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5
votes
3answers
363 views

How exactly do logic and mathematics interact and what happens when we change the logic?

[Note: this question turned out to be pretty huge, so if you think it would be better to split it up into smaller questions, please comment. The questions here are quite conceptually intertwined and ...
49
votes
6answers
2k views

Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
25
votes
3answers
2k views

Why are smooth manifolds defined to be paracompact?

The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
6
votes
2answers
240 views

How do I investigate the metamathematics of Euclid's proof of infinitude of primes?

Is primeness a predicative property? Earlier this year, I jotted down some thoughts in a paper whether Euclid's proof of infinitude of prime numbers is tautological arguing that prime numbers are ...
3
votes
1answer
289 views

Properties of computable numbers

If we enumerate* all the computable numbers, those for which there exist a turing machine that outputs its digits to arbitrary precision. What is known about the asymptotic density of rationals, ...
25
votes
3answers
914 views

Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
2
votes
1answer
128 views

Bounding projective spaces

For which $n$ does there exist a (topological, smooth, PL, complex) manifold $M^n$ such that $\partial M = \mathbb{R}\mathbb{P}^m$. Obvously, $m = n -1 $ (at least an in the real case). There are a ...
16
votes
4answers
1k views

Advanced algebraic topology topics overview

Recently I became very much intrigued by algebraic topology and am spending quite some time learning it. My reasons are three-fold: it's a beautiful theory; it gives geometric justification to (or ...
5
votes
1answer
403 views

Complex Analysis and Algebra

There are two results in Complex Analysis that have a counterpart in Algebra: -If we consider the ring of holomorphic functions in an open set $\mathcal H(U)$ with the usual sum and product, every ...
11
votes
16answers
582 views

Landmarks of subjects of mathematics

In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it). For example in ...
5
votes
2answers
494 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...
9
votes
2answers
1k views

Categories of mathematics

I am interested in understanding how mathematics is divided into many categories, such as what categories are particular cases of what, what categories do not or have little overlap with what. This is ...
26
votes
5answers
2k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
10
votes
4answers
1k views

What's more general than category theory?

First there was arithmetic with numerical calculations (i.e., one unknown on one side of an equation). Then algebra with manipulations of variables (many unknowns anywhere in an equation). Then ...