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The Utility of Universal Algebra

Alexander Macfarlane wrote a review of Alfred Whitehead’s Universal Algebra which was published by Science on March 3, 1899. Macfarlane concurs with Whitehead on the admissibility of multiplicative systems that fail the axiom of associativity. Nevertheless, his review of Whitehead is critical of an excess of formalism and the deferral of the particulars of quaternion and matrix algebra to a second volume (which never arrived).

Through an experiment of the 1890’s called “hyperbolic quaternions”, Macfarlane had found reason to consider a non-associative system, and also reason to trim expectations so that associativity might be regained in another system. Today the distinction is made between associative and non-associative algebraic rings. Macfarlane says near the end of his review: “Multiplication does not necessarily follow the commutative and associative laws, that is, ab = ba and (ab)c = a(bc) are laws of special branches only. It has been maintained by followers of Hamilton that the associative law is essential to multiplication. It is true of spherical quaternions [and exspherical coquaternions] , but is not true of the complementary branch of vector analysis [with cross product].”

Today A. N. Whitehead is seen as one of the foremost Christian theologians, given his books Religion in the Making (1926) and Process and Reality (1929). Therefore his philosophy of mathematics is read through the pages of A Treatise on Universal Algebra. Consequently, with this critical review of a seminal work of Whitehead, Macfarlane’s views in the philosophy of mathematics gained readership. As a steward of the academy, serving students in the electrical labs at Lehigh University, students studying to comprehend the physical laws that are the foundation of industrial work, Macfarlane expected mathematical textbook writers to be practical and helpful in the development of technical skill and insight. Instead of harsh words, Macfarlane asks these rhetorical questions:

1. Is geometry a part of pure mathematics ?
2. Are the definitions of ordinary algebra merely self-consistent conventions [without existential import]?
3. Are its propositions merely formal without any objective truth?
4. Are the rules by which it proceeds arbitrary selections of the mind?
5. If so, what is the chance of their applying to anything useful? [utility criterion]
6. How can [the] perfect correspondence [of a mathematical model] be secured, except by the conventions being real definitions, the equations true propositions, and the rules expressions of universal properties?

Utility is bound up with stewardship, as in the council J. W. Gibbs gave to young Edwin Bidwell Wilson: “one good use to which anybody might put a superior training in pure mathematics was to study the problems set us by nature.” For Macfarlane, the electronic era was dawning. A student of P. G. Tait, a physicist, then a mathematical-physicist, Macfarlane poured himself into linear algebra and vector methods, contributing to the development of mathematical tools for grasping the dynamics suitable to electronics. For instance, he used this review to disseminate the idea of hyperbolic versors, a concept he carried forward, from James Cockle and W.K. Clifford, to develop his Space Analysis.

Macfarlane's leadership becomes more and more apparent as spacetime foundations are investigated. But he did not restrain himself from entering the sociological polemics of religion; for instance his biography of William Kingdon Clifford, delivered in 1901 but not published until 1916, gives key details of Thompson and Tait's The Unseen Universe.

It is a recurrent problem in mathematical communications that there are too many generalities and not enough particulars. Such could be a slant on Macfarlane’s review. But not completely, for he has opportunity to say that “It is satisfactory to find that Mr. Whitehead adopts the latter [non-associative] view, and, indeed it is involved in his detailed exposition of vector analysis in the concluding book of his first volume.”

Read the Alexander MacFarlane Homepage to learn about his contributions to science.
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