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65
votes
31answers
7k 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 ...
65
votes
29answers
9k views

What is the single most influential book every mathematician should read?

If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
65
votes
12answers
5k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
64
votes
15answers
5k views

A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of ...
63
votes
22answers
6k views

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...
62
votes
32answers
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 ...
62
votes
10answers
7k views

Complex math problem that is easy to solve

I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math ...
62
votes
10answers
10k views

Results that came out of nowhere.

Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. ...
62
votes
14answers
6k views

Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?

I'm a second year math student. And I've the following problem. When I prepare myself for an exam, I can distinguish two phases. First I'm mainly interested in whatever is necessary to pass the ...
61
votes
23answers
10k views

Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique ...
61
votes
6answers
4k views

Strategies for Effective Self-Study

I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied ...
60
votes
21answers
12k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
60
votes
19answers
12k views
60
votes
8answers
3k views

Why are all the interesting constants so small?

A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is ...
60
votes
5answers
2k views

How are mathematicians taught to write with such an expository style?

I wasn't sure if this question was appropriate for MSE. One of the major complaints we see in industry is a person's ability to communicate which includes writing. We see the same thing on questions ...
58
votes
7answers
5k views

What made you choose your research field?

I was wondering how come modern mathematicians do not seem to discover as many theorems as the older mathematicians seem to have done. Have we reached some kind of saturation limit where all commonly ...
58
votes
8answers
3k views

How to make notes when learning a new topic

At the moment I'm trying to teach myself Riemannian geometry by reading some books. I take notes by hand but I find that I forget it all and worse, either lose the sheets of paper or don't want to ...
58
votes
6answers
3k views

How Do You Actually Do Your Mathematics?

Better yet, what I'm asking is how do you actually write your mathematics? I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up ...
57
votes
7answers
4k views

Why do books titled “Abstract Algebra” mostly deal with groups/rings/fields?

As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem ...
57
votes
4answers
7k views

Books that every student “needs” to go through

I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on ...
55
votes
12answers
8k views

Can the golden ratio accurately be expressed in terms of e and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ...
55
votes
4answers
6k views

How do I convince someone that $1+1=2$ may not necessarily be true?

Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
54
votes
6answers
26k views

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?
54
votes
3answers
5k views

Mathematical research of Pokémon

In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some ...
53
votes
8answers
2k views

Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, and then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, ...
53
votes
3answers
3k views

Getting Students to Not Fear Confusion

I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro ...
52
votes
12answers
4k views

Is it morally right and pedagogically right to google answers to homework? [closed]

This is a soft question that I have been struggling with lately. My professor sets tough questions for homework (around 10 per week). The difficulty is such that if I attempt the questions entirely ...
52
votes
20answers
4k views

Is there a great mathematical example for a 12-year-old?

I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets. Why? Because the maths department at her school is outrageously good, and set ...
52
votes
21answers
3k views

Your favourite maths puzzles

Okay, so this question was bound to come up sooner or later- the hope was to ask it well before someone asked it badly... We all love a good puzzle To a certain extent, any piece of mathematics is a ...
51
votes
18answers
6k views

If there are obvious things, why should we prove them?

Obviously, there are obvious things in mathematics. Why we should prove them? Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$? Prove that $f(x)=x$ is continuous on $\mathbb{R}$? $\dotsc$ Just ...
51
votes
12answers
2k views

I lost my love of math; I'm getting it back. How can I determine if math is actually right for me? [closed]

This question has been on my mind for a very long time, and I thought I'd finally ask it here. When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. ...
51
votes
9answers
3k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
50
votes
28answers
6k views

“Simple” beautiful math proof

Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
50
votes
19answers
18k views

How do I convince my students that the choice of variable of integration is irrelevant?

I will be TA this semester for the second course on Calculus, which contains the definite integral. I have thought this since the time I took this course, so how do I convince my students that for a ...
50
votes
12answers
19k views

Why study linear algebra?

Simply as the title says. I've done some research, but still haven't arrived at an answer I am satisfied with. I know the answer varies in different fields, but in general, why would someone study ...
50
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 ...
49
votes
16answers
6k views

Interesting “real life” applications of serious theorems

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ...
49
votes
16answers
3k views

Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious. Can you give me examples of conjectures/theories that seemed true but through rigorous ...
49
votes
10answers
5k views

Is learning (theoretical) physics useful/important for a mathematician?

I'm starting to read The Princeton Companion to Mathematics, at the beginning it says: A proper appreciation of pure mathematics requires some knowledge of applied mathematics and theoretical ...
49
votes
5answers
5k views

Getting better at proofs

So, I don't like proofs. To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert. Oftentimes the proof in the book is something that I ...
49
votes
5answers
3k views

Is there any branch of Mathematics which has no applications in any other field or in real world?

Is there any branch of Mathematics which has no applications in any other field or in real world ? for instance , maybe : number theory ? mathematical logic ? is there something like this ?
48
votes
9answers
8k views

Advantages of IMO students in Mathematical Research

Everyone in this community i think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries ...
48
votes
12answers
3k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
48
votes
7answers
3k views

Intuition in algebra?

My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) ...
47
votes
9answers
6k views

Does it ever make sense NOT to go to the most prestigious graduate school you can get into?

I'm a senior undergrad at a top-ish(say, top 15) math school. I'm a solid, not stellar, student. This year I'm taking the qualifying exam grad courses in algebra and analysis and have been taken aback ...
47
votes
10answers
12k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
47
votes
4answers
4k views

Mathematicians don't quit, they fade away

Edit: This question is now closed for being not related to math, but many people pointed out that becoming an actuary is one of the most viable career path for someone with skills in pure math. Noone ...
47
votes
7answers
2k views

How to explain to the layperson what mathematics is, why it's important, and why it's interesting [closed]

A mathematician walks into a party. No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. ...
47
votes
2answers
3k views

Sheaf cohomology: what is it and where can I learn it?

As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology. I have two questions about the matter. ...
46
votes
49answers
4k views

What was the book that opened your mind to the beauty of mathematics?

Of course, I am generalising here. It may have been a teacher, a theorem, self pursuit, discussions with family / friends / colleagues, etc. that opened your mind to the beauty of mathematics. But ...