Identity in Tractatus

In Tractatus, identity of object Wittgenstein expresses by identity of sign, and not by using a sign for identity (5.53). He speaks roughly that (5.5303):

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 at all.

Then he also says at 5.535 that Russell's "Axiom of Infinity", which says that there are infinitely many objects (at 4.1272, Wittgenstein says one even cannot say so), would express itself in language through the existence of infinitely many names with different meanings. In fact, as Russell says in his introduction to Tractatus, the rejection of identity removes one method of speaking of the totality of things.

To Wittgenstein, tautologies also say nothing at all (4.461), although they are not nonsensical (4.46211). Of course, everyone absolutely agrees that tautologies like "if it is raining, then it is raining" really say nothing at all. However, the "problem" of human beings (or of the world) is that the world is so complicated that we cannot determine many tautologies at first sight.

Identity is a relation in Mathematics. There are occasions at which one object may have more than one name / sign - signs are used to refer to the descriptions of some objects, and eventually it is found (proved) that these descriptions have the same reference. In the real world, we might call the murderer of some case of murder X, and eventually we found that he was the man called A: A = X. The introduction of the identity-sign simplifies our deduction.