“Mathematics is an esoteric art.” One day in my first year IB mathematics class, this is what Mr Davies stated. I thought it pretty cool, mysterious and attractive. Three years later, studying theoretical physics, I was asked to prepare a presentation to my tutorial group. When I read over it, I can see hints of having been an IB student who had been exposed to thinking about knowledge in a TOK kind of way. I would like to share here those thoughts of mine from 1994 on the nature of mathematics, only slightly edited to change it from a talk to the written form.
Some years ago my mathematics teacher told me: Mathematics is an esoteric art. I thought that interesting, so there the saying was, on the first page of my maths folder. It was not until I started to study physics at Imperial College that the phrase came back to me. Since physics is nearly exclusively written in the mathematical language, the notion of mathematics being an art form started to concern me.
Then when I read a few of Georg Cantor’s letters I got worried. At first for this presentation, I was looking at Transfinite Number Theory, of which Cantor is the founding father. Cantor and another great mathematician, Leopold Kronecker, were in a bitter dispute about the handling of infinities [the 2007 documentary Dangerous Knowledge by David Malone looks in some depth at Georg Cantor, I like to show this to my mathematics as well as TOK class and one can find this documentary online]. I found these disagreements about the foundation of the most certain science not only surprising but, to put it mildly, disconcerting. Unlike physics, which evolves, mathematics is entirely cumulative, it is an ever widening network of logical consistency. So the foundations are very important indeed. A dispute about the foundations of mathematics, due to its close interdependence, would equally have an effect in physics.
Let me give an example. Five men, shipwrecked, find themselves on a deserted island with a monkey, and a pile of coconuts as the only edible thing. They agree to split the coconuts into five integer lots, any remainder will go to the monkey [this is a famous problem in mathematics, in fact, Martin Gardner starts with it in his collection of writings in The colossal book of mathematics (http://goo.gl/KJbBr3) ]. In the middle of the night, the first man wakes up hungry. He decides to take his share of the coconuts, divides the pile into five finding one coconut remaining. He eats his lot and gives one to the monkey, then returns to sleep. Later the second man wakes up. Feeling hungry, he, too, divides the coconuts into five lots, giving the monkey the remainder, which again is one coconut. So do the remaining three men. The next morning, too embarrassed, nobody says anything about their nightly escapades. They divide the remaining coconuts into five lots, giving the remaining one coconut to the monkey. Find the initial number of coconuts.
After some fiddling, you find the simultaneous equation [there is a better explanation in Martin Gardner’s book]
x – 6 = (15625 f + 5385) / 1024
where x is the initial coconuts and f is the share everybody gets in the morning division. There are many, in fact infinite, solutions, but the smallest number is 15621. However, soon after this solution was given [this is what I wrote in 1994, but I read in Gardner’s book that the following is not verified], Paul Dirac came up with the solution of -4, which is clearly a solution! This shows, I think, not only some interesting characteristics about mathematics, namely its independence of reality, but also the potential pitfalls a physicist may encounter when purely relying on mathematics.
So lets take a closer look at it.
There are essentially four interpretations of mathematics: Platonism, Conceptualism, Formalism and Intuitionism. Due to the limited time [I believe, judging by my notes, that the presentation had to be no longer than 12 minutes] I will discuss Platonism only, for reasons which hopefully will become clear.
At the birth of science as we know it today, stood undoubtedly Galileo [a bit of a simplification, but I was young]. A remarkable feature of his thinking was his belief that Nature was ‘a book written in mathematical characters’. This, we will see, is perhaps the single most influential and fundamental idea behind Platonism, and heavily relied on in modern science.
The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious. Some, for example Nobel Prize winner Eugene Wigner, suggest that there is no rational explanation for this. The mathematical formulation of the physicist’s often crude experience leads to many cases of amazingly accurate descriptions of a large class of phenomena. Newton established the Law of Gravity, but could only verify it with an accuracy of about 4%. Today it has proved to be accurate to less than 1 part in 10,000.
[skipping some physics here]
The Platonists would explain that mathematics is so successful in capturing the workings of Nature by saying that Nature is mathematical. Plato’s philosophy is often dubbed as “idealism” and this is represented in how it deals with mathematics. Most remarkable is the assertion that pi really is in the sky. That is, mathematical concepts “exist out there”, independently of you and me. Plato’s belief is that of forms, invariant blueprints from which we can only observe shadows. And so mathematics is discovered rather than invented.
Mathematical ideas like the number ‘seven’ are regarded as immaterial and immutable ideas that exist in some abstract realm, whereas our observations are of specific secondary realisation, like seven dwarfs. Thus, we seem to be able to conceive of perfect mathematical entities -like straight lines or right angles- yet all the examples we see of them are imperfect in some respect -I cannot draw perfectly straight lines or exact right angles [any of my students will tell you that!]. The particulars that we witness in our world are each imperfect reflections of a perfect exemplar or form. These forms exist somewhere and, most strikingly, Plato believes that place is not the human mind; hence, we discover, rather than invent mathematics.
Here perhaps we encounter the most obvious hurdle for this interpretation of mathematics. One might wonder how to clarify the relationship between the universals and the particular examples of them. For as far as our minds are concerned, the universal blueprint is just another particular. This was recognised early on by Aristotle. Aristotle argued that if all men are alike because they share something with the ideal and eternal archetype Man, how can we explain the fact that one man and this archetype Man are alike without assuming a second archetype. And would the same argument not require a third, fourth in an endless regression of ideal worlds?
Nevertheless, the Platonic picture of mathematics is popular among physicists, though less so amongst mathematicians, and it is almost universally regarded as irrelevant by consumers of mathematics such as psychologists, sociologists or economists [who says scientists regard themselves so highly?]
One of the most extreme manifestations of the Platonic assumption of the universality of mathematics is in the search for extraterrestrial intelligence in the universe. For years terrestrial radio telescopes have broadcast signals in the direction of nearby star systems; and equally search for incoming signals. These broadcast signals are all mathematical codes, believing that these ETs would have, if anything, in common mathematical knowledge [three years later the movie Contact came out, you can see the trailer here where it, in fact, mentions this underlying idea of the universality of mathematics
As we were a tutorial group under the wings of a cosmologist [Andy Albrecht, now at UC Davis http://goo.gl/IePyoh], it was of interest to consider the following. We can observe different quasars (http://en.wikipedia.org/wiki/Quasar) so widely separated that there has not been time for light to travel beteween since the “beginning”. Thus they are truly causally separate phenomena and cannot have had mutually influence in any known way. Yet we find that the detailed aspects of the spectrum of light that they emit adhere to identical mathematical analysis. This should surely give us confidence in the existence of some universal substructure that is mathematical in character. Hence, cosmology is the most vivid expression of the Platonic doctrine, assuming that mathematics is something larger than the causally linked physical universe.
So to make up the balance, the advantages of the Platonic picture of mathematics are obvious. It is simple. It makes the effectiveness of mathematics a thoroughly predictable fact of life. It removes human mathematicians from the centre of the mathematical universes where otherwise they must cast themselves in the role of creators. And it explains how we can be so successful in arriving at mathematical descriptions of those aspects of the physical world which are furthest removed from direct human experience.