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In My Philosophical Development Russell wrote,

I come next to what Wittgenstein had to say about identity, which has an importance that may not be obvious at once. To explain this theory, I must first say something about the definition of identity in Principia Mathematica. Among the properties that an object may have, Whitehead and I distinguished some as what we called 'predicative'. These were properties which did not refer to any totality of properties. You may say, for instance,'Napoleon was Corsican' ..., and, in saying such things, you do not refer to any assemblage of properties. But if you say 'Napoleon had all the qualities of a great general' or 'Queen Elizabeth I had all the virtues of her father and grandfather and the vices of neither' you are referring to a totality of qualities. Properties that in this way refer to a totality we distinguished from predicative functions in order to avoid certain contradictions. We defined 'x is identical with y' as meaning 'y has all the predicative properties of x', and, in our system, it followed that y had any property that x had, whether predicative or not. To this, Wittgenstein objected as follows: 'Russell's definition of "=" won't do; because according to it one cannot say that two objects have all their properties in common. (Even if this proposition is never true, it is nevertheless significant.)

Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing (Tractatus, 5.5302 and 5.5303)

At one time I accepted this criticism, but I soon came to the conclusion that it made mathematical logic impossible and, in fact, that Wittgenstein's criticism is invalid. This appears especially if we consider counting: if a and b have all their properties in common, you can never mention a without mentioning b or count a without at the same time counting b, not as a separate item but in the same act of counting. You could, therefore, never conceivably discover that a and b were two. Wittgenstein's position assumes that diversity is an indefinable relation, although I do not think that he knew he was making this assumption. But if he is not making it, I do not see on what grounds he can say, as he does, that it is significant, to say that two objects have all their properties in common. If, however, diversity is admitted, then, if a and b are two, a has a property which b has not, namely, that of being diverse from b. I think, therefore, that Wittgenstein's contention as to identity is mistaken. And, if so, it invalidates a large part of his system.

I don't understand how Russell's argument (second paragraph) refutes that of Wittgenstein's. Even in view of Russell's arguments, I think Wittgenstein's arguments make sense. One may or may not "conceivably discover" that a and b are distinct, but that doesn't alter the fact that in fact they are (if it's possible at all).

Also, it seems to me that the definition of "=" could be made more precise if we would rephrase it as,

x=y iff y has all the predicative properties of x and conversely.

If this is granted then I think that Wittgenstein's argument makes even more sense. If for example there were two things x and y such that, x=y then we can say that x and y are same object, i.e., there are in fact not two things but only one and it is surely nothing significant to say that two identical things are identical.

Can anyone help me in explaining it?

  • Feynman once idly speculated that all electrons and positrons were the same single particle, just whizzing back and forth in time (in the mathematical view a particle going back in time appears as the forward particle's anti-particle, e.g. electron becomes positron). One problem with that, of course, is that then there should be the same number of electrons and positrons. It was not entirely serious. :-) – Cheers and hth. - Alf May 29 '15 at 5:38
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    There is a "way out"; when we assert an identity A=B we are asserting that the two names "A" and "B" denote the same object. This is way an assertion like "Scott is the Author of Waverley" is not a vacuous tautology. – Mauro ALLEGRANZA May 29 '15 at 12:56
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    I must have been having a bad day. – Russell McMahon May 29 '15 at 14:26
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    Not necessarily. But there are multiple meanings of "identity" : the logical one that I commented above, and anotehr one, more concerned with ontology; see The Identity of Indiscernibles. – Mauro ALLEGRANZA May 29 '15 at 19:39
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    @Adam. I am sorry that I had had to delete your answer because it needs elaboration and a tighter argumentative structure. But don't be discouraged. You are new to PSE and we all have to learn the ropes. I look forward to your next answer. Best - Geoffrey – Geoffrey Thomas Nov 16 '18 at 14:09
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The key point is the sentence:"Even if this proposition is never true, it is nevertheless significant", I italicized "significant", as Russell does in his text. Russell is talking about manipulating objects in a formal system (of Principia Mathematica). What he says is that while objects can be "actually" distinct if they are "equal" in his system this fact is insignificant (irrelevant), contrary to Wittgenstein. Because one can "never conceivably discover that a and b were two" without creating a "diversity" property, which would make them un"equal".

This becomes clear from the following paragraphs, where Russell describes his definition of 2 as a class with non-"equal" x, y, one of which is always "equal" to any z in this class, and then writes:"if two things have all their properties in common, they cannot be counted as two, since this involves distinguishing them and thereby conferring different properties upon them". In other words, his definition of 2 still works even if the defining class "actually" has more than two objects.

This is now well understood in mathematical model theory. Peano arithmetic admits non-standard models, for example, with "infinite" numbers, despite the induction axiom suggesting that they are not there. How does this happen? There is no way to distinguish between "finite" and "infinite" numbers in first order logic, so whether they are there or not is "insignificant" as far as arithmetical properties are concerned.

Wittgenstein was fond of mixing language with meta-language, even inclined to admit contradictions resulting from that, see Berto's The Gödel Paradox and Wittgenstein’s Reasons. But Russell points out that Wittgenstein's alternative of always using "different letters for different objects" is technically unworkable, and would make mathematical logic "impossible". "To say of one thing that it is identical with itself is to say nothing, and to say of two things that they are identical is nonsense" is a nice quip, but the second part is questionable, and both parts are beside the point of "equality". In interesting cases we do not know, and in some cases we can never know, if "equal" things are identical or not, e.g. when manipulating formal expressions representing them. And even the first part needs some twisting to accomodate examples like "the evening star is the morning star".

  • But Russel's equal and Wittgenstein's identical mean two totally different things. The former is an equivalence class and the second is an identity relation. There is a real point to distinguishing conventional equality from actual identity. This is a matter of Russel simply imposing his standards on someone who has a different set. – user9166 May 30 '15 at 17:37
  • Each of these perspectives does not make the other impossible, you can imagine all sorts of distinctions that are real but cannot be captured in language, or you can decide they don't really exist because they "can't matter". Model theory works either way. Russel is simply wrong to say that other views cannot be made to support logic. – user9166 May 30 '15 at 18:04
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    @jobermark From meta-language point of view there can be no "true" identity in mathematics, it is always only equivalence. It follows from Skolem's theorem that first order language can not fix a model, we can not even be sure what x and y in x=y represent, let alone if they are truly identical. So Wittegnstein's dream of a first order language that serves as its own meta-language is as impossible (because of Skolem) as Hilbert's dream of first order theory that proves its own consistency (because of Gödel). – Conifold Jun 2 '15 at 1:14
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    @jobermark Higher order theories are no good either, as Russell found out the hard way before his axiom of reducibility. Now we know that they are not even recursively axiomatizable, i.e. one can't even tell axioms from non-axioms. And from context they were both discussing a language for mathematics (mathematical logic). Of course we can always extend the language to make "unseen" distinctions seen, but by Skolem some always remain unseen. Russell is right that Wittgenstein's idea of representing true identity in language is completely unworkable. – Conifold Jun 2 '15 at 1:19
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    @jobermark Wittgenstein said "this won't do" specifically about Russell's "equality" in Principia Mathematica, i.e. about representing identity in mathematical logic, not about its Platonic "existence". And this is in line with Wittgenstein of Tractatus. We also know that "later Wittgenstein rejected many of the assumptions of the Tractatus, arguing that the meaning of words is best understood as their use within a given language-game", i.e. he agreed with Russell, and even went further. And "significant" identity is the only way it works in modern mathematical logic. – Conifold Jun 3 '15 at 2:11
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I'm not going to say much about Wittgenstein and Russell because I haven't read them. However, I will say something on the narrow issue of whether two things can be distinct but also identical. If identical means having every measurable quality in common, it is possible for two things to be identical but also distinct in the sense that there is more than one of them.

In quantum mechanics, if you have an object in the state |0> you can write that state as 1/2(|0>+|0>) or 1/3|0>+2/3|0> and so on. Any instance of |0> produces exactly the same expectation values and there is no measurement that can distinguish them. Nevertheless, you can divide the instances of a system in that state up in different ways, so they are distinct. And you can't do without being able to add up states in this way since the relevant numbers are used to make all of the probability predictions of quantum mechanics when you add up non-identical states.

For more detail see "The Beginning of Infinity" by David Deutsch, Chapter 11 and

http://conjecturesandrefutations.com/2015/04/04/fungibility-in-quantum-mechanics/.

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