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3
votes
2answers
473 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 ...
9
votes
3answers
989 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
455 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 ...
52
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 ...