Articles of soft question

Is the exclusion of uncountable additivity a drawback of Lebesgue measure?

A friend and I were having a discussion about Lebesgue measure. I attempted to be profound by making the following points: Analytic geometry has been a fantastic tool, but the concept of representing a continuous “object” as a collection of points is inherently contrived (with a negative connotation). Immediately we run into the paradox that […]

Why is full- & faithful- functor defined in terms of Set properties?

Wikipedia entry or Roman’s “Lattices and Ordered Sets” p.286, or Bergman’s General Algebra and Universal Constructions, p.177 and in fact every definition of full and/or faithful functor is defined in terms of the Set-theoretical properties: surjective and injective on (compatible) arrows. Why aren’t full-/faithful- defined in terms of epic and monic, in other words, in […]

What are some good examples for suggestive notation?

Motivation: Today I first wondered about and later remembered why the set of all functions from a set $X$ to $Y$ is denoted $Y^X$. They wikipedia page gives the explaination “The latter notation is motivated by the fact that, when $X$ and $Y$ are finite and of size $|X|$ and $|Y|$, then the number of […]

Size Issues in Category Theory

Barr and Wells state in their text Toposes, Triples and Theories (pdf link) It seems that no book on category theory is considered complete without some remarks on its set-theoretic foundations. The well-known set theorist Andreas Blass gave a talk (published in Gray [1984]) on the interaction between category theory and set theory in which, […]

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta “function” is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to0^+.$$ Obviously, this sequence of functions converges to $0$ when $x\neq0$ and diverges otherwise. As far as I know, what is literally meant is that $$\int_{\mathbb{R}}f(x)\delta(x)dx\text{ is defined as }\lim_{a\to0^+}\int_{\mathbb{R}}f(x)\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}dx\text{, when $f$ is well behaved.}$$ I’ve read the wikipedia page […]

Algorithm for multiplying numbers

Background Today I had to explain to some kid how to multiply numbers with multiple digits in them. Then I recalled, that some other day I answered this question describing one of the numerous so-called vedic math methods. Essentially, the method sets up a scheme for multiplying digits by drawing crossing lines so that for […]

Mere coincidence? (prime factors)

Whether some things in mathematics are mere coincidences might keep philosophers busy for 100,000 aeons, but maybe when such a coincidence gets exploited then it’s not a “mere” coincidence any more. So time for a somewhat imprecise question: a list of prime factorizations shows us this: $$ 1445=5\cdot17\cdot17 $$ and if you were doing factorizations […]

Why heat equation is not time-reversible? (Time arrow in mathematics)

Inspired by a question I asked here, I am rethinking about a question: Why heat equation is not time-reversible? I don’t know too much about PDE and physics but I guess there should be some “time arrow” in mathematics. Consider the following initial value problem: $$ \begin{cases} \Omega: (x,t) \in \mathbb{R} \times (0,+\infty) \\ u(x,0) […]

Better Notation for Partial Derivatives

I’m constantly seeing questions here where people are confused about the notation $\frac {\partial f}{\partial x}$ or $\frac {\partial f}{\partial x} (x,y)$ or $\frac {\partial f(x,y)}{\partial x}$. Is there some better notation which exists for partial derivatives? If not, can anyone suggest one? Problems with this notation: Is the derivative evaluated at the point $(x,y)$ […]

What happens after the cardinality $\mathfrak{c}$?

While having measure theory this year the following came in my mind: When we go from finite objects to infinite we “lose” a lot of properties. For example the summation isn’t well defined anymore, when the sum doesn’t converge, and operations which are abelian in the finite case are not abelian in the infinite case […]