Simon Weisgerber

One of the general criteria G. H. Hardy identifies and discusses in his famous essay A Mathematician’s Apology (Cambridge University Press, Cambridge, 1940) by which a mathematician’s patterns must be judged is seriousness. This article focuses on one of Hardy’s examples of a non-serious theorem, namely that 8712 and 9801 are the only numbers below 10000 which are integral multiples of their reversals, in the sense that 8712 = 4·2178, and 9801 = 9·1089. In the context of a discussion of generality, which he considers an essential quality of seriousness, he explains that there is nothing in this example which “appeals much to a mathematician” and that it is “not capable of any significant generalization.” Interestingly, since the publication of the Apology, more than a dozen papers—including one by the renowned mathematician Neil Sloane—have been published that discuss generalizations of Hardy’s example. By identifying the most important aspect of Hardy’s notion of generality, it is argued that, contrary to the views of several researchers, Hardy’s claim regarding the non-capability of any significant generalization is still tenable. Furthermore, this case study is presented and discussed as an example of the multifaceted nature of mathematical interest.

A passage from Jody Azzouni’s article “The Algorithmic-Device View of Informal Rigorous Mathematical Proof” in which he argues against Hamami and Avigad’s standard view of informal mathematical proof with the help of a specific visual proof of 1/2+1/4+1/8+1/16+⋯=1 is critically examined. By reference to mathematicians’ judgments about visual proofs in general, it is argued that Azzouni’s critique of Hamami and Avigad’s account is not valid. Nevertheless, by identifying a necessary condition for the visual proof to be considered a proper proof in the first place, and suggesting an appropriate way to establish its correctness, it is shown how Azzouni’s assessment of the epistemic process associated with the visual proof can turn out to be essentially correct. From this, it is concluded that although visual proofs do not constitute counterexamples to the standard view in the sense suggested by Azzouni, at least the visual proof mentioned above shows that this view does not cover all the ways in which mathematical truth can be justified.

A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly within the framework of progress Maddy offers, is developed with a special focus on mathematicians’ peer-review practice.