Are These the Last Digits of Pi?

Composite of src1, src2

Thomas Keller and Heiko Rölke led a team at the University of Applied Sciences in Graubünden, Switzerland, that set a new record for the computation of last August. They computed 62.8 trillion digits of . The last ten digits they obtained are 7817924264.

Today, we wish people Happy Pi Day amid wishes for happier days overall.

Pi Day needs the day to be written American-style as 3/14/22. In international style it would be 31/4/22, but April does not have 31 days. This year involves the numerator of the simplest serviceable approximation to pi:

Because is irrational, any finite fraction or number of digits is an approximation. Because is computable—indeed highly efficiently computable in senses we have covered—we can adduce that the code for doing so represents exactly. Furthermore, the symbol lends itself to many other calculations that yield exact finite results.

The digits, however, have their own mystique. We still do not know whether they are normal in any base, let alone base ten. Speaking as mathematical Platonists, we regard the infinite sequence as a completed, unchanging entity—one for which assertions like “it is normal” have currently-definite truth values.

This is what events of the past few weeks have prompted Dick and me to question. If our world presently stops existing, 7817924264 will be the last digits of that we know.

Etsy Math Mug source

Not Just WW III

I already have a story of impermanent truth rooted in Russia. On my phone I have a free app for the Lomonosov Tablebases, which give the perfect result of all chess positions with 7 or fewer pieces. Those tables took years to compile and exist only as 100+ terabytes on a machine at the Lomonosov Moscow State University computer center.

Sometime over the new year, I noticed that the server stopped working. It was reportedly the victim of a ransomware attack. Is it meaningful to say that the tables currently exist? Mathematically, yes, and physically likely also yes—assuming the bits were merely blocked and not altered on the storage plates.

Happily, there is a second 7-piece table called Syzygy, which uses only about 18 terabytes and exists in multiple dowloaded copies. However, this leads us to another question about correctness. The Syzygy computation reproduced some key extremal features of the Lomonosov compilation, such as the position with the longest number of moves needed to win. Not all have been verifiable, because Syzygy counts the time needed to make concrete progress in the form of a capture or pawn advance rather than the time to give checkmate. What I don’t know—and maybe now won’t know—is:

Has Syzygy been used to verify every win/draw/loss (WDL) verdict computed by Lomonosov, and vice-versa?

Doing so would require cross-checking many trillions of chess positions. Of course, we should expect that the algorithms used to produce these tables have been verified as completely correct.

That is worth saying again: The chess algorithms have been verified in themselves. This is not a case of a shock I had before Christmas, when a module in my own chess software threw an error for the first time since I wrote it in 2014, having worked perfectly on over 100 million moves in several million games. The function in question records not only the exact source and destination squares of a move but also the minimum information required by short-form notation systems to disambiguate it from other pieces that could move to the square. There was one game where the players horsed around until one side had 5 queens that could all move to the same square, and that was 1 more than my scheme had presumed possible. My code is thus not-quite correct.

But even analytical correctness fails in cases of hardware error. A cosmic ray temporarily changed the outcome of an election in Belgium. Fortunately, the systems have cross-checks to catch these events. When the subject is mathematical truths, however, how are we checking? On what basis can we be satisfied by such checks?

Emergence

Despite our Platonist convictions, as practitioners of mathematics we experience its truths as emergent. By “emergent” we mean more than saying theorems are unknown until the point in time where they are clearly proved. Pace our claimers, P versus NP has not been proved either way, and we live in a world where even those who strongly believe will aver it is unknown. We covered this recently.

Our sense of emergent meets at least the “weak” criterion enunciated here by David Chalmers, a sense of unexpectedness that we have often discussed going back to the beginning of the blog.

Chalmers’s strong sense, when applied to mathematics, leads into independence results of the kind effected by Kurt Gödel, this blog’s eponym. We realize that none of our 1,000+ posts has yet attempted a deep appraisal of what these independence results mean. We will not do so now because we are questioning something more basic: Discussion of Gödelian independence is with regard to a truth value that is presumed to exist. What if it doesn’t exist?

We are not just catching the tension of Platonism with logical positivism and related paths to asking, what is science? We are asking whether reality aligns with the position that we already feel is best for practice.

“Emergent Mathematics” is a teaching movement that, in the words of one prominent paper, gets away from mathematics courses that “are focused on completed results that often hide the messiness and complication that led to their production.” In trying to explain much data showing that the most experienced mathematicians are not the most accomplished teachers, the paper’s two authors seem to identify the former with the position of putting emphasis on completed results. They seek the best attitude for childhood learning of mathematics, and believe it to be orthogonal to that of presenting finished mathematics. But going another 90 degrees around the dial, perhaps their compass needle’s other end points to the best philosophical position for creating mathematics.

Open Problems

I think we would all still agree that the next trillion digits of currently exist. The tougher question is whether it is scientifically meaningful to postulate knowledge of them, without knowing them. We may get an opinion on that from a regular friend late Wednesday into Thursday.