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Any student of modern math must know what it feels like to drown in a well of telescoping terminology.
For a high-profile example, let’s take the Calabi-Yau manifold, made famous by string theory.
A Calabi-Yau manifold is a compact, complex Kähler manifold with a trivial first Chern class.
Before you could even guess what that definition might mean, you would need to find another source to define a Kähler manifold:
A Kähler manifold is a Hermitian manifold for which the Hermitian form is closed.
After which you would need a third source to define a Hermitian manifold:
A Hermitian manifold is the complex analogue of the Riemannian manifold …
And you’re down the rabbit hole. When everything is named for its discoverer, it can be impossible even to track the outline of a debate without months of rote memorization. The discoverer’s name doesn’t tell you anything about what the landscape is like, any more than the “Ackerman” in Ackerman’s Island helps to convey a sandbar in downtown Wichita. Except in a few one-hit-wonder situations where a famous mathematician had extremely narrow tastes (like an Ackerman who, as everyone knew, could only live on sandy substrates, and never left the state of Kansas), their name gives no mnemonic boost whatsoever. Whatever faint associations it might once have held fade away, especially when the discover was neither famous nor narrow, and the reader is several generations removed.
Some very nice names have sprung up with no clear first use, like “pair of pants” and the Hairy Ball Theorem.
This nesting of proper nouns helps to make higher math impenetrable not just to outsiders, but also to working mathematicians trying to read their way from one subfield into another. The venerable Bill Thurston was known to complain about the perversity which, by the end of his career, had produced Thurston’s theorem, which says that Thurston maps are Thurston-equivalent to polynomials, unless they have Thurston obstructions. Every field has terms of art, but when those terms are descriptive, they are easier to memorize. Imagine how much steeper the learning curve would be in medicine or law if they used the same naming conventions, with the same number of layers to peel back:
A Thurston tumor is a benign Thurston growth in the bones of patients with type-1 Thurstonism.
A Thurston homicide requires a finding of Thurston recklessness and is a Thurston-class felony.
The Ancient Greeks were better about this. Euclid’s Elements is full of common, descriptive names, even though he was drawing on discoveries made by many different people. If he needs a term for something like a triangle with two sides of the same length, he calls it “isosceles,” literally “equal-legged” in Greek. A triangle with sides of all different lengths is “scalene,” or “unequal.” Euclid doesn’t even name the Pythagorean Theorem we all learn in school after Pythagoras, preferring just to state it plainly. In ancient Greece, it was polite for students to attribute their work to their teachers rather than themselves, if attribution was needed at all, so in the same way that Plato credited his own insights to Socrates, the eight or more objects now named after Pythagoras on Wolfram MathWorld might well be due to his students.
Things seem to have gotten out of hand after the Renaissance. Pierre Fermat’s name is on not just his Last Theorem and his Little Theorem, but on points, primes, pseudoprimes, polynomials, conics, spirals, a principle in optics, and a method for factoring odd numbers. Henri Poincaré, working at the end of the 19th century, has at least 21 mathematical entities named after him. It looks to me as though Bernhard Riemann might have as many as 82.
The average number of coauthors on math papers has gone up since 1900. So has the number of working mathematicians in the world, which raises the odds of independent rediscoveries, separated in time or space. These two trends have opened the door to triple and even quadruple hyphen situations, as in the Albert-Brauer-Hasse-Noether Theorem and the Grothendieck-Hirzebruch-Riemann-Roch Theorem.
Imagine how much steeper the learning curve would be in medicine if it used the same naming conventions.
Names may get even longer if Vladimir Voevodsky carries the day and modern math becomes dependent on computer-verified proofs. Papers published through big collaborations on shared technologies in other fields now have thousands of authors, but a theorem cannot have a thousand hyphens. We could pack in more people if we used initials, as with HOMFLY polynomials, named for their six co-discovers (Hoste, Ocneanu, Millet, Freyd, Lickorish, and Yetter) and sometimes even called HOMFLYPT polynomials, to mete out credit to Przytycki and Traczyk as well.
That, or mathematicians could take a pass on immortality and introduce their new objects with sensible, semantically parsable names instead.
For role models in the modern age, we look first to John Horton Conway, lately lost to COVID-19, whose many amazing names include the Monster for the largest sporadic simple group (with over a thousand octillion elements) as well as Monstrous Moonshine for that group’s totally unexpected connection with modular functions. More recently in topology, I have enjoyed Josh Greene’s changemaker vectors, whose components can sum to any integer less than their total value, as if making exact change with cash. David Wolpert and Bill Macready proved the No Free Lunch Theorems often cited in machine learning, which hold that every improvement in an optimization algorithm in one domain must come at the expense of worse performance in another, although Wolpert attributes their name choice to David Haussler. Surely we can agree it is a much better name than “Wolpert-Macready-Haussler Theorems” would have been.
Of course, some credit or blame must lie with the collective response to a new result, not just the individual who presents it. Poor Riemann did not name Riemannian manifolds after himself. Names like that emerge in the wave of secondary scholarship reacting to a new idea, and emergent names aren’t always bad. Some very nice names have sprung up though diffuse consensus with no clear first use, like the term “pair of pants” for a sphere with three holes in it (which I can’t trace further than a Bourbaki Seminar in 1978) and the Hairy Ball Theorem, which says that every vector field on a sphere with even dimensions must have a zero point, so any hairy billiard ball must have a cowlick (whose trail runs cold with Morris Hirsch’s Differential Topology textbook in 1976). But in general, mathematicians seem to feel a Scout’s Honor to name new things after their creators, unless those creators act like Conway and make concerted, repeated efforts to give descriptive names to their own objects instead.
In the last decade, the field of algebraic geometry was set on fire by “perfectoid spaces” rather than “Scholze spaces” because Peter Scholze kept on calling them that in his talks and papers. Like Conway and Wolpert, he put his descriptive name into the titles of his work, not just the body. That seems to help. By contrast, Shing-Tung Yau says in his autobiography that the Calabi-Yau manifold was given its name by other people eight years after he proved its existence, which Eugenio Calabi had conjectured some 20 years before that. Calabi and Yau would have had more right than anyone to interject and suggest something else, but as Yau tells it, they were both proud and happy to watch their names spread together through scholarly publications and through popular culture. What we have now is a system which gives naming rights to the discoverers in an implicit way, where your contributions will bear your names by default, unless you decide to agitate for something else.
Why do mathematicians continue to proffer and accept this courtesy, when it increases their own mental load and makes their own work more opaque?
The worst answer I can imagine is the one Pope Gregory VII gave for refusing to let the Holy Scripture be translated out of Latin: “… [I]f it were plainly apparent to all men, perchance it would be little esteemed and be subject to disrespect; or it might be falsely understood by those of mediocre learning, and lead to error.” The memory-intensive naming schemes in modern math may have the result of boxing out the laymen, but we must hope the priests of the academy are not doing it on purpose.
A more sympathetic answer would be that mathematicians want the glory of seeing their names outlive themselves as a reward for the long, solitary hours they labor to produce their results. In law or in medicine, research has a practical object, often with money attached. Can we count on the pleasure of finding things out to sustain work in pure math, if we eliminate this appeal to the ego?
I hold out the hope that research culture would be better off without it. One of the most compelling reasons Grisha Perelman gave for refusing his Fields Medal and his Millennium Prize was the unfairness of singling out one person as the progenitor of a 100-page proof, which necessarily represents a stitching together of many people’s breakthroughs, made over many decades of work. Changing the name scheme of modern math might imply a change in motive force, but if that change discouraged some people, it might welcome others in.
Laura Ball is a journalist-in-residence at the Kavli Institute for Theoretical Physics, a Thiel Fellow, and an alumna of the Math Prize for Girls program. She spent the last two years researching computational morality at Mila, the Quebec AI Institute.
Lead image: Ollyy / Shutterstock
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