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Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.

63 votes
Accepted

Why do categorical foundationalists want to escape set theory?

I don't agree that this is what (most) categorists who are interested in foundations are doing. It is true that Lawvere in the mid-60's (and perhaps to this day) wanted to develop a theory of catego …
Todd Trimble's user avatar
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29 votes

An example of a proof that is explanatory but not beautiful? (or vice versa)

A proof that many people say they find beautiful, but in my view is not at all explanatory, is Zagier's one-sentence proof of the sum of two squares theorem.
25 votes

When have we lost a body of mathematics because errors were found?

Volume II of Frege's Grundgesetze der Arithmetik (Basic Laws of Arithmetic) had already been sent to the press when Bertrand Russell informed him that what we now call "Russell's paradox" could be der …
21 votes
Accepted

What's special about the Simplex category?

Intuitively, I see the product-preservation or indeed finite limit preservation of geometric realization $\hat{R}: [\Delta^{op}, \mathbf{Set}] \to \mathbf{kSpace}$ as lifting (through the forgetful fu …
Todd Trimble's user avatar
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12 votes

What if Current Foundations of Mathematics are Inconsistent?

Voevodsky is not the only one who hopes for a proof of inconsistency (as mentioned in Dick Palais's answer): see Conway and Doyle's Division by Three, bottom of page 34, where they express the same ki …
12 votes

Logic in mathematics and philosophy

A more recently developed candidate might be Linear Logic, which is a successful formalization of modes of reasoning of considerable philosophic interest. I highly recommend Jean-Yves Girard's inimita …
Todd Trimble's user avatar
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11 votes

Why hasn't mereology succeeded as an alternative to set theory?

I think one could argue that just as there are categorical versions of set theory, for example Lawvere's Elementary Theory of the Category of Sets, there are analogous categorical versions of mereolog …
Todd Trimble's user avatar
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11 votes

Is PA consistent? do we know it?

I am a little baffled by some of this discussion. It seems everyone agrees that consistency of PA is a theorem, if you accept some stronger system, such as ZFC. So, PA is consistent relative to ZFC. J …
8 votes

Variable-centric logical foundation of calculus

In one of my comments over at the other thread (this other thread), I had mentioned some discussion at the $n$-Category Café about a differential $\lambda$-calculus (from the looks of it, different to …
Todd Trimble's user avatar
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7 votes

Supervenience in mathematics

When I read To me it seems unconceivable that a mathematical object can have any extra properties, let alone in a supervenient manner, that is, two isomorphic objects would have to share them. …
Todd Trimble's user avatar
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5 votes

Examples of abstractions that did *not* turn out to be useful

I don't have an opinion of my own about Fuzzy Set Theory, but someone whom I respect, Saunders Mac Lane, seemed to think it wasn't a very fruitful development. (I freely acknowledge he could be opinio …