Hamilton’s inspiration

Sir William Rowan Hamilton was “unquestionably Ireland’s greatest man of science” in the opinion of Stuart Hollingdale, author of Makers of Mathematics (Penguin, 1989). Bearing in mind recent posts on this blog and elsewhere about creativity in mathematics, I think one of Hollingdale’s paragraphs is illuminating

Hamilton regarded his discovery of quarternions, after a gestation period of some fifteen years, as his greatest achievement. He describes his ‘flash of inspiration’ in a letter to one of his sons:

On the 6th day of October, which happened to be a Monday, and Council day of the Royal Irish Academy, I was walking to attend and preside, and your mother was walking with me along the Royal Canal; and although she talked with me now and then, yet an undercurrent of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw immediately) of many long years to come of definitely directed thought and work . . . Nor could I resist the impulse - unphilosophical as it may have been - to cut with a knife on the stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols i, j, k: namely i²=j²=k²=ijk=-1, which contains the solution of the problem.

(p.341 ibid.). Hollingdale describes the ideas as “gestating” for fifteen years, exactly as Guy Claxton argues ideas should in Hare Brain Tortoise Mind (Fourth Estate 1998). Hamilton’s own language is illuminating in that he characterises his own role in the process as that of a passive observer: inspiration “flashes”, thought is an “undercurrent”, a “circuit seemed to close”, and a “spark flashed forth”. Immediately it seems like he could perceive where the work would take him, and what would need to be done to make rigorous and clear his inspiration - exactly the transition from “undermind” to “d-mode” Claxton describes.