Articles of soft question

What pattern does set theory study?

If mathematics is the science of patterns, what pattern does set theory study? My thoughts so far: set theory studies the pattern of relationships between members of a collection and between collections. Background: This isn’t a homework question, I’m starting on Introduction to Mathematical Thinking and ensuring I have a good grounding in the fundamentals […]

Idea behind the factorization of the matrix $\operatorname{diag}(a,a^{-1})$ in algebraic K-Theory

If $a \in S$ is some invertible element in a ring $S$, then a computation shows $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} \pmatrix{1 & a \\ 0 & 1} \pmatrix{0 & -1 \\ 1 & 0}.$$ If $R \to […]

Connections between number theory and abstract algebra.

I haven’t taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat’s little theorem, the law of quadratic reciprocity, etc. rely upon techniques found in abstract algebra? Thanks.

Differential Geometry without General Topology

I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: “oh, but what’s the problem ? First learn General Topology, and you’ll understand Differential Geometry even better!” and I agree with that, but my point is: I’m a student of […]

How many different proofs can a theorem have?

I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?

How does one know that a theorem is strong enough to publish?

Question. How does one know that a theorem is strong enough to publish? Basically, I have laid out a framework in which many theorems may be proven. I’m only 18 and therefore lack knowledge of whether this framework and the theorems sprouting from it are trivial along with the theorems. What is a good indicator […]

Soft Question: Weblinks to pages with explanation on quadratics.

I recently placed a question based on quadratics and received a few valuable answers. One of them was a comment in an answer with a link in it which I found useful. But unfortunately the webpage (of which the link was sent) was in Russian (which is totally a foreign language to me) and so […]

What would happen if we created a vector space over an integral domain/ring.

I’ve always been curious about this, why do we use fields as the only algebraic structure to put vector spaces over? It seems a bit arbitrary to me, so I was wondering what would happen it we replaced the requirement with something less structured like a ring (with unity, we don’t want to violate the […]

Book recommendation for Putnam/Olympiads

I have been concentrating on olympiad questions, and PUTNAM exams, Putnam is my main focus. Can you suggest a book from one of these: Problem Solving Strategies By Arthur Engel Putnam and Beyond by Andreescu Titu et. al First, can you tell me which one is better from the above? I only have one choice? […]

Sphere homeomorphic to plane?

I just took a course in general topology about a month back, and I was wondering whether it was possible to explain why the Earth seems flat from our point of view but is in fact a sphere using the concept of a homeomorphism? Is it the fact that the sphere and plane are homeomorphic […]