Is God a Mathematician_ - Mario Livio [106]
In spite of this pre-eminence, the first significant appraisal of mathematics was occasioned only recently by the advent of non-Euclidean and four-dimensional geometry. That is not to say that the advances made by the calculus, the theory of probability, the arithmetic of the infinite, topology, and the other subjects we have discussed, are to be minimized. Each one has widened mathematics and deepened its meaning as well as our comprehension of the physical universe. Yet none has contributed to mathematical introspection, to the knowledge of the relation of the parts of mathematics to one another and to the whole as much as the non-Euclidean heresies.
As a result of the valiantly critical spirit which engendered the heresies, we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed. Yet this is what Pythagoras would have thought—and Descartes, along with hundreds of other great mathematicians before the nineteenth century. Today mathematics is unbound; it has cast off its chains. Whatever its essence, we recognize it to be as free as the mind, as prehensile as the imagination. Non-Euclidean geometry is proof that mathematics, unlike the music of the spheres, is man’s own handiwork, subject only to the limitations imposed by the laws of thought.
So, contrary to the precision and certitude that are the hallmark of statements in mathematics, here we have a divergence of opinions that is more typical of debates in philosophy or politics. Should we be surprised? Not really. The question of whether mathematics is invented or discovered is actually not a question of mathematics at all.
The notion of “discovery” implies preexistence in some universe, either real or metaphysical. The concept of “invention” implicates the human mind, either individually or collectively. The question therefore belongs to a combination of disciplines that may involve physics, philosophy, mathematics, cognitive science, even anthropology, but it is certainly not exclusive to mathematics (at least not directly). Consequently, mathematicians may not even be the best equipped to answer this question. After all, poets, who can perform magic with language, are not necessarily the best linguists, and the greatest philosophers are generally not experts in the functions of the brain. The answer to the “invented or discovered” question can therefore be gleaned only (if at all) from a careful examination of many clues, deriving from a wide variety of domains.
Metaphysics, Physics, and Cognition
Those who believe that mathematics exists in a universe that is independent of humans still fall into two different camps when it comes to identifying the nature of this universe. First, there are the “true” Platonists, for whom mathematics dwells in the abstract, eternal world of mathematical forms. Then there are those who suggest that mathematical structures are in fact a real part of the natural world. Since I have already discussed pure Platonism and some of its philosophical shortcomings quite extensively, let me elaborate a bit on the latter perspective.
The person who presents what may be the most extreme and most speculative version of the “mathematics as a part of the physical world” scenario is an astrophysicist colleague, Max Tegmark of MIT.
Tegmark argues that “our universe is not just described by mathematics—it is mathematics” [emphasis added]. His argument starts with the rather uncontroversial assumption that an external physical reality exists that is independent of human beings. He then proceeds to examine what might be the nature of the ultimate theory of such a reality (what physicists refer to as the “theory of everything”). Since this physical world is entirely independent of humans, Tegmark maintains, its description must be free of any human