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23
votes
3answers
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
3answers
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
4answers
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
5answers
6k 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
4answers
478 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
2answers
876 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
4answers
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
2answers
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
8answers
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
2answers
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
3answers
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
2answers
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
2answers
457 views

what is logic without a proof system [closed]

Is there a proof for no proof ?
13
votes
7answers
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
10answers
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 ...