Enumerative Induction in Mathematics
From Firenze University Press Journal: Journal for the Philosophy of Mathematics
Alan Baker, Swarthmore College
In 1919, Hungarian mathematician George Polya was pursuing questions concerning prime factorization andbegan noting down, for each of the first few natural numbers, whether a given number had an even number ofprime factors or an odd number of prime factors. What Polya was counting was not distinct prime factors, butjust the raw number of prime factors. So, for example,10=2.5has an even number of prime factors,11has anodd number, and12=2.2.3has an odd number. Next, Polya recorded, for each given number,n, how manynumbers in the set{1,2,3,…n}have an odd total number of prime factors and how many have an even total number. Looking at all the numbers up to 100, Polya noticed that at every point at least half of members of theset{1,2,3,…n}had an odd total number of prime factors. Intrigued, Polya extended his calculations up ton= 1,500 and found that the pattern continued to hold. At this point, Polya conjectured in print that this holdsuniversally. In other words, for alln, at least half of the numbers less thannhave an odd total number of primefactors. This came to be known asPolya’s Conjecture(PC).Polya presumably made the conjecture because he believed that it was likely to be true. What grounds didhe have for this belief? According to the narrative outlined above, Polya’s belief in the truth of PC was basedexclusively on enumerative induction. 1,500 instances from a larger domain were examined and found to fitthe hypothesis, and on this basis the hypothesis was conjectured to hold universally. As such, this seems likea particularly pure case of enumerative induction, with few extraneous complicating factors. But was Polyarationalin taking the results of the first 1,500 cases to lend support to the universal hypothesis in this way?More generally, can enumerative induction lend genuine support to mathematical claims? It is these questionsthat I shall be taking up in what follows.
Scene Setting
Before proceeding it may be helpful in clarifying the scope of our inquiry to distinguish the questions I am askingfrom some other, related (and also interesting) questions, and also to make explicit several presuppositions thatI will be taking for granted.Firstly, I am not asking whether enumerative induction in mathematics can yieldknowledgeof general mathematical claims. My focus here is on justified belief, not knowledge, and I will be taking no stand on thepotential link between enumerative induction and knowledge. Secondly, I am not asking whether enumerative induction in mathematics can (or should) ever be on a par with deductive proof. As it pertains to mathematicalmethodology, everything I say is compatible with deductive proof continuing to be the gold standard fordemonstrative mathematical reasoning.1Thirdly, I am not asking whether enumerative induction can providecompellinggrounds for belief in a general mathematical claim. Rather, I am interested in whether enumerativeinduction can achieve the more modest goal of providing some positive support.As for presuppositions, the work being done in this paper will be carried out against the background ofa general anti-scepticism about induction. In other words, I will be presuming that enumerative induction inthe empirical sciences is generally (other things being equal,ceteris paribus) a rationally acceptable tool foracquiring justified belief. Also, although I will continue to talk sometimes about “enumerative induction inmathematics” my focus will almost exclusively be on enumerative induction in number theory. The examples Iwill be discussing will all involve claims about the totality of the natural numbers. Putting together these twopoints, the question I asked at the end of Section 1 can be precisified as follows: is there something distinctivelyproblematic about using enumerative induction to bolster beliefs in arithmetical claims?
DOI: https://doi.org/10.36253/jpm-2931
Read Full Text: https://riviste.fupress.net/index.php/jpm/article/view/2931
