Articles of soft question

Generic Points to the Italians

When I first learned algebraic geometry, I naturally wiki-ed the subject and there was a line there that said the old school Italians used the notion “generic points without any precise definition.” Now that I know what a generic point is, I am curious as to how did the Italians think about a generic point […]

Is there anything special about a transforming a random variable according to its density/mass function?

Lets say that $X\sim p$, where $p:x\mapsto p(x)$ is either a pmf or a pdf. Does the following random variable possess any unique properties: $$Y:=p(X)$$ It seems like $E[Y]=\int f^2(x)dx$ is similar to the Entropy of $Y$, which is: $$ H(Y):=E[-\log(Y)]$$ It seems like we can always make this transformation due to the way random […]

Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there’s one thing that I need to clarify and that is the properties of real numbers and their proofs. Closure Laws For all $a,b \in \mathbb{R}$, $a+b$, $a-b$, $ab$, $a/b$ are real numbers. Thus […]

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 […]