Proving a pint

The BBC News website today reports a further contribution to that perennially popular topic, the mathematics of food and drink. Specifically, a group of researchers from the University of Limerick have produced a study that claims to settle the question of why the bubbles in a freshly-poured pint of Guinness travel downwards rather than rising — as we’d expect them to do, given that they’re less dense than the surrounding fluid.

Quotation for the week: Aristotle

A quotation this week from the great Greek philosopher Aristotle, with a sting in the tail concerning mathematicians who try to push their luck outwith mathematics:

… when the subject and the basis of a discussion consist of matters that hold good only as a general rule, but not always, the conclusions reached must be of the same order… For a well-schooled man is one who searches for that degree of precision in each kind of study which the nature of the subject at hand admits: it is obviously just as foolish to accept arguments of probability from a mathematician as to demand strict demonstrations from an orator.

(Book 1 of the Nicomachean Ethics 1094b24, tr. Martin Ostwald)

A severely mistranslated version of this quotation has been doing the rounds for a while, but this version mentions mathematicians so it’s clearly preferable. You might also like to consider what Aristotle might have had to say about the overconfidence of contemporary mathematical modellers…

(DP)

Going underground

You may have spotted a recent article on the BBC News website reporting research that analysed the structure of subway networks from a mathematical point of view. In Glasgow, we have one of the oldest but also one of the simplest subway systems in the world; the subway systems for larger cities such as Moscow or Paris tend to be considerably more complicated and to have developed episodically over many years. Nevertheless, the study reported by the BBC [full paper here] claimed to find some interesting regularities in their structure.

Quotation for the week: Dee

Dr John Dee is one of the most perplexing figures of the Elizabethan period: a classical scholar, a mystic and astrologer, and a mathematician — not a combination we’re used to seeing today. For Dee, as for many of his contemporaries, religion, mathematics and magic didn’t exist in separate categories. Indeed, as he makes clear in his Mathematicall Praeface to Euclid’s Elements, mathematics occupied a position somewhere between the natural and the supernatural:

A meruaylous newtralitie haue these thinges Mathematicall, and also a straunge participatiõ betwene thinges supernaturall, immortall, intellectual, simple and indiuisible: and thynges naturall, mortall, sensible, compounded and diuisible.

Dee enjoyed (or endured) a reputation in his lifetime as a magician — he may have been the inspiration for Shakespeare’s character Prospero — and it persists to this day in some circles. You can decide for yourselves how widely mathematics is still regarded as a department of the occult…

(DP)

Quotation for the week: Shimura

The Japanese mathematician Goro Shimura is best known for his work with Yutaka Taniyama, in particular the Taniyama–Shimura conjecture, the (partial) proof of which by Andrew Wiles also supplied the proof of Fermat’s last theorem. In an interview (exerpted in this PBS documentary), he gives an interesting perspective on working with a brilliant mathematician like Taniyama:

Taniyama was not a very careful person as a mathematician. He made a lot of mistakes, but he made mistakes in a good direction, and so eventually, he got right answers, and I tried to imitate him, but I found out that it is very difficult to make good mistakes.

Indeed.

(DP)

Engineers’ induction and the Borwein integral

You’re probably already familiar with the story about the mathematician who tries to explain proof by induction to two friends: a physicist and an engineer. After some deep thought, the engineer announces that he’s found a proof by induction that all odd numbers are prime: “Three is prime; five is prime; seven is prime; so all odd numbers are prime.” Trying to be tactful, the mathematician asks gently what happens if he tries to extend the sequence a bit further. The physicist chips in: “That’s easy. Three is prime; five is prime; seven is prime; nine isn’t prime; eleven is prime; thirteen is prime; so all odd numbers are prime to within experimental error…”

Although we laugh at “engineers’ induction” — extrapolating a general statement from the pattern you see in a finite number of cases — most of us have been guilty of it at some point. It can, of course, be a good way to generate conjectures which can then be proved or disproved properly, but it’s always dangerous to put too much faith in patterns.

A classic and well-known example is Moser’s “pizza-slicing” problem, which generates the sequence 1, 2, 4, 8, 16 — and then, unexpectedly, 31. Here, courtesy of a passing comment on Spiked Math, is a less well-known example.

Quotation for the week: Arnold

The text of a 1997 address “On teaching mathematics”, by the great Russian mathematician V. I. Arnold who died in 2010, opens with a provocative statement:

Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.