I came across an interesting passage from Prof. Paul Davies’ book The Mind of God which I thought would be worth sharing. In this section of the book, Davies is discussing the apparent independent, stable, ‘reality’ of the landscape of mathematics – the so-called ‘Platonic Realms’, or what Rudy Rucker labels the ‘Mindscape’ – and the mystery of why and how the human brain has evolved extraordinary abilities that allow comprehension of seemingly useless (in an evolutionary sense) concepts such as abstract mathematics:
The mystery becomes even deeper when we take account of the existence of mathematical and musical geniuses, whose prowess in these fields is orders of magnitude better than that of the rest of the population. The astonishing insight of mathematicians such as Gauss and Riemann is attested not only by their remarkable mathematical feats (Gauss was a child prodigy and also had a photographic memory), but also by their ability to write down theorems without proof, leaving later generations of mathematicians to struggle over the demonstrations. How these mathematicians were able to come up with their results “ready-made”, when the proofs often turned out to involve volumes of complex mathematical reasoning, is a major puzzle.
Probably the most famous case is that of the Indian mathematician S. Ramanujan. Born in India in the late nineteenth century, Ramanujan came from a poor family and had only a limited education. He more or less taught himself mathematics and, being isolated from mainstream academic life, he approached the subject in a very unconventional manner. Ramanujan wrote down a great many theorems without proof, some of them of a very peculiar nature that would not normally have occurred to more conventional mathematicians. Eventually some of Ramanujan’s results came to the attention of Hardy, who was astonished. “I have never seen anything in the least like them before,” he commented. “A single look at them is enough to show that they could only be written down by a mathematician of the highest class.” Hardy was able to prove some of Ramanujan’s theorems by deploying the full range of his own considerable mathematical skills, but only with the greatest difficulty. Other results defeated him completely. Nevertheless, he felt they must be correct, for “no one would have the imagination to invent them”. Hardy subsequently arranged for Ramanujan to travel to Cambridge to work with him. Ramanujan unfortunately suffered from culture shock and medical problems, and he died prematurely at the age of only thirty-three, leaving a vast stock of mathematical conjectures for posterity. To this day nobody really knows how he achieved his extraordinary feats. One mathematician commented that the results just seemed “to flow from his brain” effortlessly. This would be remarkable enough in any mathematician, but in one who was largely unfamiliar with conventional mathematics it is truly extraordinary. It is very tempting to suppose that Ramanujan had a particular faculty that enabled him to view the mathematical Mindscape directly and vividly, and pluck out ready-made results at will.
Scarcely less mysterious are the weird cases of so-called lightning calculators – people who can perform fantastic feats of mental arithmetic almost instantly, without the slightest idea of how they arrive at the answer. Shakuntala Devi lives in Bangalore in Inida but regularly travels the world, amazing audiences with feats of mental arithmetic. On one memorable occasion in Texas she correctly found the twenty-third root of a two-hundred-digit number in fifty seconds!
Even more peculiar, perhaps, are the cases of “autistic savants”, people who are mentally handicapped and may have difficulty performing even the most elementary formal arithmetic manipulations, but who nevertheless possess the uncanny ability to produce correct answers to mathematical problems that appear to ordinary people to be impossibly hard. Two American brothers, for instance, can consistently outdo a computer in finding prime numbers even though they are both mentally retarded. In another case, featured on British television, a handicapped man correctly and almost instantly gave the day of the week when presented with any date, even from another century!
We are, of course, used to the fact that all human abilities, physical and mental, show wide variations. Some people can jump six feet off the ground, whereas most of us can manage barely three. But imagine someone coming along and jumping sixty feet, or six hundred feet! Yet the intellectual leap represented by mathematical geniuses is far in excess of these physical differences.