In theory, mathematics is supposed to be "pure," "rational," unsullied by the fact that it is human minds that invent it, and maybe belong to a
Pure Land Plane of Pure Platonic Objects. It can be beautiful.
On the other hand, many definitions seem to have been arrived at only through long struggle—trial and error. Also, even within the nice final definitions we include lots of horrible objects and it seems there is some amount of struggle between wanting a clean definition or a clean theorem and wanting to be able to apply theory in general.
The case for mathematics being a pure land is just standard math-is-beautiful stuff. All that is great! I have no problem with it. Math can be beautiful! On the other hand, math can also be disgusting and horrible, not to mention contingent. That seems important, too.
It could be interesting to explore this properly, but I don't care enough right now, so instead I'm going to list a bunch of stuff that evokes for me the charnel ground nature of math:
- Imre Lakatos' Proofs and Refutations and "monster-barring"; the geometry examples from that book centering on Euler's theorem for polyhedra, if I recall correctly.
- Euclid's fifth postulate and spherical and hyperbolic geometries.
- Real analysis; continuous but not differentiable functions; Thomae's function; Cantor's set; Cantor's function; lots more if I reviewed e.g. Understanding Analysis for just the "special topics" chapters and the "ugly examples" bits and exercises.
- Probably could find more topology examples—spaces that are not Hausdorff, or that break various other conditions. (If I recall correctly, T1-separability is weaker than Hausdorff.)
- The finding that any system equivalent in power to arithmetic cannot prove its own consistency, which Ted Chiang's "Division By Zero" plays with. (Did I write that correctly?) Gödel's incompleteness theorems.
- Mochizuki's claimed proof of the abc conjecture and the resulting controversy/discussion among mathematicians.
- The story about how Bill Thurston killed a whole field (foliation theory) by writing up a bunch of stuff and not explaining it properly, from "On Proof and Progress in Mathematics." Suggests impermanence of math.
- I tend to flinch away from "ugly" math. For me that includes differential equations. I find my sense of "ugly" frequently but not necessarily includes applied math. The "ugliness" has a vibe of dealing with details using specific, tailor-made approaches—it is not elegant or even remotely one-size-fits-all. There are flowcharts for dealing with different objects or different kinds of problem. There is a different tactic or approach for every (currently solvable) kind, which a flowchart can systematically guide someone to use. It's full of special cases. You cannot just memorize one simple formula and call it a day; you have to remember fifteen different formulas and how and when to use them, each of which may come with its own tricks, tweaks, and extra steps. (This to me feels like the most charnel ground part of math!) It likely has no grand elegant unifying theories, and exacting proofs of what theorems it has only seem to confuse the subject more for a first-time learner. What it does have is lots of known solutions that work in specific cases.
- Calculating the determinant of a matrix by hand, on paper, as an unsimplified expression involving three or more levels of square roots... thanks, Dr. Manon. 🤮
I'd love to know what else might belong on this list.
These items cover impermanence, inseparability (of math from the world), discontinuity ("ugly" math?), and lack of definition. Is there also an example that centers insubstantiality? The examples I've seen of insubstantiality all seem inseparable from impermanence, so maybe the above list covers it? What could insubstantiality mean in the context of mathematics other than impermanence?