# Value Judgments in Mathematics: G. H. Hardy and the (Non-)seriousness of Mathematical Theorems

## TL;DR

This paper examines G. H. Hardy's view on mathematical seriousness using an example of non-serious theorems and explores whether generalizations of these theorems undermine his claim.

## Contribution

The paper argues that Hardy's claim about the non-generalizability of certain theorems remains valid despite later mathematical generalizations.

## Key findings

- Hardy's example of non-serious theorems (8712 and 9801) is revisited in light of later generalizations.
- The paper concludes that Hardy's notion of generality remains a valid criterion for mathematical seriousness.
- The case study illustrates the complex nature of mathematical interest and value judgments.

## Abstract

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 \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$8712=4\cdot 2178$$\end{document}8712=4·2178, and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$9801=9\cdot 1089$$\end{document}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.