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Results tagged with mathematical-philosophy
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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.
34
votes
Abstract thought vs calculation
The first proof I ever saw of the orthogonality relations for characters of finite groups was computational: it did a lot of matrix computations and manipulations of sums, which I didn't like at all. …
25
votes
What does T+non-Cons(T) mean?
For mathematicians of an algebraic bent it may be helpful to think of a nonstandard model of, say, Peano arithmetic as a very funny sort of ring (really a semiring), obtained by starting from $\mathbb …
- 118k
19
votes
What's special about the Simplex category?
This is mostly a response to the title question and the third question in the body; I have nothing intelligent to say about finite products. The point of view I want to defend here is the following:
…
- 118k
15
votes
Age of Stochasticity?
Here is a result that gives the flavor of the kind of thing along these lines I hope to see in the future. Recall Tarski's undefinability of truth: under suitable assumptions, a formal system can't be …
- 118k
11
votes
Math History Question about the exponential function
Short answer: Most likely undefined.
Long answer: The "naive" definition of $f(x) = a^x$ where $a, x \in \mathbb{R}$ and $a > 0$ is as follows. You know how to define $f(n)$ where $n$ is an intege …
- 118k
10
votes
Why do Groups and Abelian Groups feel so different?
Here's a totally different suggestion, which is why I'm writing it separately. As has already been remarked, Ab has the really nice property that it's enriched over itself and monoidal closed. We do …
- 118k
4
votes
What are trivial objects, in general?
I think nearly everything I call trivial is the initial or terminal object in some category (but not necessarily a zero object, as many familiar categories don't have zero objects). Let's go through a …
3
votes
Broken Symmetry
I don't agree with your "always." It is certainly possible to write many proofs in linear algebra or involving an equivalence relation that don't require the choice of a basis or a representative of …
- 118k