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18
votes
0answers
500 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
17
votes
0answers
260 views

To what extent were mathematicians in previous centuries aware of the lack of rigour in their methods?

By modern standards, much of pre-modern mathematics isn't rigorous. Famous examples include Euler's solution to the Basel problem or literally anything involving sets before Cantor and Russel came ...
16
votes
0answers
733 views

Is this $really$ a categorical approach to $integration$?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
16
votes
0answers
504 views

Irrationality of e

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
15
votes
0answers
1k views

What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
14
votes
0answers
441 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
10
votes
0answers
113 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
9
votes
0answers
124 views

Is Category Theory geometric?

In "From a Geometrical Point of View" (http://www.amazon.com/gp/aw/d/1402093837?pc_redir=1407132421&robot_redir=1) Marquis states that category theory is thoroughly geometric. Could someone ...
9
votes
0answers
123 views

Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the ...
9
votes
0answers
2k views

Using distance formula to find slope, any reason to use the concluding equation?

So, today I was observing a class that I will be a TA for this semester and the professor started to talk about the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Well, my mind wandered a little ...
8
votes
0answers
145 views

A question on popularization of math: inspiring the beauty of mathematics while making New Year's wishes

Many fellow students of mine today shared by various means the following picture: . I was told that this picture is supposed to communicate the beauty of math in a funny way, but I really can't ...
8
votes
0answers
157 views

Intuition behind Erdős proof of the infinitude of prime numbers

Suppose by contradiction that there are finitely many primes, namely $p_1, p_2,...,p_k$, where $k$ is a natural number. Now consider another natural number $n$, and all natural numbers $m \leq n$. ...
8
votes
0answers
124 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ...
8
votes
0answers
285 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
7
votes
0answers
109 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
7
votes
0answers
193 views

How do mathematicians know what is known?

How do mathematicians know that what they are researching has not been already know for 200 years? Obviously if they are researching something that is cutting edge it is not a problem, but if one is ...
7
votes
0answers
97 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
7
votes
0answers
96 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
7
votes
0answers
137 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
7
votes
0answers
178 views

Reference suggestion: eigenvalues of tridiagonal matrices

I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals). I have seen authors use continued ...
7
votes
0answers
280 views

(Mathematical) Applications of Quantum Group

What are some mathematical applications of Quantum Groups? I have tried researching and found that it is used to find solutions to Yang-Baxter equation. Any other applications? Thanks a lot.
7
votes
0answers
684 views

How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?

I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
6
votes
0answers
66 views

Can you build metric space theory without the real numbers?

Every metric space $M$ has a completion $\widehat M$, that is, it can be embedded as a dense subset in a complete metric space. When I first came across this theorem, I thought "well, that's amazing, ...
6
votes
0answers
166 views

Most efficient way to learn mathematics

So there's a lot of advice on how to learn mathematics most efficiently, and it mostly revolves around doing problems, asking questions, and considering all possible generalizations. However, I was ...
6
votes
0answers
45 views

Exponential fields as structures with three binary operations.

The exponential rings and fields are usually studied as structures with two binary operations $(+,\cdot)$ and one unary operation $\exp(x)$ defined on a set $K$. Why not consider the exponential as a ...
6
votes
0answers
73 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
6
votes
0answers
79 views

Why is topological group not a popular topic?

In Japan, there are many universities with a formal course about topological group using the classic by Pontryagin. Yet topological group is not studied in a formal course in many other countries, ...
6
votes
0answers
44 views

Discrete analogue of Green's theorem

Following formula concerning finite differences is in a way a discrete analogue of the fundamental theorem of calculus: $$\sum_{n=a}^b \Delta f(n) = f(b+1) - f(a) $$ We can think about the Green's ...
6
votes
0answers
137 views

Is there really anything wrong with Bourbaki's Set Theory?

Recently I have started reading Bourbaki's Theory of Sets on my own. Regarding one of the explanations of a concept when I went to a Professor of our college, he asked me why I was wasting my time ...
6
votes
0answers
103 views

Legitimate papers refuting the significance of the golden ratio in art?

I'm not sure this is the right place to ask about this, but is there any legitimate peer-reviewed paper refuting the significance of the golden ratio in art? I can find numerous websites and blogs ...
6
votes
0answers
134 views

How to take limit *along a path*

So in multivariable calculus for a limit of a function to exist, the limits of the function along all possible paths must exist and equal the same value. But how does one calculate the limit along a ...
6
votes
0answers
86 views

Strategy for self-studying after M.S.

For someone who has finished their M.S. degree in pure mathematics, what is a good way to keep learning mathematics within your specialization? Would you suggest reading research articles from ...
6
votes
0answers
48 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
6
votes
0answers
147 views

Help needed with Masters' Thesis

My brother is at the very end of his Masters Program at a well-known University (Math, of course, hence my inclusion of this question on this site) and he is totally done with course work, but is ...
6
votes
0answers
172 views

Math Behind the Dragon Illusion!

Dragon illusion has been one of the items presented in the 3rd "Gathering for Gardner". This video shows the illusion. What does it have to do with mathematics?
6
votes
0answers
120 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
6
votes
0answers
181 views

Comparing the U.S. undergraduate math education to the French “classes préparatoires”

Could anyone comment on how the math track of the "classes préparatoires" compares to the U.S. undergraduate major? I took a look at some of the French entrance exams and was rather intimidated. How ...
6
votes
0answers
189 views

The high road to learn algebraic geometry

Suppose that a student has a basic knowledge in commmutative algebra at the level of the Atiyah-MacDonald. What do you think about the following steps to learn algebraic geometry? step 1: read ...
6
votes
0answers
315 views

formulating theories in math

One of my profs mentioned that sometimes people formulate theories about some type of object, but then later realize that those objects do not exist. Can someone given me an example of such a theory? ...
6
votes
0answers
430 views

Why didn't Cartan-Eilenberg develop homological algebra on sheaf theory?

Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at ...
6
votes
0answers
152 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
6
votes
0answers
301 views

homology groups, physical interpretation

One can roughly interpret Betti numbers as the number of n-dimensional holes of our space. Is there a reasonable interpretation for torsion coefficients? Moreover, for 'physical' applications, are ...
5
votes
0answers
65 views

Proofs shorter than the statement of the theorem

In Postnikov's first book in his Lectures in Geometry Series, Analytic Geometry, he states and then proves the Desargues theorem. Then he writes (in my English translated copy) "The proof has turned ...
5
votes
0answers
56 views

Why should the open mapping theorem be expected?

Soft question alert. I want to know why to expect the open mapping theorem to be true. My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of ...
5
votes
0answers
83 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
5
votes
0answers
87 views

Apparent Arbitrariness in Mathematics

Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain: 1st - ...
5
votes
0answers
66 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} ...
5
votes
0answers
71 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
5
votes
0answers
72 views

Math opportunities for a graduate

Are there any opportunities for young mathematicians who have finished their undergraduate education and are taking a year before going to graduate school, but want to do something math related? To ...
5
votes
0answers
135 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...