I’m usually quite eager on using technology wherever sensibly applicable, however whenever I make some calculations I still end up using a pen and paper, by now resulting in an entire pile of sheets with 50% crossed out and obviously lacking any searchability. Every now and then I try to re-order this stuff by LaTeX-ing […]
I notice that Russell’s paradox, Burali-Forti’s paradox, and even Cantor’s paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking if it wouldn’t be a good way to stop the paradoxes, to just prohibit sets containing themselves, via a modification in the axiom […]
I read the Mathematical Red Herring principle the other day on SE and wondered what some other good examples of this are? Also anyone know who came up with this term? The mathematical red herring principle is the principle that in mathematics, a “red herring” need not, in general, be either red or a herring. […]
Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a category you could say that for the arrow $f$ in $P\left(a,b\right)$ we have: $$f=g\circ h\Rightarrow g=1\vee h=1$$ It reminds […]
Is there some interconnection between these two topics? A sort of classification of the possibile types of nested radicals and maybe some way (hopefully bijective, in some sense) to pass from a nested radical to a partial fraction and vice versa? I know this is vague, but I didn’t found nothing about it.
From time to time or when reading a paper I hear the term “bootstrap” or “the bootstraping technique” or similar terminology. I cannot find a concise reference or explanation as to what is this method since when I google it I find all kinds of different things that are named as such. Is there a […]
Allegedly, Cauchy mistakingly “proved” that pointwise convergence of continuous functions is continuous. I saw this somewhere in a book, and it is also in wikipedia: Uniform convergence. In his Cours d’analyse of 1821, Cauchy “proved” that if a sum of continuous functions converges pointwise, then its limit is also continuous. However, Abel observed three years […]
On long car journeys with kids we are all familiar with “I spy” or “Twenty questions”. What math related games can one play on a car journey instead that are fun and offer similar variety?
So, I’m wondering why mathematicians use the symbols like $\mathbb R$, $\mathbb Z$, etc… to represent the real and integers number for instance. I thought that’s because these sets are a kind of special ones. The problem is I’ve already seen letters like $\mathbb K$ to represent a field in some books just to say […]
In an intuitive sense, I have never understood why a power series centered on $c$ cannot converge for some interval like $(c-3,c+2]$. Also, I have had a few professors casually mention that a series converges for a disk in the complex plane, centered on $c$ and with the radius of convergence as its radius. Is […]