11

Because he reads a and b occurring in the atomic proposition f(a,b) (e.g. "a is to the left of b") as referring to two different objects. 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 ...


3

It helps to put this in a concrete example. If we have two apples sitting on the table before us, it is self-evidently true that they are not the same apple, so saying apple A equals apple B (a=b) is nonsense. If we have one apple sitting on the table before us, then it would be pointless to say that apple A equals apple A (a=a); that is a best a trivial ...


2

I think the title and body questions are subtly different. Here I'm going to address the title question, which I'll paraphrase for clarity as: What sort of "mathematical truth" can a non-Platonist make sense of? I think this is less strange than it may first appear, since there is an existing parallel: "sharp" vs. "fuzzy" referents in natural language. ...


1

In his book Inconsistent mathematics, Chris Mortensen introduces complement-topoi and closed set sheaves - which I think do what you want. Unfortunately, there's some sloppiness in his presentation. Here's how I would choose to describe the situation: We define "complement-topoi" and "closed set sheaves" and show (i) how complement-topoi can be built from "...


1

I recommend the paper: B. Buldt, The Scope of Godel’s First Incompleteness Theorem, Log. Univers. 8 (2014), 499–552 , especially pages 530 - 531.


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