This evening, Jonathan Jedwab gave a public lecture, in celebration of the fact that it is Jacques Hadamard‘s 150th birthday.
The title of the lecture was “What is a research mathematician?”, and the hook linking the two was Hadamard matrices.
Jonathan’s aim in the lecture, which was very successfully realised in my view, was to show us (and members of the public, teachers and children present) that a research mathematician is a wonderful job, and involves being a combination of problem solver, nitpicker, big picture thinker, experimental scientist, aesthetician, and dreamer.
Sylvester had considered Hadamard matrices (and devised the doubling construction for them) 26 years before Hadamard’s work (which shows, among other things, how careless mathematicians are about giving names). Hadamard invented the more general product construction which produces matrices of order mn from matrices of orders m and n. It was Paley who made the conjecture we now refer to as the Hadamard matrix conjecture (or, sometimes, just the Hadamard conjecture), asserting that Hadamard matrices of all orders divisible by 4 exist. He showed us a timeline of the moments when the smallest unsolved case had jumped up, and when computers began to be used on this. (At the end, someone asked why you couldn’t settle the first open case just by throwing it onto a big fast computer.) He also led us down the garden path by proofs that the order of a Hadamard matrix must be even (refuted by the matrix of order 1) and that an order greater than 1 must be divisible by 4 (refuted by the matrix of order 2), to show us the importance of nitpicking.
Incidentally, the current smallest unsolved case is 668, if you want to have a go.
We also had a couple of proofs from “the Book”:
- How many breaks are required to divide a 4×7 block of chocolate into single squares?
- Can the seven Tetris pieces be used to cover a 4×7 rectangle?
I am very much convinced of the truth of Hadamard’s four-stage method of mathematical discovery (preparation, incubation, illumination, and verification), having used it repeatedly in my month-long battle with the alternating group in Aveiro recently. But I appreciated the fact that Jonathan embedded this into a cycle of conjecture and proof.
As a professional nitpicker, I had to find fault with something in the lecture. At the end of the talk, one of the questions was about the contrast between proof as the “gold standard” in mathematics and the subjective nature of proof (Jonathan defined proof as “an argument that convinces someone who knows the subject”, or as I used to put it to challenge my Maths and Philosophy students in Oxford, “a proof is something you can get past a referee”. His answer went into Gödel’s Incompleteness Theorem, which I thought was a bit off the point. I very much doubt that I could have done better if put on the spot. But after a few minutes’ reflection, I think my view is that we believe (or hope) that, if we were really up against the wall, we could translate our proofs into a formal system and have them verified by a computer-aided proof checker. Jonathan made the good point that standards of rigour have changed, even since Newton’s time.