The New Math in Paradise

Eric Milner gave the popular lecture "The New Math in Paradise", to the Singapore Mathematical Society on 19 June 1991. We give below a few extracts from this talk.

The New Math in Paradise.

A popular lecture in mathematics has one disadvantage compared to similar talks in other subjects like medicine, geology, zoology, physics etc. - we do not have nice pictures or experiments to capture the visual attention of our audience (although one striking exception is the subject of chaos or fractals). Also, mathematics is not really a spectator sport, to understand even the most elementary parts of our subject, it is essential that the listener should understand and follow most of the small details. On the other hand, we do have one great advantage over other subjects, and that is that we do not often have to revise our lecture notes! The proof of the following theorem (which is a favourite for many mathematicians) is almost exactly the same that Euclid might have given over 2,000 years ago (Book X, Proposition 117).

Theorem l. √2 is irrational (i.e. is not a fraction).

Suppose the theorem is false. Then √2 = p/q where we may suppose that p/q is a fraction in lowest terms (i.e. p, q are positive integers having no common divisor). Then p2 = 2q2 is an even integer. Since the product of two odd integers is odd p must be even - say p = 2r where r is also an integer. But then q2 = 2r2 is also even, and so q is even. Therefore, p and q are both even - and this is a contradiction. Therefore the theorem is true. ■

This proof is an elegant illustration of the so-called method of reductio ad absurdum. The result is a little bit surprising since the rational numbers are all that one needs for measurements in physics; they are dense in the number line, i.e. in any interval, no matter how small, there are infinitely many "fractions". The theorem shows there are gaps in the rational line. The result is attributed Hippasus of the Pythagorean school. But the rest of the school were not too pleased with the discovery since it seemed to represent a flaw in their basic philosophy. They believed that everything could be explained in terms of whole numbers (which had mystical properties) - for example, they had a very successful theory for music based upon fractions. It is said that the Pythagoreans were at sea when Hippasus discovered his beautiful theorem, but instead of celebrating the discovery he was thrown overboard and the result kept secret for many years!

I understand that several of you in the audience today are senior high-school students who have shown exceptional promise in mathematics, and you are competing for the honour of representing Singapore in the next International Mathematical Olympiad. As you know, the questions in that competition are very difficult - especially with the time constraints imposed by an examination! You might be amused to know that many very knowledgeable professors of mathematics would also have great trouble solving these problems in the allotted time. Fortunately, in the real world, speed is not all that important. But it does illustrate another feature of our subject. To solve some problems, it is not always necessary to have a great knowledge of the subject (although that is not a disadvantage!), instead one needs some fresh idea - and, of course, young people often have many fresh ideas! It is my hope that some of you will continue the serious study of mathematics, you will find it rewarding.

In my talk today, I want to describe some of the bold ideas of the German mathematician Georg Cantor (1845-1920) concerning the mathematics of the infinite, or the theory of sets. Most of modern pure mathematics is based upon this theory. In fact, some years ago educationalists who wanted to update the school curriculum, seeing that many graduate courses in mathematics began with a review of the axioms of set theory, concluded that this was the real stuff, and that we should prepare students for it in our schools. So the "New Math" was introduced into the school curriculum, even at the elementary level. In my view this may have been a mistake, since the emphasis seems to have been placed more upon the use of certain words and definitions rather than providing students with more powerful tools to actually solve problems. However, this is a debatable point and should not distract us here - I mention it simply to explain the use of the words in the title of this talk.

The infinite has always fascinated mathematicians, and has been the source of many puzzles. Some of the earliest, and most debated, were due to the Greek philosopher Zeno (~450 BC) of Elea in southern Italy. He gave several paradoxical arguments to show that motion is impossible. His purpose was rather to show that both the opposing views of time and space held at that time were untenable. For example, one paradox goes like this. In order to go from A to B (on a line) one must first go from A to the mid-point B1. But before that one must go from A to the mid-point B2, and so on. Thus, if space is infinitely divisible, so that a finite length contains infinitely many points, then it is impossible to cover a finite length in a finite time. These so-called paradoxes of Zeno were debated in many philosophical discussions through the centuries.

[Here is] a story about the physicist Crookes (of Crookes tube fame) which Paul Erdős likes to tell. Crookes noticed that when unexposed photographic plates were placed in a drawer near to one containing radium, the plates became useless. Instead of earning even greater fame, he left it for Madame Curie to discover X-rays several years later; he simply left laboratory instructions that (quite rightly!) photographic plates should not be stored near radium! For success in Science or Mathematics a certain amount of luck is needed, but more importantly, you also need an open mind.
... it seems that in mathematics we can do no better than to believe (or hope) that our subject is consistent, there can be no formal proof that this is so. However, as F De Sua (1956) pointed out:
Suppose we loosely define a religion to be any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics for example would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.

JOC/EFR January 2020