## Exploring University Mathematics

This week, a book called Exploring University Mathematics, 1 rose to the top of the pile on my bookshelf. Now there is no shortage of books which do what the title of this one promises, for non-mathematicians, or school pupils thinking about a mathematics degree. But I want to discuss this one because it shines a little light on aspects of social history and my personal history.

The book is based on lectures given at Bedford College, London, in 1965, and was edited by N. J. Hardiman and published by Pergamon Press. It has the following chapters:

• G. T. Kneebone, Sets and functions;
• J. R. Ellis, Special relativity and some applications;
• S. J. Taylor, Some properties of integers and primes;
• J. H. E. Cohn, Square Fibonacci numbers;
• M. Levison, Digital computers and their applications;
• H. G. Eggleston, The isoperimetric problem.

I can’t recall exactly when I got this book. But inside the front cover, the price \$2.50 is written in pencil, in what looks to me like an Australian hand; this suggests that I bought it between February 1966 (when Australia adopted decimal currency) and July 1968 (when I left for Britain). So I was probably a university student when I got it.

I studied in Oxford for my D.Phil. for three years, and then held a Junior Research Fellowship for three years. In 1974, I was appointed to my first permanent teaching job, a lectureship at Bedford College. Jessie Hardiman and Geoffrey Kneebone were among my colleagues there, and the department still put on lectures for school pupils and teachers; I spoke at one of these occasions, on Ramsey theory.

There are several references in the book to the importance of rigorous reasoning. Kneebone describes the problems that arose from thinking of a function as something with a smooth curve satisfying the Intermediate Value Theorem, and gives two definitions of the natural numbers, from Peano’s axioms and in terms of set theory.

Eggleston devotes a lot of time to Steiner’s arguments that if there is a simple closed curve of length L which is not a circle, then one can find another curve of the same length enclosing a larger area, but explains clearly why this does not solve the isoperimetric problem, and illustrates with the argument, which he attributes to Perron, that 1 is the largest natural number:

We consider any [positive] integer n and transform it to n2. Now n2 ≥ n and if n ≠ 1 then n2 > n. Hence no integer, except possibly 1, is the largest. Therefore 1 is the largest positive integer. This nonsense follows from the false assumption that there is a largest possible integer.

He says very clearly,

The [isoperimetric] problem had occurred to the Greek geometers of over 2000 years ago but its solution was not discovered until the 1880s. This solution, which is that the curve must be a circle, was known to the Greeks, but it is one thing to know what the solution of a problem is and quite another to establish this solution in an adequately convincing form or, as we say, to prove it mathematically.

Could one address comments like this to students and teachers now? I would like to think so!

Levison describes a couple of things that computers did in those days (payroll calculations, computing orbits of spacecraft, and playing simple games), and speculates on what they might do (machine translation, improving their game-playing skill by learning from experience).

Cohn gives the complete proof of a theorem which he had proved only a couple of years earlier: the only Fibonacci numbers which are perfect squares are 1 and 144. The proof is elementary but ingenious, relying on results of Fermat.

The chapter on computers is the one which can most obviously be mined for social history, though the level of rigour and amount of proof included in the book also carries a message. One other thing that stands out is the fact that there are several references to “schoolboys”. Fifty years ago, many fewer students went to university to do a mathematics degree, and it was the case that they were predominantly male. This is perhaps the biggest social change in my business.

I count all the things that need to be counted.
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### 4 Responses to Exploring University Mathematics

1. Robin Chapman says:

http://archive.org/details/ExploringUniversityMathematics1
(as ever with copyright material I am uncertain of the legality of this).

A reminder of more optimistic times, and a time when Bob Maxwell’s
Pergamon Press was a major mathematical publisher!

• The book was in the Commonwealth and International Library, Mathematics Division. One of the two general editors of the Mathematics Division was E. A. Maxwell. I don’t know whether this was a relative of Cap’n Bob (who is listed as Publisher, with the letters M.C., M.P. after his name).

• Robin Chapman says:

I presume E. A. Maxwell is the author of “Fallacies in Mathematics”,
“An Analytical Calculus” and many other textbooks, and not a relative of
Captain Bob. In those days Robert Maxwell was Labour MP for
Buckingham, a place that no longer returns Labour members!

2. Neil Calkin says:

Interesting to hear of Eggleston’s discussion of this: I shall have to check it out. I recall very vividly the first lecture I attended at Cambridge, in Analysis I, delivered by Bela Bollobas: he gave a series of examples of deeply flawed proofs, all of which were of false results. He followed this up with Steiner’s argument, and challenged us to find the flaw.
He refused to tell us what the flaw was, and it was (to my shame) long, long after the lecture that I understood it. I’ve since dangled this flawed proof in front of many good students, including many incoming graduate students in need of reminding that proofs need to be checked:-)

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