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50
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31answers
4k views

What are some 'conceptualizations' that work in mathematics but are not strictly true?

I am having an argument with someone who thinks that it's never justified to teach something that is not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is ...
1
vote
0answers
66 views

Why do some universities offer mathematical logic in different departments?

I'm thinking of pursuing mathematical logic after my undergraduate work and I have noticed that some universities offer mathematical logic in their philosophy department while others offer it in their ...
1
vote
1answer
27 views

The subcategory of commutative $R$-algebras is denoted by ???

Let $R$ be a commutative ring (with identity) and let $R\mathbf{Alg}$ denote the category of $R$-algebras. My question: is there a suitable notation for the full subcategory of commutative ...
1
vote
1answer
21 views

Is it possible that a randomized recursion has a nonzero probability of either converging or diverging?

I have very little "hands-on" experience with probability, but here is my context: I was looking at the random Fibonacci sequence: $$f_0=f_1=1, f_n=f_{n-1}+Xf_{n-2}$$ where $X$ is chosen randomly ...
60
votes
31answers
5k views

Theorems with an extraordinary exception or a small number of sporadic exceptions

The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case. Another example is the classification theorem ...
0
votes
0answers
40 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
2
votes
0answers
117 views

Are there any good documentary films about the continuum hypothesis?

Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ...
0
votes
0answers
31 views

Formal Trigonometric Refrence

I'm Using a textbook for mathematic which is produced to learn for normal students. Here I'm giving the link of chapter of trigonometric functions of my textbook : ...
6
votes
1answer
94 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
7
votes
1answer
73 views

Applications of Geometry to Computer Graphics

How is differential geometry (or any type of theoretical math) related to computer graphics and/or computer programming? A friend of a friend of mine has only a bachelors degree in pure math and got ...
10
votes
1answer
108 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
4
votes
2answers
235 views

How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest ...
1
vote
1answer
80 views

Alternative set therories?

Is there a version of set theory that allows the existence of a set that does not admit the empty set as a member? I.e., reject the axiom $A\cup \emptyset = A$
1
vote
1answer
35 views

Software to easily draw 3d plots from functions

my problem is that I need a way to quickly check results of my, that is to say, homework. I think that the best way to do this is to draw a plot of a function to quickly see whether my solution is ...
431
votes
134answers
26k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of Mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
6
votes
1answer
467 views

Is calculus not rigorous?

While studying single and multivariable calculus during my first year some people complained that calculus wasn't rigorous enough, when I asked about this no one seemed to be able to really specify ...
1
vote
0answers
68 views

Why are Lie Groups so “rigid”?

This is probably a naive question, but here goes. To motivate my question, I'll consider a unit circle in $\mathbb C$ or $\mathbb R^2$. This is a compact Lie group equipped with the usual exponential ...
0
votes
2answers
44 views

Which letters to use as index in sequences?

Usually the latin letters $i,j,k,l,m,n$ are used as indexes in sequences or sets with $k$ elements ($A = \{ a_1,...,a_k \} $). But when we already used all these letters is there any recommendation? ...
0
votes
0answers
28 views

Uniform distribution on convex hull

Let $X=\{ x_1,\dots,x_n \} \subset \mathbb{R}^m$. Let $H(X)$ be the convex hull of $X$. Assume that $X$ is a convexly independent set, i.e. none of the $x_i$ are a convex combination of the others. ...
0
votes
0answers
27 views

general local to global principle

Consider the Diophantine equation $f(x)=0$, where x is a vector of integers and $f: \mathbb Z^n \rightarrow \mathbb Z$ is a polynomial function. Is the following statement true? The structure of the ...
0
votes
0answers
29 views

International scholarship for maths [on hold]

Do you mathematicians know if there is a scholarship (or something similar) for maths awarded at international level regardless of the country and/or university from which you come?
18
votes
1answer
1k views

Mathematical Intuition Behind Schizophrenic Numbers?

Schizophrenic numbers (A014824) are numbers whose square roots "look" like rational numbers. They were first discussed in 2004 by Darling in the Universal Book of Mathematics (page 282), and I ...
29
votes
10answers
2k views

Real life usage of Benford's Law

I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great.
2
votes
1answer
70 views

How to see 4th dimension? [on hold]

Anyone has tricks or methods? Great help, thank you. P.S.: Abstractly of course, not actually seeing 4D, which I don't think is possible.
30
votes
5answers
6k views

How hard should a mathematician work?

Of course I would never compare myself to, and don't expect to be, one of the great mathematicians. However, I am curious as to how my work habits, dedication and passion differs from theirs. How ...
6
votes
1answer
111 views

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite ...
0
votes
0answers
46 views

A city wants to encourage downtown

could you please help me with this ( part d ) A city wants to encourage downtown employees to use public transportation. Below is the time in minutes to get to work on one morning according to ...
13
votes
5answers
431 views

Inspiring a new generation

Tomorrow is a very special day for both my students and I: we will be starting a calculus course. I'm looking for some nice quotes to read to them to convey just what a complete game changer calculus ...
9
votes
0answers
133 views

How much math is there?

Among other things I teach high school-level math, and one question that often comes up is: "How long would I have to study math in order to know all of it?" I usually tell them that it's like ...
15
votes
7answers
7k views

Online Math Degree Programs

Are there any real online mathematics (applied math, statistics, ...) degree programs out there? I'm full-time employed, thus not having the flexibility of attending an on campus program. I also ...
5
votes
0answers
44 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 ...
0
votes
0answers
49 views

Is it necessary to know all the details in proofs of theorems you study in PDE's?

I've been studying PDE from the book by L.Evans for some time now. I came across some statements in the proofs which I couldn't justify. But to complete the exercises I didn't need to know all these ...
0
votes
2answers
38 views

Infinitesimal Unit of Measurement

This is just a question that popped into my head which I lack the knowledge to answer (or even to know whether there is an answer, honestly). Does the idea of an infinitesimal unit of measurement even ...
1
vote
1answer
66 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier stuff. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. (see ...
9
votes
4answers
895 views

Why are nets not used more in the teaching of point-set topology?

I just finished working through a proof of Tychonoff's Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components ...
12
votes
5answers
202 views

Scholarly work on the beauty of math

When reading mathematical books written for a general audience, or even searching questions on this site, the adjective beautiful is often used to describe mathematics. My question is whether there ...
14
votes
3answers
329 views

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I ...
0
votes
0answers
24 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
1
vote
0answers
59 views

How long should you study Mathematics each day if you want to get into a graduate school? [on hold]

I'm a university student (Junior now!) and I was wondering how many hours fellow undergraduates and graduate students study a day. I hear posts about how time doesn't really matter and that it's about ...
6
votes
2answers
112 views

The best balance in studying Mathematics?

I'm a student studying Mathematics at a university level. I've completed Single Variable Calculus, Differential Equations, Multivariable Caculus, Real/Complex Analysis, and Linear Algebra and I've ...
0
votes
0answers
29 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
1
vote
0answers
102 views

Missing connection in real analysis?

I have long thought of real analysis as the consequences of mapping numbers to real numbers, thereby creating infinite sequences and series. From there, I can get to fairly elementary applications ...
0
votes
0answers
43 views

Self-Contained Books / Series / Lectures for Comprehensive Introduction to College-Level Math for Someone with VERY Poor Math Foundation?

I've long been interested in various math related subjects (technology, philosophy, sciences, computer science, languages, etc.) without really invested time to actually any learn any of them. I ...
17
votes
3answers
1k views

Career advice in mathematics

First of all, I don't know if this is the right place to ask about careers. If not, please direct me somewhere I can get more help. So, I'm in my first year of university, majoring in applied ...
4
votes
2answers
111 views

Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc

Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc. to represent the real number system, rational number system, natural number system respectively?
1
vote
0answers
32 views

best introductory intuitive books for learning ODE

I want to know best introductory intuitive books for learning ODE (mainly interested in Picard' theorem, Gronowall's inequality and most importantly stability). I started with Philip Hartman. Not ...
4
votes
1answer
99 views

Russian Texts on Geometry

I recently saw a question today pertaining to Russian mathematics and I have a similar question but of a slightly different flavor. I've always heard that the Soviet Union had a history of producing ...
34
votes
16answers
4k views

What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend ...
34
votes
5answers
2k views

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of ...
3
votes
2answers
271 views

Interesting Problems for NonMath Majors

Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer ...