The math-philosophy tag has no wiki summary.

**8**

votes

**3**answers

1k views

### categorification of logic

has there be an effort to categorify first order logic? More particularly, structures in the sense of logic.
If so, then every structure of a first order theory is a category. so in particular, the ...

**7**

votes

**4**answers

1k views

### Alternative axiom to induction

Is anyone aware of alternative axioms to induction? To be precise, consider peano axioms without induction PA-. Is there any axiom/axiom schema that is equiconsistent to induction, assuming PA-? If ...

**28**

votes

**5**answers

5k views

### Categorical foundations without set theory

Can there be a foundations of mathematics using only category theory, i.e. no set theory? More precisely, the definition of a category is a class/set of objects and a class/set of arrows, satisfying ...

**1**

vote

**4**answers

474 views

### Does the Golden Ratio Apply to Timing as Well? [closed]

I've seen the golden section applied to art, but does it apply to sound/timing as well? Just curious.

**13**

votes

**2**answers

847 views

### synthetic differential geometry and other alternative theories

There are models of differential geometry in which the intermediate value theorem is not true but every function is smooth. In fact I have a book sitting on my desk called "Models for Smooth ...

**4**

votes

**4**answers

4k views

### Badiou and Mathematics [closed]

Does anyone have an opinion on Alain Badiou's use of set theory? Is there anything interesting mathematically there? Also could anyone shed any light on the comment in the Wikipedia article link text ...

**3**

votes

**2**answers

476 views

### Broken Symmetry

I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I ...

**17**

votes

**8**answers

1k views

### The Importance of ZF

It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...

**8**

votes

**2**answers

2k views

### Math History Question about the exponential function

While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...

**10**

votes

**3**answers

1k views

### Is formal proof (formalized mathematics) interesting to practicing mathematicians? To educators? [closed]

Formalizing mathematical proofs so that they can be checked for correctness and manipulated by computer is a recurrent proposal, most notably stated in the QED manifesto (1994). The December 2008 ...

**6**

votes

**2**answers

2k views

### How platonistic is your attitude towards mathematics? [closed]

A discussion in the n-category cafe
about Manin's 'emotional Platonism' made me wonder how such a perception of mathematics is distributed among mathematicians and how that influences attitudes ...

**-3**

votes

**2**answers

457 views

**13**

votes

**7**answers

2k views

### Is no proof based on “tertium non datur” sufficient any more after Gödel?

There are many proofs based on a "tertium non datur"-approach (e.g. prove that there exist two irrational numbers a and b such that a^b is rational).
But according to Gödel's First Incompleteness ...

**53**

votes

**10**answers

5k views

### Why do Groups and Abelian Groups feel so different?

Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly.
However, this heuristic ...