Is God a Mathematician_ - Mario Livio [107]
Tegmark’s argument for a mathematical reality is certainly intriguing, and if it were true, it might have gone a long way toward solving the problem of the “unreasonable effectiveness” of mathematics. In a universe that is identified as mathematics, the fact that mathematics fits nature like a glove would hardly be a surprise. Unfortunately, I do not find Tegmark’s line of reasoning to be extremely compelling. The leap from the existence of an external reality (independent of humans) to the conclusion that, in Tegmark’s words, “You must believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure,” involves, in my opinion, a sleight of hand. When Tegmark attempts to characterize what mathematics really is, he says: “To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them.” But this modern logician is human! In other words, Tegmark never really proves that our mathematics is not invented by humans; he simply assumes it. Furthermore, as the French neurobiologist Jean-Pierre Changeux has pointed out in response to a similar assertion: “To claim physical reality for mathematical objects, on a level of the natural phenomena we study in biology, poses a worrisome epistemological problem it seems to me. How can a physical state, internal to our brain, represent another physical state external to it?”
Most other attempts to place mathematical objects squarely in the external physical reality simply rely on the effectiveness of mathematics in explaining nature as proof. This however assumes that no other explanation for the effectiveness of mathematics is possible, which, as I will show later, is not true.
If mathematics resides neither in the spaceless and timeless Platonic world nor in the physical world, does this mean that mathematics is entirely invented by humans? Absolutely not. In fact, I shall argue in the next section that most of mathematics does consist of discoveries. Before going any further, however, it would be helpful to first examine some of the opinions of contemporary cognitive scientists. The reason is simple—even if mathematics were entirely discovered, these discoveries would still have been made by human mathematicians using their brains.
With the enormous progress in the cognitive sciences in recent years, it was only natural to expect that neurobiologists and psychologists would turn their attention to mathematics, in particular to the search for the foundations of mathematics in human cognition. A cursory glance at the conclusions of most cognitive scientists may initially leave you with the impression that you are witnessing an embodiment of Mark Twain’s phrase “To a man with a hammer, everything looks like a nail.” With small variations in emphasis, essentially all of the neuropsychologists and biologists determine that mathematics is a human invention. Upon closer examination, however, you find that while the interpretation of the cognitive data is far from being unambiguous, there is no question that the cognitive efforts represent a new and innovative phase in the search for the foundations of mathematics. Here is a small but representative sample of the comments made by the cognitive scientists.
The French neuroscientist Stanislas Dehaene, whose primary interest is in numerical cognition, concluded in his 1997 book The Number Sense that “intuition about numbers is thus anchored deep in our brain.” This position is in fact close to that of the intuitionists, who wanted to ground all of mathematics in the pure form of intuition of the