Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

338 jeremy avigadwish to pause here to reflect on what it would take to verify these inferencesaxiomatically.Before discussing that issue, let us consider a second example. The followingfact comes up in a discussion of the Ɣ function in Whittaker and Watson(1996).Lemma 1. For every complex number z, the series ∑ ∞n=1 | log(1 + z n ) − z n | converges.³Proof. It suffices to show that for some N, thesum ∑ ∞n=N+1 | log(1 + z n ) − z n | isbounded by a convergent series. But when N is an integer such that |z| ≤ 1 2 N,we have, if n > N,(∣∣log 1 + z )n− z ∣ =n ∣ − 1 2z 2n + 1 z 3 ∣ ∣∣∣2 3 n − ... 3≤ |z|2n 2 {1 +| z n |+|z2 n 2 |+... }≤ 1 N 2 {1 + 1 4 n 2 2 + 1 }2 + ... 2≤ 1 N 22 n . 2Since the series ∑ ∞n=N+1 {N 2 /(2n 2 )} converges, we have the desired conclusion.□Once again, the text is only slightly modified from that of Whittaker andWatson (1996), and the chain of inequalities is exactly the same. The equalityinvolves a Taylor series expansion of the logarithm. The first inequalityfollows from properties of the absolute value, such as |xy| =|x|·|y| and|x + y| ≤|x|+|y|, while the second inequality makes use of the assumption|z| < 1 N and the fact that n > N.2Inequalities like these arise in all branches of mathematics, and each disciplinehas its own bag of tricks for bounding the expressions that arise (see,for example Hardy et al., 1988; Steele, 2004). I have chosen the two examplesabove because they invoke only basic arithmetic reasoning, against some generalbackground knowledge. At present, providing enough detail for a computerto verify even straightforward inferences like these is a burdensome chore (seeAvigad et al., 2007).In sufficiently restricted contexts, in fact, decision procedures are oftenavailable. For example, in 1929 Presburger showed that the theory of the³ Here we are taking the principal value of log(1 + z n ).