Tony’s Take January 2022

This month’s topics:

Gödel’s incompleteness theorem in The Guardian

Alex Bellos’s Monday puzzle in The Guardian for January 10, 2022 was derived from explanations of Gödel’s incompleteness theorem due to the logician Raymond Smullyan. (Smullyan published Forever Undecided: A Puzzle Guide to Gödel in 1987). The setting for the puzzle, as Bellos presents it, is a hypothetical island he calls “If.” Natives of If are either “Alethians” or “Pseudians;” they are indistinguishable except that Alethians always tell the truth, while Pseudians always lie. This sounds like the traditional Liar Problem, but there is a wrinkle: on the island is a ledger where every native is listed, along with his or her tribe. Anyone can consult this ledger. You arive on If and a person, Kurt, comes up to you stating: “You will never have concrete evidence that confirms I am an Alethian.” The puzzle: is Kurt an Alethian, a Pseudian or neither? Think about it before checking Bellos’s solution and before reading on.

Now comes the connection with Gödel’s incompleteness theorem, which states, as Bellos puts it, “that there are mathematical statements that are true but not formally provable.” Suppose you are the first non-native ever to visit If, so you know that everyone you meet is Alethian or Pseudian. Kurt pops up and says, just as before, “You will never have concrete evidence that confirms I am an Alethian.” But now just as he speaks the Ledger burns to ashes.

Where are we? Kurt cannot be a Pseudian, because with no Ledger that statement has to be true. So you know Kurt is an Alethian. But you can never have concrete evidence of that fact because if you did, his statement would be false, and it can’t be false since he is Alethian. Think about it. [Thanks to Jonathan Farley for bringing this item to my attention. -TP]

“Fun with Math” in The New Yorker

Dan Rockmore’s contribution to “Talk of the Town” in the January 17, 2022 issue of the The New Yorker was an item titled “Fun with Math.” He recounts attending “a recent evening of math dinner theater” organized by Cindy Lawrence, director of New York’s National Museum of Mathematics. The event featured Peter Winkler, a mathematics professor at Dartmouth and expert on math puzzles. Among the puzzles and phenomena that Winkler served up for discussion during dinner:

  • “On average, how many cards does it take to get to a jack in a shuffled deck of fifty-two cards?”
  • “What’s the best way to use two coin tosses to determine which of two coins, one fair and one ‘biased,’ is fair?”
  • Simpson’s paradox in statistics, best described by an example: for Berkeley’s graduate programs in 1973, overall “men were admitted at a higher rate than women, but, program by program, women were admitted at a higher rate.” (See MinutePhysics for a more detailed explanation.)

Apropos of Simpson’s paradox Marilyn Simons, a guest with a PhD in economics, remarked, “I think that, to a lot of us who even think we know statistics, the way we process statistics is not deeply informed.” Elsewhere Rockmore quotes her as saying that her husband Jim (identified as “a financier and a former mathematician”) doesn’t like puzzles: “He says that if he works that hard he wants to get a theorem out of it.”

Rock-paper-scissors and evolutionary game theory

“Non-Hermitian topology in rock-paper-scissors games” by the three Tsukuba physicists Tsuneya Yoshida, Tomonari Mizoguchi and Yasuhiro Hatsugai was published January 12, 2022 in Scientific Reports. This is a physics article, but it applies a nice piece of mathematics, evolutionary game theory, to the familiar rock-paper-scissors game.

The game consists of two players; at a signal each shows a clenched fist (“rock”), a flat hand (“paper”) or a vertical hand with the first two fingers displayed (“scissors”). The winner (rock smashes scissors, scissors cut paper, paper covers rock) gets one point, the loser loses one. If both players show the same symbol, each gets zero.

The article contains this image:

RPS trajectory: equilateral triangle with red arrows spiraling counterclockwise around the center point.
This image and part of the one below are from the open access journal Scientific Reports, 12, Article number 560. Reproduced under a Creative Commons license.

Here R, P and S have to stand for rock, paper and scissors, but how is this diagram related to the game? We need to make a detour into evolutionary game theory. This is a method for simulating the process of evolution in populations. Here the population is split among three subspecies; let’s call them Ravens, a fraction $s_1$ of the population, Penguins with $s_2$, and Swifts with $s_3$, where the fractions $s_1, s_2, s_3$ add up to 1. These correspond to the three “pure strategies” in the game: at every encounter, a Raven will play “rock,” a Penguin will play “paper” and a Swift, “scissors.” The state vector ${\bf s}=(s_1, s_2, s_3)$ encapsulates the current mix in the population.

Diagram of a triangle with red arrows spiraling around the center point.
Since $s_1,s_2,s_3$ are all positive and add up to 1, the state vector ${\bf s}=(s_1,s_2,s_3)$ for evolution based on the RPS game is a point in the triangle where the plane $s_1+s_2+s_3=1$ intersects the first octant $s_1, s_2, s_3 \geq 0$. The corners are the points where one of the $s_i$ is 1 and the others are 0. The point $(\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ is in the center.

Evolution occurs in time. Suppose the population is in state ${\bf s}$ at some moment. Where will it be just a litle later? For example, suppose ${\bf s}=(\frac{1}{2},\frac{1}{2},0)$. That means half the population are Ravens and half are Penguins. So a Raven will meet a Penguin with probability $\frac{1}{2}$, and can expect to lose half a point. Likewise a Penguin will meet a Raven with probability $\frac{1}{2}$ and so can expect to gain half a point. The Penguins have an advantage. If the object is to model evolution, then the Penguin’s advantage in that state should translate into their population increasing at the expense of the Ravens. That gives a clue to the meaning of the red arrow at the point $(\frac{1}{2},\frac{1}{2},0)$: the population mix at that point is shifting to the right. More Penguins, fewer Ravens.

To make this more precise, keeping the language of evolution, we measure the fitness of one of the groups at some state ${\bf s}=(s_1,s_2,s_3)$ of the population by the expected gain or loss in points at the next encounter. So the fitness of the Ravens at state ${\bf s}$ will be the probability of meeting a Penguin times $-1$ plus the probability of meeting a Swift times 1. We write this as
$F(\mbox{Ravens}|{\bf s})= -s_2 + s_3.$ Similarly $F(\mbox{Penguins}|{\bf s})= s_1 – s_3$ and $F(\mbox{Swifts}|{\bf s})= -s_1 + s_2.$

Finally we set up a dynamical system by stating that the proportion of the population in any group will increase or decrease exponentially with growth coefficient equal to the fitness of that group (which can be positive or negative) at that instant in time. Writing that statement as a differential equation gives the replicator equation for rock-paper-scissors:
$$\frac{ds_1}{dt}= s_1(-s_2 + s_3), ~~\frac{ds_2}{dt}= s_2(s_1 – s_3), ~~\frac{ds_3}{dt}= s_3(-s_1 + s_2).$$
In vector form, the equivalent equation is
$$\frac{d{\bf s}}{dt}= (s_1(-s_2 + s_3), s_2(s_1 – s_3), s_3(-s_1 + s_2)).$$

Now we can interpret the first image in this item, which shows the state space for rock-paper-scissors as an evolutionary game. The arrows represent the direction of evolution, with the magnitude encoded by color saturation. The central cross marks the equilibrium $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$; the blue loop is the solution curve obtained by numerically integrating the replicator equation starting at the point ${\bf s} = \frac{1}{3}(1-\delta, 1+\delta/2, 1+\delta/2)$, with $\delta=0.1$.

For this game, all the solution curves are closed loops. In fact, they are level curves of the function $f(s_1,s_2,s_3)=s_1s_2s_3$. This follows from the equality $\displaystyle{\frac{d{\bf s}}{dt}\cdot \nabla f= 0}$, which is actually fun to check. Try it.

The evolutionary game described here is just the starting point for the article by Yoshida et al. They consider perturbations of this system that break its symmetry—surprisingly, the resulting phenomena have parallels in condensed-matter physics.

“What physics owes to math”

On January 12, 2022 Le Monde ran a guest article with the title “What physics owes to math” (text in French) written by two physicists, Jean Farago and Wiebke Drenckhan, from the Institut Charles-Sadron in Strasbourg. The authors begin with the famous quote from Galileo about the Book of Nature being written in the language of mathematics, and go on to observe: “The millenia elapsed between the birth of mathematics and its use in physics demonstrate that this contiguity between natural phenomena and the mathematical laws of our human rationality was far from being obvious.”

Farago and Drenckhan mention that one of the most antonishing examples of the “intimate” relationship between mathematics and physics comes from complex numbers. Starting in the 16th century, mathematicians found that calculating solutions to polynomial equations with whole-number coefficients required the use of an ‘imaginary’ number $i$ with $i^2=-1.$ “How could anything be more abstract than this fictitious number, given that ordinary numbers always have a positive square ($2^2=(-2)^2=4$)!” But fast-forward to 1929 and Schrödinger’s equation $i\hbar\partial_t\psi=H\psi$, which doesn’t work without it. We read that no one was more surprised by “this irruption of $i$ in the corpus of physical laws” than Schrödinger himself, and that he described his reaction, in a footnote, by quoting an unnamed Viennese physicist, “known for his ability to always find the mot juste, the cruder the juster,” and who compared the appearance of $i$ in that equation to one’s involuntary (but welcome) emission of a burp. Our authors add: “This shows us that a contiguity can sometimes also exist between humor and physics.” [My translations. -TP]

“What physics owes to math” could have mentioned an article from Nature last month: “Quantum theory based on real numbers can be experimentally falsified,” written by an international team with corresponding author Miguel Navascués (IQOQI, Vienna). Physical experiments are expressed in terms of probabilities, which are real numbers. So why can’t there be a “real” quantum theory? The authors show that complex numbers are actually needed, by devising “a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.”