I'm reading Chihara's: Constructibility and Mathematical Existence. It says:

An even more radical view rejects the assumption that mathematics is true—at least in the straightforward way that mathematics is believed to be true by the Literalist philosophers. At one point, Hilary Putnam espoused such a view of mathematics. Making use of some ideas of the early Bertrand Russell, Putnam argued that “pure mathematics consists of assertions to the effect that if anything is a model for a certain system of axioms, then it has certain properties” (Thesis, p. 294).

According to this, rejecting that mathematics is true has something to do with using if-then. But at least intuitively, I don't see the use of if-then as something not being true, but as something that could be true or something that could be false. So what would be truth? I guess that the idea of truth would have something to do with tautologies (in which it could not be false).

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    $\begingroup$ What does it mean that mathematics is true? It doesn't make sense to me. $\endgroup$ – user2345215 Jan 11 '15 at 0:03
  • $\begingroup$ And ....when we assert that if anything is a model for a certain set of axioms, then it has certain properties ... are we allowed to care whether that assertion is true? $\endgroup$ – WillO Jan 11 '15 at 0:10
  • $\begingroup$ @user2345215 I suspect that it means roughly that when you put two and then three apples in an ordinary bag, then the bag will contain five apples. Contrast it with the same statement prefixed with "If natural numbers are an accurate model of apples in a bag, then ..." $\endgroup$ – dtldarek Jan 11 '15 at 0:12
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    $\begingroup$ For Putnam's thesis, see his Mathematics, Matter, and Method (1975), page 12-on. $\endgroup$ – Mauro ALLEGRANZA Jan 13 '15 at 15:06

In the sentence "if anything is a model for a certain system of axioms, then it has certain properties" you are looking at the if-then implication, but you should be looking at the rest of the statement. Without getting into the endless philosophical discussion of what is mathematics and what is mathematical truth in any detail, let us consider two approaches. One is that mathematics is the study of mathematical objects and that it is the quest to discover true properties about these objects. So, we may imagine that we are studying perfect circles, triangles, various other geometric objects, more abstract things, or whatever, and that we are finding out more and more true things about them.

The other point of view (expressed in the quoted sentence) is that there aren't any mathematical objects at all. All we do is say: here is a model of a certain system of axioms, and now just because of that assertion that object possesses certain properties (ones that are shared by all models of that same system of axioms). So, we may imagine that we are studying perfect circles, but we are in effect only studying models of the axioms of perfect circles, and we imagine that they are indeed perfect circles. Same goes to any other study of anything. So we are not studying objects, we are only studying logical consequences of axioms.

I hope this clarifies things.


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