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Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.

9 votes

How can a mathematician handle the pressure to discover something new?

When someone discovers "new mathematics", they have in essence discovered how to think precisely about a concept genuinely foreign to the human race. An essential piece of this process is the formal …
14 votes

The relevance of knowing "just facts"

Acquisition of knowledge progresses approximately the same for everyone: Encounter new subject, be completely ignorant. Study new subject, become less ignorant. After many years, become one of the le …
Alec Rhea's user avatar
  • 10.1k
9 votes

Most intricate and most beautiful structures in mathematics

The Surreal numbers as constructed by Conway. They contain a copy of many objects already on the list (N,On) and so surpass them in complexity, and they are recursively defined from the …
1 vote

Why are finiteness conditions important (and how to recognize them)?

Although induction was already mentioned in the comments and Martin's answer, I would like to add a little extra. Induction can certainly be preformed over larger ordinal classes than the natural num …
8 votes

Topology in non-mathematical literature

This one is a bit of a stretch, but I've always associated the Egyptian deity Apep with the long line. A depiction from the tomb of Ramesses I: Keeping in mind that hieroglyphics are a pictorial lang …
7 votes

Research in mathematics without a degree in mathematics

Some concrete advice for anyone attempting to pursue research mathematics without the infrastructure of a PhD: Use math.stackexchange for the wonderful resource it is. I cringe thinking about trying …
3 votes

What is the best way to read advanced textbooks in Pure Mathematics (PhD Level)?

Remembering the ‘core ideas of proofs’ essentially amounts to remembering all the relevant definitions and understanding how they relate to one another; this is essentially the content of any given th …
Alec Rhea's user avatar
  • 10.1k
9 votes

Is pure mathematics useful outside of mathematics itself?

It seems to me that in all honesty, pure mathematics does not immediately benefit the population at large in a direct and obvious way. At best benefits are usually theoretical (e.g., "These methods …
11 votes

Pressure to defend the relevance of one's area of mathematics

Back when I was an undergraduate, I took two ‘special study’ semester classes in surreal analysis that were structured as two-on-one meetings once a week in my (excellent) analysis professors office, …
13 votes
Accepted

How professional mathematicians deal with discouragement?

It helps me to remember that none of this stuff really matters. That might sound discouraging at first; why do something if it doesn't matter? Because nothing really matters, and we have to occupy our …
1 vote

The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?

It is my experience that physics motivates and allows the consideration of objects, with some degree of rigor, whose nature would be very difficult to understand mathematically with no motivation and …
4 votes

Each mathematician has only a few tricks

In all 200+ pages of my category theory notes, there were essentially three tricks I used in proofs: Prove that two arrows are both the arrow induced by a universal property, so they're the same arro …