Forms and Formulae: Proof and Presupposition


A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the sixth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This installment’s mathematics article is entitled ‘Geometry’, and the poetic form is anecdote. This poem tells a true story I was reminded of by the discussion of the many attempts to prove Euclid’s parallel postulate from the other postulates, before people finally considered what would happen if it were false, opening up whole new geometries. This anecdote is not directly analogous, however, since I actually proved a statement to be false rather than proving it to be independent of the other axioms and then investigating what would happen if it were false.

A statement that the learned man had tried for days to prove
was set for students as a test
for four points extra credit,
to boost percentage marks assessed
of anyone to get it.

I mined brain gold with mind-brainpan, but things did not improve.
My efforts could not beat a path
from axiom to conjecture.
I sighed, and then let go of math
and headed to a lecture.

As I was sitting on the can, the shit began to move.
I saw the field with eyes anew
and found a boundary sample
that proved the statement was not true —
an outright counterexample.

To draw for years a foregone plan, for sure does not behoove
explorers hoping quests provide
not just what’s sought, but more.
Perhaps the field was opened wide,
but I scored one-oh-four.

I’ve been sitting on a draft of this one for a while, because, as noted above, disproving something is not the same thing as proving that one axiom can neither be proven nor disproven from the others, and then launching new fields of mathematics in which the axiom is taken to be false. Besides that, it’s a poem mentioning poop (though written before Shit Your Inner Voice Says), and it has a really weird rhyme scheme and awkward rhythm, for no good reason. Then again, I did once credit my short-story-writing success to the mention of toilets.

It is a true story; my abstract algebra professor at university set a couple of problems he hadn’t managed to prove himself for extra credit, and after proving problem number one I happened to think of a counterexample for problem number 2 while doing number 2s, and ended up scoring more than 100% for that class. I felt like I couldn’t make up an entirely fictional anecdote (though that is allowed, according to to the encyclopaedia) and while I’m sure I could write all sorts of other poems about geometry (on top of at least one I already have), I don’t have a lot of anecdotes about it.

Unimpressed as I am by this particular effort, I have to publish this to get onto the next Forms and Formulae, which will be… oh, for the love of Gödel — a national anthem for the development of abstract algebra?! What have I let myself in for?! It will take a while, because I’m heading to a programming conference followed by a translation conference soon, and then I’ll probably have to exercise my fledgling musical skills again.

Meanwhile, you can enjoy the highlights videos from Open Phil, an awesome open mic night in Vienna, where I’ve been practising reciting my poetry for audiences, and other people have been doing amazing musical things and other performances. Also, here‘s a very Vi-Hart-esque video I found while searching to see whether Vi Hart had anything to say on non-Euclidean geometry:

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