In the very near future I’m going to be writing a paper on reciprocity theorems in the history of algebra. I really ought to describe a bit of that history here now, because it’s a really great story, but I’m more interested in trying to figure out how the modern approach to the subject looks. [...]

I just found this linked from Wikipedia, and wanted to share/tag it for myself.
This describes Moebius transformations, which are maps of the complex plane which have the form

for fixed . This video really emphasizes how anything having to do with the complex numbers is really about rotation.

I don’t think we have a plug-in for [...]

In this post I gave a brief sketch of what I claim is a better approach to modelling the continuum. In this post I am going to develop the ideas a little bit more thoroughly and also explain why I this is a better approach than the usual set-theoretic definitions of the real numbers.
Here are [...]

This post, and a few to follow, aren’t going to be about mathematics at all, but rather another subject close to my heart: education. I am going to share some thoughts on education in general and, in particular, the sorry state of public education (in this country and elsewhere, and from kindergarten through college). In [...]

I’ve recently finished a draft of the first chapter for my masters thesis, on the topic of mathematical structuralism. I argue that the philosophical discussion of mathematical structuralism has, by and large, misrepresented mathematical attitudes on the subject. This dialogue, beginning with Benacerraf’s classic What Numbers Could Not Be, has focused on the ontology of [...]

For my first few blog posts here at unwanted capture, I’m going to talk about a subject that I will return to often: rethinking our mathematical foundations aesthetically. This time, I want to look at our definitions for the continuum; I think that a non-standard approach can give us a cleaner presentation which is closer [...]