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A Quaternion Journey - The Abstractor
Joseph Johnson

William Rowan Hamilton was born on 4 August, 1805. Like many other famous mathematicians, he grew up as a Wunderkind, having learned scores of languages by the age of 10. His scientific mind was so respected by his peers, that he was named Astronomer Royal of Ireland before he even received his college degree. Not surprisingly, Hamilton became one of the pre-eminent mathematicians of his era, bringing prestige to his career employer, Trinity College in Dublin. Hamilton lived a productive life in his career pursuits, having contributed to theories of dynamics, optics, and algebra. His personal life was characterized by a series of ups and downs.

Regarded as one of the great ``abstractors" of algebra, Hamilton pioneered a slew of concepts. Some of these insights have stood the test of time. For example, the tex2html_wrap_inline84 operator of analysis is his construct [Hankins 92]. His notion of ``vector," without refering specifically to 2- or 3- tuples, was a first step in the subsequent abstract definition of a vector space [Fraleigh 405].

Still some of his ideas are as arcane today as they were in his day. In reading some of Hamilton's philosophy, one may get the impression that nothing short of a complete unification of the sciences would satisfy him.

Power, acting by law of Space and Time, is the ideal base of an ideal world, into which it is the problem of physical science to refine the phenomenal world, that so we may behold as one, and under the forms of our own understanding, what had seemed to be manifold and foreign [(via) Graves 87].

Hamilton was truly devoted to the marriage of philosophy and mathematics. In fact, he wrote extensively on how time and space are related to the idealisms of Immanuel Kant and Rudjer Boskovic. A consistent atomistic theory of nature beckoned Hamilton; so much so, that it formed the basis for his ideas of number and algebra. Regarding real numbers as ``moments" in time, he invented the idea of number couples. Number couples represented steps in time and could be added and multiplied (by basically using normal definitions for the operators). This attempt was a ``failure from the modern perspective [Katz 683]."

One biographer has described these philosophical forays thusly:

The most significant contribution of his philosophical studies was to confirm him in his search for the most general application of mathematics to the physical world. It was this high degree of generality and abstraction that permitted him to include wave optics, particle optics, and dynamics in the same mathematical theory [Hankins 87].
Of those notions for which Hamilton's abstraction was essential, the theory of complex quaternions stands out.

| The Abstractor | Elementary Notions | A Fourth Dimension | A Division Algebra
| Closure | An Example of Hamilton's Work | Bibliography | Back to the front page