After nearly three years of not writing in this blog, I was reminded just now of another essay I’d written in college, this one about math history, and was wondering whether to give it the “Are Video Games Ambient?” treatment and post a revision of it here. Well: after reading through “Apollonius and the Pervasiveness of Conics in Classical Greek Geometry” afresh, I think it’s safe to say it’s not going to make it to this blog any time soon. For that matter, at this point in time I found even “Are Video Games Ambient?” itself difficult to understand, the confusion, I believe, due mostly to the original draft of the essay, rather than my revision efforts (my 2020 postscript I found clear enough). But I think some of the ideas in my math history essay are, at least, worth preserving.
The idea that the conics, or conic sections — circle, ellipse, parabola, hyperbola — are “pervasive” in the geometry of the ancient Greeks, and my attempted investigation of why that might be the case, formed the backbone of my essay, and yet, as such, was met with a fair amount of skepticism from my professor. What does it mean, that these shapes are pervasive? Why is that interesting? But: the beliefs I didn’t state outright probably contained a strong clue as to what intrigued me most about this so-called pervasiveness of conic sections.
One of my grand ideas at the time, I believe, was that mathematics could be helped along in its progression, or at least the solution of mathematical problems could be aided, by metamathematics: that is, the mathematics underlying the structure of mathematics itself. In particular, I wanted to look at mathematics through the lens of the complexity of mathematical structures, where complexity was a particular characteristic that could be quantified somehow, perhaps in something like a computational-complexity sense. The idea was that the solution to a math problem should not have to be more complex than the problem itself, that it should flow, in some direct way, from the statement of the problem. The fact that this idea is clearly false (for example, in the case of the four-color map theorem) only made me want to restrict my attention to a small “toy domain” within mathematics, for which my idea would stand a better chance of holding up: as this domain, then, I picked Euclidean geometry.
I was, perhaps, aware that, despite the prevalence of Euclidean geometry in the highest levels of mathematical competitions around the world (an area in which I’d also had more than a passing interest), it’s on pretty much the opposite shore, within the dominion of math, of anything approaching the current or cutting-edge. This style of geometry has been so played out, it’s become a game. But: that was part of why it was so fascinating for me. In a world dominated by circles and straight lines, what were these shapes — ellipses, parabolas and hyperbolas — that kept popping up? They seemed inordinately complex. Well… how complex are they? Could there be some primal definition of conic sections — the simplest possible way of describing them — from which all the manifold ways of looking at these shapes would be natural consequences?
(I was also heavily into the idea that some consequences, some deductions, would be more natural — simpler, in a hopefully technical sense — than others. I related this to the divide made between “analysis”, the process through which a proof is created, and “synthesis”, the most direct way of stating the proof. It seemed to be a big problem that, in modern mathematical presentation as in the expositions of the ancient Greek mathematicians, synthesis would always seem to push analysis by the wayside. Happily, I have since run across some exceptions, such as this blog post from mathematician and Fields Medalist Timothy Gowers about the cubic formula.)
Suffice it to say, I failed to find such a primal definition. The part of my essay that impressed my professor the most was the section (and link to a webpage) about Dandelin spheres, which to me still do seem to inform the most natural argument I know of linking the conception of conics as cross-sections of a cone with their more familiar planar properties (foci, directrices). In the essay, I made the argument that, although the discovery of Dandelin spheres didn’t take place until the nineteenth century, it was not inaccessible to ancient Greek geometrical modes of thinking. Although that, at least, still seems reasonable to me, the fact that conic sections also appear in projective geometry, for example, deserved a better explanation than I gave.
Ironically, it wasn’t until later on that I learned about so-called quadric surfaces, any cross-section of any of which is a conic, which I believe finally can give some substance to my nebulous idea of conic sections’ naturalness and, thus, pervasiveness. I still have a sort of fantasy about teaching a course (now that I am, you know, a math teacher) about conics and the different geometrical contexts that give rise to them, and there are ideas in my essay that I continue to think are worth exploring. It’s going to take a lot of work, however — whether in this blog or outside of it — to unpack them!
Scribblings Towards an Incoherent Philosophy 