[T]he visible world supposes an invisible world as its interpreter, and […] in the application of the mathematics themselves there must (if I may venture upon the word) be something meta-mathematical. Though the senses may make known the phenomena, and mathematical methods may arrange them, yet the craving of our nature is not satisfied till we trace in them the projection of ourselves, of that which is divine within us
From a lecture to astronomy students given by William Rowan Hamilton in 1833 (Graves, 1882-9, vol. 2, p. 68).
Hamilton differed from other mathematicians of his time in his focus on abstract mathematical laws, and in finding inspiration in idealist philosophers such as Immanuel Kant and Samuel Taylor Coleridge, as can be seen from the quote. “Hamilton often argued that certain metaphysical views were the primary motivation for his work in mathematics” (Attis, 2004).
Hamilton had started exploring idealism in 1830 by reading the works of Berkeley. In 1831 he was reading Coleridge, and was particularly impressed with The Friend and Aids to Reflection. Coleridge was a supporter of Kantism, and Hamilton started reading Kant’s Critique of Pure Reason in October 1831. (He met Coleridge in 1832.) Hamilton described his reading of Kant as recognition rather than discovery – Kant’s ideas dovetailed with his own. By 1834 he was a confirmed idealist.
His idealism led Hamilton to see mathematics as discovering the internal laws of the mind, which would then (since they were what shaped our experience) correspond with the external laws of nature. (For Kant, time and space are categories of the mind, with time related to mathematics.) The aim of science was to “refine the phenomenal world, that so we may behold as one, and under the forms of our own understanding, what had seemed to be manifold and foreign” (letter to Logan, 27 June 1834).
Thus, even in work on optics or dynamics, Hamilton concentrated on the abstract over physical theories, as pointed out in a 1869 piece celebrating Hamilton’s life. While others were disputing the value of mathematics to exercise the mind:
Sir William Rowan Hamilton, the Dublin Professor of Astronomy, was constructing out of pure metaphysics some of the most marvellous mathematical edifices of which this century can boast
The most marvellous of those mathematical edifices is probably quaternions, discovered in 1843.
The piece also mentions Boole operating in the opposite direction, “indirectly proving that two out of three of the principles of algebra were also principles of his logical calculus” and showing the intimate connection between classical logic and mathematics. This “reconciliation of the two great scientific factions – the metaphysical and the mathematical” was taking place in Dublin and in Cork at the same time. Sadly, their correspondence was intermittent and awkward, and they never worked together.
Sources and Further Reading
David A. Attis (2004) HAMILTON, Willam Rowan in Thomas Duddy (ed) The Dictionary of Irish Philosophers.
O’Cairbre, Fiacre (2000) William Rowan Hamilton (1805-1865): Ireland’s greatest mathematician. Riocht Na Midhe, 11. pp. 124-150. (Maynooth University eprints).