for. If you have the descriptive arsenal before-hand, you'll recognize and thereby predict easily the behavior of matters with fewer observations because it would look familiar to what you have experienced in say the platonic world about approximately similar behaviors. So of course we don't need any understanding of the mechanisms which lead the planets to move the way they do, we already had the descriptions of their orbits hundreds of years before we discovered that they are moving approximately along them. Notice that we didn't coin the mathematical structures describing orbits (ellipses, hyperbola,etc..) after the observation had been made, we actually imagined it starting form more trivial observations made in our planet, like of approximately circular objects, then we freely contemplated more alternative variety of similar structures in the platonic imaginary world, this free contemplation is what made us arrive at those orbit mathematical structures way before any application or observation was discovered. And I think that's one of the most important jobs of mathematics, to supply such descriptive arsenal that objects in our world may *possibly* follow. I'd say perhaps, the particle physics objects move along some paths that we don't have the mathematical descriptive arsenal necessary to match them with, that's why we remain in ignorance about them.
Actually, what I described is exactly the case with Planck time. The limit (Planck time) is the product of the theories being used. This is from Wikipedia: "The Planck time is the unique combination of the gravitational constant G, the special-relativistic constant c, and the quantum constant ?, to produce a constant with dimension of time.".
You might say that these theories represent something real in the universe, but they only do so to the extent of our understanding. Any misunderstanding creates a limit to the mathematics which is not representative of a real limit in the physical universe. When this is the case, application of the mathematics to observations will produce an abnormal occurrence of infinities, (as we see in quantum physics) as the things being observed go beyond the limits created by the lack of understanding expressed by the theor.
Quoting Zuhair
I don't see how this notion of "before-hand" can be realistic. Before-hand, there are infinite possibilities for spatial shapes. So it could not be practical to produce all these possible models prior to observations, then after observing, attempt to fit a model to the observation. What is really the case, in practise is that we see something, observe it and take notes, then we create a model to represent it. So we work from the purest form of mathematics, simple numbers to represent observed occurrences of events, with the most primitive spatial representation of those events, toward creating a more complex spatial form, or pattern, which fits to those occurrences.
I agree that it is necessary to keep our minds open to "all possible structures" but to approach a problem with all possible structures already apprehended, and developed on paper, is not practical because unrestricted possibility approaches infinity. Therefore we take the information presented to us by the particular problem, and create structures as possible solutions, according to what is required, striving to keep our minds open to many possibilities because once we accept one we tend to close our minds to others. And this is not good, because we never actually obtain "the ideal."
Quoting Zuhair
This is not true, because the "descriptive arsenal" would have to contain all of the countless possibilities. Then, you would have to compare the observations with each of those countless possibilities to determine which description is the best. This is highly impractical, and not representative of the way that we actually proceed. in reality we create the "platonic world" to represent what we have observed.
Quoting Zuhair
I don't think that this is true either. Kepler noted that planetary orbits were not eternal perfect circles as postulated by Aristotle. This knowledge was produced by inconsistent positioning. Kepler approached this problem with numerous possible curves, and found the elliptical orbit to be most suitable. But I don't see any indication that there are any elliptical orbits available for observation on our planet, from which Kepler could have copied the design, and no indication that the design was created for anything other than the purpose of modeling planetary orbits..
Quoting Zuhair
I agree that we are very close to complete agreement on these issues, that's why I have pointed out the specific places of disagreement with "not true", hopefully to help you see that my perspective is better suited. Though you might bring me around to your perspective instead.
So we're back to this question of art (beauty, aesthetic), or utility. Do mathematicians create all sorts of shapes, forms, and structures simply because they are beautiful, and have them lying around for possible use, or do they create them to serve as solutions to particular problems. You seem to choose the former, that mathematicians create a whole arsenal of beautiful shapes, simply because they are beautiful, then physicists and cosmologists might choose from this collection of designs, those which are suitable to them. I think that mathematicians create their forms with purpose, as potential solutions to particular problems.
Honestly I think its both cases. Some structures were actually contemplated due to their own beauty in a platonic world, while others raised secondary to observations and need for application as you depicted. I in some sense do agree with you that we'll have infinite possibilities if we were to contemplate just purely, but there are definitely some scenarios that are more attractive platonically speaking than others.
Example of "mathematics prior to observation" is that the orbit of planets which suits more of an ellipse. Ellipses where there on board since ancient Greek, and their study didn't arise from contemplating planet orbits as you think. No they actually were studies on our earthly structures which are simply about inclined sections of cones. Then Kepler picked what is already available and matched it with observations about planets movements.
Other examples include Riemannian n-dimensional geometry, this was contemplated before relativity theory and other recent theories of physics which use many dimensions. Also non-Euclidean geometry was long contemplated by Al-Tusi and also by various mathematicians long before relativity theory called for their use, and they did arise from the pure study of geometry in the platonic realm, mainly becuase of the non-proof of parallel postulate. Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures.
On the other hand I agree with you that there are other situations where the mathematics had been created AFTER the observations had been made, i.e. "observation prior to mathematics" direction: like mathematics about the DNA double-helix structure, and also like quantum mechanics, Dirac delta function, etc...
So in real practice both lines are occurring, the pure investigation of those entities in the platonic laboratory and on the other hand the on-demand construction of mathematical entities to match needed application. We can say that mathematics can work to enrich our knowledge about the world by detecting behaviors in the later that we already knew of in the platonic world (in approximate manner), and also the other direction is also true, that observation in our real world as the source and the motive to contemplate certain platonic structures, so our world enriches mathematics also. It is a bilateral movement. And I think this bilaterality is important. And it (the bilaterality) should be observed if we are to have mathematics help enrich our knowledge about our world.
That said. I think we need to unleash both directions!
I wonder if this is even true. Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem. I suppose that it would be very difficult to distinguish whether the structure was created purely for beauty, or for utility. And, if you were to go and create one right now, saying you created it purely for beauty, I would argue that you did it for the purpose of your argument. So we might leave this point as unresolved, or even unresolvable. However, we might still argue our opinions, in an attempt to get the other into our own metaphysical camp.
Quoting Zuhair
OK, I think you're right here, but this does not exclude the possibility that the ellipse was created for another purpose. So it doesn't really force the conclusion that the model was produced prior to having an application. We just might not be a ware of the application it was first designed for.
Quoting Zuhair
Again, we cannot really resolve the question this way, because there would always be a reason for speculating about non-Euclidian geometry. You say that it is because the non-proof of the parallel postulate, but as I think I indicated earlier, Pythagoras was dissatisfied with the irrational nature of the square, and we also have the irrationality of pi. These are all good reasons to speculate about non-Euclidian geometry, and it would be difficult to prove that utility is not at the base of this dissatiisfaction.
Quoting Zuhair
So let's assume as we would agree, that pure Platonic contemplation is not random. How could we assign anything other than utility (what Plato calls "the good") as the thing which delivers us from randomness? If we were to contemplate pure beauty, completely devoid of utility, wouldn't this be randomness itself?
Quoting Zuhair
I'm not convinced that there is such a thing as pure investigation in the platonic laboratory. I think this would require that we totally remove ourselves from the necessities of life, and the constraints of the physical world, and this is impossible. That is why Plato himself settled on "the good", as that which makes the intelligible objects intelligible. The good inheres within the essence of the intelligible object therefore, as what gives it the characteristic of being intelligible. If we remove this good, we are overwhelmed by randomness. And randomness might itself be the most beautiful thing there is, such that we would be overcome by the beauty of pure randomness, but such beauty would be inherently unintelligible because of the nature of randonmness, and therefore impossible to be the source of any type of structures.
I didn't claim an absolute platonic approach to mathematics, definitely not. But it appears to me that once we get to think about some mathematical problem, even if that raised within the context of solving a problem about our world, or certain application, etc.., we can easily figure out many offshoots that can be developed on purely theoretic basis, those offshoots can later on have applications, but I would guess that many were coined even before seeking some application that was raised to instigate them. Lets take a very simple example, lets take the negative integers, those were contemplated by the Chinese as well as the Greek mathematician Diophantus way before the Arabs made full use of it in commerce and related business. Actually Diophantus objected to their existence as being "false", this shows that he knew them but saw no application for them and rejected them along philosophical lines, much as many people rejected imaginary numbers, irrational numbers, and "transfinite" numbers, non-standard naturals etc.. Yes you can start with a shape in application and then try to figure out its rules in the platonic world then problems will raise and their solution can be approached in the mathematical realm and many offshoots of that approach may later turn to have applications. As I said Non-Euclidean geometry was an offshoot of the challenge caused the the "parallel postulate" of Euclid, now this is a pure mental problem, it was not related to utility, many geometricians in their endeavor to prove the fifth postulate actually discovered the roots for non-Euclidean Geometry. The equi-interpretability of non-Euclidean Geometry with the Euclidean was established therefore establishing the independence of the fifth postulate (the parallel postulate), this is a pure platonic problem, however it turned to have a utility later one in Einstein relativity theory. Same to be spoken about Riemannian n-dimensional Geometry, which raised from within solving problems that seem to me to be purely related to theoretic mathematics, rather than being instigated by some particular application in the real world, later on Relativity theory used this multi-dimensional space of Riemann. Regarding shapes ellipses, and others might have been raised from some problem about the real world as contemplating the idea of a shape nearer to that of an egg, or the inclined cut section of a cone, etc.. but that also can be seen as an offshoot of contemplating closed figures beginning from one with ideal symmetry, i.e. the circle, to ones less symmetrical and so on, one can Platonically think of a whole spectrum of these.
I think (though I don't have a proof ) that many mathematics even if initiated in application, would have a pure platonic intermediates bringing many possible structures, then many of those would fade away because they don't have applications, while those that have, will continue also raising problems about them in the platonic world leading to many offshoots, some of which would have applications, and so on..
I don't think the whole of mathematics, i.e. every step of it, was instigated by some utility in a direct manner to solving a problem in the real world, neither do I maintain that the whole of mathematics did or even could have proceeded in an absolutely purely platonic manner. Its a mixture of both that we have.
You ask, and it shall be so . . .
I work with this sort of thing all the time.
Later: Interesting, my image came up perfectly, then when I checked part of the code had been deleted (by whom?). It's coming up now. Let's see how long it will last this time.
Well, it didn't last. There must be a rule against linking to images on this site.
What is it? I can't open it.
For me and my colleagues, |z| getting larger and larger without bound means z -> infinity. An actual point at infinity is irrelevant in practice. If I think of time going to infinity, I mean it in this sense. If you look at the projective plane sitting below the Riemann sphere, you can see z moving further and further out, without bound, and as it does so its projection on the sphere moves closer and closer to the north pole, but never reaches it.
https://en.wikipedia.org/wiki/Riemann_sphere#Extended_complex_numbers
You can't even do links here it seems. This is a page on my website, and it comes up first time, then won't connect. It keeps returning to this forum. I don't know what is going on. The Wikipedia site keeps coming up. Not mine.
I think we've covered much ground in this thread so it would be difficult to summarize. The pivotal issue seems to be the reality of Platonic objects. So we had an extensive discussion concerning what various mathematical symbols are representative of, whether they represent objects, if so, what kind of objects, and particularly the identity of the objects. The law of identity was prominent. . It appears like axioms which treat an infinite collection as an actual object, require the reality of Platonic objects. However, the point I argued is that mathematical objects do not have an identity which is consistent with the law of identity. .
No, we didn't really discuss the nature of reality. We discussed the difference between trying to make true descriptions of objects, and creating imaginary figures. Both of these, I would say, are part of reality. The issue I think is whether the imaginary figures qualify to be called objects. So the question would be to define "object", and this is why I turn to the law of identity. An object has a unique identity.
The principle of induction is intimately related to the recursion theorem. The principle of induction states that if [math]\phi[/math] is a predicate such that [math]\phi(0)[/math] and [math]\phi(n) \rightarrow \phi(n + 1)[/math] for all [math]n \geq 0[/math], then [math]\phi(n)[/math] is true for all natural numbers [math]n[/math]. A proof of the induction principle is a recursive function that transforms a proof of [math]\phi(0)[/math] and a proof of [math](\forall n \geq 0)(\phi(n) \rightarrow \phi(n + 1))[/math] into a proof [math](\forall n)\phi(n)[/math] for any predicate [math]\phi[/math].
The induction principle is interesting because it allows us to conclude that [math]\phi(n)[/math] is true for every natural number [math]n[/math] without checking that [math]\phi(n)[/math] is true for every natural number [math]n[/math]. In other words, it allows us to conclude that every member of an infinite set of objects possesses some property by performing a finite amount of work. There are analogous induction principles for other infinite sets. For example, some proof assistants automatically generate induction principles for arbitrary datatypes satisfying certain technical conditions.
In other words, there is a systematic relationship between certain infinite sets and inductive proofs (recursive functions), the latter of which are "potential infinities" in your terminology. In other words, induction principles (recursive functions) systematically translate between statements about "potential infinities" and statements about the elements of certain infinite sets. This has nothing to do with curly braces or other arbitrary syntactic phenomena unless you are committed to some kind of hardline formalism about mathematics.
Here is another example: consider the "potential infinity" defined by the Fibonacci sequence. You can generate every Fibonacci number using a recursive function defining the sequence. In other words, the recursive function defines the first, second, third, and so forth, Fibonacci number. However, you can always consider the collection of elements generated in this way by saying: "suppose that [math]n[/math] is a number in the Fibonacci sequence." What you are talking about, in the latter case, is an infinite set of objects - there is no limit to the number of objects that satisfy this condition, although there are restrictions on the kinds of objects that satisfy the condition.
The nature of reality is not an issue here. The nature of an object is. "Reality" is the more general concept, so there is more to reality than just objects. What we are interested in here, is objects.
Quoting quickly
Now the issue, which we discussed already in the thread, is whether or not a written numeral necessarily represents an object. In actual usage, the numeral might be used to represent an object, or it might not. If it doesn't represent an object, then any supposed count is not a valid count.
Your example seems to create ambiguity between the symbol, and the thing represented by the symbol. So you would have to clarify whether there is actually existing numbers, existing as objects to be counted, otherwise the claim of "an infinite set of objects" is false. As proof, it doesn't suffice to say that it is possible that a numeral represents an object And actual usage of symbols demonstrates that it is possible that the symbol represents an object, but also possible that it does not. To present the symbol as if you are using it to represent an object, when you really are not, is deception.
I don't understand why skepticism about the meaning of mathematical language is relevant to the discussion about potential and actual infinities. The notion of potential infinity is the notion of a process that can be repeated indefinitely. For example, one can always consider successively larger values of the Fibonacci sequence. The language of sets (types, classes, etc.) provides one way of talking about the elements generated by such processes. The notation used is entirely irrelevant.
You've described a potential infinity, but not an actual infinity. To understand an actual infinity we need to understand the actual existence of the elements represented by mathematical language.
I read a little of the Stanford Encyclopedia of Philosophy article on the topic.
It seems that Aristotle thought of an actual infinity to be akin to the infinite divisibility of, for example, a finite length in that it is "complete" and potential infinity to be something like the non-terminating process of adding 1 to any number and getting the next greater number.
When you say "the set of natural numbers," you mean "the set of all objects that can be generated from zero and the successor function and which satisfy the Peano axioms" (or something similar). In other words, I can use the language of sets to talk - all at once - about every object satisfying these conditions. If that doesn't satisfy your condition that someone "understand the actual existence of the elements represented by mathematical language," then perhaps we simply have different conceptions of the purpose of mathematical language (e.g., the language of sets).
Imagine the real numbers in the interval [0,1]. Is there an actual or potential infinity of them?
Well one answer (the wrong answer) is we can go on dividing forever by 2 (say) so there must be an actual infinity of reals in the interval.
But we can only go on dividing forever in our minds - if we tried this in reality, we'd never finish dividing (process goes on forever - we'd never finish) - so the possibility of infinite division is just a figment of our imagination (like its possible to levitate in your imagination - but not in reality).
There are therefore (according to Aristotle and I agree with him), a potential infinity of real numbers in the interval [0,1]. When we perform division by 2, we actualise the number 1/2. When we perform division by 2 again, we actualise 1/4 and 3/4. And so on. At no point in this process is there ever an actual infinity of real numbers.
So in summary, actual infinity is a purely imaginary concept . It is sometime mentally convenient to regard (say) the set of reals as actually infinite - but that is not telling us anything about reality any more than our ability to imagine a square circle or a fairy.
I'll attempt to answer the actual question.
In my opinion it's 37.54, but my uncle here says 289, and my aunt insists it's only 9.
You are rejecting the idea of "forever" which translates to "infinitely long time". So you reject the very concept of infinity because your premise rejects the very concept of infinity.
We've been down this road once before. ("What time does the clock show that has been going on since infinite time.")
If you can't imagine it, fine. But arguing that infinity does not exist because you can't imagine it is on hand a weak argument, on the other hand, a subjective argument.
I would suggest you instead focus on elapsed time. Is there, for example, an actual infinity of moments in a second? That is a more interesting question. A few thoughts:
- There must be a temporal difference between 'now' and 'then' else 'now' would be 'then'
- The temporal difference can’t be zero / infinitesimal else 'now' would be 'then'
- So there is a finite difference between 'now' and 'then'
- Could that finite difference be infinitely divisible?
A similar question is: 'when you move your hand, do the particles of your hand pass through an actually infinite number of positions?'. Or do they do something similar to a quantum jump of an electron, on a tiny Planck level scale?
Obviously, this is all related to Zeno's paradoxes, which I think are indicative that time and space are discrete. I can't prove it though obviously.
We have been considering models of infinite divisibility here:
https://thephilosophyforum.com/discussion/7320/continua-are-impossible-to-define-mathematically/p1
It seems to me that there are no sound mathematical models of infinite divisibility? So that may lend some weight to the idea that space/time/motion are discrete?
I've read through much if not most of your writings in this thread. I think I understand where you're coming from. I found two remarks I can get some traction on.
Quoting Metaphysician Undercover
ZFC is unsound. Well yes. Of course ZFC is unsound. Its deductions are valid, meaning that each theorem follows from the axioms and prior theorems according to the purely syntactic rules of deduction.
But it's unsound, in the sense that its premises are not necessarily true. It's not even clear that they are meaningful. Personally I don't think they are. I don't personally believe that sets, as understood in ZFC, exist in the real world. You're arguing that they don't but of course they don't. I'm in total agreement with you on this point.
I'll go you one better. ZF is unsound. No need to invoke the axiom of choice. In ZF we have the axiom of infinity, which states that there is an infinite set. Nothing in the physical world corresponds to that as far as we know. So ZF is unsound.
So ok. ZF[C] is unsound. This is a commonplace observation. Might you be elevating it to a status it doesn't deserve? Are you thinking of it as an endpoint of thinking? What if it's only the beginning of philosophical inquiry?
Here's what Bertrand Russell, who knew a thing or two about both mathematics and philosophy, said in 1910. This quote is usually given as a one-liner but the entire paragraph deserves repeating.
https://www.goodreads.com/quotes/577891-pure-mathematics-consists-entirely-of-assertions-to-the-effect-that
Since this understanding is over a century old, it seems reasonable to take it as a starting point for the study of the philosophy of mathematics; and not the endpoint, as you have done. Mathematics is not literally true. It's a language for expressing physical theories; and it's a discipline unto itself that must be taken on its own terms.
Let's take Russell's quote one step farther. In 1960 the physicist Eugene Wigner published an article called, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
The title says it all. Mathematics is so fundamentally divorced from reality that it is unreasonable that it should have anything to tell us about the real world.
And yet it does.
That, I submit, is one of the key issues of the philosophy of math. We agree that math isn't true, but it nevertheless seems intimately related to the real or the actual.
This is what I mean when I say that "ZCF is unsound" is the starting point of philosophical inquiry into the nature of mathematics; not the end of it. I get the feeling you think it's the last word. It's only the first.
Next: Part 2.
Quoting Metaphysician Undercover
A lot here to work with. Let me break it down a little at a time. I've already made many of these points in other threads lately so I'll try to keep this short.
Quoting Metaphysician Undercover
Yes. This is true. Euclid's version of the proof, from around 300BC, shows that there are no two integers whose ratio squared is 2.
It shows that the square of the ratio of two integers is never 2. This is a result in number theory. It's indisputable.
That it comes up so naturally, as the diagonal of a unit square, shows that (at least some) irrational numbers are inevitable. Even if we're not Platonists we suspect that a Martian mathematician would discover the irrationality of [math]\sqrt 2[/math]. There's a certain universality to math. Euclid's proof is as compelling today as it was millenia ago. The philosopher has to acknowledge and account for the beauty, simplicity, and power of Euclid's proof, undiminished for 2400 years; not just deny mathematics because some number doesn't happen to be rational.
Quoting Metaphysician Undercover
No. No no no no no no and no. I've hit this point several times in other recent threads, and already enumerated these bullet points over in the other thread. Let me just recapitulate these points briefly.
* A real number is not its decimal representation. [math]\sqrt 2[/math] happens to have an infinite, nonrepeating decimal representation. That's an artifact of decimal notation, not a characteristic of the real number [math]\sqrt 2[/math] itself. There are many finite characterizations of [math]\sqrt 2[/math].
* [math]\sqrt 2[/math] is a computable real number in the sense of Turing 1936. Therefore even a diehard constructive mathematician would gladly accept the existence of [math]\sqrt 2[/math]. From now on when I say existence I mean mathematical existence. I'll stipulate that we can't measure [math]\sqrt 2[/math] in the real world. As long as you'll stipulate that we can't measure 1 either. So the real world doesn't even come into play here. Measurement's approximate. Mathematical numbers are exact, but not necessarily aspects of the real world. I gather this is much of your thesis anyway. I agree.
* [math]\sqrt 2[/math] has a repeating continued fraction representation. I won't explain what that is but the Wiki article's pretty good.
Now the continued fraction representation of [math]\sqrt 2[/math] is [math][1; 2, 2, 2, 2, ...][/math], or "1; followed by all 2's." You can't get any more finite that that. That phrase completely characterizes [math]\sqrt 2[/math] as a continued fraction. There's no priority between decimals and continued fractions. It's just that one's taught in high school and the other in math major number theory class. They're both equally valid ways of representing a real number.
3) Euclid's proof depends only on Peano arithmetic (PA). In terms of proof strength, PA is the same as ZF without the axiom of infinity. So a diehard finitest would accept the irrationality of [math]\sqrt 2[/math]. A finitist accepts the existence of each of 0, 1, 2, 3, ..., but not a completed set of them. You don't need a completed set of natural numbers to prove that the square root of 2 is irrational. It's a finitary fact, not an infinitary one.
So far we've seen that both constructivists and finitists would have no problem accepting [math]\sqrt 2[/math]. If we informally rank real numbers based on believablity, the noncomputables are the most unblievable reals. Those are the numbers that genuinely encode an infinite amount of information. Next in believability are transcendentals like [math]\pi[/math]. They're computable, but they're mysterious. Their existence wasn't even proven till the 1840's.
The most believable irrationals are simple algebraic numbers like [math]\sqrt 2[/math]. Numbers that are the roots, or zeros, of polynomials with integer coefficients like [math]x^2 - 2[/math]. Those are super-easy to construct without using infinitary methods; and as the quadratic formula from high school shows, the ones that are the roots of quadratic polynomials are easy to determine.
[math][/math]
* [math]\sqrt 2[/math] lives in a finite extension of the rationals. This is the abstract algebraic approach I've talked about recently in another thread.
But there's a much more down to earth way to explain this point. The idea is to view our ever-expanding number systems as the solution to the problem of solving some type of equation.
So in the beginning (again this is thematic, not necessarily historical) we have the positive integers 1, 2, 3, ... We can solve equations like [math]x + 2 = 3[/math] with no problem, the answer is [math]1[/math].
But what if we try to solve [math]x + 3 = 2[/math]? We're stuck. We haven't got a number to solve this equation. So we invent one. In fact we invent zero and all the negative integers just so we can now solve more equations.
Now later on this invention gets formalized into the abstract algebra that I've presented. But the essential idea is just solving as many equations as we can, even if we have to make up numbers to do it. And over time, everyone comes to believe in these made up numbers.
So ok, now someone gives us the equation [math]5 x = 7[/math]. Now we're stuck, there's no integer, positive or negative, that will work. So we have to invent the rational number [math]\frac{7}{5}[/math] to solve the equation.
So @Meta you see that the rational numbers, which you think are so sacred, are themselves just made-up fictions that at one time were controversial and not widely believed in.
Ok now we have the rationals. What do we do with an equation like [math]x^2 - 2 = 0[/math]? We have to invent [math]\sqrt 2[/math]; and by extension all of the other real numbers of the form [math]a + b \sqrt 2[/math] where [math]a, b \in \mathbb Q[/math].
Now of course we need to solve [math]x^2 = 3[/math] and all the other quadratics; and we have to solve all the third degree equations, and so forth. There is a class of real numbers called the algebraic numbers that contains all the solutions of all the polynomial equations.
It's tricky though because what about the simple quadratic [math]x^2 + 1 = 0[/math]? We have to invent a new kind of number, the imaginary unit [math]i[/math] and the complex numbers, the set of all [math]a + b i[/math] where [math]a[/math] and [math]b[/math] are real.
And this process keeps going. There are quaternions and octonians and sedonians and more. There are weird numbers like the p-adics. There are Cantor's tranfinite ordinals and cardinals. There are matrices and tensors, beloved by the physicists.
Every time mathematicians need a new kind of number they invent one; and then after a passage of time, people come to believe in these new numbers.
There was a time when people believes in the integers but not the rationals. You believe in the rationals but not the irrationals. But there's nothing special about the rationals. They were once a fiction too. Your choice is arbitrary and not based on anything but the era in which you were born, and your level of mathematical education.
Quoting Metaphysician Undercover
But no. It's not bad. It just is. And remember, if you lived in a different age you'd think negative numbers were bad. Then you'd think zero is bad. Then you'd think rational numbers are bad.
Your feeling about what numbers are real and which aren't is purely a matter of accident based on the age you live in and your level of mathematical knowledge.
I hope you can get outside yourself to get this point. There's nothing special about the rational/irrational jump. It's one jump in a long historical line of mathematical sophistication from putting a mark on a cave wall every time you kill a mastodon; to doing the most advanced research mathematics and phyiscs. You can't draw a line that says this step is real and the next step isn't. The rationals are just as fake -- or just as real -- as the irrationals.
So it's not "bad" that [math]\sqrt 2[/math] is irrational. It just is. Some numbers are rational and some are integers and some are p-adics and some are quaternions. Mathematicans have lots and lots of numbers. They all have mathematical existence.
Quoting Metaphysician Undercover
Of course a unit square is not impossible, it's the most obvious geometric figure there is after the equalateral triangle. Speaking of which, an equalateral triangle with sides of 1 has an altitude of [math]\frac{\sqrt 3}{2}[/math]. Can't avoid those pesky irrationals, they are literally all over the place. Everywhere you look.
But to say a square is "impossible" simply because you don't like the great discovery of Pythagoras ... that seems a little beyond the pale for serious discourse in my opinion.
Quoting Metaphysician Undercover
Equivocating irrational as in "not a ratio [of integers]" with irrational as in cray-cray. Come on, man, you can do better than that.
If the sides of a square are rational the diagonals are irrational. Man get over it. It just is, like the sun rising in the east. A fact about the world. That's one of the curious things about abstractions. Even though we make them up in our minds, they still bear definite truth values. 5 is prime even if you don't believe in the Platonic existence of 5. As a philosopher of math you need to account for that, not deny it because you don't like the Pythagorean theorem.
Quoting Metaphysician Undercover
I'm not sure what you mean but from what follows I'm guessing you mean squares, cubes, hypercubes, and in general n-cubes; which, by the way, are perfectly well understood in math.
Quoting Metaphysician Undercover
That's just silly. There's nothing fundamentally unsound. The diagonal of a unit square can't be expressed as a ratio of integers. It's just a fact. It doesn't invalidate math. Even the Pythagoreans threw someone overboard then accepted the truth.
The tl;dr on all this is that you're just mistaken that there's anything special about rational numbers, other than the fact that (by definition) they're ratios of integers. Lots of naturally occuring mathematical constants turn out to be irrational.
Among the irrationals, the very simplest are the quadratic irrationals like [math]\sqrt 2[/math], meaning that they're roots of quadratic equations. They're so easily cooked up using basic algebra, or Turing machines, or continued fractions.
The bottom line is that your unfamiliarity with certain aspects of math is leading you to philosophical errors. [math]\sqrt 2[/math] has a very strong claim to mathematical existence and it's a finitary object, not an infinitary one. Likewise [math]\pi[/math] is computable, hence encodes only a finite amount of information. Decimal representation is not determinative of a real number's infinitary nature.
Thank you fishfry. I'm vey impressed that you actually took the time to read and try to understand what I was saying. Most just dismiss me as incomprehensible or unreasonable and go on their way.
Quoting fishfry
As Aristotle explained, logic proceeds from the better known, toward understanding the lesser known. The premises are the better known, the conclusions lesser known. A conclusion requires multiple premises so if there's a possibility that each premise is incorrect, the probability is multiplied in the conclusion. Therefore the "endpoint" of thinking is more uncertain than the beginning point. The engineer applies mathematics as if there is a high degree of certainty in the axioms (premises), and uses these toward producing an understanding of what is until then unknown.
The principles we apply in any philosophical inquiry must be of the highest degree of certainty in order to give any credibility to the conclusions of the inquiry. Science and engineering apply mathematics with a confidence that the underlying axioms are sound. If the engineers thought that the mathematical axioms were unsound, they would request better ones. A philosopher such as myself may approach the axioms with skepticism. But the skepticism must be justified by underlying principles with higher certainty. If one were to cast doubt on any particular axiom, that philosopher must appeal to further principles which are known with an even higher degree of certainty. That's what I believe I am doing, referring to principles with a higher degree of certainty, such as the laws of identity and non-contradiction, to cast doubt on these mathematical axioms.
What Russell says is true to an extent, people who apply mathematics do not generally question the accuracy of the axioms. However, in the passage quoted he fails to mention the reason for this. The reason that they do not question the accuracy of the axioms is that they have confidence in those axioms. So yes, applying mathematics is no different from applying other logic, we proceed with the underlying knowledge that if the axions are true then so will be the conclusions, when the logic is properly applied. However, what is not mentioned in the passage is that when we apply mathematics we have a high degree of confidence in the truth of the premises, and that's why we are using mathematics.
Quoting fishfry
Perhaps you and I can find an agreeable starting point for a philosophical inquiry, here. We agree that the axioms of mathematics are not true, so we can now examine the confidence with which mathematics is applied. Mathematics does not consist of "true principles", so the confidence is not based in truth. I suggest that the confidence is based in utility. It has worked in the past, it works today, therefore it will work in the future. Mathematics is reliable, therefore we have confidence in it.
Now, this reliability indicates that mathematics is, as you say "intimately related to the real or the actual". I have no problem agreeing with you on this. So we can ask, what produces this reliability, what is the nature of this relationship with the real or actual. I propose that the reliability is produced by approximating the truth. Reliability does not require that we know the absolute truth concerning the matters we are involved with, but it does require that we have an approximation of the truth, and this approximation lowers the probability of mistake, increasing reliability. The nearer we can approximate the truth, the higher will be the reliability. In this way we remove "truth" from the absolute logical categories of is/is not, placing it into the relative, degrees of probability. Propositions are not judged to have truth or falsity in an absolute way, they judged for probability of truth.
Quoting fishfry
Didn't you just agree that in a very fundamental way, mathematical axioms are unsound? And you supported this with the quote from Russel. Or do you think it's just some axioms which are unsound? ZF is unsound but Euclidian geometry is sound. What about the parallel postulate? Anyway, if some axioms are sound and others unsound, we'd have to revisit all the axioms anyway, to distinguish the degree of soundness.
Sure, it's "a fact" that Euclidian geometry produces points which have a distance between them as immeasurable, but it's also a fact that this is a problem which has not been resolved. We've agreed not to speak in terms of soundness or truth of the axioms, so all we can look at is whether the axioms are problem free, and these geometrical premises create problems, i.e. that there are points with immeasurable distance between them.
Quoting fishfry
Let me apply the principle described above, to ?2 now. Truth is relative, not absolute. First, it is probably not true that a square is natural. The right angle is artificial. It was produced for various purposes, surveying plots of land, establishing parallel lines, etc.. So it is highly unlikely that ?2 comes up naturally, it comes up as a result of our desire to create parallel lines, or whatever other purposes we use the right angle for. Therefore, our desire to create parallel lines and such, has produce the right angle, which is extremely reliable for these purposes, so it must approximate the truth about "space" to a fairly high degree.
However, there is this problem which the right angle creates, and that is that it allows us to very easily make two points with an immeasurable distance between them. The "circle" and "pi" has a very similar problem, so this geometrical system based on a point with degrees of angles around the point is suspicious. Now, I see two ways we could go with this inquiry. We could follow principles like Peirce's and say that's simply the way space is, there's a vagueness about it which incapacitates us in this way, making it impossible to measure the distance between these points. The nature of space is such that we cannot determine points in space, only vague infinitesimals. That assigns "the fault" to the object, space. Or, we could proceed in the way that I recommend, and consider that our desire to make parallel lines and such, has misguided us relative to truth, such that we modeled space with right angles etc., which was an approximation to the truth, but maybe not accurate enough for the purposes we now have. This assigns "the fault" to the subject, the model. The problem being that the circle and the right angle do not properly represent the spatial area around the point. Notice that the latter way, which I propose, gives us inspiration to delve into our spatial models, divulge our past mistakes and produce a better approximation of space. The former way suggests that it is impossible to model space more accurately, because we cannot model a precise point. And that's just the way that space is, impossible to model in such a way as to rid ourselves of the conclusion that if locations are represented as precise points, there will be points with an immeasurable distance between them. So we are uninspired to look for that better way of modelling space. Therefore, if we are inspired to increase the degree of reliability and certainty with which we apply mathematics, we need to revisit this model of space, to see where it leads us astray.
Quoting fishfry
That irrational numbers are "easily cooked up" from our axioms is clear evidence of weakness in the axioms. As explained, irrational numbers represent things which cannot be measured using the existing axioms. If more and more axioms need to be layered on in a seemingly endless process, to create the appearance that these immeasurable things actually can be measured, this is simply closing the barn door after the horse has run away.
You seem to be lacking in reading skills tim, it's no wonder you're so confused. I was talking about what someone can do with Euclidean geometry, not about what someone can do walking on the ground. So your example is way off track.
If you want to argue that mathematical objects and geometrical constructs are not real objects, then I'm in agreement with you there, and that's the position I've taken on this thread. And, the fact that we can create an idealized object (created through the use of defined terms, rather than drawing, or walking on the ground), like a circle or a square, and make these objects such that there is immeasurable distances within the objects, is further evidence that these are not truly "objects". My argument has been, that a mathematical object, like what is referred to by "2" or "3" does not qualify as an object in any way consistent with the law of identity. And set theory is based in, and requires, this false premise that these are objects. In other words, set theory violates the law of identity.
If you're interested, go back and read the thread, Zuhair and I covered much ground. Fishfry was in and out, and for the most part did not keep up with the discussion, but now seems to have a renewed interest.
Sorry, I didn't understand your challenge, you said something obscure about making you go away. I'm afraid you have your own free will and I can't make you go away.
Now that you've made you question clear, I'll answer it. But it should be obvious from what I already wrote, so I don't understand why you're confused. Construct a square according to the rules of Euclidian geometry, the opposing corners are an immeasurable distance apart.
Thanks man. I was going to say the same but @Metaphysician Undercover and I have reached a point of mutual civility and I'm trying to keep that going. So I didn't say what you just did, though I endorse and agree with it.
I have been reading @Meta's post and it's hard to know where to start. @Meta, do you agree that math isn't physics? You complain that we can't measure [math]\sqrt 2[/math] but we can't measure 1 either. All physical measurement is approximate and never exact. You don't seem to appreciate this point.
When you say "squares are impossible" you seem to be denying abstraction. Of course there are no perfect right angles in the world, but there are in abstract reasoning. You can't deny abstractions, civilization runs on them.
Quoting Metaphysician Undercover
Hey it's great. @Meta is polite to me now and insulting others. Thanks @Tim for bearing this burden on my behalf.
You're welcome.
Quoting Metaphysician Undercover
No, really?
Just kidding!! I enjoy the new civility and hope to perpetuate it.
I couldn't tell who was crazy, you or me. You were so sure of yourself when you said I didn't understand anything about anything. I wanted to find out what was going on. I believe I did. You think math should be physics. You think perfectly accurate distances exist in the real world. You seem to deny abstraction, as in your claim that squares are "impossible," your word. You did not yet take my point that rational numbers are just as questionable as irrationals. I mean, what the hell is [math]\frac{7}{5}[/math], anyway? You can't measure it. You can't show it to me. You can't evenly divide a pie into seven pieces. [Insert clever pun on pi here]. So in fact your psychological belief that rational numbers are more deserving of mathematical existence as irrational ones, is simply that: a psychological belief.
I have explained in recent posts (here and the other thread) that:
* The number [math]-4[/math] is a made-up mathematical gadget that allows us to solve the equation [math]x + 5 = 1[/math].
* [math]\frac{7}{5}[/math] is a made-up mathematical gadget that allows us to solve the equation [math]5 x - 7 = 0[/math].
* The number [math]\sqrt 2[/math] is a made-up mathematical gadget that allows us to solve the equation [math]x^2 - 2 = 0[/math].
* The imaginary unit [math]i[/math] is a made-up mathematical gadget that allows us to solve the equation [math]x^2 + 1 = 0[/math].
In each case, mathematicians of a given historical age become interested in an equation they can't solve. They say, "Well what if there were a funny gadget that solved the equation?" Then after a few decades the made-up gadget becomes commonplace and people come to believe in it as a first-rate mathematical object, and not just a convenient fiction. Eventually we can construct these gadgets both axiomatically in terms of their desired properties; and as explicit set-theoretic constructions. They do indeed become real. Mathematically real, of course. Nobody ever claims any of this stuff is physical. You are fighting a strawman.
Note also that new classes of numbers can be explained as algebraic phenomena, not just geometric ones. The diagonal of a square is one vision; the solution to an equation is another. The fact that we have multiple ways to arrive at the same place is a clue that we are dealing with truth. Not physical truth. Abstract mathematical truth. If the geometers hadn't discovered the square root of 2, the algebraists would have.
So this is where I want to engage. You seem to want math to be what it can never be. You want an abstract, symbolic system to be real, or actual. That can never be. Your hope can never be realized.
As far as the larger philosophical issue:
We agree math isn't literally true about the world. We wonder what its status is. Well, there are things that are physical and things that are abstract. Physical things are presumed true. That's a rock on the ground, if I pick it up and drop it, it accelerates toward the earth at 32 feet per second[math]^2[/math].
But abstractions can be real as well. Driving on the left or right is a social convention that varies by country. It's artificial. Made up. It's nothing but a shared agreement. It could easily be different. Yet it can be fatal to violate it. Driving laws are social conventions made real. See Searle, The Construction of Social Reality.
I submit that 5 is prime and the square root of 2 both exists and is irrational. Those are abstract truths. They are not physical. But they are true. [Typo fixed].
We can go further. "5 is prime" is not like traffic lights. In chess we can make up variants of the game where the pieces are arranged differently or there are pieces with different types of moves. Math is a formal game as well, but we are NOT FREE to say that 5 is not prime. This I think is the core mystery of math. 5 is prime even though there is no 5 and there are no primes.
I am satisfied with this principle if we can apply it consistently. We do not measure mathematical "objects", they are tools by which we measure objects. That's why I argued that they are not proper "objects". Now let's apply this to set theory. Cardinality, for example is a measure. If the applicable principle is that we do not measure mathematical "objects", then why allow this in set theory? It's inconsistency.
So either we can measure mathematical objects, like squares, and the sides of squares, just like we can measure the cardinality of sets, or we cannot measure these so-called mathematical objects. I'm fine with the latter principle so long as we maintain consistency. But if we allow that we can measure these so-called objects, then we can measure a square, and find that the diagonal cannot be measured.
Quoting tim wood
It's what we call an "irrational number", implying an immeasurable length. Are you familiar with basic geometry?
Quoting fishfry
This is not at all what I've been saying, so I think we might not really be making any progress.
Quoting fishfry
Neither you nor I is talking about physical objects here. What we are talking about is the "made-up gadgets" which you describe here. You seem to imply that there is a difference between these funny gadgets, and "first-rate mathematical objects" I deny such a difference, claiming all mathematics consists of made-up gadgets, and there is no such thing as mathematical objects. But this is contrary to set theory which is based in the assumption of mathematical objects. If you really think that a "funny gadget" becomes a "mathematical object" through use, you'd have to demonstrate this process to me, to convince me that this is true.
Quoting fishfry
How can you not see that this is a problem for set theory? Set theory assumes that it is dealing with real, actual mathematical "objects". That is a fundamental premise. Now you agree with me, that mathematics can never give us this, real or actual things being represented by the symbols. So why don't you see that set theory is completely misguided?
Quoting fishfry
So your argument is that the "funny gadget" gets made into a "first-rate mathematical object" through convention, just like driving laws. But those are ";laws", not "objects". Let's suppose that the mathematical symbols referred to conventional laws instead of "objects", as this is what is implied by your statement. How would this affect set theory? Remember what I argued earlier in the thread, sometimes when a symbol like "2" or "3" is used, a different law is referred to, depending on the context.
Quoting fishfry
I don't see how "the square root of 2 exists" could possibly be true, It is an irrational ratio which has never been resolved, just like pi. How can you assert that the solution to a problem which has not yet been resolved, "exists"? Isn't this just like saying that the highest number exists? But we know that there is not a highest number, we define "number" that way. Likewise, we know that pi, and the square root of two, will never be resolved, so why claim that the resolution to these problems of division "exist"?
What am I missing here? :chin:
I'm making the point that the facts of math are not quite as arbitrary as the facts of other formal games such as chess. That's something peculiar or interesting about math. We can't make up a form of math where 5 isn't a prime (in the usual integers). Yet 5 and "prime" are abstract and somewhat fictional entities.
I'm not entirely sure what your question was. But the point is that math isn't like driving laws or chess. Some things in math are objectively true yet not facts about the physical universe.
The issue though, is that we are talking about measuring the ideal square, just like set theory talks about measuring the ideal numbers. We are not talking about measuring a representation of the ideal square, which is written on the paper, just like set theory is not concerned with measuring the numerals, it is concerned with "the numbers" represented by the numerals.
Quoting tim wood
This is absolutely false. The reason why the "numeric representation" of irrational numbers is a problem is because there is no number to be represented. How it is proven that the square root of two is irrational is a demonstration that it breaks the rules of "rational numbers". Pythagoras proved that the square root of two is not a rational number. This means that if we're using the rational number system, the square root of two falls outside of that system, there is no number for it. In the other thread, fishfry called this "a hole" in the rational numbers, but I disagreed with that term.
In reality therefore, an irrational number, has a numeric representation, we clearly have a numeric representation of pi, and square root two, but there is no corresponding number for these representations. As fishfry indicated in the other thread, we might create a new number system (the real numbers) and try to include what is represented by irrationals, within that system, as numbers. I do not understand the construction of the real numbers, but I am willing to argue that this is a flawed approach. Instead of addressing the real problem, which is the fact that we can produce spatial representations (circles and squares) where numbers do not apply, and adjusting these representations accordingly, the mathematicians have created an extremely complex number system, which simply veils this problem.. In other words, instead of addressing the real problem, which is a feature of our faulty spatial representations, and trying to solve that problem, the mathematicians have just hidden it under layers of complexities.
Now obviously, our visual impression of a square is vague to a certain extent, as is any physical measurement of a square, and if we repeatedly measured the diagonal of an actual square with very high precision we would obtain a range of (rational valued) measurements. In other words, by labelling the diagonal of a square with, say, sqrt(2), we are using Sqrt(2) not as denoting an algorithm, but as denoting an arbitrary rational or distribution of rationals, that is yet to be determined through a process of (repeated) measurement.
This is another example of why game-theory is a good model of mathematics use.
There is no rational number which is equivalent to what is represented by "square root of two". So the problem is not as you say, one of numerical representation, it is that there is no number for what is represented. There is a numerical representation, commonly used, but no number which is represented by it.
"I submit that 5 is prime and the square root of 2 both exists and is rational. . . . But they are true."
The square root of two is rational? Am I misreading your sentence? :worry:
Oh that's a typo, sorry. Is that what you were asking earlier? Yes typo of course. I'll go back and fix it.
That sums it up pretty well. Irrational ideas are incoherent, so there is good reason to rid our conceptions of such things. What we like in our conceptions is the quality of being rational and we are wise not to accept irrational ones conceptions.
Quoting tim wood
No, I do not allow that there is such a thing as the square root of two, that's the whole point, and it is what I just explained to you in the last post. Saying "the square root of two" is really a matter of saying something, which represents nothing real, just like "pi" is a matter of saying something which represents nothing real. The mathematics clearly demonstrates that there is nothing real represented by these expressions. So it really doesn't make sense to say that there is "such a thing as the square root of two", just like it doesn't make sense to say that there is such a thing as pi. These are symbols which we use because they are extremely useful, but we ought to respect the fact that there really is nothing which is represented by them. You might imagine an ideal square with a diagonal line bisecting it, or an ideal circle with a line bisecting it through the middle, but the mathematics demonstrates that these are incoherent images.
Quoting tim wood
A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? We can speak of an indefinite quantity as a quantity to be measured, when we assume that the thing is measurable, but to say that a quantity has a number is to say that it has been measured and is no longer indefinite. To say that a quantity has a number, but that the number is indefinite, is contradiction pure and simple.
In the example of sqrt(2), Alice creates a computer program f(n) for calculating sqrt(2) to n decimal places and sends f(n) to Bob. Bob receives f(n), then he mysteriously decides upon a value for n, then runs f(n) and sends the result to Alice.
I can't relate to your analogy. We haven't established that there is such a thing as a "numberline". I think the other thread demonstrated that there is afundamental inconsistency between numbers, which mark non-dimensional points, and a line which has spatial dimension. The issue is how is the incremental increase (discrete quantity) between two numbers, accurately represented by a line?
My argument is that both of these problems, the incompatibility between points and a line, and the irrational nature of the square root of two, are each a part of the same problem. That problem is the way that we represent space, in dimensions. The spatial representation is incompatible with the numeric representation. The trend in mathematics, at least from Descartes onward, has been to adapt the numerical system to account for the problems encountered by this incompatibility. I am arguing that this is the wrong approach, what is really at fault here, is our spatial representation, and this is what needs to changed.
Do abstractions exist at all? I suggest they don't. A number line "exists" only as an abstraction, but this is not true existence. It's just a concept, in which a set of logical/mathematical properties are considered abstractly. The same is true of numbers, whether rational or irrational. "3" doesn't exist, but collections of 3 objects exist - so we can think abstractly about 3-ness. Neither does Pi exist; nevertheless we can abstractly consider the fact that all "circles" (another abstraction) have Pi as the ratio between their circumference and diameter.
As I have pointed out in other recent threads, mathematics is the science of drawing necessary inferences about hypothetical states. Consequently, mathematical existence does not entail metaphysical actuality, only logical possibility in accordance with a specified set of definitions and axioms.
That sounds reasonable. In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"?
Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former.
I wonder if I understand that. Potential infinity is often taken to be the collection (but not set) of the natural numbers 0, 1, 2, 3, ... It's potential in the sense that given n you automatically have n+1; but you never have all of them taken together at once in a set.
But how is that metaphysical actuality? There's nothing in the physical world that corresponds to the endless sequence of natural numbers. It seems to me that 0, 1, 2, 3, ... and {0, 1, 2, 3, ...} are equally abstract. One can accept or reject the axiom of infinity; but either way you end up with a structure whose "actually" is certainly in question. Yes? No? I have no idea.
It is not; as I said, mathematical existence--including the potential infinity of the natural numbers--is not metaphysical actuality, it is logical possibility. From your other comments, I think that we agree on this; perhaps you misread my previous post.
I think I still don't know what actuality means.
But you said that potential infinity has metaphysical actuality. Don't the natural numbers (as modeled by the Peano axioms) contradict that?
No, I said exactly the opposite of that.
Quoting Relativist
Quoting aletheist
Actual infinity corresponds to metaphysical actuality, while potential infinity and mathematical existence correspond to logical possibility.
Since I have found you to be a normally clear-headed and insightful participant here, I fear that your persistence in dealing with @Metaphysician Undercover lately may be producing some unfortunate side effects. :grin:
Metaphysical Actuality: The philosophical position that thought becomes actual by becoming concrete. Subjectivity, the "I" has constitutive validity, having sole omnipotence.
This seems to me an extreme position. How then does it interact with historical actuality? :chin:
That is not what I mean by "metaphysical actuality." I just mean the modal property of being actual, rather than merely possible or strictly necessary, such that something possessing it acts on and reacts with other things in the environment.
LOLOL.
So, combining "metaphysical" with "actual" means someone is thinking a metaphysical thought? Or does the expression imply an interaction with physical reality? I am going on a classical definition of the expression. What do you really mean? Please clarify with examples.
I am not a philosopher.
Thanks. :chin:
I didn't understand a word @aletheist wrote but I was embarrassed to admit it.
I do not understand what metaphysical actuality is. Are the natural numbers metaphysically actual? The completed set of natural numbers? The square root of 2? Chaitin's constant, which is known to be noncomputable? Is a brick metaphysically actual? How about an electron? A quark? A string?
What does thinking have to do with anything? Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it from "possible" and "necessary."
Quoting jgill
That is closer, since whatever is physical is actual in the relevant sense. However, just to be clear, I hold that reality is not coextensive with actuality; there are also real possibilities and real necessities.
Quoting jgill
If I say that I have an apple, what I usually mean is that I have an actual apple. If I posit a set of apples in the strictly mathematical sense, then I am talking about something that is logically possible, but not (necessarily) actual. :smile:
"Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it"
Speak to metaphysics, please. Define "actual" in that context. :nerd:
No, these are all numbers; and again, existence in mathematics entails only logical possibility, not actuality in metaphysics.
Quoting fishfry
Yes, in accordance with how I was using that term.
Quoting fishfry
That depends on whether one is a scientific realist about each of these entities. I am currently inclined to say yes, probably, and maybe.
Quoting aletheist
The actual is that which acts on and reacts with other things.
On the other hand, for a little clarity:
Des Bosses to Leibniz (1700s): "Monads are metaphysical actualities."
Now, that makes sense. :cool:
That's right, we covered this earlier in the thread. We need to distinguish between three things, the symbol, what the symbol means (abstract concept, what you called "3-ness"), and what the symbol is being used to refer to (physical object, and groups of physical objects). There appears to be an inclination in this thread, to conflate the latter two things, and claim that what the symbol means, and what it refers to, are one and the same thing, i.e. the symbol simply refers to a concept. This is an ontological error, meaning is not a thing.
Quoting fishfry
Alethiest and I go way back. We're not too far apart metaphysically, only disagreeing on some finer points. But in relating ontology to mathematics, altheist employs intentional vaguery and ambiguity in terms, as well as outright contradiction to support unreasonable mathematical principles. So we part here.
LOLOL.
It's a good thing we all have a sense of humour here. LOLOL!