This is the first 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, even though the Companion already contains plenty of poems. The first entry in the former is entitled ‘What is Mathematics About?’ and the first entry in the latter is abecedarius.

The following is an abecedarius of what mathematics is about — an ABC of mathematics, if you like. You can also try reading it along to ’88 Lines About 44 Women’ (which you might be familiar with from The Brunching Shuttlecocks’ ‘88 Lines About 42 Presidents‘ or the great Luke Ski’s ‘88 Lines About 44 Simpsons‘) though the rhyme scheme is different. It only coincidentally has a similar meter, but once I saw it I decided to go along with it.

**A**xioms are how you ask ‘what if'; just pick some — you decide.

**B**reak it down and every branch of math(s) depends on these.

**C**alculus will help you count the branches that you can’t divide,

**D**ifferentiating the conditions at the boundaries.

**E**lements of Euclid was a textbook for millennia.

**F**unctions follow formulae to map domain to range.

**G**ödel showed some true things can’t be proven, but still many are,

**H**eld without theology as truths that never change.

**I**nconsistent axioms will prove all and its opposite,

**J**eopardising hopes the formal system will be sending forward

**K**nowledge for deriving knowledge-prime or knowledge-composite.

**L**ogic’s only limits are the ones that something’s tending toward.

**M**anifold(s) are ways to bring such limits to geometry.

**N**umerous are non-numeric methods that we use.

**O**ften are two manifolds the same, up to isometry,

**P**roving that(,) there’s gobs of generality to lose.

**Q**uod Erat Demonstrandum quoth inerrant understander,

**R**igorously rational and rooted in the real,

**S**ymbol-shuffling spanning such solution sets with candor,

**T**heorem after theorem or conjecture from ideal.

**U**niversal sets have mathematicians quite inside themselves;

**V**ector spaces set a basis they can build upon.

**W**olfram’s Weisstein’s MathWorld’s website rivals books on many shelves.

**X** rules the domain that functions are dependent on.

**Y**‘s home on the range is the solution set that many seek.

**Z**eno cuts each line in half so drawing it is undefined.

**Alpha**bet is insufficient;

**Beta** hurry onto Greek.

**Gamma** raises complex powers.

**Delta** changes Zeno’s mind.

**Epsilon**‘s so small that

**Zeta** covers the prime landscape sole.

**Eta**‘s very many things;

**Theta**‘s varied just by one

**Iota** in the calculus where

**Kappa** played a founding role.

**Lambda** has a calculus.

**Mu** (micron)’s small, but not-none.

**Nu** math(s) is Tom Lehrer’s nightmare.

**Xi**‘s that universal set.

**Omicron**‘s a small big-O.

**Pi** squares circles’ radii.

**Rho**‘s a row (zeros-out) rank.

**Sigma** sum is all you get.

**Tau** is sometimes phi, 2pi.

**Upsilon**, we wonder, ‘Y?’

**Phi**‘s the golden ratio.

**Chi**-squared ballpark’s on the ball.

**Psi**‘s a polygammous one.

**Omega**hd, there is no end;

**Aleph**-null can yet extend;

**Aleph** one is still too small;

**Beth** one, too, still isn’t all;

**Beth**-two, one can yet transcend.

**Gimel** still can bring you some,

**Daleth** beats continuum.

Now you know your ABC(-Omega-Aleph-NOP)

Out you go to maybe see (oh, mathematicality!)

That math(s) is an infinity (for all things there exists a key!)

And cast it as a trinity (a singular plurality!)

When I decided to do this, I don’t think I realised how many Greek letters there were. In the time it would have taken to finish a normal abecedarius, I was only halfway there, and further motion seemed impossible. Luckily, Zeno was there to sympathise. I also didn’t realise any Hebrew letters after bet were used in mathematics. Apparently Cantor used gimel and daleth for yet bigger infinities. I hope to write a new Forms and Formulae each week, so the later forms had better not be this long. I didn’t always stick to things from the ‘What is Mathematics About’ article, or even that subject. However, I think I conformed to the abecedarius form fairly well; the abecedarius is often used for religious purposes, and I was able to work in that mathematics requires no faith (‘held without theology’) and extends beyond alpha and omega, and also that the differing ways of abbreviating the word in different countries (with or without ‘s’) makes it similar to the three-in-one Christian trinity.

I had many ideas for the ‘pi’ line, pi being one of the most famous numbers amongst non-mathematicians. The irrational thing is, it’s famous for the most unremarkable things about it. Pi is famous for being irrational, even though almost all numbers are irrational. It’s often touted as being remarkable for containing any sequence of digits you care to imagine, although for this to be true, pi would have to be what’s called a normal number (again, like most other numbers) and this has not even been proven to be the case.

Don’t get me wrong; pi is a very useful number, for much more than just circles, but celebrating its irrationality or conjectured normality would be like exalting the fact that a Nobel laureate has toenails. Numbers with these properties are far more common than the special, rational numbers that we usually deal with, but they are ignored because they don’t have cute names or easy-to-explain uses. So I wanted pi’s general celebrity and big-headedness to get across. I quite liked this one as a variation on holier-than-thou:

Pi is piouser-than-i

But I wanted ‘big’ in there to go with the previous line:

Pi’s a big-shot (in the sky)

Or (after I changed the previous line to its current ‘small big’ form)

Pi’s a little big and high

Alas, I couldn’t find a good word for big-headed that had the right rhyme at the end. Finally, I realised I could rhyme the word radii, bringing in pi’s well-known association with circles:

Pi is big on radii

I finally got rid of the ‘big’ when I realised I could also briefly imply that this overhyped number can square the circle (something which mathematicians tried to do for hundreds of years before proving it impossible.)

Pi squares circles’ radii

So there you go. That’s how many times a line has to be revised without even making a full circle. I’m still fond of the first version, but it isn’t fair to suggest pi has personality problems without even hinting at why, or saying anything nice about the number. I have grapheme → colour synaesthesia, not ordinal linguistic personification.

I thought far too much about the F-line too. Functions follow formulae, sure, and maybe this series should be called Form and Functions (but no, formula is more general) but do I mean that mathematical functions proceed according to formulae, or that practical applications come after formulae? I wanted to say ‘that map ideas to use’ or something like that, but it turned out that the ordinary mathematical idea of mapping domain to range rhymed with what I wanted to say in the fourth line, so I refrained from shoehorning in another obscure pun.

Another thing I thought a lot about changing was the Tom Lehrer line. New Math sounds like it was a teachers’ nightmare as well, so I wondered if just putting ‘a Lehrer’s nightmare’ would be enough to evoke both Tom Lehrer’s song and the German word for ‘teacher’. In the end, I decided not enough readers would know both those things.

With enough constraints, poetry is like sudoku, really. You have to fill a line with a certain number of syllables in a certain rhythm, and you know you have to have this rhyme here (unless you change another line) and that letter there, and for bonus points maybe you want to echo something on another line, or add extra assonance or alliteration somewhere in the middle, and you want to allude to x different uses of that letter and meanings of that word at once… in the end there aren’t very many valid solutions, and you’ll know when you’ve found one. I’ve never actually played sudoku; it’s not as much fun to read.

Cutting the lines in half for the Greek letters gave those new half-lines a sense of self-importance, and they went and grew syllables that the distant rhymes hid behind, but I think it was an okay compromise. I added a few extra rhymes to compensate, because you can never have too many. You can probably have too many shoehorned math(s) puns. Maybe if I use clown shoes there will be more room for puns in them alongside the poetic feet.

And with that, I think I’d better shoe myself out.