Mathematical structures, 3

Week 3 is over. The notes are on the web, but I didn’t actually cover everything in the notes. The plan was that part of notes 2 (the counting results such as Inclusion–Exclusion and the Binomial Theorem) would spill over into week 3, and give me two clear lectures to talk about infinite sets.

But the plans changed. I had an email from Apostolos Doxiadis (author of Uncle Petros and Goldbach’s Conjecture and Logicomix) saying that he was currently in London and suggesting that we meet up. So we met in a pub in Fleet Street last week, and he offered to come and address my students. How could I refuse?

So he came along for the Tuesday lecture; after a bit of trouble finding the place, and a bit more trouble getting his computer to talk to the projector (I thought those problems were more or less sorted, but I’d reckoned without the idiosyncrasy of Macintoshes), gave a memorable lecture about mathematical travels.

He began by showing my three objectives for the course:

  1. to introduce the basic objects of mathematics (numbers, sets and functions), and their properties;
  2. to emphasize the fact that mathematics is concerned with proofs, which establish results beyond doubt, and to show you how to construct proofs, how to spot false “proofs”, how to use definitions, etc.;
  3. to get you involved in the excitement of doing mathematics.

He said he wasn’t talking about the first, but would show how the second and third are related. He showed us Raphael’s famous picture of Plato’s academy, and circled in red the three people who are responsible for the modern view of mathematics as too difficult and esoteric for ordinary people: Pythagoras, who restricted his instruction to the initiates, and introduced strange symbols and an air of magic and mystery; Plato, who moved mathematics from the real world of surveying and accountancy to the heaven of archetypes; and Euclid, with his book full of proofs, which tortured students for nearly two millennia.

We also met Henri Poincaré, finding the proof of one of his great theorems while stepping onto the bus for an excursion (after a lot of hard work in preparation); Évariste Galois, the most romantic mathematician ever, but one whose letter (written the night before he was killed in a duel) contained lots of abstract technical mathematics; Andrew Wiles, spending seven years in his attic fulfilling his childhood dream by writing a paper full of Selmer groups and cohomology (we were shown an extract). The students were remarkably quiet, so I think they were taking it in.

So the week was reduced to two lectures; I finished infinite sets but didn’t manage to say all there was to say about counting, some of which will have to be deferred to next week.

Apart from this change of plan, the course is still going well. There have been some problems with tutorials, none too serious, but most tutors are still happy with the way they are going; those students who bother to turn up are happy to discuss their ideas, even in front of the board. However, they are a bit reluctant to hand in written solutions. I think that writing mathematics clearly is a different skill from either understanding it or explaining it at the board; they will not learn that unless they practice and get feedback.

Today’s lecture was about the uncountability of the real numbers. I had intended to go further; but the students were up for discussion, and we spent the whole hour on it (and another quarter of an hour discussing it with some of them afterwards).

In Hilbert's hotel

The supplementary material this week contains a short account of Hilbert’s hotel. I took the opportunity to use in it Neill‘s illustration below of Kirkman and some of his fifteen schoolgirls arriving at Hilbert’s hotel:

Kirkman's schoolgirls arrive at Hilbert's hotel

This picture is in the 1996 edition of my Combinatorics book; but if you buy a copy from Cambridge University Press now, they print on-demand the 1994 edition, so you won’t get this picture. So I will take the opportunity to show the other picture in the front matter of the book: this shows Euler’s officers crossing one of the bridges of Königsberg:

Euler's officers crossing a bridge in Koenigsberg

Functions next week, then onward to the natural numbers!

PS I haven’t pointed you to the Logicomix website; it has been hacked by some organisation selling pharmaceuticals on-line …

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About Peter Cameron

I count all the things that need to be counted.
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One Response to Mathematical structures, 3

  1. Walter Sinclair says:

    “I finished infinite sets but didn’t manage to say all there was to say about counting…” I would say you did quite well, under the circumstances.

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