Articles of soft question

Elementary geometry from a higher perspective

I’m searching for some references that deal with topics from “elementary geometry” analysing them from a “higher” perspective (for example, abstract algebra, linear algebra, and so on).

What does “working mathematician” mean?

What does “working mathematician” mean? Is this term derogatory? What properties of a “working mathematician” are considered undesirable, and what attitude contrasts them?

Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) \rightarrow (\exists f : X \rightarrow Y)(\forall x:X)P(x,f(x))$$ Note the that converse is a theorem of ZF, modulo certain notational issues. Anyway, […]

making mathematical conjectures

If a non-mathematician wanted to conjecture something and had strong numerical evidence to support the conjecture, how would he/she go about doing so? Would the mathematical community (a) take it seriously? (b) even look at it?

Whatever Happened to Nearness Spaces?

I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange concerning Nearness Spaces. Has it hit a dead-end, been replaced by a better approach, simply forgotten, or called by a different […]

Why are they called “Isothermal” Coordinates?

If I understand correctly, Gauss proved that given any oriented Riemannian surface, one can find a complex structure on the surface so that the metric on the charts is just $f|dz|$, where $f>0$. I’ve heard these coordinates referred as “conformal coordinates,” which makes sense, but I’ve also heard of them referred to as “isothermal coordinates.” […]

Calculating sin and cos based on combination of exponentiation and power series?

Taylor series for $\sin(x)$ and $\cos(x)$ around $0$ are well known and usually exercised in beginner calculus courses. However as we know Taylor series are very localized, ultimately fitting only one point and a series of limits of that one point (differentials of integer orders): $$\begin{align}\sin(x) &= \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{(2k+1)!} \\\cos(x) &= \sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(2k)!}\end{align}$$ We also […]

Do expressions like $(-1)^{2/3}$ show up naturally in pure or applied math?

Background. Let $x$ denote an arbitrary real number. Then $x^n$ can be defined for each $n \in \mathbb{N}$ as follows: $$x^n = \underbrace{x \times \cdots \times x}_n$$ If $x$ is furthermore non-zero, then we can extend the above definition by declaring that $x^k$ makes sense for each $k \in \mathbb{Z},$ by defining: $$x^{-n} = \underbrace{(1/x) […]

What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been trying to build a quaternion extension of $\mathbb{Q}$. I have been pursuing an overly optimistic method, which has not quite been […]

What does “rigorous proof” mean?

I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don’t know what does it really mean and what differences makes it to be more ‘rigorous’. My understanding of a proof is that a proof is some explanation to convincing others that a statement is […]