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This question ended up longer than I intended (though most of the bulk is interesting remarks by Hamilton), so I thought it might be good to include my question at the beginning before the admittedly-lengthy background:

Question: Why did Hamilton view the scalar part of a quaternion as representing time? Does the modern viewpoint of quaternions in physics admit an interpretation that involves time but does not require relativity and related thoughts as a prerequisite?

Background: It strikes me as remarkably ahead-of-his-time that Hamilton preferred to think, or perhaps insisted on thinking, of algebra as the study of a time variable. In fact, while I'm certainly no math-historian, by my reading he is actually quite uncomfortable with the relatively newfound spread of abstraction in algebra, particularly in terms of imaginary numbers. He laments on the chasm between this abstraction and the firm footing of science:

Yet a natural regret might be felt, if such were the destiny of Algebra; if a study, which is continually engaging mathematicians more and more, and has almost superseded the Study of Geometrical Science, were found at last to be not, in any strict or proper sense, the Study of a Science at all....

...and later...

The author acknowledges with pleasure that he agrees with M. Cauchy, in considering every (so-called) Imaginary Equation as a symbolic representation of two separate Real Equations: but he differs from that excellent mathematician in his method generally, and especially in not introducing the sign $\sqrt{-1}$ until he has provided for it, by his Theory of Couples, a possible and real meaning, as a symbol of the couple (0, 1).

As a solution to his quandry, Hamilton postulates that the interpretation of algebra as the study of time is the way to base algebra with imaginary numbers on a scientific footing, writing:

It is the genius of Algebra to consider what it reasons on as flowing, as it was the genius of Geometry to consider what it reasoned on as fixed.

In his treatise on the subject: "Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as The Science of Pure Time," he develops a tremendous amount of basic algebra (from addition and ordering to indeterminate forms and exponentiation) through this lens. The sticky part is that he does not seem (to me, at least) to resolve the issue at hand; that of providing an intuitive formulation of algebra in which one can relate time and imaginary numbers, at least beyond that of Cauchy's theory of couples referenced above. And yet he himself, however, declares victory on the matter, writing that this "Theory of Couples is published to make manifest that hidden meaning." He is so taken by this point of view that he later interprets quaternions as a "scalar plus vector" as a "time plus space" element of spacetime:

Time is said to have only one dimension, and space to have three dimensions. […] The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions.

Ever since Einstein and Minkowski (and others), it is quite commonplace to think in terms of spacetime (and indeed the concept apparently dates back to d'Alembert in 1754), but without relativity/Lorenz metrics/etc. at one's disposal, it is striking how dedicated Hamilton was to the point of view of relating time and imaginary quantities.

Question (redux): It is really Hamilton's strikingly-modern interpretation of the scalar part of a quaternion as representing time that is the basis for this question. Why did he do this? Does the modern viewpoint of quaternions in physics admit an interpretation that involves time but does not require relativity and related thoughts as a prerequisite?

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Are you curious about this because of the recent reinterpretation of the "Mad Tea Party" of Alice in Wonderland as a critique of Hamilton's quaternions? –  stankewicz Dec 6 '10 at 18:01
To save others the trouble of googling, Keith Devlin writes on Melanie Bayley's take on Alice here maa.org/devlin/devlin_03_10.html and there's a copy of Bayley's original article here mssia.wordpress.com/2010/03/09/alices-algebra –  j.c. Dec 7 '10 at 1:39
Is there a modern viewpoint of quaternions in physics that you have in mind? There are many uses in physics if one allows that SU(2) is the unit quaternions, but I don't see these as directly admitting interpretations involving time. –  j.c. Dec 7 '10 at 1:45
I don't think it is correct to think of Hamilton's idea as "ahead of his time" just because it formally resembles Minkowski's idea of a spacetime. Hamilton, and several generations of Irish mathematicians after him, tried hard to find possible applications of quaternions. Why did he do this? Guinness and Whiskey, I would guess. –  Franz Lemmermeyer Dec 7 '10 at 15:47
It would be good to say why you cite d'Alembert in 1754. On p. 1010 of volume 4 of Diderot's Encyclopedie (1754), d'Alembert writes "A clever gentelman with whom I am acquainted believes that nevertheless, one could view duration as a fourth dimension and that the product time by solidity would be somehow a product of four dimensions." I learned this from pp. 5--6 of Lang's Calculus of Several Variables. Lang remarked that the clever gentelman is probably d'Alembert himself, who was hesitant to attach himself too closely to (quoting Lang) "what must have been at the time a far out idea". –  KConrad Feb 27 '11 at 6:22
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1 Answer

I believe that he may have been influenced by Kant. According to a quote from this website: "Kant maintains that geometry discovers the universal laws of space, and algebra discovers the universal laws of time. Space and time are "pure intuitions" by which perception can take place, so they are a priori and universal."

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Thanks. I had found this connection as well, but didn't (maybe couldn't) track down anything more precise. For example, it seems like interpreting Hamiltonian's scalar as a 4th spatial dimension would equally satisfy these intuitions. –  Cam McLeman Dec 6 '10 at 17:24
Quaternions can be used to describe rotations in three dimensional space in which the real term is the cosine of half the angle and the other three coordinates represent the axis of rotation so that could be motivation for thinking of three of the coordinates spatial and the fourth as non-spatial. –  Kristal Cantwell Dec 6 '10 at 20:26
0 Try these: Bloor, D. (1981), 'Hamilton and Peacock on the Essence of Algebra', in H. Mehrtens, H. Bos and I. Schneider (eds.), Social History of Nineteenth Century Mathematics, Boston: Birkhäuser, 202-32. Hendry, J. (1984), 'The evolution of William Rowan Hamilton's view of algebra as the science of pure time', Studies in History and Philosophy of Science, 15 (1), 63-81. –  David Corfield Dec 7 '10 at 10:41
I'd found the latter, but not the former. Thanks for the reference. –  Cam McLeman Dec 7 '10 at 14:09
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