**7**

votes

**5**answers

3k views

### Models of ZFC Set Theory - Getting Started

For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)?
Provided:
I can think of models of any ...

**1**

vote

**4**answers

3k views

### Are all mathematical theorems necessarily true?

Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by ...

**44**

votes

**2**answers

5k views

### Lawvere's “Some thoughts on the future of category theory.”

In Lecture Notes in Mathematics 1488, Lawvere writes the introduction to the Proceedings for a 1990 conference in Como.
In this article, Lawvere, the inventor of Toposes and Algebraic Theories, ...

**5**

votes

**1**answer

915 views

### Co-Objects are better [closed]

This is a rather vague question, but perhaps we can talk about it.
There are two types of mathematical objects (which don't exclude each other):
A) There is a good description of morphisms defined ...

**1**

vote

**2**answers

428 views

### In what sense Fraissean view point shows Model Theory can be done without any formal syntax and deduction rule?

In this post I want to look at an issue I was in doubt when looking at the comment of F. G. Dorais in the post In model theory, does compactness easily imply completeness?
F. G. Dorais remark was:
...

**11**

votes

**2**answers

820 views

### Where are we working when we prove metamathematical theorems?

I am posting my comment from this question as a separate question, as was recommended to me.
(EDIT: I'm sorry if it ended up being too similar a question, I just wanted to phrase it in the ...

**32**

votes

**5**answers

3k views

### Were Bourbaki committed to set-theoretical reductionism?

A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent ...

**14**

votes

**10**answers

1k views

### Can you prove equivalence without being able to calculate it?

In mathematics we often seek to classify objects up to an equivalence relation, where two objects A and B are said to be equivalent if there exists a map $f:A\rightarrow B$ satisfying certain ...

**7**

votes

**3**answers

2k views

### randomness in nature [closed]

What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...

**24**

votes

**3**answers

4k views

### Category of categories as a foundation of mathematics

In
Lawvere, F. W., 1966, “The Category of
Categories as a Foundation for
Mathematics”, Proceedings of the
Conference on Categorical Algebra, La
Jolla, New York: Springer-Verlag,
1–21.
...

**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 ...

**31**

votes

**5**answers

7k 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

481 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.

**14**

votes

**2**answers

901 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

5k 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

480 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 ...

**10**

votes

**3**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

458 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 ...

**60**

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 ...