Exactly what was Wittgenstein's argument against identity?

Question

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)

Like Russell said, "Wittgenstein announces aphorisms and leaves the reader to estimate their profundity as best he may. Some of his aphorisms, taken literally, are scarcely compatible with the existence of symbolic logic."*

Russell seems to understand Wittgenstein's point. I've read both W's Tractatus and Ramsey's "The Foundations of Mathematics," but I still couldn’t understand Wittgenstein's argument against identity.

*Russell, Bertrand. "The Impact of Wittgenstein." My Philosophical Development. New York: Simon and Schuster, 1959. 126 Print

Answer 1

It seems to me that it is a sort of blunder from Wittgenstein.

Wittgenstein criticizes the logical rules for identity already in 5.434, becuase they are not expressed with a "correct logical notation".

It seems to be a critique of Frege's and Russell's theory of quantification and identity. Wittgenstein’s approach seems to be that no adequate logical notation would include the identity sign, and to claim that the "=" sign becomes unnecessary if stay consistent with the use of names:

5.53 Identity of object I express by identity of sign, and not by using a sign for identity. Difference of objects I express by difference of signs. [see also 2.0233]

Thus - it seems - a logical perfect language will uses different signs for different objects, like the numeral 1 to denote the number 1 and the numeral 2 to denote the number 2. If so, a formula like 1 ≠ 2 is a non-sense, or at least "useless".

According to Wittgenstein, the only legitimate use of the sign of identity is at a meta-level, in order to talk about the use of signs, and not to assert anything substantive about the world. Thus he says:

4.241 When I use two signs with one and the same meaning, I express this by putting the sign ‘=’. between them. So ‘a = b’ means that the sign ‘a’ can be substituted for the sign ‘b’. (If I use an equation to introduce a new sign ‘b’ laying down that it shall serve as a substitute for a sign ‘a’ that is akeady known, then, like Russell, I write the equation - definition - in the form ‘a = b Def.’ A definition is a rule dealing with signs.)

4.242 Expressions of the form ‘a = b’ are, therefore, mere representational devices. They state nothing about the meaning of the signs ‘a’ and ‘b’.

This account of the role of the identity sign is in contrast to that of the mature Frege, who initially adopted something like Wittgenstein’s view himself in his Begriffsschrift (§8) :

Identity of content differs from conditionality and negation in that it applies to names and not to contents.

But later Frege rejected it in Über Sinn und Bedeutung. This rejection were motivated by the fact that the metalinguistic account made identity statements in general (and mathematical equations in particular) into relatively trivial linguistic truths, whereas really they were capable of expressing "real" knowledge.

We can consider the logical axiom for identity:

∀x (x=x);

according to the standard semantics for first-order language, it express the "trivial" fact that "every objcet is equal to itself".

By way of the quantification axiom : ∀x α → α[t/x], where t is a term, we can derive e.g. its "arithmetical" instance : 1 = 1. Again a "trivial" true sentence of arithmetic : "the number 1 is equal to itself".

But the properties of identity are used also in the arithmetical axioms [see Peano axioms] for the successor function S and for the sum (+) and product (binary) functions.

With them, and the usual abbreviation for the numerals : 1 for S(0) and 2 for S(1), i.e. S(S(0)), we can derive the formula :

1 + 1 = 2.

This formula can be "read" at the meta-level (as Wittgenstein do) as expressing the identity of reference between two terms (two names).

But it express also an arithmetical fact (as Frege stressed) that is not a "linguistic" one, but a "real" piece of arithmetical knowledge.

Answer 2

The extract in the question doesn't posit an argument, but an assertion; item 5.302 in the Tractatus has:

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).

According to this, is the number one equal to one? But is this what W is asking, when he is asking about objects - are they objects of the world or objects in some platonic realm?

For example, there are two cups on the table next to me; can I say that cup=cup? When I examine them they are in fact different, one having spots and the other one stripes...

Answer 3

I would argue that Wittgenstein's statement is common sense; he returns to this point in the Philosophical Investigations:

(216) “A thing is identical with itself.” -- There is no finer example of a useless sentence, which nevertheless is connected with a certain play of the imagination. It is as if in our imagination we put a thing into its own shape and saw that it fitted.

and:

(253) “Another person can’t have my pains.” -- My pains -- what pains are they? What counts as a criterion of identity here? Consider what makes it possible in the case of physical objects to speak of “two exactly the same”: for example, to say, “This chair is not the one you saw here yesterday, but is exactly the same as it”.
...
I have seen a person in a discussion on this subject strike himself on the breast and say: “But surely another person can’t have this pain!” -- The answer to this is that one does not define a criterion of identity by emphatically enunciating the word “this”. Rather, the emphasis merely creates the illusion of a case in which we are conversant with such a criterion of identity, but have to be reminded of it.

(254) The substitution of “identical” for “the same” (for example) is another typical expedient in philosophy. As if we were talking about shades of meaning, and all that were in question were to find words to hit on the correct nuance.

I understand this roughly as this: there is a sense to the word identical, according to which a thing cannot be identical with anything else, and to say it is identical with itself is to say nothing:

(279) Imagine someone saying, “But I know how tall I am!” and laying his hand on top of his head to indicate it!

It seems common sense.

Answer 4

To say something is identical, means they are of the same thing. To say a thing is like itself, is to say nothing. You do not add something, or explain what that the thing is. You simply state: Here is a thing. Obvious, trivial and explains nothing. A simple observation at best.

One thing can be, falsely, thought of as two things. Same reference different sense as Frege would say. Here identity, or '=' show that relationship between expressions of symbols.

'1+1' and '2' has the same reference but different sense. You have not acquired any metaphysical knowledge. You have only disguised the reference with yet another sense. Here is a thing: '1+1 = 2' Here is the same thing again: '0.5 + 0.5 + 0.5 + 0.5 = 1+1'. This also reference the thing: '(1+1 = 2) = (0.5 + 0.5 + 0.5 + 0.5 = 1+1)' What are we doing? Nothing. You simply state: Here is a thing. The meaning of the symbols are arbitrary, because they could easily mean something else entirely.

Answer 5

What he means is that all equivalence is imposed, not naturally occurring. No two things are the same thing, except under a specific notion of sameness. If they were actually the same, we could not identify them, in order to then identify them. Every thing has the property of being itself, which no other thing shares.

Throughout mathematics, for example we create equivalence classes by deciding what does and does not matter, and we ignore what we have decided to ignore. Two pairs of triangles may be equal in arithmetic, where all pairs are instances of two, and unequal in geometry where the fact they do not coincide is important.

We do the same thing in a less clear manner throughout the rest of life. But just as it is in mathematics, that is always a choice. There is nothing natural about any specific notion of equality. We just have strong conventional agreement about what differences should be ignored under what circumstances.

Answer 6

In his introduction to Wittgenstein's Tractatus Bertrand Russell writes the following about Wittgenstein's issue with identity: (page 14)

The definition of identity by means of the identity of indiscernibles is rejected, because the identify of indiscernibles appears to be not a logically necessary principle. According to this principle x is identical with y if every property of x is a property of y, but it would after all be logically possible for two things to have exactly the same properties. If this does not in fact happen that is an accidental characteristic of the world, not a logically necessary characteristic, and accidental characteristics of the world must, of course, not be admitted into the structures of logic.

This view leads Wittgenstein to "banish identity" and consider different letters to mean different things.

Russell however offers a need for identity in logic:

In practice, identity is needed as between a name and a description or between two descriptions.

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Russell, B. Introduction. In Wittgenstein, L. Tractatus Logico-Philosophicus Retrieved from Internet Archive at https://archive.org/details/WittgensteinLudwig.TractatusLogicoPhilosophicus19222019/page/n13