Why x=x ?
"an apple is an apple", but why? I do not get why any certain thing called 'x', should be 'x'.
I know that, proving 1+1=2 is hard, whilst it is so simple(logically and practically), but I do not see anyone trying to prove x=x, because it may seem so simple and obvious, as it may look pure stupidity to question it, but that is absolutely my point. The simpler it gets, the complexer explaining it.
Can someone help me out please?
I know that, proving 1+1=2 is hard, whilst it is so simple(logically and practically), but I do not see anyone trying to prove x=x, because it may seem so simple and obvious, as it may look pure stupidity to question it, but that is absolutely my point. The simpler it gets, the complexer explaining it.
Can someone help me out please?
Comments (194)
Well, no, it isn't.
With S the successor function, and x+1=S(x), and 1=S(0), we can see that 1+1=S(S(0))=2. You can trivially prove it by using PA's rewrite rules for addition. (PA is standard number theory)
Quoting Monist
PA's second axiom:
The fact that PA axiomatizes this rule for natural numbers means that it is not provable in PA.
In fact, you can generalize this remark: the defining rule for equality cannot be proven but only be defined in the mathematical theory.
For example in ZFC (=standard set theory), axiom 1, i.e. "the axiom of extensionality", defines the equality of sets:
Again, ZFC's logic for extensionality is an axiomatic starting-point rule that cannot be proven. It can only be used to prove derived rules (=theorems).
The starting-point rules, i.e. the system-wide premises, in a mathematical theory are always arbitrary, unexplained and unjustified beliefs. That is simply the essence of the axiomatic epistemology.
1+1=2 is a more meaningful proposition than simply 1=1. It tells us something about the definition of each quantity in the relationship and something about the relationship itself. As with the x=x proposition it makes an assumption that numerical values remain constant, that 1 today is the same as 1 tomorrow and unlike the x=x proposition, this is a significant assumption.
"An apple is an apple".
For this to mean anything, it would require that each instances of the word "apple" to have a unique meaning.
Or else you are just saying "This is an apple, and its also apple".
What is an apple?
An apple.
Its a way of avoiding the real question. Beyond the appearance of an apple, what is there....what is it really.
Well, if its real reality, then yes. If its fake reality, then no.
The answer is to clarify the question.
"Is what we think of as reality, the real reality, or a fake reality".
I would say, what we think of as reality is a fake reality.
Something is axiomatically true, we usually say, if its true by definition.
An apple is an apple because we have defined an apple as an apple.
But how do we know if a definition is correct or complete.
That is a difficult question, I would say.
Sorry if I'm turning the question into something bigger the intended scope.
And sorry for making three separate posts.
The thing you need to get, I think, is that X=X says something about the language we use to describe the world, and not necessarily about the world itself.
In the world nothing is perfectly identical to some other thing. Every apple has some, even if only miniscule, difference compared to another apple. But they are similar enough that we can abstract away from raw sense-data and make up categories and use those to be able think further than mere experience of fleeting moments.
It's not a perfect tool, and that is important to realise so you don't expect things it can't deliver. But at the same time, it's still the best tool we have even if imperfect.
So... to answer your question, this is not about proving x=x, x strictly speaking doesn't equal x. It's about utility, we want x to be equal to x so can get along with whatever it is we want to infer from that.
Meaning auto-supposes self-identity.
When talking about the soccer match, thinking of the neighbor, etc, we automatically go by identity.
Abandon identity and the posts here are meaningless.
:up:
I really don't get the reason why anyone would ever use that phrase, "An apple is an apple.", unless they're just playing words games, which isn't a complex thing at all.
How is using that phrase different than saying, "An apple" while pointing at an apple? Is your pointing the equivalent of = ?
[math]X\ne X[/math]
See Banno's Game.
That’s why we don’t play that game.
Agreed.
Mathematical sentences, such as X=X, are constituent rules of the mathematical theory that defines them, and are true only in the model(s) for that theory. Such model is never, ever the real, physical world.
For example, if we are talking about natural numbers, we can say that the sentence is true in the standard model for number theory (=PA):
PA [math]\vDash[/math] X=X
The models for number theory (=PA) are NOT the physical universe.
The physical universe itself is a model of the elusive and inaccessible theory of everything (ToE). We do not know if the following sentence is true, because for that we would need to know the TOE:
ToE [math]\vDash[/math] X=X
We simply have no clue as to whether the sentence mentioned above is an axiom in the ToE, or otherwise a theorem provable from axioms in the ToE.
Mathematics does not prove anything, or asserts any truth, about the physical universe because the model(s) for any available mathematical theory, such as number theory, or set theory, and so on, are not the physical universe.
As others have noted, you need to first make clear to yourself what question you are actually asking. Once you do that, the answer may become apparent.
Well, it was much harder for Russell and Whitehead, PA did not satisfy me for many reasons.
Quoting alcontali
Thank you for this reply, it helps me a lot, but does not solve my problem. Aren't axioms, self-evident assumptions? If so, when can we accept self-evident beliefs, just when they are practical? Do we have to analyse the relation between truth and practicality then?
Quoting Mike Radford
Under what conditions is x=x true, when we just accept it? I might be persuaded that x=x.
Quoting Mike Radford
The proposition is simply: A thing resembles itself. The question is, "what is the proof?"
Quoting Mike Radford
Lets imagine that x is a variable, then again we end up with x=x. Thank you for your perspective.
Quoting Yohan
Exactly, I am trying to find a ground to stand on. Starting points known as axioms, simply suck. :-)
Quoting jorndoe
The Law of Identity states that a certain thing is identical to itself, and I ask why.
Quoting Mike Radford
1=1 may or may not be meaningful, but is seems to be true, but why? (It is meaningful in the sense of understanding the internal relation of any quantity; the Law of Identity)
Quoting Harry Hindu
Instead of 'apple' try 'thing'. Saying "a thing" while pointing at the thing does not explain why the thing identical to the thing. It does not explain the relation between the thing and the thing. x=x does, it simply tells that the thing, is itself. The point is, why? :-)
You might be interested in the idea of why we have a system of counting to ten, switch, then repeat because we have 10 fingers.
The Law of Identity is quite easy to understand.
The Law of Invoking Sense?
Which i believe is a natural law.
Originally, in Greek antiquity, over 2500 years ago, i.e. primarily in Euclid's Elements, axioms were meant to be self-evident. For a long time, classical Greek geometry was the core foundation of mathematics. This is no longer the case. From within mathematics, axioms are nowadays considered arbitrary starting points. Especially the formalist philosophy views it like that.
There may still be a link -- outside the realm of mathematics -- that views our most important axiomatizations, i.e. number theory and set theory, as concepts that are inspired by our innate intuition and by the nature of the universe that surrounds us. From within mathematics, however, it is wrong to view it like that, because the epistemology of mathematics does not allow us to make that kind of claims.
Hence, axioms are best viewed as arbitrary, unexplained, and unjustified beliefs.
Quoting Monist
Mathematics does not seek to be practical. On the contrary, the desire for ever-increasing abstraction leads us to make sure that mathematics is preferably meaningless and and useless:
So, mathematics is preferably unrelated to the physical world and therefore meaningless. It can certainly not be applied directly. It must go through downstream user domains such as science, engineering, and so on, which are empirical and reintroduce the physical universe, with a view on harnessing meaningfulness and usefulness.
Consequently, in and of itself, mathematics is not just meaningless but also useless.
The ontology of mathematics is a set of arbitrary, unexplained, and unjustified core beliefs (=axioms) from which we derive new beliefs (=theorems) that are purposely meaningless and useless.
Quoting Monist
The term "truth" in mathematics are facts (=data) in a given structure, i.e. a model, that satisfies a set of logic sentences derivable from a particular theory. It has nothing to do with truth in the physical universe. As I have written before, such mathematical model is never the physical universe, simply because we do not have a copy of the theory of everything.
So, mathematics is not real-world true. As I have mentioned above, mathematics does not seek to be practical either.
What is the relation between the practicality of Base 10 and the necessity of The Law of Identity?
Decimal systems have the use of just naming quantities(switching to Base 12, does not change any quantity), Law of Identity does not deal with the names of a quantity, but the definition, it deals with the property of x.
Can you explain why you gave the example of our counting system? I compeletely miss the point.
For me, it is NOT QUITE EASY to understand, nor is any 'thing' quite easy for me to understand.
Quoting Per Chance
Off-topic
You'll have to excuse me. I'm not well versed in Logarithms or the Law of Identity.
Quoting Monist
I guess i was just fed up of laymen's evoking of mathematical mysticism in philosophy.
Quoting Monist
Such as the above statement/s for example.
Well, what would be the alternative? A thing that is not what it is? An apple that is not an apple?
I accepted the risk of being an idiot when starting the topic, and I see the risk is true, as I see answers underestimating the question. Partly it is my fault. My ability to address the problem may be weak because this is not my native language, I will do my best to re-write my question.
+ To establish any knowledge, I have to believe in it
+ I am not able to believe and/or have no ground to build on with certainty.
+ There is some certainty in the language of math itself and is applicable to the world.
+ The basic idea of math is that any value has an identity(is equal to itself). And here the problem for me starts.
+ When applied to the real world: what is, is(even simpler: `being` or `thing`). It seems to me as a good starting point, because to believe anything, you should believe the thing(being) first.
But then, how do I believe a thing, is a thing.
Why x=x ?
If I can explain the necessity of identity, I can prove a thing can exist.
Why not?
Because if such an alternative existed, it wouldn't exist.
I see a conclusion, but no premises. How and why?
There is no such thing as a thing that has a relationship with itself. Things establish relationships with different things.
Again I don't even understand the point of your question. I dont understand the differece between x and x=x. Why not just say x? In saying x, you are saying what it is. x=x is just redundant information and therefore useless.
Quoting Harry Hindu
Why is identity necessary?
Quoting Harry Hindu
Simple, x is an unknown value, x=x is the principle that it is a value.
Quoting Harry Hindu
If x=x is useless because it is redundant information, is first-order logic useless as well? Because, it literally depends on this axiom. For every x | x=x
Does identity exhaust what it is for the thing to be itself? Isn't an identity a label? A label is not the thing. Words are not the thing itself. Are you talking about the thing, or what we call it? Both are different things that have a relationship with each other.
Quoting Monist
If the value is unknown, how can you say they are equal? It seems that you need to know what x entails for the = to be useful. x does not equal x because both x's are on opposite sides (they occupy different space and are typed at different times on the screen (one is after the other) of the =, so I don't know what you mean for two different x's to be equal.
Any word can be counted as a label then, that perspective does not help much.
And again, we are not talking about the x`s on your screen, which have different locations :-)
We are talking about a thing, being itself.
I should have used the word variable instead of unknown, in my language, we call that variable idea unknown. Semantic problems...
Well, first things first: X is just a stand-in symbol for any object or number you wish it to be. But I think you got that far yourself already.
To a more specific example with your apples. Everything is what it is and not some other thing. If you have an apple--like a real world apple, it can't be anything else at the exact time you are holding it. It has to be itself. What else could it be? If your apple were to be a not-apple, say a banana, then it wouldn't be apple=banana. It would be banana=banana.
You can even expand the equation: apple=apple=not-banana. Or x=x=~y. Or apple=apple=nothing that is not apple. Or x=x=~(~x).
Five minutes from now it can be mush in your stomach, and five hours from now it can be released as energy and water and air, etc. Five months ago it was in the clouds and the earth and the tree. But right here, now, it is just itself.
Let's try it this way:
Apple=Banana is true, if the properties of apple and banana are completely identical. Certainly, they are not. So Apple?Banana. It would explain why an apple couldn't be not an apple. But this logic is only true, if the properties of apple are identical to themselves.
If I start counting the properties of Apple now.......yes, they do meet the properties of Apple. As far I could observe the properties of x, they are identical to the properties of x, therefore x=x is true. This is inductive reasoning which does not guarantee anything.
It follows from the contradiction "X is not X", because from a contradiction, anything follows:
https://en.wikipedia.org/wiki/Principle_of_explosion
But even without explosion, the original assumption that an object is not what it is, is absurd enough.
Thank you so much!
For now, I am satisfied with the answers. But not with the constitution of beliefs on these given Axiomatic Laws and Logical Principles. So I will continue my journey with questioning them.
And absurd is okay.
If you are using mathematical symbols, then the above should suffice. If you are arguing why a thing is the same as itself, then jump into the deep end of the metaphysical pool and splash about.
:yawn:
Actually the definition of 2 in the dictionary is 1+1. the definition of 10 is 9 + 1
You can actually get good at math just by memorizing definitions.
10 ofcourse to some extent requires knowing the defintion of 1 through 9.
The reason 2 was ever defined is because humans are much deeper thinkers than all known animals.
I'm sure some animals understand what 2 is to some degree.
(I suppose your inquiry itself is meaningless, if identity was abandoned; not just your inquiry, every inquiry.)
There cannot be anything in particular prior identity.
Quoting Monist
It's not. (Necessity presupposes identity, if we're talking modal logic at least.)
Ah, I think I see a problem here. You're introducing an observer into the mix. Our observations of the world are always (to some extent) inductive. (And, as an aside, people often demand too much of inductive reasoning. You can be justifiable and reasonably sure of something without having to be 100%, infallibly sure of it, but that's a subject for a later day.)
The point that the law of identity simply makes is that any object, like (let's call it, for sake of specificity, and to make clear we're not taking about general apples, but of one specific apple in the time-space continuum) Apple#1 is the same as Apple#1 because Apple#1 DOES have the exact same identical properties, including their location in the space-time continuum as Apple#1. Whether or not you the observer can verify that the apple you're holding is actually the apple you're holding is not as certain, because there is always the off-chance that you're insane/a brain in a vat/under some magicians illusion.
You just pushed this back from X to properties of X, but that doesn't really change anything. Just do a variable substitution: let Y designate what used to be called X, and let X now be a property of Y. Everything that was said of X would still apply, since it does not actually depend on the meaning of X - it's a purely syntactical exercise. Metaphysics doesn't come into this.
My two cents...
Imagine x = God
Then, by definition, x is all good, all powerful and all knowing
If x is NOT = x then x is NOT God and so x is not all good, not all powerful and not all knowing
So we get the following contradictions:
x is all good AND x is not all good
x is all powerful AND x is not all powerful
x is all knowing AND x is not all knowing
These contradictions arise because we assumed x is not = x
Therefore, x = x
In more general terms every x has a set of properties and let's call it P so Px (x has properties P) is true
If x is not x then x will not have properties P i.e. ~Px
Px & ~Px is a contradiction
Therefore x is not = x is false i.e. x = x
In argument form:
1. If x = x is false then x has property P AND x doesn't have property P
2. It is false that x has property P AND x doesn't have property P (contradiction)
Ergo
3. x = x is true (modus tollens)
Whatever an x is, x has a set of finite properties, say Ax, Bx and Cx.
If x is not x then one of the following will be true: ~Ax or ~Bx or ~Cx or all.
We then have a contradiction: Ax & ~Ax or Bx & ~Bx or Cx & ~Cx.
So, it's false that x is not x or it's true that x = x
An argument:
Px = x has all properties that x has
Ox = x doesn't have at least one of the properties x has
x = x the law of identity
~(x = x) the law of identity is false
1. Px
2. ~(x = x) > Ox
3. Ox > ~Px
4. ~(x = x).....assume for reductio ad absurdum
5. Ox.....2, 4 MP
6. ~Px.....3, 5 MP
7. Px & ~Px....1, 6 conj (contradiction)
8. x = x 4 to 7 reductio ad absurdum
Awesome, but excuse me please, can you explain why Ox > ~Px ?
Well, if x lacks one property from a set of properties P that defines x then it follows that it's false that x has all the properties P. Right?
Why? Words are things. If x is a variable, then x can be anything, including a word.
Quoting Monist
I doesn't make a difference. If you don't know what x is, then how can you say it is equal?
Why talk about a thing being itself? What problems do you hope to solve, or answers you expect to get?
Why not just talk about the thing, as opposed to the thing being itself? How is that any different than talking about the thing being itself?
Quoting Monist
No it's not. Philosophy is rife of absurd questions. Some questions just aren't worth asking.
x=x being true, can be put in words as 'the necessity of identity.'
If that is understood, now;
A. I did not understand the underlying reason of your statement:''words are not the thing itself'', how did you come to the conclusion that I am talking about the words. And how did you relate that conclusion with the word 'identity', without considering the concept of identity.
B. Words are things. Agreed. x can be anything, including a word. Agreed. What has this to do with the context here?
C. Where did you see that I am defending that x=x is true, I do not know. I ask why it is necessary to be so, since axioms are based on this very idea.
D. It is important to talk about a thing being itself, to understand what constitutes the being of a thing. For example; if identity is false for things, nothing may even exist. I try to catch what is going on, and why I do that is explained in my other reply.
E. You can talk about many things about the thing, one of them is their identity.
F. The absurdity of rational thought constituted on axioms is okay, but me questioning it is not... I prefer being free.
Quoting Harry Hindu
G. It reminds me of a Greek tossing Socrates on the street :-) I wish I could have that certain statements, without knowing the answers :-)
Thank you,
Methinks it is axiomatic that everything is self-identical. No proof is required or possible -- hence all of the so-called proofs above simply assume the principle of identity holds for each of their premises.
I think there's always a reason a la Liebniz.
Does this spot [O] ‘fit’ into its white surrounding? - But that is just how it would look if there had at first been a hole in its place and it then fitted into the hole. So when we say “it fits”, we are describing not simply this picture, not simply this situation. “Every coloured patch fits exactly into its surrounding” is a somewhat specialized form of the law of identity."
(Wittgenstein, PI, §216)
Right. There is at least no deductive, non-circular proof for x=x, because a deductive proof requires formal logic and all of formal logic rests on the assumption that x=x is true/valid.
You can only inductively confirm x=x is true. And you can reason that any other premise makes no sense, goes nowhere, is incomprehensible, and can nowhere be found to be true in thought or reality.
To quote Aristotle (and I'm just quoting it for the fun of the quote, not because I'm trying to say anything about anyone involved in this conversation):
"Some people, through their lack of education, expect this principle, too, to be proved; for it does show a lack of education not to know of what things we ought to seek proof and of what we ought not. For it is altogether impossible for there to be proofs of everything; if there were, one would go on to infinity, so that even so one would end up without a proof; and if there are some things of which one should not seek proof, these people cannot name any first principle which has that characteristic more than this."
One has to wonder what this could possibly mean. Abstracted from the idle speculations of bad philosophy, if one were to be asked in conversion: "do you know you are yourself?" the only possible reply is "what on earth are you on about?".
I don't understand how x=x means "necessity of identity". What are you trying to say when you write x=x. or "necessity of identity"? What kind of identity are you talking about and why is it necessary?
Quoting Monist
Because you are talking about "identities" and identities are words that refer to some thing's attributes for the purpose of categorizing those attributes under one word.
Quoting Monist
Then, you tell me because I still don't understand the point of your question, "Why x=x?" It doesn't have to be a word we are talking about. What does anything have to do with the context here?
Quoting Monist
How about in the very same post you replied to me?
Quoting Monist
Quoting Monist
How is talking about a thing being itself different than just talking about the thing to understand what constitutes the thing. You seem to be hooked on this word, "being". What do you mean by, "being". How is it different than saying a thing has these particular attributes that we group under one word, - it's identity. Being Monist entails being conceived by their parents and being raised in the very place they were raised. What new knowledge have I acquired about Monist that I already didn't know? To say that Monist is being Monist doesn't give anyone anything useful to explore. x=x is simply redundancy and redundancies are not useful. It seems that identities are useful when talking about some thing without having to talk about all of its attributes. We talk about its attributes when we use the word that we have agreed upon that refers to all of those attributes.
Quoting Monist
Is it? Is the label that others put on you, part of what makes you you, or are you you prior to being labeled by others? It seems to me that identities are what one places on another. You are you prior to being identified and identification is useful when you don't want to spend time talking about attributes. Others can describe you. Their description is your attributes. Your identity is a word that refers to all of those attributes.
Quoting Monist
Rational thought can't be absurd, or else it's not rational.
What do you mean by identity? x=x is a mathematical equation and mathematics deals with numbers, which are not identities in the way you seem to be using the word. In 32 = 32, what is the identity? What are we talking about here? What is 32?
It appears that you are confusing biology with mathematics.
Really, show me a excerpt from a book on evolution by natural selection that refers to the reflexive property of x=x.
x=x is the reflexive property. x+0=x is the identity property. Maybe this whole thread is based on a misunderstanding of algebraic properties.
Again: what does this mean? Babies certainly develop a sense of self, but an 'awareness that they are themselves'? What else would they be? What else could they even in principle develop an awareness of? Does this question have any stakes? Seems like wordplay to me.
If anything, self-awareness arises out of a certain recognition of invarience in relation to an environment; distinction and not identity is primary.
X=X is a representation of the fact... of distinction?
That people have tried to wring conclusions from tautologies has always puzzled me. I remain puzzled.
tldr: While self-identity does not have any meaning as a stand-alone principle, if we need to formalize the concept of equality, we have to state self-identity explicitly as part of its definition.
There's a need to pretend profundity. It's an odd thing. Is it pretence? Perhaps.
Except this is totally wrong. An apple is an apple. It is also a fruit. That something is itself says nothing. It is semantically empty. As are all tautologies. Tautologies cannot be exclusionary because they are not even inclusive: they say nothing.
The most you could say is that an apple being an apple precludes it from being a not-apple (¬X), but this too says nothing at all. It is an extentional and not intensional distinction.
The only way around this is Leibniz's way, which was to pull the entire world into the 'identity' of any one thing, such that relations of inclusion and exclusion are relations of compossibility and incompossibility of entire worlds. Leibniz needed God to balance the whole equation, so that seems like a bit of a lost cause too.
Totally. Insofar as one is setting out a grammar for logic, this makes total sense. But this speaks to a certain use of language - it certainly doesn't amount to some kind of biological or even metaphysical principle.
I suppose, extrapolating on this a little, that the only real occurences of 'is' are those that can be denoted by the '=' sign, in, for example, mathematical formulae. In such contexts, x=x has a degree of certitude that can never be conferred on the declaration that an individual particular is such-and-such. For example, things that are named as part of a class of things ('this is an apple') are always subject to further qualification (like, what kind of apple, perhaps it's a replica, and so on.)
So this is why abstractions (such as signs and numbers) possess the intrinsic intelligibility that material particulars do not. I also think it is why the ability to grasp the meaning of the equals sign, that X=X, is essential to the formation of intelligible ideas and language.
Note the relationship between Spir's idea, and the notion of the reality of intelligibles, which is fundamental to pre-modern philosophy.
-------
* I can't help but notice the resemblance of this overall idea to neo-platonism.
See. This is what happens when one hews to the principle of identity as some kind of metaphysical postulate. You have to leave the world behind. Religious tripe. All the more reason to be suspicious of metaphysical extrapolations with respect to it. The closer you get to Platonism, the more off-track one is.
Yeah and I'm pretty sure Nietzsche didn't see it that way either. That the empirical world is ever changing, is not a reason to conclude that it is not real... it's the other way arround. We should should be suspicious of knowledge taken to far on the back of axiomatic principles like the law of identity, because the world does in fact seem to be ever changing. Only when we freeze time does the law really hold up.
And I take that to be the question of the OP, what then is the justification of it? It can't be proven because it's the basis of the whole system, and only if things are frozen in time do they remain the exact same thing. So I think the justification is not because it's true, proven or follows from our experience of the world. The justification is utility, because it works... for our purposes. Things remain constant enough, or change slow enough, that we can assign categories and work with them. It's the best and only thing we have to attain some kind of knowledge about the world.
He clearly wants to replace religion by a simplistic and childish exercise in infinite regress, the kind of which has never produced anything of value or even worth knowing. His views are therefore worthless.
Just extend his sentiments to "x" OR "apple" OR whatever :smile:
Well, you have to start somewhere, I suppose.
What do you think of X=X+1 ? :gasp:
Necessary truths are not ‘in time’. I think you're making the mistake of believing they're real in an objective sense, when they're actually transcendental or necessary truths - ‘true in all possible worlds’.
In matter-form dualism, the form is what makes particulars intelligible. It is also what brings order out of chaos, as matter in itself is unintelligible until it takes form. That is why, for classical dualism, the form or principle of a particular is what is real, as it is grasped intuitively by the intellect rather than mediated by sense. The form is also what a particular truly is, whereas this or that instance is accidental and temporal.
Don’t overlook the original impetus of philosophy was to identify an unchanging reality in the flux of experience.
Quoting alcontali
He was known as a philosopher, rather than a religious writer, although it seems very little of his work has been translated.
Yes, in mathematics, the syntactic entailment of the "=" operator is defined -- axiomatized really -- for each different data type. For example, you will find an axiom for the meaning of equality in number theory and in set theory ("extensionality").
Even when you create your own custom data types, you will have to provide such definition by yourself. Otherwise, the system does not know how to evaluate equality.
This principle is reflected all the way down to practical applications in programming languages.
For example, you can turn standard tables into custom data types in the lua programming language. You can do this by attaching a metatable with functions, one of which is supposed to define what equality means.
Quoting Metatable Events, __eq section
Not one formal system "knows" what equality is supposed to mean, unless you explicitly tell it.
I don't think they are real in an objective sense, in fact I don't think the word true applies at all to 'necessary truths'. Truth value comes from verification, you check to see if a statement is true or not by looking. Logical necessity is only truth-preserving if you will, i.e. 'if the premise is true, then what follows logically is also true'.
This all seems completely backwards to me. Ideas are not real, precisely because they don't exist in time. That's what it means to be real, to exist in space and time. Only particulars exist, and universals are abstractions of those. They enables us to 'abstract away from reality' to gain more general applicable knowledge. But that knowledge is not reality itself. I mean, that's like saying the map is more real that the world it is based on.
And yes, the original impetus of philosophy was mistaken.... a mistake we struggled the next couple of millennia to get away from. The fact that we have a need for certainty or permanence is no reason to assume that that can be attained. There is no unchanging reality is what our senses tell us.
Of course. You're modern.
Yeah I guess I was born that epoch.
Quoting Wayfarer
A snide reply doesn't really deserve an answer. But I will say this, Socrates and his followers weren't exactly classical either, they were just wrong. They represented a break from and a decline for ancient classical Greece. Only then did they turn away from the sensual to the abstract, when Greece (Edit: Athens I should say) was already in decline.
It's interesting to reflect on Buddhism in this regard. Buddhism famously says that everything is anicca, impermanent, and that there is no permanent essence, substance (in the philosophical sense) or self. The principle is that all of the testimony of the 'five heaps' (skandhas) which are the aggregates of sense + the organ of mind (manas) does not contain anything which is self, which is permanent, and which is not intrinsically unsatisfactory (dukkha).
However - this is always the case, it is utterly invariable - which I think is interesting. I mean, it's not as if the Buddha's teaching was true in India circa 450 b.c.e. but has since been superseded by something else. No, what Buddhists teach is that it's always true but that people fall away from being able to grasp it due to the predominance of ignorance in the 'dharma-ending age'.
So there too, I aver, there is actually a quest for what is imperishable, what does not pass away, the deathless, the not-subject-to corruption. But the way this is conveyed in Buddhist philosophy is by way of negation, by showing that everything that is sensed and perceived 'arises depending on causes and conditions'. When that is abandoned, which is precisely the aim of the Buddhist discipline of renunciation, only then is 'the deathless, the unconditioned, the unborn' realised.
And I wonder if there is not some resonance between that principle, and Spir's axiom that 'the empirical is unknowable.'
Russell argued that this is a primitive proposition that must be assumed true without proof.
Why, you ask, we do that? Well, it is hard to see what alternative we have in regards to X =X. Is it possible for a thing to not be itself?
I don't see how X=X represents how we identify things. This is essentially saying Mac=Mac. What does this tell me that I don't already know? If I were to ask, "What is Mac?" and you reply Mac=Mac, then is that all there is to Mac? Or does Mac=Human being + English speaker + X + Y, etc.? Identity, in the way you are using the term (that has nothing to do with the mathematical property of identity so I don't understand why you're using a different mathematical property to argue your point) is a way of taking all of your Xs, Ys and Zs and putting it under another symbol, "Mac", to make communicating all of your Xs, Ys and Zs more efficient.
When it comes to being you, you are much more than a name, right? You're more than just some scribbles on a screen. Scribbles on a screen are observed and interpreted just as your body and behavior are. I can distinguish scribbles from bodies. Names are not bodies and their behaviors. Names refer to, or are about, those bodies and their behaviors. So, identity is not a case of x=x, it is a case of x=a + b + c. My identity would be y = a + b + d, and so on.
Intuition.
Some things just make innate sense, we say they are self-evident. That is the answer you are looking for. There is no deeper, more specific, or more meaningful reason.
Because traditionally it has been taken as one of the _starting assumptions (premises), which are called axioms or postulates.
https://en.m.wikipedia.org/wiki/Law_of_thought
Aristotle, for example, considered it to be the primary axiom for deriving the very concept of truth and falsity, thus consequently upon which all the rest of logic depends.
Yes, axioms are self-evident assumptions, but being self-evident is supposed to upgrade its status from “mere assumption” to “something better”.
To be practical in the useful sense of that word entails reliability and consistency, which really is ‘validity’ in the most meaningful, i.e. practical sense.
But by practical I do not just mean necessarily useful, I mean our ability to interact with physical reality on a common ground where we proclaim ‘objectivity’ for any given claim, and thereby confirming, or at least increasing our confidence in axioms and hypotheses.
An apple is an apple can refer to a whole fruit category or species, meaning apples generally share common properties that differentiate them from strawberries, crocodiles, and everything else. It is really about similarity rather than identity.
More pragmatic is referring to actual apple out there in the world, and then identity has to do with location. The proof is empirical observation that only one physical particle can exist at any given point in space at any given point in time, which basically means any actual or physical thing, as opposed to virtual or imaginary thing, can only be what it is when it is, i.e. it can only be itself and nothing else.
You are pretty much correct but why is x=x a problem? Sure x = a number of combined variables. But that also means that x is a particular thing. And no other thing is that particular thing so in making this distinction we must also accept that x is unique; x is itself. Which, again, can be a culmination of other variables, maybe infinite amounts. It is easier and still works formulaically to use the shorthand, x=x. If x were not equal to itself then a harder question would arise: "Does x exist independently of itself?"
My biggest question to you is "why should x=x be something you don't already know?" That's the point. I brought biology into the conversation because of its relevance to corresponding mathematical models. x=x is obvious to us because we evolved for it to be. Otherwise we would not have survived in the same way.
It also states that a certain thing is different to every other thing, in a way. But in any case it is vague and ambiguous statement, especially since there is indeed more than one valid context in which the law can, and has been interpreted.
Words, being ontologically virtual, a form of embedded information, like all the other imaginary entities from mental realm are not identical with things they represent, and a single word can represent multiple things, or even change meanings depending on various factors.
In this context physical entities, in contrast to those abstract ones, actually do stand for what they are. And in that sense we can say physical entities are identical to their ontological manifestation or existence within space and time, as opposed to be a representation or pointer to something else.
Semantically it is about coherence, and the law states that during a reasoning the meaning of any term must remain constant. It originated in this context. Here, the reason why is because otherwise conversation would be meaningless.
Mathematically it is about interchangeability, and the law states algebraic manipulation of variables around equal sign preserves equation validity as long as both sides compute the same value. In this case the reason why is because it makes no difference to the calculation result.
Physically it is about persistence, and the law states something does not cease to exist or become something else for no reason, it remains to be that which it was, uniquely defined by its spatial location and geometrical morphology at any given point in time. The answer to the why question here is because we observe it to be so.
This last paragraph above I guess is the context you are asking about, and of course quantum mechanics has its own take on the fundamental nature of existence, but then I do not hear anyone is saying QM makes sense or follows rules of logic, so I suppose we can simply ignore it.
Many axioms aren't self-evident in any fashion.
For example, take the axiom of regularity in ZFC set theory. The reason why it is there, is because in 1917 Dmitry Mirimanoff started writing lengthy rants on the existence of sets that are not "well founded". Imagine that you define:
A = { A }
This construction is not stable, because it is equal to:
A = { { A } }
A = { { { A } } }
and so on
...
So, it is not allowed to construct a set that looks like A = { A }. Therefore, the axiom of regularity is some kind of syntactic bug fix.
The same holds true for the axiom of specification. It prevents unrestricted comprehension which can otherwise cause Russell's paradox. Hence, it is another bug fix.
The language, i.e. the set notation itself, along with the language of first-order logic allow for specifying contradictions. Hence, one way to alleviate the problem -- but not really to solve it completely -- is to add rules (as axioms) that stamps out the most obvious ones.
So, axioms are also mere syntax restrictions for the language in which the theory is being elaborated. That type of axioms cannot be self-evident in any fashion, because you will first have to discover the problems that you will need to fix, by actually using the language.
Furthermore, what is there self-evident about the other axioms in ZFC?
For example, the axiom of power set. They happened to need power sets for Cantor's infinity theorems, but they could not guarantee that these power sets always exist. So, they ended up axiomatizing their existence. Problem solved. Is that kind of origin for an axiom, something to be considered "self-evident"? I don't think so ...
To get the conversation back on track - the problem here is that metaphysics, generally, is the discipline of reckoning 'what must be the case' in order for things to exist in time and space. 'Metaphysics anticipates the general structures of reality by formulating the way our knowing operates. Science actually works out the explanation of the data by a never-ending process of research.' ~Lonergan.
So, such things as logical principles, scientific laws, mathematical objects, are all essential to empirical science, but they don't necessarily exist in time and space either. Rather, they form part of the architecture of reason, by means of which judgements about time and space are arrived at.
That is why, for instance, there are many intense and unresolved metaphysical arguments about (for instance) the meaning of quantum physics, whether there are multiple universes, whether life and mind are ultimately reducible to organic chemistry - and so on. And only certain aspects of many of these arguments comprise objects that exist in time and space. Empirical theories need to be validated against observational evidence - although even that is now being disputed - but metaphysical postulates cannot.
My remark about you being modern, is not a pejorative, but it was a bit snide. But it is natural for us moderns to think that ideas exist 'inside' the mind, or a 'product' of the brain. We carry around this world-picture comprising us, as intelligent subjects, who model or represent the world in our highly-evolved forebrain, largely along darwinian lines. But we don't see that this model is also a construction in the sense understood by critical philosophy (such as Spir's). We are conditioned into naturalism and scientific realism by consensus, and it's often hard to question.
It would be interesting to hear if there are any non-self-evident axioms outside abstractions in the field of mathematics.
Yeah, exactly. That's why the idea self-identity speaks to anything at all without further specification is so silly. That it has some kind of univocal extension, ranging from math to biology as if some kind of meme, is so egregiously bunkum as to not warrant serious response. As if one could simply say "a thing is identical to itself" and think one has said anything meaningful at all. Those who talk of 'proving' or 'disproving' that x=x simply know not of what they talk.
One needs first to clarify what 'x=x' means.
What the claim that x=x means is that in a strictly logical system the symbol 'x' can be 'replaced with the symbol 'x' without altering the truth value of the system ( ie whether the sequence of steps is 'true' within the system or 'false' within the system.)
Since the two symbols are in fact the same, it is unlikely, if not impossible for the truth value if the system to be altered.
In the same way, the claim that 'x=y' is the claim that the symbol 'x' can be replaced with the symbol 'y' in a particular logical system without altering the truth value of the sequence of logical steps.
Hope this helps.
Nowhere in Buddhism itself is Nirv??a described as 'a state of mind' but a psychologized and subjective interpretation is popular in the West.
I think the point of 'self-identity' is not the obvious truism that something is 'equal to itself'. I take it to be a reference to 'what truly is'. I think, in Greek philosophy since Parmenides, it has been understood that individual particulars - the inhabitants of the sensory domain - are a combination of 'is' and 'is not', an admixture of what is real (changeless, enduring) and what is corruptible. But I will need to do some more reading on that point.
Actually, this is the reverse of what is the case. Aristotle formulated the law of identity so as to put the identity of a thing within the thing itself. This was done to fight against logical sophism which proceeded from the premise that a thing is what we say it is, how we identify it. So the law of identity, when properly formulated, is meant to produce a healthy respect for the difference between what we say about the world, and the way that the world actually is, by emphasizing that the real identity of a thing is within the thing itself, rather than in our identification of the thing.
Accordingly, "x=x" is a poor representation of the law of identity. If "x=x"is meant to represent "a thing is the same as itself", it only brings us further from the real world, by adding an extra layer of representation. Even the saying "a thing is the same as itself" cannot properly represent the law of identity, because to make the law of identity a representation like this, is to say something about the thing, implying that a thing's identity is in something that we say about the thing. But this is exactly what the law of identity is intended to avoid.
Therefore we cannot take the law of identity literally. We must look at what it means, what it is intended to tell us, rather than what it literally says or represents. If we look at it in the latter way we will tend to laugh at it and make fun of it. "A thing is the same as itself" doesn't say anything about anything, so it must be a useless statement and therefore the basis for a joke. But it's not meant to say anything about anything, it's meant to be a general statement, saying that no matter what the identity is that we hand to a thing, it's not the thing's true identity.
However, we can see that the law of identity provides us with a separation between the metaphysical (true) way of looking at things and the epistemological (representative) way of looking at things, described by Plato in the cave allegory. Truth is found in the way that things really are, not in representations (what human beings say about things).
For something to exist or be real does necessarily mean it is manifested in time and space. To exist out of time is to exist never. To exist out of space is to exist nowhere. To exist nowhere or never means not existing at all. Ok?
Unicorn and number 3 do exists in my mind and they are real as electrochemical dynamics of my brain. Pacman exists and is real, both abstractly or virtually as electrodynamics in electronic components of its arcade machine, and actually in its physical form on the display screen.
For something to be real or to exist means it is causally relevant, or measurable. In other words, it exists if it matters. Virtual existence is material existence as well, and thus causally relevant. Only virtual things do not exist materially in their actual form, but virtually like any other form of information, indirectly embedded in the physical morphology of the material world, within space and time.
Not at all. Numbers are not real as ‘electrochemical dynamics’. Here you’re mistaking an event for a representation. Neural dynamics don’t ‘represent’ anything, they’re not signs. Science has sought to understand the neural events triggered by simple leaning tasks through scans, and no regularities or patterns can be found at all. It’s not as if some pattern of neural events ‘stands for’ a number or other kind of concept. This idea that concepts are neural events is the myth that underlies materialism, but it’s not true.
All you’re expressing is the belief that ‘everything exists in time and space’. But you’re not seeing that time and space themselves are co-created by the observing mind, they don’t have a reality independent of cognition. This is one of the cardinal points of Kant’s Critique of Pure Reason.
The Buddha is (among other things) ‘lokuttara’, meaning ‘above the world’. I think, to be candid, that is a synonym for ‘supernatural’, although that word is a taboo in modern culture, isn’t it? (This is something that ‘secular Buddhists’ struggle with.) In any case, in the Tibetan depictions of the wheel of life and death, the Buddha and bodhisattvas are represented as being outside the cycle of transmigration, indicating transcendence. Whereas for naturalism ‘the cycle’ is all that is real, there is no conceptual equivalent to ‘outside’ it. The import being that Nirv??a is a soteriological concept - the ending of all suffering (not in a state of non-existence.)
Your English dictionary module seems to be broken. That is not an event, but process, and also collection of states it produces. Namely the process of thinking and states of mind with content such as ideas or mental pictures.
What in the world are you even trying to say? You asserted a lot of “it is not true”, but forgot to explain any, and also never mentioned what do you believe is true instead.
Ughh. I’m simply unpacking the meaning of the words defined in English dictionary, and you can not continue this argument without redefining those terms first or you will end up contradicting yourself and basically not really speaking English language but gibberish.
1. To exist out of time is to exist never.
2. To exist out of space is to exist nowhere.
3. To exist nowhere or never means not existing at all.
Where exactly and why do you believe to see a fallacy?
I'm not going to engage with you if you keep slinging schoolyard insults.
As another poster mentioned, it is in our limited way in which we describe our world and existence via language. We must remember that definitions of words change over time. Words are an extension of life, so there is variance. Just as there is variance in things. We have to be careful not to attribute so much value to 'naming' (nominalism), because in doing so, we discard our understanding of how things are only recognized to by way of their 'relation' to other things. This is the core concept of 'Logos'. Naming should only be a survival tool to help us differentiate. When we apply nominalism as a scientific tool, we are disregarding essential components. (Do a search for what Charles S. Peirce said about Ockham and a ship wreck.) X is only X when you are talking about one thing, and even then you are identifying it in relation to what it is not.
I think that all axioms are non-self-evident, especially in mathematics.
The idea that axioms would have to be self-evident stems from classical Greek geometry, of which the axioms were eminently visible objects, i.e. points, lines, triangles, circles, and so on.
Of course, that kind of axioms were "self-evident": Just "have a look" by yourself.
This approach did not keep flying, however.
First, Euclid's Elements started getting competition from language-only mathematics in the Algorithmi's Liber Algebrae (12th century). Algorithmi did not draw anything at all. There was absolutely nothing to "see". At the same time, algebra was much, much more powerful than classical Greek geometry.
The final blow to classical Greek geometry and its straightedge and compass constructions came when Gauss algebraically expounded its fundamental limitations:
Quoting Wolfram
It wasn't just Gauss' genius but especially Gauss algebra tools that allowed him to run circles around the ancient Greek. If everything you do, has to be "self-evident" -- like the ancient Greek wanted -- then you are not going to get particularly far ...
Sure. X is your name. But then what if someone else is named Mac? The differences between the two Macs is their differing combined variables, not their names. Which Mac are we talking about? The one with X, Y and Z as opposed to X, Y, and B. What makes you unique isn't your name, it is your combined variables.
Quoting Mac
If I know x, or if I know your name is Mac, then why would I need to know x, or your name again? I need to know what x, or Mac, entails to know what is unique about x, or Mac.
If x is not x, then it's a contradiction.
X is x because I perceive it to be.
Quoting Wayfarer
Quoting Wayfarer
I'm a bit hesitant to reply to this, because you touch on a lot of things, and I'm not sure I can do justice to the all issues being raised. But anyway, with that caveat out of the way...
My first general remark would be, aren't we always conditioned into something? I get that you are not a big fan of scientific realism and naturalism, but it's not as if past times were free of conditioning, to put it mildly. At least the scientific method comes with the tools to question itself. And for me that is important, because, what can I say... I like clear skies.
Concerning metaphysics, I kind of agree with Nietzsche's view on that, namely that most of it springs from the psychology and the moral views of people. Absent any way to verify it, what informs those metaphysical views really? Reason you might say, but reason doesn't inform us about anything absent empirical data, it needs something to work with. Take for instance quantum mechanics, one of the reasons why there are so many perfectly reasonable interpretations right now is because there is no empirical data to rule out any of them. Reason alone doesn't get you there if there is no data. And then there's a history of metaphysics being used to ground all kinds of moral theories. So yeah, metaphysics I try to stay away from as much as possible.
About the last quote, of course the language we use to describe the world doesn't exist in space and time. It's merely a description of world, not the world itself. The laws themselves do not exist, right? Take for instance the second law of thermodynamics, entropy never decreases over time. The universe doesn't behave like that because there exists a law that makes or causes the universe to behave that way. Rather the universe behaves that way because there are a lot more high entropy configurations of the universe then low entropy states, and statistically, given enough time, it will therefore naturally end up higher entropy. The law is an abstraction and description of that process, not the cause.
And finally, I don't think logical principles, scientific laws, mathematical objects, etc... have historically been developed apart from the empirical sciences, in a pure reason kind of way, so that we can use them. They have being developed in concert with each other, the one pushing the other and vice versa. They are tools, and people don't care about developing tools that have no use... usually.
What 'the unconditioned' is, is a very hard thing to articulate in the modern philosophical lexicon. But I think that whole question of what you're conditioned to believe or accept, and the meaning or significance of 'the unconditioned', is (or should be) central to philosophy. I think that originally, important elements from Greek philosophy that became absorbed into Christian theology were concerned with 'the unconditioned'. That appeared as the One ('to hen') in neoplatonism. Even materialism sought the unconditioned, which was the basis of atomism - that the atom, meaning 'indivisible' or 'uncuttable', represented the philosophical absolute, appearing in such a way as to give rise to phenomena. It's still a very influential idea, even if it's been undercut by later physics. But in neoplatonism, realising the nature of the unconditioned or the one, consisted of something very much like mystical union (henosis), which again is alien to modern thought. I'm not saying that therefore it's superior or that modernism is therefore wrong or inferior, but what's important to see is that modernism doesn't really understand the issue at all, the entire domain of discourse and the concept of 'the unconditioned' is no longer significant to it. (Note this vivid statement of 'the unborn', which I take to be analogous to 'the unconditioned', in the early Buddhist texts.)
Quoting ChatteringMonkey
But, does it? First it makes some really basic metaphysical assumptions about the nature of reality, which is that what is real, or at least what might be considered, is that which is amenable to measurement and mathematical analysis. It takes a stance regarding what constitutes proper knowledge, and then forgets that it has taken that stance. This is how methodological naturalism, which is to bracket out the subjective, morphs into metaphysical naturalism, which then makes statements about the nature of reality beyond the scope of what is the proper subject of science. And that happens a lot.
Quoting ChatteringMonkey
I am highly sceptical about Nietzsche, although never having done a unit, I'm hesitant to comment on him directly. But consider some of the philosophical traditions which he saw fit to criticize (if not rubbish). Neo-thomism is situated within a domain of discourse within which critical analysis of metaphysical conceptions is alive and well. I'm thinking of recent analytical Thomist philosophers like Bernard Lonergan, John Haldane, Jacques Maritain, and others of that ilk. (See Philosophy lives, John Haldane, for a sample.)
Now, of course, you can say 'well they're all Catholic, and I don't accept the fundamental premise of their work, which is the existence of God', which is fair enough. I'm not myself Catholic, nor could ever consider converting to Catholicism, but I think their writing preserves elements of a perennial philosophy which are again generally absent from modern philosophical discourse. (See also Does reason know what it's missing.)
Quoting ChatteringMonkey
Consider the expression 'not the world itself': per Kant, we don't know 'the world itself'. It's not as if the world exists objectively apart from us as subjects. Subject and object are co-arising or co-defining. Theories, language, number, mathematics, and so on, are all part of the way we bring order to experience, but they also enable us to discover many things we couldn't otherwise know (which is one of the reasons Kant's 'synthetic a priori' was considered by him to be so important.)
Switching registers a little, the objective or external world does not have the inherent reality we generally attribute to it. That's not to say that it's merely unreal or a fantasy, not at all. But all our judgements as to 'what is real' have a subjective pole or aspect which is not itself apparent to the workings of science (except for science has now been forced to recognise that through the 'observer problem' in physics, per the blind spot.)
I think, by way of summary, you're operating from an instinctive position of scientific naturalism - which is fine, I'm not saying that's inherently a problem. It only becomes a problem when it's taken to be something it's not, which is a critical philosophy, because philosophy is critical in a way that naturalism is not.
Methinks you got it!
And this is true of entities that defy the winds of time, like 2=2. But introducing a time gap when speaking of physical objects is different. X(t1)=X(t2) ? Is one PM today the same as one PM tomorrow? Is anything physical the same from day to day? One could argue that my car yesterday is the same as it is today, in a rough sense, but of course it isn't.
Like I said, that doesn't show us anything that we didn't already know. x = x is no different than just stating x.
"Is" and "is not" are exhaustive and comprehensive operators, there is no third option, which means that x must be x, since it cannot be not x because that undermine all syntactical reasoning.
Showing something that we already know is redundant. Redundant information is not useful.
Harry Hindu: Hello, I'm Harry. Who are you?
Mac: Hello, Harry. I'm Mac.
Harry Hindu: What is it like to be Mac?
Mac: It's like being like Mac.
Harry Hindu: What is Mac?
Mac: Mac is Mac
They are different statements though. One is the statement of x, the other is describing something about x. It's the difference between saying dog and saying a dog is a dog.
Merely declaring so is much like saying the Moon didn't exist until onlookers noticed it in the sky.
We differentiate perception and the perceived; always elevating their relation to existential dependency is poor philosophy.
You know that Albert Einstein famously asked that very question. The exact quote is:
As recalled by his biographer Abraham Pais.
[b]Why did Einstein, of all people, feel obliged to ask that question?[b]
They're both saying the same thing. The latter sentence just says it twice. Redundant.
That sounds to me like a rhetorical or incredulous question meant to convey Einstein's opinion that he thinks the moon does exist when it's not looked at.
In what English class did you learn that "x=x" or "it is itself" counts as a description of something, and not earn you a detention for being cheeky? This is a misuse of the English word 'description'.
The issue is, why Einstein, and why that question. It's related to a point that Jorn Doe picked up on, but is not really related to this thread, so I will not elaborate right now.
Because he thought Pais was nuts. Anyway, physicist David Mermin has since resolved the question to everyone's satisfaction. :-)
Quoting Boojums All the Way Through - N. David Mermin
If in the course of reducing a logical expression one obtained the opposite, all this would mean is that one's use of terminology isn't consistent. The solution is either a wholesale adjustment of the axioms that defines one's terminology, or to forbid on a case-by-base basis any derivation that leads to contradiction. For example, if Peano arithmetic was discovered to be accidentally inconsistent, a possible solution is to retain it's rewriting axioms for arithmetic, but to forbid any derivation beyond a certain size.
Philosophers have an unfortunate tendency to mistake ordinary uses of equality as denoting a physical relation between things rather than as being a linguistic relation between terms. For example, if the word "Now" is considered to refer to presently moving objects we arrive at the Hegelian contradiction "Now isn't now". But all this means is that our definition of 'now' is inconsistent. The contradiction is removed by replacing each and every use of "now" with a unique and new term, such that we are never tempted into equivocating one "now" with another.
This is why "x=x" is not a good way to express the law of identity. It really doesn't serve that purpose.
:up:
Maybe if you have a bad teacher, but think you just don't fully understand the word "description."
Quoting Harry Hindu
One is saying x and the other is saying something about x.
No seriously - if someone says: "describe this dog to me", and you reply "it's a dog", there are a few possibilities - you misheard the question; you were being cheeky; its so obvious what the dog looks like that it'd be redundant to describe it any further; you don't understand English; you're unacquainted with the dog so are unable to elaborate. What you have not done, is give a description of the dog.
Yes, as you know, Mermin was referring to Bell's Theorem which shows that the predictions of quantum mechanics are inconsistent with local realism (where realism, in the sense used here, refers to counterfactual definiteness - the ability to speak "meaningfully" of the definiteness of the results of measurements that have not been performed).
If we reframe the question to be about photons instead of angels, it turns out an unlimited number can because particles with integer spin, such as photons, are not subject to the Pauli exclusion principle. :-)
Quoting Pauli exclusion principle
Quoting Saphsin
While x=x is almost always not a satisfactory answer to the question "what is x?" in most circumstances of non-philosophical discussion, it does tell us more about x than simply stating "x" would. It tells us that x is self-identical. Self-identical is an attribute. That it happens to be an attribute all things in the universe share makes it no less an attribute. Thus, calling attention to said attribute is, by any definition of description of which I know, a description.
I'm in agreement with you both that it's not even a very interesting attribute in most cases of any practical matter. It's important, though, that we know that x=x is an attribute all things share, because that's where formal logic starts. If you go through a formal proof which says otherwise, i.e. x=~x, you're in trouble.
This seems like linguistic sommersaults to me. A distinction without a difference for no purpose.
Except if you want a foundation for formal logic :)
Well, wrong and wrong, but I guess at this point all is left is to agree to disagree. :chin:
Cool story bro.
Um....:
Quoting StreetlightX
Okay then.
:snicker:
Yes, saying something redundant about x.
X is a symbol and symbols are already about something. A symbol can't be about itself. Then its not a symbol, but the thing itself.
"Artemis" is a string of symbols that is about something that isn't a string of symbols. It is about what it is to be Artemis. "Name" is a string of symbols that is about the string of symbols, Artemis, not about what it is to be Artemis.
The movie, The Time Machine, taken from Wells' novel, shows the moon that is not there breaking apart in a catastrophic sequence, which, apparently, is not there as well. :scream:
The catch is, we also know that it is nearly impossible for “nobody” to be “looking”, because anything at all counts as “somebody” and any kind of interaction at all counts as “looking”. So this boils down to saying the moon demonstrably does not exist when it stops interacting with the rest of the universe, which is in turn a reasonable definition of nonexistence, making the claim rather trivial.
It's the redundancy of x=x that tells us something about x explicitly that just x does not. X is self-identical. X doesn't tell us that. Although common sense might suggest it, it's not actually expressed through just X. It's only expressed by x=x.
Also, x=x is three symbols, even one of them is repeated.
You don't know what redundant means.
What are you saying when you say x? Are you just making a sound, or does the sound symbolize something that isn't you just making noises with your mouth?
*Sigh*
I'm just trying to phrase this in a way that will get through to you by using your own words. I.e., explaining that the very aspect you think is superfluous and repetitive is the key to understanding why it's actually saying something new.
Quoting Harry Hindu
And if you're still stuck on how x is a symbol for anything and/or everything in both math and logic and can't move beyond that to see how its being applied here, I'm not sure the conversation can go anywhere.
Then why form your reply as an argument rather than an agreement. If what you are saying is that we are saying the same thing differently, then just say so.
Yes, I agree that x=x is saying something new. It is saying it is redundant.
Quoting Artemis
I asked you a question. I'm asking what you mean by just x. From there, I might be able to understand what you mean by x=x. If we're not talking about symbols, or meaning, then we're just talking about scribbles.
Now you're just totally confused.
Quoting Harry Hindu
Of course x is a symbol, but in this discussion it is not meant to stand for anything specific. It's a placeholder for anything and/or everything in the universe. Saying "X" (i.e., just x) is different than "x=x" (which is more than just x).
But if it helps you, we can use a specific example, like apple. "Apple" is different than "Apple = Apple."
This was the point I was trying to make before, less bluntly. Thanks.
Leibniz: "It is what it is."
Clinton: "It depends on your definition of 'is'."
:chin:
??? I hope this is a joke, and that you have not received a fake message from someone who has hacked my info! :gasp: