Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [soft-question]

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

For questions whose answers cannot be objectively evaluated as correct or incorrect, but which are still relevant to mathematics.se.

12079 questions
359
votes
23 answers

Why don't we define "imaginary" numbers for every "impossibility"?

Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such…
lily
  • 3,727
265
votes
10 answers

How to read a book in mathematics?

How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs? I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around…
Pax
  • 5,762
153
votes
8 answers

Do most mathematicians know most topics in mathematics?

How many topics outside of his or her specialization is an average mathematician familiar with? For example, does an average group theorist know enough of partial differential equations to pass a test in a graduate-level PDE course? Also, what are…
Sid Caroline
  • 3,729
132
votes
5 answers

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 get if I study, but hard to come up with on my own.…
bobobobo
  • 9,502
128
votes
13 answers

Advantages of Mathematics competition/olympiad 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 participating from around the world. What's…
user9413
121
votes
2 answers

What makes a theorem "fundamental"?

I've studied three so-called "fundamental" theorems so far (FT of Algebra, Arithmetic and Calculus) and I'm still unsure about what precisely makes them fundamental (or moreso than other theorems). Wikipedia claims: The fundamental theorem of a…
beep-boop
  • 11,595
117
votes
13 answers

What are some interpretations of Von Neumann's quote?

John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. Did he mean that with experience and practice,…
NebulousReveal
  • 13,935
116
votes
14 answers

Can you give an example of a 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 problems (proofs, probably) that on the surface…
Judy
  • 1,271
107
votes
6 answers

How to learn from proofs?

Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or give you a better understanding of concepts and…
Roun
  • 3,017
92
votes
8 answers

Will it become impossible to learn math?

I was thinking about this today and it seems like a good question. Assuming mathematics will keep on expanding, do you think it will ever become impossible for a beginner to learn all the known material on a subject (such as mechanics), simply…
Ninja Boy
  • 3,133
80
votes
13 answers

Why there is no sign of logic symbols in mathematical texts?

Either in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, etc. Instead, I just see their "read as" or…
MKR
  • 1,088
75
votes
3 answers

how to read a mathematical paper?

I hope that this question is on-topic, though it is not quite technical. I am curious to hear from people how they approach reading a mathematical paper. I am not asking specific questions on purpose, though at first I had a few. But I want to keep…
normvector
  • 713
69
votes
12 answers

Is it morally right and pedagogically right to google answers to homework?

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 on my own, I usually get stuck for over 2 hours per…
yoyostein
  • 19,608
65
votes
10 answers

Is it not effective to learn math top-down?

By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving basic results. Eventually you'll get there but is…
Daniel Donnelly
  • 21,342
65
votes
6 answers

What does strength refer to in mathematics?

My professors are always saying, "This theorem is strong" or "There is a way to make a much stronger version of this result" or things like that. In my mind, a strong theorem is able to tell you a lot of important information about something, but…
Zachary F
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