A view of the bridge at Broombridge, featuring the plaque commemorating the discovery of quaternions by Hamilton.

William Rowan Hamilton and Kant

I have written before about Hamilton’s interest in philosophy, particularly Berkeley and idealism. This post will give some detail of William Rowan Hamilton’s interest in and understanding of Kant.

In this post on the discovery of quaternions I mentioned that the project which had his children asking each morning “Well, Papa, can you multiply triplets”? started with a paper : “Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time”, read to the Royal Irish Academy on November 4th, 1833, and June 1st, 1835, and published in 1837 (pdf from TCD).

Those of a philosophical bent may hear echoes of Kant in that long title. Kant argued that space and time were “pure intuitions”: not concepts in the mind, but existing prior to experience and structuring our experience. Mathematics was the expression of these pure intuitions, with geometry relating to space and algebra to time. This similarity is underscored by Hamilton’s assertion in the first section of the work that:

“The argument for the conclusion that the notion of time may be unfolded into an independent Pure Science, or that a Science of Pure Time is possible, rests chiefly on the existence of certain a priori intuitions, connected with that notion of time, and fitted to become the sources of a pure Science; and on the actual deduction of such a Science from those principles, which the author conceives that he has begun. “

The obvious explanation is that Hamilton was inspired by Kant in taking this view. However, as Michael J. Crowe points out.1 Hamilton does not mention Kant in the Essay. In addition, even earlier in 1827 Account of a Theory of Systems of Rays(pdf, TCD) Hamilton wrote that “The sciences of Space and Time (to adopt here a view of Algebra which I have elsewhere ventured to propose) became intimately intertwined and indissolubly connected with each other.” This expresses a similar view to the 1830s paper regarding the roots of algebra and was written, according to Crowe, four years before Hamilton started reading Kant.

It seems Hamilton independently came to a similar conclusion as Kant regarding the basis of mathematics in the structure of the mind. In 1834 in a letter to Lord Adare, Hamilton states that he enjoys reading Kant but “a large part of my pleasure consists in recognising through Kant’s works, opinions, or rather views, which have been long familiar to myself, although far more clearly and systematically expressed”. Hamilton also suggests Kant owes a debt to George Berkeley, with whose work Hamilton was well acquainted (and which perhaps had led his mind in this direction): “Kant is, I think, much more indebted than he owns, or, perhaps knows, to Berkeley, whom he calls by a sneer, ‘GUTEM Berkeley’. . . as it were, ‘good soul, well meaning man’”2.

Of course, it is possible that Hamilton picked up elements of Kant’s thought in the intellectual circles in which he moved, perhaps without even being aware of it. It is in any case interesting that this philosophical insight led to a long mathematical quest, in which Hamilton persisted despite ongoing failure and which finally bore fruit on the 16th October 1843 with the discovery of quaternions.

A view of the bridge at Broombridge, featuring the plaque commemorating the discovery of quaternions by Hamilton.Featured Image: Broombridge. (c) William Murphy/Flickr (CC BY-SA 2.0)


  1. Michael J. Crowe (1994) A History of Vector Analysis: The Evolution of the Idea of a Vectorial System Courier Corporation, pp. 24-5.
  2. Letter quoted in R. S. Ball (1895) The Great Astronomers available online at Project Gutenberg
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