William Rowan Hamilton and Algebra as Time

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were the destiny of Algebra; if a study, which is continually engaging mathematicians more and more, and has almost superseded the Study of Geometrical Science, were found at last to be not, in any strict or proper sense, the Study of a Science at all....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

• 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
• 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
• (Upvoting the following would be useful: the maa-link at 2010-12-07 01:39:33Z seems defunct. I am quite sure that the content in question is at maa.org/external_archive/devlin/devlin_03_10.html.) – Peter Heinig Feb 24 at 17:38

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."

• 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

Hamilton had started down this very abstract path of showing that geometry described space and algebra described time, but was forced to abandon it after realizing it couldn't work. He was then attracted by the notion of algebra as a language — and specifically a system of symbols. If you read on in the source for that quote about "space plus time", he summarizes the whole idea with this couplet:

And how the One of Time, of Space the Three,
Might in the Chain of Symbol girdled be.

He thought of the scalar part of the quaternion and the "vector" part as related but separate symbols, just as he considered geometry and algebra to be related but separate. He was not unifying them.

Thus, it seems to me that this question rests on two mistaken notions. First, it doesn't give enough credit to the long history of dynamics and the treatment of space and time on similar (though not precisely equal) footings — or even Hamilton's own important place in that history. Second, it gives too much credit to the fact that the separate components of a quaternion are all bound up in a single object, while ignoring the fact that they are fundamentally different types of elements within that object. That is, Hamilton was trying to draw parallels between space and time, but he was still clearly trying to keep them separate. So I will argue that Hamilton's interpretation is actually not strikingly modern, because it was not unifying space and time. As for its place in modern physics, I think this answer on physics.se explains it in pretty good detail.

While Einstein and Minkowski famously treated space and time as four-dimensional, this was not an innovation that they introduced; it already had a long history in physics. Instead, their innovation was to unify space and time, so that they can transform into one another via what we now call a Lorentz transformation.(1) Prior to them (and Poincaré, Lorentz, and maybe even FitzGerald), space and time were always kept quite distinct; they could be considered at the same time, but would never mix. So we can distinguish between the ancient parallel treatment of space and time, as opposed to the modern unified treatment of spacetime.

And this restricted combination had a long history already, starting at least with Descartes who treated time with its "durations" in the same way as space with its "extensions" — which are notions we now call dimensions. Of course, Newton, Kepler, Galileo (and surely the list goes on) used four coordinates for space and time. KConrad's comment points out the d'Alembert had already been quite explicit about combining space and time into four dimensions, and even taking their product — which makes sense since his operator combines them as in $$\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2$$. But of course, Lagrange is the one we now think of as Hamilton's most direct influence.(2) And Lagrangian mechanics are much like Hamilton's own mechanics in that they are all about the relation between evolution through time and space.

(1) Note that Lorenz and Lorentz are two different people. Though both names appear in relativity and closely related fields. Lorentz is the one you want to associate with metrics.

(2) That's not to dispute Kristal's assertion that Kant influenced Hamilton — his "Pure Time" essay reads like a text generator that was fed the Critique of Pure Reason. But Kant was in no way combining the two — never mind aunifying them. As far as I can tell, the reason Kant spoke about both of them was because he regarded them as distinct.