Unfortunately, the Tao Manifesto is full of selective bias in order to convince readers of the benefits of τ over π. They pinpoint formulas that contain 2π while ignoring other formulas that do not. We demonstrate below that when making the change to τ, there are lots of formulas that either become worse or have no clear advantage of using τ over π. Tauists also claim that their version of Euler's formula is better than the original, but we will see that it is in fact weaker. The benefits of τ only appear when viewing πfrom a narrow minded two dimensional geometrical point of view, but these benefits disappear when looking at the bigger picture. We will see how the importance of π shines through as it shows up all over mathematics and not just in elementary geometry.
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.