Articles of soft question

How to conceptualize unintuitive topology?

I found Project Origami: Activities for Exploring Mathematics in my university’s library the other day and quickly FUBAR’d (folded-up beyond all recognition) the couple sheets of paper I had with me at the time. I showed the book to a couple friends who have studied some topology, and we looked at one project with the […]

What does it mean for a theorem to be “almost surely true”, in a probabilistic sense? (Note: Not referring to “the probabilistic method”)

I recently came across this paper where the Goldbach conjecture is explored probabilistically. I have seen this done with other unsolved theorems as well (unfortunately, I cant find a link to them anymore). The author purports to bound the probability of the Goldbach Conjecture being false as $\approx 10^{–150,000,000,000}$ What is mathematics to make of […]

Why Do Structured Sets Often Get Referred to Only by the Set?

Why do structured sets, like (N, +) often get referred to just by their set? Under this way of speaking, where N denotes the natural numbers, + addition, and * multiplication, (N, +, *) and (N, +) both can get referred to as N. But, due to our ancestors we can readily talk about (N, […]

Reference Text that develops Linear Algebra with Knowledge of Abstract Algebra

Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I’ve worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), […]

Is there a 'Mathematics wiki' analogous to 'String theory wiki'?

I came across this site and am wondering if there is a similar page for Mathematics or its sub-areas. Would be very nice if there is one such site which provides ‘canonical’ references for each sub area and preferably is editable like the Wikipedia system so that it reflects entire community’s opinion and not just […]

What is a vector?

What is a vector? As the question says what is a vector and what are its uses or, I mean, when should we use vectors? Is this a branch of geometry or algebra or trigonometry?

Will computers one day start creating and proving math for us?

I suspect in the future we might be able to build computers that research math for us. And I also suspect they will probably be way more efficient at doing research than we are. I do think this question is relevant to this site, even though it doesn’t have a definitive answer (hence the tag: […]

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?

Beautiful proof for $e^{i \pi} = -1$

To celebrate the recent neuroscientific study that shows the beauty of math is in the mind, what is your most beautiful proof that $e^{i \pi} = -1$?

The sections of the projection $\bigsqcup_{i:I} X_i \rightarrow I.$

I just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn’t necessarily surjective, of course, because one or more of the $X_i$ may be empty. Anyway, I noticed that the set $\prod_{i:I} X_i$ can be identified with the set of sections of […]