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I'm trying to understand Frege's argument for the existence of mathematical objects. Specifically, I'm trying to understand the premises of classical semantics and truth.

Classical Semantics. The singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects.

[...]

Truth. Most sentences accepted as mathematical theorems are true (regardless of their syntactic and semantic structure).

(From the Stanford Encyclopedia of Philosophy.)

First, doesn't classical semantics presuppose the existence of mathematical objects? Frege's argument ends at the conclusion that there exist mathematical objects, but this seems to be circular reasoning since classical semantics states that singular terms of mathematics refer to mathematical objects. (Perhaps I should be thinking of it less like an existential and more like a hypothetical. Does classical semantics merely state that mathematical statements purport to refer to mathematical objects without asserting the existence of mathematical objects?)

Now, for truth: why does it only assert that most theorems are true? In order for a sentence to be accepted, wouldn't a proof have to exist? Does Frege just mean that there are some theorems that we believe to be true but aren't true?

Thanks.

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    Please, give a link to these Stanford articles.
    – Mr. White
    May 26 '20 at 1:59
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    I've added a link and tweaked the citation a bit. May 26 '20 at 3:52
  • It is similar to the argument from perception: the object I see in front of me appears to be an apple, I have no reason not to take what I see at face value, therefore the object I see is an apple. Mathematics taken at face value (classical semantics) appears to involve mathematical objects, therefore they exist. One can argue that we do have reasons not to take mathematics at face value, this is what fictionalists argue, and Frege's argument is often supplemented by Quine-Putnam's indispensability argument to address that. But even supplemented is it not as persuasive as it once seemed.
    – Conifold
    May 26 '20 at 5:19
  • This is not Frege's argument: Frege tried to prove the existence of mathematical objects: numbers. May 26 '20 at 6:15
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    What Frege stated was one of the cornerstone of semantics of modern mathematical logic: in a (formalized) language, names must refer to objects. A name without reference is not allowed in a formalized language. The idea of a "logical perfect language" was developed by Russell and Wittgenstein (in the Tractatus). May 26 '20 at 6:17

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