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.

Axioms are how you ask ‘what if’; just pick some — you decide.
Break it down and every branch of math(s) depends on these.
Calculus will help you count the branches that you can’t divide,
Differentiating the conditions at the boundaries.

Elements of Euclid was a textbook for millennia.
Functions follow formulae to map domain to range.
Gödel showed some true things can’t be proven, but still many are,
Held without theology as truths that never change.

Inconsistent axioms will prove all and its opposite,
Jeopardising hopes the formal system will be sending forward
Knowledge for deriving knowledge-prime or knowledge-composite.
Logic’s only limits are the ones that something’s tending toward.

Manifold(s) are ways to bring such limits to geometry.
Numerous are non-numeric methods that we use.
Often are two manifolds the same, up to isometry,
Proving that(,) there’s gobs of generality to lose.

Quod Erat Demonstrandum quoth inerrant understander,
Rigorously rational and rooted in the real,
Symbol-shuffling spanning such solution sets with candor,
Theorem after theorem or conjecture from ideal.

Universal sets have mathematicians quite inside themselves;
Vector spaces set a basis they can build upon.
Wolfram’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.
Zeno cuts each line in half so drawing it is undefined.
Alphabet 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.
Omegahd, 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.

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