Way of the Tao

Professor Terry Tao (born 1975) is an Australian mathematician now working in the US. Tao was a child prodigy who taught himself arithmetic at age 2. He received his PhD at age 20, was appointed a full professor at UCLA at age 24 and won the Field’s Medal in mathematics at age 31. This award is something analogous to a Nobel Prize in maths. (There is no Nobel Prize in mathematics though this is not because Nobel’s wife ran off with a mathematician).

Gregory Mankiw (and Joshua Gans) recently discussed a delightful version of the ‘airport problem’ a problem Tao conceived while manoeuvring around a long airport terminal. The apparently self-evident answer to the first part of this easy-to-state problem* is wrong but thinking clearly about the issue clearly resolves things quickly**.

Anyway I happened to see some classes given by Tao that have been recorded on YouTube. He is not only an obviously brilliant mathematician but an extremely able teacher. This introductory class on the prime numbers is beautifully presented.  Poetic, insightful, simple.

It is a pity – from Australia’s perspective if not his own - that he is not working in Australia building up a great mathematics department and attracting scholars of renown here. I’d make the same observation about notable scholars in my own field of economics. That’s definitely not a criticism of any individual – purely an observation that academe here would be better-off with their presence.

By the way Terry Tao is also a blogger – I added his site to my list of blogs a few weeks ago.

*You wish to get from one end of a long airport terminal to the other in minimum time by walking and taking motorised walkways where you can also walk. But you need to stop to tie your shoelaces. Should you stop on or off the walkway?

** Consider two people taking the walking journey. They leave at the same time and one stops to tie his shoelace just before he gets on the walkway while the other takes one more step and gets on the walkway where he then ties his shoelace. It is easy to see who gets to the other end first!

William's maths question

I have always enjoyed maths and used to try to solve elementary maths problems as a school kid. I can remember with joy finally solving a simple-to-state problem after hours of thinking about it. Often the solution popped out of my head almost involuntarily after I had spent a lot of time thinking about it* and then almost abandoning the effort. I recalled some history here in an early post that aroused no interest at all.

My son William out of the blue this evening asked me 'are there fewer prime numbers than natural numbers?'. Firstly, I told him that Euclid had proved that the number of primes was infinite in 300BC (a modern proof is here). Then I told him that 'counting' elements in infinite sets was analogous to but an extension of counting elements in finite sets. I then tried to tell him the little I remembered about Cantor's theorem of countable sets which intrigued me as a kid.

I told him if you can put a set into 1 to 1 correspondence with the natural numbers - a bijection - then a set is countable. In this sense there are as many even numbers as natural numbers and, as a consequence of one of Cantor's main theorems, as many rational numbers as natural numbers. There are as many primes as there are natural numbers since the nth prime number can be put into 1:1 correspondence with the number n for n=1,2,3,...... Thus the set of primes is countable.

This I hope answered his question although, to be honest, that a 10 year old asked it was far more interesting than the answer I had to scratch around to recall.

* I think I spent more time trying to think things through then than I do today. My knowledge is greater today but my brain often switches to autopilot mode and I apply knowledge rather than think.