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  • Hardy's Mathematics
  • Tim Armstrong (bio)

In his poem "He Revisits His First School," Thomas Hardy imagines a mathematical haunting at the Stinsford and Bockhampton National School:

Yea, beglimpsed through the quaint quarried glassOf green moonlight, by me greener made,When they'd cry, perhaps, "There sits his shadeIn his olden haunt—just as he was     When in Walkingame he     Conned the grand Rule-of-Three     With the bent of a bee."

(VP, p. 512)1

The reference is to Francis Walkingame's famous textbook The Tutor's Assistant: being a compendium of practical arithmetic, published in many editions with updates by different hands from 1751, and often handed down within families.2 In his autobiography Hardy remembers that he was still using the textbook in his three years at Isaac Last's Academy, aged 13 to 16: "his course of instruction included elementary drawing, advanced arithmetic, geometry, and algebra, in which he was fairly good, always saying that he found a certain poetry in the rule for the extraction of the cube-root, owing to its rhythm, and in some of the 'Miscellaneous Questions' of Walkingame. In applied mechanics he worked completely through Tate's Mechanics and Nesbitt's Mensuration."3

What would it mean to take that "certain poetry in the rule" seriously? My proposition is that there is something in mathematics which appealed to Hardy, perhaps above all in the mapping of relations which it offered between placeholders (by which I mean symbols, and by extension words, conveying unspecified values). Verse form is one example of such relations, and Hardy's relation to what he called the "verse skeletons" of meter and stanza form has often been discussed.4 One can also relate mathematics to the peculiarly abstract quality of Hardy's work. There are poems in which people have names like "who?" "Him," "her," or "a man," "the woman" (as in "A Man Was Drawing Near to Me"). These pronouns are not unusual in themselves, but in Hardy they often seem to carry an abstract value, hinting at a particularity negated in seeking more general formulas. In what follows, I will concentrate on two [End Page 557] topics: firstly, ratio or the "Rule-of-Three" (that is, with the mechanics of calculation and the question of placeholders like x and y in mathematics); and secondly the question of series and sets, including issues of ordering, inclusion, and multiplicity (mathematically, this includes the idea of ordinality, or sets that have an order like the natural numbers 1, 2, 3, 4; and of cardinality, which is a measure of the total size of finite and infinite sets).

In an earlier article I tentatively attempted to link Hardy to the mathematics of series and infinitesimals—a branch of mathematics undergoing a revolution in his lifetime in the hands of Richard Dedekind, Georg Cantor, and others.5 Their work addressed the ancient problem of the number line, or "continuum" as mathematicians were beginning to call it, and implicitly the problem of formalizing the continuity of nature. But that material was very much at the borders of his later philosophical reading, rather than being something to which he had a direct relation. What I want to do in this essay is look more closely at the school mathematics Hardy actually described, but also reflect on the possible relations between his mathematics and Alain Badiou's thinking. Badiou is important here because of the foundational status he assigns to modern set theory, and to the idea that mathematics tells us about the world.

Hardy's Schoolboy Mathematics

Hardy's schooling has been described by Michael Millgate and others, and its background in national and local education has been described by Jane Mattison. Mattison describes midcentury educational problems, especially among the poor: low attendance, untrained teachers, and a resistance to science. The inspectorate set up in 1839 by the Education Select Committee of the Privy Council reported that Dorset schools were particularly weak at arithmetic. Hardy went to the village school at eight, and then to Isaac Last's Nonconformist day school in Dorchester, avoiding both the old grammar school (Hardye's, in decline) and William Barnes's school, which may...

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