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By Root 689 0

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 mathematics” may represent just one possibility out of a rich variety of “flavors” of mathematics. For instance, instead of using rules based on mathematical equations to describe nature, we could use different types of rules, embodied in simple computer programs. Furthermore, some cosmologists have recently discussed even the possibility that our universe is but one member of a multiverse—a huge ensemble of universes. If such a multiverse indeed exists, would we really expect the other universes to have the same mathematics?

Molecular biologists and cognitive scientists bring to the table yet another perspective, based on studies of the faculties of the brain. To some of these researchers, mathematics is not very different from language. In other words, in this “cognitive” scenario, after eons during which humans stared at two hands, two eyes, and two breasts, an abstract definition of the number 2 has emerged, much in the same way that the word “bird” has come to represent many two-winged animals that can fly. In the words of the French neuroscientist Jean-Pierre Changeux: “For me the axiomatic method [used, for instance, in Euclidean geometry] is the expression of cerebral faculties connected with the use of the human brain. For what characterizes language is precisely its generative character.” But, if mathematics is just another language, how can we explain the fact that while children study languages easily, many of them find it so hard to study mathematics? The Scottish child prodigy Marjory Fleming (1803–11) charmingly described the type of difficulties students encounter with mathematics. Fleming, who never lived to see her ninth birthday, left journals that comprise more than nine thousand words of prose and five hundred lines of verse. In one place she complains: “I am now going to tell you the horrible and wretched plague that my multiplication table gives me; you can’t conceive it. The most devilish thing is 8 times 8 and 7 times 7; it is what nature itself can’t endure.”

A few of the elements in the intricate questions I have presented can be recast into a different form: Is there any difference in basic kind between mathematics and other expressions of the human mind, such as the visual arts or music? If there isn’t, why does mathematics exhibit an imposing coherence and self-consistency that does not appear to exist in any other human creation? Euclid’s geometry, for instance, remains as correct today (where it applies) as it was in 300 BC; it represents “truths” that are forced upon us. By contrast, we are neither compelled today to listen to the same music the ancient Greeks listened to nor to adhere to Aristotle’s naïve model of the cosmos.

Very few scientific subjects today still make use of ideas that can be three thousand years old. On the other hand, the latest research in mathematics may refer to theorems that were published last year, or last week, but it may also use the formula for the surface area of a sphere proved by Archimedes around 250 BC! The nineteenth century knot model of the atom survived for barely two decades because new discoveries proved elements of the theory to be in error. This is how science progresses. Newton gave credit (or not! see chapter 4) for his great vision to those giants upon whose shoulders he stood.