Is God a Mathematician? Read Free Page B

material reality that surrounds us. The working mathematician can be likened to an explorer who sets out to discover the world. One discovers basic facts from experience. In doing simple calculations, for example, one realizes that the series of prime numbers seems to go on without end. The mathematician’s job, then, is to demonstrate that there exists an infinity of prime numbers. This is, of course, an old result due to Euclid. One of the most interesting consequences of this proof is that if someone claims one day to have found the greatest prime number, it will be easy to show that he’s wrong. The same is true for any proof. We run up therefore against a reality every bit as incontestable as physical reality.
    Martin Gardner, the famous author of numerous texts in recreational mathematics, also takes the side of mathematics as a discovery . To him, there is no question that numbers and mathematics have their own existence, whether humans know about them or not. He once wittily remarked: “If two dinosaurs joined two other dinosaurs in a clearing, there would be four there, even though no humans were around to observe it, and the beasts were too stupid to know it.” As Connes emphasized, supporters of the “mathematics-as-a-discovery” perspective (which, as we shall see, conforms with the Platonic view) point out that once any particular mathematical concept has been grasped, say the natural numbers 1, 2, 3, 4,…, then we are up against undeniable facts, such as 3 2 4 2 5 2 , irrespective of what we think about these relations. This gives at least the impression that we are in contact with an existing reality.
    Others disagree. While reviewing a book in which Connes presented his ideas, the British mathematician Sir Michael Atiyah (who won the Fields Medal in 1966 and the Abel Prize in 2004) remarked:
    Any mathematician must sympathize with Connes. We all feel that the integers, or circles, really exist in some abstract sense and the Platonic view [which will be described in detail in chapter 2] is extremely seductive. But can we really defend it? Had the universe been one dimensional or even discrete it is difficult to see how geometry could have evolved. It might seem that with the integers we are on firmer ground, and that counting is really a primordial notion. But let us imagine that intelligence had resided, not in mankind, but in some vastsolitary and isolated jelly-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.
    Atiyah therefore believes that “man has created [the emphasis is mine] mathematics by idealizing and abstracting elements of the physical world.” Linguist George Lakoff and psychologist Rafael Núñez agree. In their book Where Mathematics Comes From, they conclude: “Mathematics is a natural part of being human. It arises from our bodies, our brains, and our everyday experiences in the world.”
    The viewpoint of Atiyah, Lakoff, and Núñez raises another interesting question. If mathematics is entirely a human invention, is it truly universal? In other words, if extraterrestrial intelligent civilizations exist, would they invent the same mathematics? Carl Sagan (1934–96) used to think that the answer to the last question was in the affirmative. In his book Cosmos, when he discussed what type of signal an intelligent civilization would transmit into space, he said: “It is extremely unlikely that any natural physical process could transmit radio messages containing prime numbers only. If we received such a message we would deduce a civilization out there that was at least fond of prime numbers.” But how certain is that? In his recent book A New Kind of Science, mathematical physicist Stephen Wolfram argued that what we call “our