I woke up this morning thinking to finish a geometry essay that has been hanging like a piano above my head this past month or so. I got out my geometry book, set up my computer, curled up with a pen and notebook, and tried writing. The only stuff that would come out were intentions to write about geometry: particularly prop. 35 of Euclid that says that “Parallelograms on the same base and in the same parallels are equal to one another.” It’s a lovely proposition — significant in that it is the first instance wherein Euclid proves equality for non-coincident figures, in that it relies heavily upon the assumption that there is only one possible line through a given point and parallel to a given line, and in that it uses the shared areas outside of the given parallelogramic figures to add and subtract areas within the given parallels. The ability to construct figures within the same parallels creates a known relationship that is not dependent on coincidence, and creates the possibility of measurements of equality or inequality between otherwise quite different figures. So I wanted (and indeed still intend) to write about what exactly is going on there; how the shift is accomplished between coincident equality and area equality; its ramifications throughout the remainder of book 1, etc.
Anyway, that’s what I’ve been trying to write about; or rather avoiding writing about. I could say that I don’t know how to write this kind of essay, that I’ve not thought seriously about math… perhaps ever, or at least not in this way, and that I’ve been putting mounds of attention into other things of more immediate importance to myself and haven’t had any intellectual energy left over to tackle a task with which I have so little familiarity. Well, alright, but I just had a thought that seems true: even if all that were the case, if this had been a literature or a political philosophy essay I still would have written it in a somewhat timely fashion. At the very least I would have had some idea what question I wanted to ask. This one’s different in that it actually hurts to try to write it. What’s with that? The geometry itself doesn’t hurt in that way: Euclid proceeds in a rational and orderly fashion, and excluding reductio proofs, which are convincing, though not enjoyable to follow, at the end E. says QED, and I’m inclined to respond “Ah! Yes! That’s very good; QED indeed!” That was especially my experience in learning prop 47 (on the squares of the sides of a right triangle).
Anyway, back to that thought I had this morning. I’m not sure of its truth, but it makes narrative sense. In my experience I have liked logical analysis, but have no background or education in it; one of my favorite classes as an undergrad. was argument analysis because, as the class name implies, we got to analyze the structure of various verbal arguments, and we could be either quite close to showing the true structure of a given argument, or else quite mistaken. I liked that a lot, and was fairly good at it. There’s something very satisfying about finding the internal order of something (I suppose there’s a similar satisfaction to be found in math, but I never arrived at a solid enough understanding of that structure to appreciate it ). So I would go around trying (though not by choice) to find out the structure of whatever I happened upon. What I mostly happened upon was educational rubbish that no matter how hard I tried I could never make any coherent sense of. That I knew already. What I hadn’t quite noticed, however, and which I now suppose to be the case, is that after spending some four years spent in looking for an internal structure to things that presented me with no such structure, berating myself for the attempt, trying not to look for any structure, not understanding what was being said, and so on, I must have suppressed both ability and desire to analyze anything in an orderly fashion. At least that’s the most reasonable explanation I can come up with.
This is, as it turns out, more than a little inconvenient for trying to write or read about mathematics. Well, OK. but then what? There’s a perfectly attractive series of geometric propositions sitting there waiting to be analyzed in some coherent fashion. I’ve got a way overdue assignment in which I’m obliged to deal with those propositions in some fashion. But what do I do with them?