This note sets up a series of related notes pertaining to my explorations in Geometry and by extension, Maths.

The explorations are my work in trying to make sense of the world through math. They are presented here, in part, to motivate me to express my thoughts in a more organized fashion than I might otherwise, and in part to share in the hopes that some small few might benefit or wish to chat about things.

So, enjoy and if you do, feel free to email me about it.

The notes to come are not mathematical treatises, they are mostly me trying to figure stuff out. I expect they will be superseded with more intelligible writings as I learn more, but they are not definitive. They are more like a journal where I lay out my thoughts of the moment.

I have always found math confusing and so much of it lacking comprehensibility or cohesion. When I came across Geometry, as part of my personal remediation plan doing Saxon Math, begun about 7 years ago. I was stunned to find something that actually made sense, was coherent and cohesive. It turns out that Euclid, writer of the Elements of Geometry, established by example, one of the greatest models of thought that the world has ever known. I dug in to his subject, Geometry, and I dug in hard. I worked through propositions referenced by Saxon, in the Elements. I was mesmerized, but not at first. At first, it seemed incredibly difficult to understand, but after many, many hours of work and contemplation, it clicked and I got it… even if I couldn’t apply it in novel circumstances without great difficulty on the drop of a dime. But, I felt like I could see where it was coming from and where it was going. Thank you Euclid and thank you John Saxon for introducing me to the subject in such an accessible way.

Before I dive into the explorations, I feel compelled to defend Euclid, who while appreciated by many, is mocked unfairly at times by folks. Euclid’s postulates are not complete. He made some assumptions that people eventually decided were unwarranted. He didn’t prove every case of every postulate. Other geometries are possible and valid. So what? Euclid wrote a work of mathematics that lives as the finest example of mathematics ever written (to date). He didn’t write a book on how to write axiomatically, or how to write with rigor. If he had, things would have been quite different. Rather, he showed the world, by example, how to write a book with axioms and rigor, that would stand as the eminent example of how to do so, for the better part of two and a half millennia.

I find it amusing and bemusing that folks spend vast amounts of time and energy learning Euclid’s work, only to turn around and smugly criticize its flaws. To my mind they are like children who want to show their independence from their parents - meanwhile, everyone can see by looking that they are their parent’s children. In a similar vein, modernists from David Hilbert to Bertrand Russell, while widely criticizing Euclid, so clearly resemble their forbear, that it beggars belief. Its shocking to me that they wouldn’t simply correct the flaws inherent in his work (as an exemplar of a mindset) and extend it, without bad-mouthing it. These geniuses, and geniuses they so clearly are, have not, with all there labor achieved a work as influential as the work attributed to Euclid.

Anyway, enough ranting, off to explore :).

– will

post added 2023-12-19 12:27:00 -0600