Is God a Mathematician_ - Mario Livio [80]
I feel very happy to find you inclined to transform yourself into a naturalist to observe the phenomena of the arithmetical world. Your doctrine is the same as mine; I believe that numbers and the functions of analysis are not arbitrary products of our mind; I think that they exist outside of us with the same necessary characteristics as the things of objective reality, and that we encounter them or discover them, and study them, just as the physicists, the chemists and the zoologists.
The English mathematician G. H. Hardy, himself a practitioner of pure mathematics, was one of the most outspoken modern Platonists. In an eloquent address to the British Association for the Advancement of Science on September 7, 1922, he pronounced:
Mathematicians have constructed a very large number of different systems of geometry. Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians’ observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics…The function of a mathematician, then, is simply to observe the facts about his own hard and intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics.
Clearly, even with the contemporary evidence pointing to the arbitrary nature of mathematics, the die-hard Platonists were not about to lay down their arms. Quite the contrary, they found the opportunity to delve into, in Hardy’s words, “their reality,” even more exciting than to continue to explore the ties to physical reality. Irrespective, however, of the opinions on the metaphysical reality of mathematics, one thing was becoming obvious. Even with the seemingly unbridled freedom of mathematics, one constraint remained unchanging and unshakable—that of logical consistency. Mathematicians and philosophers were becoming more aware than ever that the umbilical cord between mathematics and logic could not be cut. This gave birth to another idea: Could all of mathematics be built on a single logical foundation? And if it could, was that the secret of its effectiveness? Or conversely, could mathematical methods be used in the study of reasoning in general? In which case, mathematics would become not just the language of nature, but also the language of human thought.
CHAPTER 7
LOGICIANS: THINKING ABOUT REASONING
The sign outside a barber shop in one village reads: “I shave all and only those men in the village who do not shave themselves.” Sounds perfectly reasonable, right? Clearly, the men who shave themselves do not need the services of the barber, and it is only natural for the barber to shave everyone else. But, ask yourself, who shaves the barber? If he shaves himself, then according to the sign he should be one of those he does not shave. On the other hand, if he does not shave himself, then again according to the sign he should be one of those he does shave! So does he or doesn’t he? Much lesser questions have historically resulted in serious family feuds. This paradox was introduced by Bertrand Russell (1872–1970), one of the most prominent logicians and philosophers of the twentieth century, simply to demonstrate that human logical intuition is fallible. Paradoxes or antinomies reflect situations in which apparently acceptable premises