The Larson Research Center

One of the most interesting of all biographies in math and science is that of Sir William Rowan Hamilton. David Hestenes relates how his life’s work in the 18th Century is misunderstood, because it is often painted in terms of his fascination with his quaternions, which were historically eclipsed by Heaviside’s and Gibb’s vector analysis. Indeed, until recently, quaternions had sunk into obscurity. Hestenes writes:

…because for the last twenty years of his life, Hamilton concentrated all his enormous mathematical powers on the study of quaternions in, as [E .T.] Bell would have it, the quixotic belief that quaternions would play a central role in the mathematics of the future. Hamilton’s Judgement was based on a new and profound insight into the relation between algebra and geometry. Bell’s evaluation [see his sketch entitled “An Irish Tragedy”] was made by surveying the mathematical literature nearly a century later. But union with Grassman’s algebra puts quaternions in a different perspective. It may yet prove true that Hamilton looking ahead saw further than Bell looking back.¹

However, the applications of quaternions, in 3D computer graphics and robotics, as an alternate, more efficient, method of image rotation and manipulation in the 21st Century, does not fulfill that destiny. The new perspective Hestenes has is that Hamilton found something fundamentally important, the “profound insight into the relation between algebra and geometry.” This new insight is that Hamilton found more than a use for four dimensional numbers, “he found a system of numbers to represent rotations in three dimensions. He was looking for a way to describe geometry by algebra, so he found a geometric algebra.”

Combining Hamilton’s operational interpretation of number, with Grassmann’s quantitative interpretation of number, led Clifford to the powerful geometric product, though his premature death almost took the secret with him. Hestenes observes:

Quaternions today [late 20th Century] reside in a kind of mathematical limbo, because their place in a more general geometric algebra is not recognized, …[but] Clifford observed that Grassmann developed the idea of a directed number from the quantitiative point of view, while Hamilton emphasized the operational interpretation. The two approaches are brought together by the geometric product. Either a quantitative or an operational interpretation can be given to any number, yet one or the other may be more important in most applications. Thus, vectors are usually interpreted quantitatively, while spinors are usually interpreted operationally.

Of course, it was Hestenes that brought Geometric Algebra, and the quaternions that are part of it, out of obscurity, during the last half of the 20th Century, but by then the institutionalization of academia and science, which might have benefited most from Hamilton’s “profound insight” into the fundamental relation between geometry and algebra, manifest in quaternions, had long since been set in the language of complex numbers and vector analysis.

Hestenes realized this, of course, and mostly turned his efforts to convincing the mathematical and academic communities of the pedagogical advantages of Geometric Algebra, due to its almost complete generalization of number with geometric magnitude. However, he also developed his 4D spacetime algebra (STA) and was able to show that it applies to a wide variety of problems in physics. Today, there are many approaches to physics based on Clifford algebras, but all are connected, more or less directly, to quaternions, emphasizing the geometric significance of vector products, while avoiding the much less intuitive concepts of matrices and tensors elements.

Nevertheless, it’s clear that Hamilton’s “belief that quaternions would play a central role in the mathematics of the future” was founded in a more fundamental vision than a basis for alternate formalisms, even if that includes providing a geometrical component to replace otherwise abstract concepts of multi-dimensional rotation. He really believed that since geometry is the science of fixed magnitudes, it follows that algebra should be the science of numbers, not just a tool, or a language, for science, but a science in and of itself.

In the preface to his “On Algebra as the Science of Pure Time,” he explains the difference between algebra as art, or language, and algebra as science, and then he explains why he feels algebra must be much more than a formulation of symbolic ideas:

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 if, in thus exchanging the ancient for the modern Mathesis, there were a gain only of Skill or Elegance, at the expense of Contemplation and Intuition.

If, in the end, Hamilton’s greatest achievement was simply the invention, or discovery, of quaternions, which form the basis for a more elegant formalism of quantum physics, or relativity theory, then, even with the new-found utility of quaternions that Hestenes has helped bring to light, the pathos of the “Irish Tragedy” is only blunted. In the final analysis, because it was not these more skillful and elegant applications Hamilton sought to gain, but the far more lofty objects of contemplation and intuition - algebraic truth - as it were, he would probably have agreed with Bell that his career was a failure. He lamented:

So useful are those rules, so symmetrical those expressions, and yet so unsatisfactory those principles from which they are supposed to be derived, that a growing tendency may be perceived to the rejection of that view which regarded Algebra as a Science, in some sense analogous to Geometry, and to the adoption of one or other of those two different views, which regard Algebra as an Art, or as a Language: as a System of Rules, or else as a System of Expressions, but not as a System of Truths, or Results having any other validity than what they may derive from their practical usefulness, or their logical or philological coherence. Opinions thus are tending to substitute for the Theoretical question,—“Is a Theorem of Algebra true?” the Practical question,—“Can it be applied as an Instrument, to do or to discover something else, in some research which is not Algebraical?” or else the Philological question,—“Does its expression harmonise, according to the Laws of Language, with other Algebraical expressions?”

Thus, he sets out on the quest to find aglebraic truth, and what we discover is that the famous moment, under the bridge, when he carves the quaternion equation in the stone, after years and years of searching, what he thought he found was not so much a geometric algebra, as Hestenes describes it, but a key to understanding the science of algebra that had started with a successful development of numeric couples, but mysteriously couldn’t be extended to numeric triplets, but only numeric quartos.

Given the cursory treatment of quaternions in academia, one is simply led to view them as natural extensions of complex numbers. If one imaginary number works such magic in the complex plane, then it’s only logical to think that adding more imaginary numbers would be even better.  Of course, this isn’t the case.  Adding one more imaginary number to a 2D complex number, to form a 3D number, doesn’t work.  You have to add two more imaginary numbers to the one of complex numbers to get the algebra to work, and even then it’s got a huge defect, in a sense, even though it’s perfectly understandable that a 4D system is non-commutative, given the directions of 3D space in terms of locations defined with four dimensions (3 space and 1 time dimension.)  Yet, this couldn’t have been pleasing to Hamilton.  Before the moment under the bridge, he wrote:

For it has not fared with the principles of Algebra as with the principles of Geometry. No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by Euclid in his Elements, two thousand years ago; though he may well desire to see them treated in a clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the plan of the argument.

Of course, this was before the days of non-Euclidean geometry (Hamilton probably turned over in his grave!), but his point is still valid: geometry is inductive. It’s an inductive science.  Algebra, on the other hand, is not nearly as satisfactory in this regard.  Hamilton goes on:

But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing. It must be hard to found a Science on such grounds as these, though the forms of logic may build up from them a symmetrical system of expressions, and a practical art may be learned of rightly applying useful rules which seem to depend upon them.

My guess is that the reason Hamilton spent the rest of his life, working with quaternions, is not because he sought to improve “the practical art” of vector analysis, but because he was convinced that there was a “science of algebra;” that is, he searched in vain for a way to make algebra an inductive science, as geometry is an inductive science, or was in his time.

As it turns out, he was really, really, close, but it wasn’t until Larson shed the light of reciprocity on mathematics and physics that we could really appreciate why.  We’re still trying to get our pea brains around it.