Math Tells
How to tell what part of math you are from
Gerolamo Cardano is often credited with introducing the notion of complex numbers. In 1545, he wrote a book titled Ars Magna. He introduced us to numbers like in his quest to understand solutions to equations. Cardano was often short of money and gambled and played a certain board game to make money—see the second paragraph here.
Today, for amusement, Ken and I thought we’d talk about tells.
What are tells? Wikipedia says:
A tell in poker is a change in a player’s behavior or demeanor that is claimed by some to give clues to that player’s assessment of their hand.
Other Tells
Ken and I have been thinking of tells in a wider sense—when and whether one can declare inferences amid uncertain information. Historians face this all the time. So do biographers, at least when their subjects are no longer living. We would also like to make inferences in our current world, such as about the pandemic. The stakes can be higher than in poker. In poker, if your “tell” inference is wrong and you lose, you can play another hand—unless you went all in. With science and other academic areas the attitude must be that you’re all-in all the time.
Cardano furnishes several instances. Wikipedia—which we regard as an equilibrium of opinions—says that Cardano
acknowledged the existence of imaginary numbers … [but] did not understand their properties, [which were] described for the first time by his Italian contemporary Rafael Bombelli.
This is a negative inference from how one of Cardano’s books stops short of treating imaginary numbers as objects that follow rules.
There are also questions about whether Cardano can be considered “the father of probability” ahead of Blaise Pascal and Pierre de Fermat. Part of the problem is that Cardano’s own writings late in life recounted his first erroneous reasonings as well as final understanding in a Hamlet-like fashion. Wikipedia doubts whether he really knew the rule of multiplying probabilities of independent events, whereas the essay by Prakash Gorroochurn cited there convinces us that he did. Similar doubt extends to how much Cardano knew about the natural sciences, as correct inferences (such as mountains with seashell fossils having once been underwater) are mixed in with what we today regard as howlers.
Every staging of Shakespeare’s Hamlet shows a book by Cardano—or does it? In Act II, scene 2, Polonius asks, “What do you read, my lord?”; to which Hamlet first replies “Words words words.” Pressed on the book’s topic, Hamlet perhaps references the section “Misery of Old Age” in Cardano’s 1543 book De Consolatione but what he says is so elliptical it is hard to tell. The book also includes particular allusions between sleep and death that go into Hamlet’s soliloquy opening Act III. The book had been published in England in 1573 as Cardan’s Comfort under the aegis of the Earl of Oxford so it was well-known. Yet the writer Italo Calvino held back from the inference:
To conclude from this that the book read by Hamlet is definitely Cardano, as is held by some scholars of Shakespeare’s sources, is perhaps unjustified.
To be sure, there are some who believe Shakespeare’s main source was Oxford, in manuscripts if not flesh and blood. One reason we do not go there is that we do not see the wider community as having been able to establish reliable principles for judging what kinds of inferences are probably valid. We wonder if one could do an experiment of taking resolved cases, removing most of the information to take them down to the level of unresolved cases, and seeing what kinds of inferences from partial information would have worked. That’s not our expertise, but within our expertise in math and CS, we wonder if a little experiment will be helpful.
To set the idea, note that imaginary numbers are also called complex numbers. Yet the term complex numbers can mean other things. Besides numbers like it also can mean how hard it is to construct a number.
In number theory, the integer complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions, multiplications, and parentheses. It is always within a constant factor of the logarithm of the given integer.
How easy is it to tell what kind of “complex” is meant if you only have partial information? We don’t only mean scope-of-terminology issues; often a well-defined math object is used in multiple areas. Let’s try an experiment.
Math Tells
Suppose you walk in log-in to a talk without any idea of the topic. If the speaker uses one of these terms can you tell what her talk might be about? Several have multiple meanings. What are some of them? A passing score is
- She says let be a c.e. set.
- She says let be in .
- She says by the König principle.
- She says is a prime.
- She says is a prime.
- She says is solvable.
- She says let its degree be .
- She says there is a run.
- She says it is reducible.
- She says it is satisfiable.
Open Problems
What are your answers? Do you have some tells of your own?
Whenever I used to pick a book on logic in a bookstore — Okay Baby, I’ll explain that later — I would turn right away to the index to see if C.S. Peirce was mentioned. That would always tell me …
I’m going to guess it’s about word problems on groups. We got the computability theory (and use of omega not N) and various group theory notions. Second guess is computable model theory
Though maybe I read that wrong and you don’t mean for there to be a single answer for all 10.
A related issue concerns when some open problems were first posed. Take for example the existence of odd perfect numbers. The oldest explicit statement of the question appears to be in Descartes. But if one looks at what older sources are doing (for example, proving that a power of a prime cannot be perfect), then it looks like it is a question they are thinking about.