Is God a Mathematician_ - Mario Livio [109]
The cognitive scientists make a fairly strong case for an association of our mathematics with the human mind, and against Platonism. Interestingly, though, what I regard as possibly the strongest argument against Platonism comes not from neurobiologists, but rather from Sir Michael Atiyah, one of the greatest mathematicians of the twentieth century. I did, in fact, mention his line of reasoning briefly in chapter 1, but I would now like to present it in more detail.
If you had to choose one concept of our mathematics that has the highest probability of having an existence independent of the human mind, which one would you select? Most people would probably conclude that this has to be the natural numbers. What can be more “natural” than 1, 2, 3,…? Even the German mathematician of intuitionist inclinations Leopold Kronecker (1823–91) famously declared: “God created the natural numbers, all else is the work of man.” So if one could show that even the natural numbers, as a concept, have their origin in the human mind, this would be a powerful argument in favor of the “invention” paradigm. Here, again, is how Atiyah argues the case: “Let us imagine that intelligence had resided, not in mankind, but in some vast solitary 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.” In other words, Atiyah is convinced that even a concept as basic as that of the natural numbers was created by humans, by abstracting (the cognitive scientists would say, “through grounding metaphors”) elements of the physical world. Put differently, the number 12, for instance, represents an abstraction of a property that is common to all things that come in dozens, in the same way that the word “thoughts” represents a variety of processes occurring in our brains.
The reader might object to the use of the hypothetical universe of the jellyfish to prove this point. He or she may argue that there is only one, inevitable universe, and that every supposition should be examined in the context of this universe. However, this would be tantamount to conceding that the concept of the natural numbers is in fact somehow dependent on the universe of human experiences! Note that this is precisely what Lakoff and Núñez mean when they refer to mathematics as being “embodied.”
I have just argued that the concepts of our mathematics originate in the human mind. You may wonder then why I had insisted earlier that much of mathematics is in fact discovered, a position that appears to be closer to that of the Platonists.
Invention and Discovery
In our everyday language the distinction between discovery and invention is sometimes crystal clear, sometimes a bit fuzzier. No one would say that Shakespeare discovered Hamlet, or that Madame Curie invented radium. At the same time, new drugs for certain types of diseases are normally announced as discoveries, even though they often involve the meticulous synthesis of new chemical compounds. I would like to describe in some detail a very specific example in mathematics, which I believe will not only help clarify the distinction between invention and discovery but also yield valuable insights into the process by which mathematics evolves and progresses.
In book VI of The Elements, Euclid’s monumental