On Nondeductive Mathematics
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On Nondeductive Mathematics ˇ IKIC´ ZVONIMIR S
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o justify nondeductive methods in mathematics, Brown [2, 3] introduces a thought experiment that assumes (for the sake of argument) that the first principles of quantum mechanics are just as certain as the first principles of Peano arithmetic. In both systems we should try to construct derivations for as many propositions as possible. But then he asks, what about the rest? In quantum mechanics it is obvious that we use nondeductive methods (experiments, conjectures, statistical arguments, etc. to get the rest), and Brown thinks that we should not treat Peano arithmetic differently. We use inductive methods to enlarge what we know about the physical realm, and we should use them in the same way to enlarge what we know about the mathematical realm. Their epistemic situations are the same, so we should have the same epistemic outlook for each. Brown offers a concrete example. He notes that the apparent randomness of the primes is reflected in the randomness of the size of gaps between them. Since there are infinitely many primes (and consequently gaps), he thinks that we can expect the number of gaps of size 2 to occur infinitely often and that in this way, the twin primes conjecture is justified by this simple and compelling inductive argument. Let me first criticize this concrete example. In 1849, Gauss wrote that as a boy, he had pondered the problem of estimating
T
pðx Þ ¼ #fp x j p is primeg and had come to the conclusion that at around x, the primes occur with density 1=log x (where log is the natural logarithm). He concluded that pðx Þis approximated by Z x dt x Liðx Þ ¼ log t log x 2 (where f ðx Þ gðx Þ means that f ðx Þ=gðx Þ ! 1 as x approaches infinity). In 1896, Hadamard and de la Valle´e Poussin independently proved Gauss’s conjecture, which has since been known as the prime number theorem: pðx Þ Liðx Þ
x : log x
It follows that the average gap between primes less than x is log x: average gap ¼
â
X pðn þ 1Þ pðnÞ x ¼ log x p ð x Þ x= log x pðnÞ \ x
(where pðnÞ is the nth prime). The average gaps rise to infinity as x increases to infinity. The individual gaps may be dispersed in such a
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way that there are infinitely many gaps of length 2, or they may be much more concentrated, with only finitely many gaps of length 2. The twin primes conjecture bets on the first possibility, but the log x average (i.e., 1=log x probability) has nothing to do with that. Since the average spaci
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