Arguments in favour of finitism.
Despite the fact that finitism is not the traditionally accepted viewpoint of modern philosophy, we can still think of its merits.
In mathematics, we often use the terms like an infinite set and we even assign cardinality to it.However, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number but sets have definite number of elements and on other hand infinity is not definite.How can we justify the existence of infinite sets.
I hope Cantor forgives me.
In mathematics, we often use the terms like an infinite set and we even assign cardinality to it.However, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number but sets have definite number of elements and on other hand infinity is not definite.How can we justify the existence of infinite sets.
I hope Cantor forgives me.
Comments (69)
- how many different lengths can have a segment?
- how many different directions there are? ( intended as different unit vectors in 3d space )
I think ancient Greek mathematicians had this point of view: there are discrete numbers used for counting, that are objects with zero dimension; there are lengths, with dimension 1, areas with dimension 2, and volumes with dimension 3. And each of these "kinds" of measures is not comparable with the others. So, the question of how many "points" are needed to form a line does not make really sense, since objects of dimension 0 and 1 are not comparable. So, no need of infinite sets.
I Think the case with vectors in 3d is the same,we can say the norm of the vector containing the sum of all unit vectors in different directions (unlimited ) will be zero, but if we can argue consider the collections of and infinite amount of vectors in opposite direction to each other, however the resultant norm cannot be determined.Similar case arises in series.
The measure of Cantor set is zero and that requires the existence of an infinite set.
Can you clarify on the representations of geometrical objects using an infinite set, I think finite sets suffice.
The behaviour of uncountable infinite set gives bizzare results as mentioned above.
If we adopt a non platonic view of mathematics, and on basis of that, we cannot justify infinity.Infinity can be idealized but we can never point to it in the universe.
Hilberts paradise I guess welcomes it.
Pardon my ignorance.
In every case the series (convergent or not) is made of a countable set of 1-dimensional segments of non zero measure. You add segments to obtain a segment, not points.
Quoting Wittgenstein
If you think of a segment as made of a set of points, it must be an uncountable set of points because a countable set of points would have zero as 1-dimensional measure.
If you instead say that points cannot be used to build a segment because points have no measure, then you agree with me that points and segments are two different kinds of "objects", so one cannot be built starting from the other
~~Mephist~~
I agree that we add segments to obtain a segment, however l have two questions :
1.If the segment is not made up of points, is it a non zero measure or something else ? ( I tend toward your view )
2.Can a divergent series be obtained from an uncountable set, ( I think it can be ) but can a convergent series be obtained from a uncountable set ? ( A sum that is definite must have a fundamental difference to a divergent one )
I think it is reasonable to take segments and point as different cause we can remove the paradox of infinite points or the use of uncountable set of points.
Having already admitted the existence of uncountably many line segments, you can take their union in one application of the axiom of union to join them. It's hard for me to understand admitting the existence of a collection of sets but denying their union. Of all the axioms of set theory that are commonly debated or questioned (choice, powerset, infinity, foundation), union is not one I've ever heard anyone question.
A segment (1-dimensional) has a non zero 1-dimensional measure, and can be only made up using objects with non-zero 1-dimensional measure. ( I am speaking about the view of mathematics that ancient Greek mathematicians - for example Archimedes - had )
Quoting Wittgenstein
If you believe in finitism (as you proposed) there are no infinite sets, neither countable nor uncountable.
Again, this is Archimedes' philosophy of mathematics, not mine.
However, a series is BY DEFINITION a sum of terms indexed by integers. An integral on real numbers instead is a sum of terms indexed by real numbers, that are an uncountable set in ZF Set theory. Then, in modern mathematics a convergent sum of an uncountable set of terms is perfectly normal.
In C programming, the equivalent symbol to infinity is the volatile keyword. When a C program declares a volatile object, say "volatile int myInteger" , the program is declaring that the value of "myInteger" isn't specified by either the programmer or the program itself, but that the value will be supplied later at an unspecified time by the environment. The specification of an 'infinite' number of iterations in a computer program, say while(true) {..}, is therefore equivalent to writing while(volatile int myInteger){....}.
Unfortunately, set theory and logic do not possess an equivalent concept. The closest axioms they possess is the Axiom of Choice, but this axiom is flawed because it is considered to be either accepted or rejected universally across all sets, and it also fails to discriminate sets which are volatile and bounded from sets which are volatile and unbounded. Consequently the status of the axiom of choice remains confusing and controversial.
In my opinion, the historical cause of debates over the existence of infinity is the result of logicians failing to recognise that the semantics of logic and maths isn't fully a priori.
Jeez that's not true. A volatile variable is one that is, for example, mapped to an external data source. Declaring a variable volatile tells the compiler that it can't depend on nearby code statements in order to optimize the variable.
This simply has nothing at all to do with transfinite ordinals and cardinals as understood in math. It's apples and spark plugs.
Quoting sime
Those terms have no referents in math. I am not sure where you are getting these notions. Your ideas about the axiom of choice and mathematical infinity are idiosyncratic to say the least.
sorry, I should have said infinity is equivalent to volatile and unbounded. I am saying volatile and unbounded is equivalent to the specification of an infinite set, considered as extension, in cases where the infinite set is not directly defined in terms of a constructive algorithm.
Quoting fishfry
Likewise, Transfinite ordinals divide into those which are specified constructively as tree-growing algorithms and those which denote unspecified trees to be supplied by the environment, whether bounded or unbounded.
You have a link in support of your ideas? They are very strange. I don't want to flat out say they're wrong, since my ignorance is vast. But I know a little about ordinals and I can't correspond your words to anything I know.
Quoting sime
I don't know what is a volatile and unbounded set. Can you provide some examples so I can understand what you're saying?
Volatile is not a term of art in math at all. And its use in C programming is very specific as I think we agree. It just tells the compiler not to optimize the variable.
The integers are unbounded because you can't draw a finite circle around them all. The unit interval is bounded since all its elements are within 1 unit of each other. Yet the unit interval has far larger cardinality than the integers. So I am not sure what you're trying to get at.
Regarding ordinals, you say that there are some which "denote unspecified trees to be supplied by the environment ..." That's .... well again it's idiosyncratic. There are ordinals which are computable and ordinals which are not. Are you thinking of the Church-Kleene ordinal or one of the other exotic countable ordinals?
I really don't know what you mean by saying (some) ordinals are supplied by the environment. That's not what ordinals are. Ordinals are order types of well-ordered sets. They don't take on values like memory locations in an executing computer program. I think perhaps you might be trying to push programming analogies farther than they can go.
Yes, any programmer's use of an infinite FOR loop. We all know in practice that infinite loops are, in a pragmatic sense, merely finite loops whose termination condition isn't specified by the program. In other words, the termination of the algorithm isn't internally constructive from the point of view of the program itself.
Quoting fishfry
Right, i'm am not so much referring to compiler mechanics, as to the logic of volatile types. Programmers use a richer notion of logic than is used by traditional set theory that equivocates internally constructed sets with externally supplied sets. The consequence of this is mistake is the kludge known as the Axiom of choice that allows the specification of arbitrary unbounded sets, but only for unbounded sets, effectively conflating arbitrariness with unboundedness [/quote]
Quoting fishfry
Whenever we refer to an integer, we are either referring to a integer which we ourselves have or will construct using an algorithm in our possession, or we are referring to an arbitrary integer that is to be delivered to us by some external source. Constructivists make the mistake of conflating existential quantification with construction. It is a mistake, because, say, we cannot run society on software that uses only predictably terminating bounded for loops. Platonists on the other hand, while rightly insisting that non-constructed sets are indispensable in practice, wrongly locate the source of that indispensibility to a priori notions of existence.
[math]\sum_{n=1}^{n=infinity}[/math] [math]\frac{-1^n}{n}[/math] , as you can see in this series we have not indexed the set using negative numbers, and l think the series will not be well defined if we do not restrict R to natural numbers.( countable )
If you want to reconcile geometrical objects with only finite sets, this approach has been used.To quote from the article mentioned below,
Further more if the galois field has characteristic 3, then 2+1=0.
There is a problem in this theory, in that x=-y cannot be thought of as a straight line but ( it is ).
I hope we can discuss this article, to make arguements move forward in the thread.
https://plato.stanford.edu/entries/geometry-finitism/supplement.html
I am unfamiliar with C programming and l would feel comfortable to relate a similar problem in applied mathematics, sometimes in mathematical modeling, where the constraints even as small as the number 7 can be treated as infinity as from 7 and onwards, the program gives the maximum output.I think the axiom of choice is similar to the Euclids parallel axiom, they should be used or ommited depending on the situation.
Btw, I have a question regarding algorithm, I read somewhere Von Nueman saying that it is impossible to have an algorithm that can strictly generate random numbers.Is there even such a thing as a strictly random number.
However, I am not surprised that there is an axiomatization of euclidean geometry that doesn't make use of infinite sets. What I said is a very simple and obvious thing: if you don't allow the existence of infinite sets, you have to treat segments as a different kind of thing than a set of points.
But in the galois field, they treat the segment made of points but don't use infinite sets, the one mentioned in the article.Can you send me any article,book recommendations that make your position clear to me cause l may be confusing your point here, I hope not.
I am terrible at explaining things but at same time I am wondering which one is that which you dont understand.
Can you quote it.
I agree that this series should be defined on natural numbers, but I don't understand to what part of my post is this related. I said that a series, by definition, is a sum of a countable set of elements. Is this an objection to this point?
I wasn't objecting to a countable set, but to an uncountable set.
Nevertheless both are infinite sets.
Can you consider this arguement against an infinite set,
What is the probability of an event happening over an infinite amount of events, it would be zero.We can go on and prove that the possibility of any event happening will be zero but that would be absurd if we applied it to the world.
Are you arguing against infinity in math? Or just in the world? It's perfectly clear that we all have a intuition of the natural numbers 0, 1, 2, 3, 4, 5, ... They are generated by the simple rules that:
* 0 is a number;
* If n is a number, so is n + 1.
Put those rules into a Turing machine and they crank out the endless sequence of natural numbers.
Nobody claims that the infinite collection [not yet a set, that requires the axiom of infinity] of natural numbers is instantiated in the natural world. It only exists as a mental abstraction, like justice or traffic laws or Captain Ahab.
Or are you arguing that you accept mathematical abstractions but denying that they're physically real? That's perfectly sensible.
But by referring to the world, you are out of the realm of the abstract and instead making a trivial point about the world. That weakens your argument considerably.
In any event, infinitary probability theory is well understood and allows for probability zero events that nevertheless may happen. For example the probability of picking a random real number and having it be rational is zero; yet the rationals are plentiful. [That's not a precise statement, but it can be made precise without loss of intuition].
How will you generate negative number using n+1 ?
( natural numbers are infinite nevertheless)
My problems is with the use of infinity as a number in certain mathematical problems, for example the lim 1/x as x approaches 0 will be written equal to infinity.But using an equal sign with infinity can be challenged even in its abstract form, l do understand the theory behind limits but in certain cases referring to infinity, mathematicians treat it as a number, not a concept.
Let's suppose in an ideal world, it is treated merely as a concept, Quine would argue that there are certain infinities allowed which find applications in science but those which are not applied to natural world should not be given equal weighting to the earlier ones.
Consider a mathematical abstraction which describes the world ( quantum mechanics for eg ), l think some mathematical abstraction can co-exist with the real world although some don't.If such an abstraction does not agree with reality as we know, we can drop them even if they are consistent mathematically.
On the other hand, l am not talking about their existence as cats or dogs exist ( that would be stupid ) but their metaphysical existence.Earlier on you mentioned generating an unending natural numbers, but can you ever list them, l am not talking about physical limitation but the nature of infinity would not allow even the fastest computer to list them all, hence it can be contended that infinity as an abstract mathematical can never be produced theoretically, hence it does not exist.
I used the probability theory as an example because it is related to the world, but since you claimed it can be made intuitive, l would like you to clear that up. If someone were to talk of negative probability ( fenyman did l think ), where we consider things we do not observe but which do occur in the real world.( l can be wrong here ), that is more understandable than the use of infinity in probability theory.There is also another problem, if all the probability are 0, then the the total probability of all events will not give 1.That is against the law of probability.Further more if you take natural numbers as the domain of probability distribution, it would be not be well defined.
I would like to quote this for explaining my point of view regarding your objection.
The number is one.
For instance, consider a sigma algebra (i.e. a sample space) denoting the set of possible outcomes for an infinite sequence of coin tosses t(1),t(2),... i.e. a coin toss process whose length is undefined a priori. Coin tosses aren't a mathematical concept but an empirical affair, and conversely mathematics isn't an empirical theory. Therefore it makes no sense to insist that the sigma algebra of infinite coin tosses must be constructive. For it might well be the case that a sequence of coin-tosses is truly random in the sense that cannot be represented by any computable function. This is the case if it is believed that for any computable binary function f there exists a subsequence of observations t(1)..t(n) that isn't equal to f(1)...f(n). Unfortunately, set theory fails to distinguish externally observed processes from internally constructed processes, hence the reason why finitists and infinitists continue to argue past one another.
" Unfortunately, set theory fails to distinguish externally observed processes from internally constructed processes, hence the reason why finitists and infinitists continue to argue past one another. "
"Well consider For it might well be the case that a sequence of coin-tosses is truly random in the sense that cannot be represented by any computable function "
I agree with the first paragraph, it is a great observation in my opinion.However for the second paragraph, can you consider two guys that have unending supply of coins, and each toss say for n times, we can calculate the probability for the next test by n/2^n.Well l dont think a coin toss can be that random, however if we were to pick a prime numbers out of real numbers, l think we would not have a comparable function for it.
Since in the end you mentioned the defect in set theory, l think we can argue for a constructive case, where a statement is either true or false.
My knowledge in this field is weak and it would be of great help if you can provide some resources on treatment of probability theory from a non classical point of view.
I couldn't get what you were saying, can you elaborate.
You take one and you start stretching it out; that's infinity.
I am a bit on the asperger side, but l can sense that you are kidding.
Are you talking about this series.
1/2 +1/4+1/8......till infinity =1
This converges.
Consider this series
-1+1-1+1.......infinity, this is not well defined.
So a bounded sequence can have a series that is not well defined.
:wink: there is no end to a joke about infinity.See what l did there.
Here's an example - take a picture, 100 by 100 pixels.
Infinitely zoom in - the picture is now infinitely large.
Infinitely zoom out - the picture is now infinitely small.
Each pixel within that picture is its own picture and has its own pixels.
In the end - it all converges in to one; one picture, one pixel.
Perhaps we have no choice but to fix some meaning for 'random.' One fascinating way to do this is:
[quote=Wiki]
Kolmogorov randomness defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. To make this precise, a universal computer (or universal Turing machine) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program whose length is shorter than the length of the string itself.
[/quote]
https://en.wikipedia.org/wiki/Kolmogorov_complexity
So, you say that infinite lines in euclidean geometry are the same thing as segments of length one, right?
I thought this is obviously absurd. OK, if I have to say why: because infinite lines always intersect if they are not parallel, and finite segments can be not parallel and not intersect
Quite.
Quoting Mephist
Yeah, I got that, but don't see the problem with it - if you look at infinite lines as stretched out finite lines.
You have two finite lines that are too short to intersect; you infinitely stretch them out and they intersect.
To illustrate, you can infinitely divide a finite thing - which provides the notion it is an infinite thing, with infinite parts.
Now let's bring up the issue at hand.
Two finite things which are according to the aforementioned - likewise two infinite things, don't intersect.
But the idea is that two infinite things must intersect.
And my solution is simple - the halves of the one intersect in sum.
They are finite and likewise infinite - according to the aforementioned, and intersect either way you look at them.
And this comes about, due to it all always amounting to one.
Sure enough, you could just draw two lines that don't intersect and showcase that, but the two non-interlockers along with their void still amount to one.
Point being, if there is one, the lines can and cannot intersect, and all parts amount to one.
I take a segment of length one and I stretch it until it becomes of length 2. But this is still the same segment, so 1 = 2. What's the difference between this and your argument? The halves of two are the same as one. Why is your argument not valid for finite lengths?
I can even prove that a segment the same as a circle: just bend it and it becomes a circle!
If I want to compare the size of two objects I can operate on the objects only with transformations that don't change their size: for example I can put them one on top of the other by moving them, but I can't split them in two or stretch them.
I think I didn't understand completely your argument, but why can't you say the same thing for two instead of one?
Sorry, I realize now that I didn't answer to this question.
I read the article that you posted (https://plato.stanford.edu/entries/geometry-finitism/supplement.html).
It's not true that "they treat the segment made of points". They say that "a point p corresponds to a couple (x,y)" and "A line corresponds to a triple (a,b,c)"; they don't say that "a line is a set of three points". But I agree that this is an example of finite models for Euclides' axioms, (by the way, in their original form Euclides' axioms are not expressed in a formal language, but I don't want to be picky on this point).
The infinite model that I was referring to is the standard one, based on the standard topology of the real line (https://en.wikipedia.org/wiki/Real_line).
I actually don't have a proof that there are no models of Euclides' axioms where a line is a finite set of points. On the contrary, I think you can easily build one by taking as space a finite 2-dimensional array of points, but this is obviously not the right model for the physical space.
So, probably I should have said that: if you want to build a model of the physical space where a segment is described as a set of points (as in standard topology), you need infinite sets. Otherwise, you have to build a model of the physical space where a segment is not a set of points (an example of this is kind, not based on set theory, is smooth infinitesimal analysis: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis).
What I've said is valid for finite lengths, but if it sounds incoherent it's due my inability to properly explain. I'm not sure how to remedy that.
The best way I can put it is that one is the whole and thusly references all lengths without exception, and all lengths individually and as a contingent amount to one. The contingency of the whole amounting to repetition of one which derives distinctions precisely from the twists you mentioned.
I'm really not sure how to put it better, ironically, as it all amounts to how you put it. :)
Here is what I think. Whenever we construct a set ourselves, we must choose the next element to include in the set, either explicitly, or by implicitly by defining a rule of selection. But in order to do so, it must be first be assumed that our elements are individuated a priori for a process of construction to make sense. Yet if we are given a set, say a parcel through the post, it's elements aren't individuated until we inspect the set. If the parcel we are given is called "infinite", all this means is that we shouldn't expect the termination of our parcel inspection to be decided by a property internal to the parcel.
Mathematics tends to call parcels with non-individuated elements "equivalence classes" of elements. Like in the above example, this allows mathematics to either construct sets in a 'bottom up' fashion from elements, or to construct elements in a 'top down' fashion from parcels.
But in my opinion the "infinity" in mathematics is not such an arcane entity as your "one" thing.
Infinitely big and infinitely small are the "simplification" of extremely big and extremely small.
So for example you can consider an infinitely long and infinitely thin line as a model of a very long and thin wire, because you want to "abstract away" the properties related to it's real length and width, and consider only the position and orientation of the wire.
The only reason that infinite numbers were ruled out from mathematics (on the contrary of infinite geometric objects, that were always considered allowed) was the idea that you cannot reason about them without getting inconsistencies in logic. But now we know how to deal with infinity using formal logic in a perfectly sound way, and nearly all important mathematics is making use of infinite and infinitesimals.
So, to me asking if infinite really exists is like asking if a plan really exists, since all "real" objects are 3-dimensional: it's only a mathematical approximation (simplification) due to the fact that you want to ignore one of the 3 dimensions. But probably this point of view is only due to my limited knowledge about philosophy..
In the viewpoint that math is about symbol manipulation formalisms, we may not even be interested in the concept "meaning" as some kind of correspondence factor with the real, physical world.
For a starters, the symbol ? is just one character and not an ever-growing sequence of characters. Hence, it is perfectly suitable for participation in symbol-manipulation formalisms. An ever-growing sequence of characters, however, would be a problem, because in that case our symbol-manipulation algorithm may not even terminate.
Secondly, the symbol ? is not irrelevant or immaterial, because it is definitely mentioned in usable reduction rules that can successfully extend classical arithmetic. For example: a + ? = ? and also a * ? = ?, with for example, a ? ?. The symbol clearly has some kind of "absorbing" effect on other elements in its domain. You can actually obtain/generate the ? symbol by performing particular manipulations on more common elements of the domain, such as 1/0 = ?.
With the various reduction rules available, the symbol ? could actually be useful when you seek to produce a closed form output result from a particular input expression. The domain does not even need to be numerical and the algebraic structure not necessarily a field. The rule templates will undoubtedly still be consistent.
For example, with a and b arrays, elements from the Array domain, and a + b defined in a particular meaningful way, and ? a meaningful extension to the Array domain, you will find that a + ? = ?.
So, it will still absorb the other elements during addition and multiplication. In that sense, it is a bit similar to the zero symbol, which also absorbs other elements after multiplication while leaving them unchanged after addition.
In other words, a field or other algebraic structure can successfully be extended with the symbol ? while maintaining consistency and while satisfying the pattern in existing rule templates for the symbol. So, infinity may indeed not be a number but it is certainly a legitimate extension element in numerical algebraic structures.
Good point. It's even in floating point systems. I like the idea of an old cash register modified to ring up [math] \infty. [/math]
I also like f([math]\infty[/math]) for the limit of f as x goes to [math]\infty[/math]. I find this kind of limit fairly intuitive. Many uses of infinity are intuitive in math. Finitism is cute, but it doesn't do justice to our intuition as a whole.
BTW, you mention math as pure reason. I relate to that formalist point of view, but I suggest that metaphor and intuition are important in doing math. I'd say that it's a kind of language.
:smile: I am late here.
Let's consider the formalist view of math. I think that mathematics is primarily based on substitution where we replace a set of symbols with another set of symbols which are equal or equivalent in some cases. How do we decide that ? By "meaning" l meant the criterion for substituting one expression to another. Formalism has axioms and there are rules of inference etc. It cannot work without them.
The problem with using the infinity symbol is that there are infinities bigger than others. It is a single character but can we substitute it with numbers ? Consider the real line, all the real number lie on it but infinity doesn't. We can by some fancy definitions extend it to hyper-real and have the rules of adding numbers to infinity like a+infinity=infinity etc. Can you generate this symbol by any finite amount of operations ? I dont think we can and in my opinion formalism is basically about operations on symbols ? Therefore by allowing infinity, we sort of compromise the formal system. This is the basic idea behind the constructivist approach, if l am wrong, you are more than welcome to correct me.
I think that when we introduce the infinity symbol, we will have to drop associative law and commutative law too. There is a theorem by rieman which says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. This doesn't apply to finite series for a reason and that is different laws regulate the symbols which have finite connotation and those which have an infinite connotation to them.
The real system has been extended to the hyper real and with it's own extended rules for operation but can we construct the equal or even equivalent of this symbol by same set of operation. By introducing the symbol into the rules and not being able to generate it from the real numbers is cheating. Is this extension valid ?
Depends on how you define the word "number". If you define it narrowly, to mean "a natural number", then you're right, infinity is not a number. But that's not how most people define it. Likewise, if you define the word "set" narrowly, to mean "a finite collection of elements", then you're right, there are no infinite sets. But again, that's not how most people define it.
Infinity is a number and infinite sets are sets and that's all there is to it.
Quoting Wittgenstein
Make sure you don't confuse conceptual existence with empirical existence. The existence of the concept of infinite set is one thing and the existence of infinite collections of physical objects out there in the world is another. The former kind of existence is clearly real, the latter can be disputed.
In case of sets, we use natural numbers to determine the cardinality but putting that aside, l would say there is 1-1 correspondence between all real numbers and a point on real number line, infinity doesn't lie there. How do you define numbers ? Most people do define a set in mathematics as a collection of well defined and distinct elements. Infinity is not an element even in the infinite natural set( or any other infinite sets). If you regard infinity as number, that implies that it is finite, since all numbers are finite.Hence a contradiction in terms.How do you define numbers ( the real numbers ) ? How can you justify infinity as a number ?
I never compared the physical world with the mathematical one. Even if you consider the "conceptual existence " we cannot construct infinity even with all the symbols and operations in a system. This is not a physical limitation but a conceptual one. The concept of infinity does not allow it to be constructed out of numbers. Consider the case of halting problem precisely that of determining, from a description of an arbitrary computer program and an input, whether the program will finish running (i.e., halt) or continue to run forever. It was proven to be impossible and this is a conceptual restraint not an empirical one. Similar case applies to infinity.
Can you justify the infinity axiom.
If there is a match between a rewrite rule and an expression, you can rewrite the expression. So, it all depends on the rewrite rules of the system. These rewrite rules have no particular "meaning".
Say that there exists the following rewrite rule in the system: kk* --> k+, with k any arbitrary subexpression, then we can rewite the expression xyabc(abc)*rs --> xy(abc)+rs. This has no "meaning". The resulting expression is just the result of the mechanical application of the rewrite rule on the original expression.
Quoting Wittgenstein
Both axioms and rewrite rules are arbitrary. For example, the SKI combinator calculus uses the following rewrite rules:
Ix --> x. Kxy --> x. Sxyz --> xz(yz).
We can use this rewrite system to rewrite input expressions to output expressions, and derive new theorems from the system. For example, since SKxy -> y, we can see that SKx = I for any arbitrary choice of x. This theorem is meaningless, because the statement, which is provable from the construction logic of the SKI system, that "?x in D: SKx = I" does not correspond to anything in the real, physical world.
Quoting Wittgenstein
Infinity itself is a Platonic abstraction that is compatible with numbers, which are themselves also Platonic abstractions. Numbers are themselves no real-world objects either. Infinity is compatible when you can extend the rules for arithmetic to support the inclusion of infinity. while not damaging the algebraic structure.
Quoting Wittgenstein
We don't care about lines in this context.
Quoting Wittgenstein
I think that you correctly depict the constructivist view on infinity. But then again, I don't read much constructivist literature, because in my opinion, they are missing the point anyway. Cantor's elaboration of the concept of infinity is nicely consistent. I have no problem with it. I really do not see what the fuss is all about.
Quoting Wittgenstein
Concerning commutativity, the problem does not seem to occur in x + ? versus ? + x, as both expressions get rewritten to ?. In my impression, the arithmetic rewrite rules handle the symbol perfectly well. Adding ? should not modify the algebraic structure. Otherwise, you should not add it to the domain that you are dealing with. So, for example, if D' extends D with ?, and
Quoting Wittgenstein
If adding it, can be done while preserving algebraic structure and therefore consistency, you can go ahead and add it. It does not even need to be about numbers.
In fact, there are situations where you must add ? in order to guarantee the consistency of field operations. For example:
For current cryptographic purposes, an elliptic curve is a plane curve over a finite field (rather than the real numbers) which consists of the points satisfying the equation y² = x³ + ax + b along with a distinguished point at infinity, denoted ?. This set together with the group operation of elliptic curves is an abelian group, with the point at infinity as an identity element.
Without identity element, point addition would not be a group operation. The domain here does not consist of numbers but of two-tuples (x,y):
The equations (for point addition) are correct when neither point is the point at infinity, (0,0). When adding the point at infinity to another point, the result is simply the other point.
Elliptic curve arithmetic has obviously nothing to do with the real, physical world. It was not abstracted away from the real, physical world. Elliptic curve arithmetic is a Platonic abstraction that has characteristics and properties that turn out to be interesting, while adding a point at infinity is not only a requirement for consistency, but it also happens to work absolutely fine.
If we look closely, you are in fact using k an arbitrary sub expression repeatedly. Let's say abcRefg where R is a relation or to simplify pRq, where the R can be an equal symbol or an inequality. Let's say we want to generate numbers from this system by a function F(x), where the input is a string of characters and the output is a number.
Can you generate infinity from that function ? I dont think so, unless you think we can construct an infinite number of characters in a string.
Let's consider your example KK*-->K+, how about this case abK*-->cd+ . Would that be rule if and only if ab=cd.
An absolute formal view will lead to many problems and there is also another problem with this language as it allows a function to take itself as an argument, that may lead to paradoxical self referential statements. Like a set of all sets for example.
By alleging infinity to be a platonic abstraction doesn't help us understand its nature at all. I don't think there is a platonic world where all mathematical ideas can be found and the existence of non euclidean geometry proves that we have to create new maths a lot times by simply dropping some axioms ( parallel line axiom) and hence modify our system.
There is a transfer principle, to extend real to hyperreal and it is actually consistent as you mentioned in your earlier posts.
I just read about elliptic curves, the point at infinity isn't something lying at infinity. They use that term when they draw an elliptic graph on a 2d plane however in 3 dimensional geometry a line does intersect the elliptic curve at 3 points, case closed. Even, the real line can have a point at infinity by simply curling it around and making it meet at and end. This is in no way related to infinity as a concept. I think there is a misunderstanding here . I don't know much about elliptic curves but consider the aymtotoes of a hyperbola(x^2/a^2-y^2/b^2)=0 and then you get two aymtotoes and we say that they intersect the hyperbole at infinity, that does not mean they do. They keep getting closer and closer. You can never give the point of intersection there.
[0,1,0] is used as the point at infinity sometimes so that the equation p+q+r=0 is satisfied or some other form. When the need arises they let p+0=0 so l dont see how the point at infinity is related to infinity that we are discussing here. We also use the word intersect in different sense sometimes, for example if l say two parallel lines intersect each other at imaginary points, l am not referring to the normal case of intersection.
Not sure. For example, if there is a rewrite rule "x/0 = ?" for x not zero, the symbol could start popping up in output expressions. If you feed that output expression into your function F(x), it depends on whether F will accept it as an argument, and if so, if can successfully associate an output to it. Not sure at all.
Quoting Wittgenstein
You will need to apply two successive rewrites:
ab(ab)* -(1)-> ab+ -(2)-> cd+
(1) rewrite rule: KK*-->K+
(2) rewrite rule: ab-->cd
Quoting Wittgenstein
Yes, Russell's paradox and Gödel's Incompleteness obviously apply. Axiomatic systems are quite powerful, but they also tend to be incomplete.
Quoting Wittgenstein
The nature of infinity is what you can do with it in your system. It will participate in rewrite rules. From there on, its nature emerges out of the rewrite rules in which it participates. If:
x & %SYMBOL = %SYMBOL
then %SYMBOL could have the role of the identity element, if domain D with operator "&" is meant to be a group. It could be part of the kernel of a homomorphism of sorts. The nature of %SYMBOL will become increasingly clear by considering the other rewrite rules.
Quoting Wittgenstein
It is just an example of an algebraic structure in which adding a concept of infinity keeps the entire system consistent. Otherwise, the domain has no identity element, and then is no longer a group. That is not allowed, because the system effectively makes use of the systemic property that the domain is an abelian group under addition. You can clearly see that in the definition of the encryption/decryption functions and of the sign/verify signature functions. When you prove that decryption is the inverse of encryption, you make use of the fact that it concerns an abelian group under addition.
This has nothing to do with the real, physical world. In this case, a point at infinity is just a tool to keep that cryptographical system consistent. This principle can be generalized. Adding infinity to a domain is just an instrument that can be used to achieve a particular purpose.
What is the constructivist alternative to doing that?
This approach is deeply embedded in existing technology nowadays:
The U.S. National Institute of Standards and Technology (NIST) has endorsed elliptic curve cryptography in its Suite B set of recommended algorithms, specifically elliptic curve Diffie–Hellman (ECDH) for key exchange and Elliptic Curve Digital Signature Algorithm (ECDSA) for digital signature. The U.S. National Security Agency (NSA) allows their use for protecting information classified up to top secret with 384-bit keys.
Therefore, it is a bit late in the game to argue whether extending a domain with the infinity symbol makes sense.
Mathematics and logic is one thing ... but reality is temporal and spacial. And if everything has always existed everywhere that just means since the beginning and in all places ... it doesn't actually imply infinity. The idea of infinite stuff seems rather odd, to me. There is however much there is. Everything is everywhere but there can't be more than there is ... and actual infinity would seem to suggest that there's more than there is... and that makes no sense.
The universe is eternal but finite, I say. Everything has always existed but that doesn't imply infinity because it doesn't imply before time because there is no time before time. There was a first moment—hence the universe is finite.
Only one part of the domain of knowledge, i.e. in Kantian lingo, "synthetic a posteriori", deals with the real, physical world. The other part, "synthetic a priori", only deals with abstract, Platonic worlds.
In my opinion, there may be too much emphasis on "reality".
In my professional activities, I have never dealt with "reality". I have only ever dealt with Platonic abstractions and their implementation in software. That is why I cannot comprehend why anybody would be so obsessed with the real, physical world to the exclusion of everything else.
Infinity is clearly a Platonic abstraction. Why would anybody try to shoehorn it into the real, physical world? For what purpose?
That's according to your own definition of the word "number". You defined the word "number" to mean "a finite quantity". That's not the standard definition.
What is the standard definition of number ?
The concept of infinity exists. That's what I meant by conceptual existence.
I hope you understand my point after reading this.
It explains my point.
Given that a mathematical extension is a symbol (‘sign’) or a finite concatenation of symbols extended in space, there is a categorical difference between mathematical intensions and (finite) mathematical extensions, from which it follows that “the mathematical infinite” resides only in recursive rules (i.e., intensions). An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms.
What is the standard definition of number ?
It won't since your language is a computer based language, you can only involve numbers or characters.
F(a)=F(b)=F(c)……………=1, F(abc)=3
F(abc) =F(a)+F(b)+F(c)
So inserting F(1/0) will be undefined not infinity.
Whether you want to replace --> with =, they are relations and quantification over infinite terms does not makes sense. That's my point.
I understand that cryptography requires an abelaian group for keeping all values of operations under the group but l can assure you the point at infinity is not related to infinity we are discussing here. There is no operation in solving weierstrass equations which involves the infinity symbol.
Finitist mathematics is not meant to discount the standard mathematics, we are just exploring a new world which is limited. Theory is more important than application
I certainly agree with that. In my impression, it is indeed not possible to use infinity in every slot where you can fit finite numbers.
Concerning "-->", yes, every arrow is fundamentally bidirectional, but when you rewrite expressions, you will generally only use one direction. So, a+b --> c obviously implies that c --> a+b. Still, only one of both will be useful in your rewrite strategy, in order to obtain a closed-form expression, or when trying to prove a theorem.
Quoting Wittgenstein
Galois fields? Pick a prime power p^n and carry out all arithmetic modulo p^n. Approximately all algebraic structure that exists over infinite/countable will turn out undamaged! ;-)
I can't give you a precise verbal description of the meaning of the word "number". What I can do is I can show you that people define the word "number" in such a way that it encompasses quantities that are not strictly finite, quantities such as Pi.
I think pi is finite, as it is less than 4, even 3.15 is greater than pi.You can give a mathematical definition of a number.
The cryptography technique using weierstrass equations that you mentioned uses finite field , like galois field.
Can l redirect your question back to you, as to suggest a cryptography system from an infinite field.
I think it is possible to use an infinite field but more difficult than a finite field.
There's a stack exchange discussion on this question, "In cryptography, why do we reduce elliptic curves over finite fields?".
ECC over infinite fields are certainly not in use.
Imagine that P = s * G is the public key computed by multiplying a secret s of arbitrary size. A first step in the protocol will be to transmit P to the recipient while keeping s secret. If P can be arbitrary size, then you would need to transmit to the recipient a message of arbitrarily long size. I think that this problem alone is already a show stopper.
If you solve the problem by computing P = s * G mod 2^m, then you can guarantee that P will never be larger than m bits. That would already be one reason -- there could certainly be more -- to restrict calculations to finite fields.
How about having a matrice A.X=Y where A,X and Y have infinite number of rows and columns,X is the solution set which contains the secret, a row or a column in the matrix can specify the operations to perform on A to obtain A inverse. That can obviously be either transmitted via matrice element position or be a common understanding between the reciever and transmitter. This sounds really naive and stupid but l hope you can suggest improvements to using matrices in cryptography.
If X is the secret and Y is the public key, then A would be some kind of generator of sorts. That would require that each non-zero element of the domain of Y can be written as A*K for some value of K.
In ECC, the variable A (aptly named G) is a/the generator of the Galois field. Its multiples can produce every element in it.
If that would not be the case, then it makes particular values of Y impossible. If it makes enough values of Y impossible, then it makes it easier to work your way back from Y to X, and solve the elliptic curve discrete logarithm problem. It would certainly damage the intractability of the elliptic curve discrete logarithm problem and hence reduce the strength of the encryption algorithm.
Adding along a Weierstrasz elliptic curve thoroughly distorts the addition operation in arithmetic. That is the first source of encryption/distortion strength in ECC. A second source of strength/distortion, is that it also operates in a finite field that automatically wraps around the result of arithmetic calculations. For example, in a mod 7 Galois field, 2+4=>6 (smaller numbers) while 3+5=>1 (larger numbers), you get the strange result that adding smaller numbers gives a larger result. As such, it destroys any expectation of monotonicity in calculations.
You would need matrix A to unleash a strong source of distortion. Also elliptic?
Quoting Wittgenstein
Daniel Bernstein is arguably the top guru in the field of practical design of algorithms. This is his home page. His most famous implementation is undoubtedly nacl.
The theoretical top number one is Adi Shamir. He came up with many of the theorems. He is also the "S" in the famous (but now probably outdated) RSA cryptosystem.
You are asking me to weigh in on stuff that is rather at their level. I am just a user of their stuff, really. I don't come up with the theorems by myself, and actually, not even the core implementations. I just build in the stuff into other things, hopefully, without creating too many additional issues ... ;-)