I’ve recently become aware of an entertaining controversy in the world of mathematics. It seems a dissident faction of mathematicians is advocating the replacement of the mathematical constant π with the related constant τ = 2π. T he self-described ‘Tauists’ are conducting their campaign with a form of ha-ha-only-serious dry humor that gently mocks the conventions of the mathematical literature, but if we receive it only as satire, we risk missing some serious and interesting issues in play.
For background, you can read The Tau Manifesto and a rebuttal at The Pi Manifesto. I’ve read some of the source arguments these were constructed from, such as Bob Palais’s original “π is wrong! “article; having done so, I think the manifestos sum up the state of play pretty well.
I was at one time a mathematician with a serious interest in foundational issues; that is, I was interested in studying the axiomatic basis of mathematics itself. One relevant question the π vs. τ dispute raises is what we actually mean when we say that “π is an important mathematical constant”. Is there any sense in which we can say that π = τ/2 is “more special” than τ = 2π, or vice-versa?
Curiously, though the disputants sidle up to the second question in a sideways manner, I haven’t yet seen anyone tackle the first one directly. Perhaps this is because it hasn’t attracted a foundationalist’s attention yet; most of the mathematical community, so far, seems to dismiss the dispute as trivial. I don’t think it is, though, and even if I’m no longer officially a mathematical philosopher myself I can at least play one on the net.
The real meat of this dispute is in how we evaluate competing notations for the same mathematical system, and to approach that question we need an explicit theory of what mathematical notation does and what it is for. The π-vs.-τ dispute has legs exactly because mathematicians in general have only a loose and implicit theory of notation; while it leads to broadly shared intuitions most of the time, it does tend in edge cases like the π-vs.-τ dispute to collapse into personal esthetic evaluations that cause a lot of argumentative heat exactly because they can’t really be logically defended.
In fact it is exactly this collapse that the Tauists are gently parodying, even as they make a serious case for τ. But the Tau Manifesto gets caught up in specific arguments about its controversy enough that it only glances at the more general question: what makes one mathematical notation better than another?
We can start by noticing that mathematical notation has two broad functions. One is to facilitate computation; the other is to help mathematicians generate intuitions about its subject matter.
The first of these is relatively easy to think about. In the past, changes in notation have brought about dramatic improvements in ease of computation, leading sometimes to very large consequences in the real world. Perhaps the most dramatic example was the shift from Roman numerals to modern positional notation during the early Renaissance. This made arithmetic so much easier that every human endeavor in contact with it got revolutionized, leading to results as diverse as double-entry bookkeeping, open-ocean navigation, and (arguably) the invention of physics. In more recent times, the invention of tensor calculus in the late 1800s proved essential for helping Albert Einstein and others perform the essential computations of General Relativity Theory.
The second use, helping generate intuitions, is much less well understood. No mathematician doubts that expressive notation is like wings for the mathematical imagination; nor that a clumsy, poorly chosen notation is like hanging weights on it. But, as in Hollywood, nobody knows what will work for audiences until it’s tried. Our evaluations of “expressive” and “clumsy” can usually be only be made after the fact and in a relatively fuzzy way.
But the most important property of good notation serves both purposes. Good notation expresses complex ideas in a simple and regular way. And this is something we can actually formalize, because human brains being what they are, “simple” unpacks to “few enough symbols to fit in the brain’s working storage”. Short formulas with large consequences are the greatest achievements of both pure and applied mathematics.
This gives us a metric. Suppose we have a list of theorems and derivations that we consider important, and two alternative notations for expressing them. There is a known way to map without loss from one notation to the other and back. Which, then, is better?
The simplest answer is, I think, the fundamentally correct one. Write them all down in both notations and count symbols. The notation with the lower symbol count wins, and not by accident but because handling it will impose lower overhead on the user.
The Tauists and pi partisans understand this well enough that they argue back and forth partly by listing important formulas or theorems that are simpler in their preferred notation. But lacking any explicit idea that mathematical notation needs to be optimized for the limited short-term memory capacity of human brains, they spend what I think is too little time on such “global” arguments and way too much on “local” ones – that is, whether the ratio that π or τ expresses is “more fundamental”.
I think this local argument really rests on a sort of lurking Platonism, a belief that a mathematical formula is a kind of statement or claim about ideal forms at least some of which have an existence independent of the formulas. Nobody in the dispute can quite bring themselves to utter the claim that π is “real”, whereas 2π is just a derivative arrangement of symbols; nor does one hear the opposite claim that τ is “real” but τ/2 is not. But that sort of essentialism is stooging around underneath the arguments the disputants do make, denying its own presence but nearly impossible to miss.
If the last century and a half of mathematics has taught us anything, though, it’s that Platonism doesn’t work. Kurt Gödel put the final bullet through its head with his Incompleteness Theorem in the 1930s, but it had been living on borrowed time ever since Bertrand Russell blew up Frege’s axiomatization of number with a simple paradox in 1902. Mathematical Platonism has since almost disappeared as a philosophical position, but not as a psychological one; I’ve noted before that mathematicians then to be formalists in theory but Platonists in practice. In disputes like τ-vs.-π the tension between these positions surfaces, because arguments about the notation of mathematics have a natural tendency to slide over into arguments about its ontology.
Having restated the underlying problem in a way that I hope clarifies the dispute, I will now take a position on the merits. I think the Tauists have the better of the argument. I don’t think I’m being influenced too much by the fact that their side gets to make clever puns about Taoism, difficult though that lure seems to be for anyone involved to resist. It really does appear to me that the τ notation yields a net simplification.
Much more importantly, though, I think the best way to resolve this dispute is to throw out all the essentialism and the arguments about what π and τ “really mean” geometrically. We should focus ruthlessly on the global question: what notation makes our formulas simpler?
That way of thinking about the problem implies an answer to the question about what we mean when we say “π is an important mathematical constant”. We mean that it shows up repeatedly in simple formulas – and that replacing it with an equivalent expression that is not one symbol (such as, say τ/2) would involve a loss in concision with no gain in expressiveness.
The Tauists claim that changing to τ would actually gain some concision. Very well then; let’s do a systematic audit. Representatives of the Tauists and the pi partisans should be locked in a mathematics library until they choose a list of books and papers that covers trigonometry, calculus, and analysis. Then, the burden should be on the Tauists to translate the entire pile into τ notation. Then, both sides should count symbols, checking each others’ work. Most compact notation wins!
Or, to put it a different way, if you’re going to get involved in the τ-vs.-π dispute, beware of circular arguments.