Because little emerges from a vacuum
WARNING : This post won’t make much sense unless you’ve read at least my first Topology As Touching post. Product Topologies and the Mystery of the Finiteness Constraint Okay, it looks like I’m back to posting about topology. As per usual, I have been slow-burn revisiting topological notions through the lens of touching , a different axiomatization of topology that seems to give me a different…
I’m probably the wrong person to listen to when it comes to anything about that extension of abstract algebra we call category theory . I’ve been trying to learn about it for literally decades , and much of that time has involved an uncomfortable feeling that something fundamental about the material is just bouncing off of me. Contrast this with many of my colleagues who find great comfort in…
WARNING: the following contains a whole lot of pedantry about proving theorems at a level of detail such that you could likely convince a computer proof assistant of their correctness. I teach a course where students learn to write such proofs without computer assistance, because doing so can profoundly affect how one perceives the possibilities for formal definition and proof. 1 Sometimes the…
\(\newcommand{\thetitle}{Topology as a theory of Touching} \newcommand{\touches}{\mathrel{\delta}} \newcommand{\ntouches}{\mathrel{\not\delta}} \newcommand{\setdiff}{\mathop{\backslash}} \newcommand{\U}{{\mathcal{U}}} \newcommand{\touchesy}{\mathrel{\delta_Y}}\) Topological spaces are monoliths, but as with many mathematical phenomena, we can talk about fragments of them. In this case the…
\( \newcommand{\touches}{\mathrel{\delta}} \newcommand{\ntouches}{\mathrel{\not\delta}} \newcommand{\setdiff}{\mathop{\backslash}} \newcommand{\U}{\mathcal{U}} \newcommand{\inv}[1]{\U \setdiff #1} \newcommand{\R}{\mathbb{R}} \) In our last episode, I introduced the idea of topological connectedness and related it to the mean value theorem from real analysis. It's kind of cool that that theorem…
\( \newcommand{\touches}{\mathrel{\delta}} \newcommand{\ntouches}{\mathrel{\not\delta}} \newcommand{\setdiff}{\mathop{\backslash}} \newcommand{\U}{\mathcal{U}} \newcommand{\inv}[1]{\U \setdiff #1} \newcommand{\R}{\mathbb{R}} \) It's been years since my last (a.k.a. first) post, where I described a presentation of some basic concepts in point set topology in terms of a concept I called "touching",…
\( \newcommand{\touches}{\mathrel{\delta}} \newcommand{\ntouches}{\mathrel{\not\delta}} \newcommand{\setdiff}{\mathop{\backslash}} \newcommand{\U}{{\mathcal{U}}} \) On the advice of a friend in grad school, I took a topology course, and to my surprise I have found many of the ideas there relevant to programming languages research. For instance, Dana Scott's theory of domains was heavily rooted in…