Saturday, August 21, 2010

Alfred North Whitehead: Mathematical Unifier?

Inspired by the underlying commonality of the existing mathematics of his day, would Alfred North Whitehead be as successful as Rene Descartes in establishing a new branch of mathematics?

By: Ringo Bones

Ever since Rene Descartes made possible the happy marriage of curves and quantities – i.e. by merging all the arithmetic, algebra and geometry of ages past into a single technique – to produce analytic geometry, many a wannabe great mathematician had tried to create their own branch of mathematics by combining existing ones. During the latter half of the Victorian Era, none got closer than Alfred North Whitehead. But today’s kids would certainly ask who the heck is he?

Alfred North Whitehead (1861 – 1947) English mathematician and philosopher, was born at Ramsgate, the Isle of Thanet, Kent, on February 15, 1861, of a family of teachers and ministers. His father was an Honorary Canon of Canterbury. Upon entering Trinity College, Cambridge, Alfred North Whitehead devoted himself to mathematics. But by being a member of the Apostles Club – philosophy, literature, history, politics and religion were all subjects of intense discussion.

In 1885, Whitehead became a Fellow of Trinity and began teaching mathematics. His Treatise on Universal Algebra led to his election to the Royal Society in 1903. A decade of collaboration with the most brilliant of his former pupils – Bertrand Russell – resulted in the publication of the monumental Principia Mathematica consisting of three volumes between 1910 and 1913.

Alfred North Whitehead’s mathematical work that gained him fame was the 1898 publication of his pioneering work called A Treatise on Universal Algebra. Now called Abstract Algebra, it was his unfinished attempt to unify “the various systems of Symbolic Reasoning allied to ordinary algebra.” The first and only volume that was published is a detailed investigation of H. G. Grassmann’s Calculus of Extension – Ausdehnungslehre in German – which was first published in 1844 but insufficiently appreciated and George Boole’s Algebra of Logic. These mathematical works had attracted Whitehead’s attention by their bold extension of algebraic methods beyond the traditional realm of the quantitative.

Whitehead restated Grassmann’s calculus and employed it to unify a variety of geometries; thus the theorems of projective geometry were exhibited as consequences of the definitions of the calculus. Some, but comparatively few, additions to the superstructure of mathematics were included – resulting in a main achievement that was more a novel unification. But this was along relatively unorthodox lines, thus the work had little influence among mathematicians.

Returning to the great Principia Mathematica, the first portion of which is a deductive elaboration of formal logic from a few axioms; the remainder is a detailed deduction, from this alone, of the basic concepts and principles, first called cardinal arithmetic and then of the other recognized mathematical sciences; and many new sciences suggested by the broad definitions being laid down.

The whole is written in exact and elaborate symbolism, taken partly from Giuseppe Peano. The work is basic for students of the foundations of mathematics, in spite of the fact that the first portion, for which Russell was mainly responsible, was involved in difficulties which have challenged experts ever since, and that the fourth volume, which was to deal with geometry alone, and was to be written by Whitehead alone, never appeared. Thus his attempts to unify various geometries were seen nothing more by mathematicians – then and now – as nothing more that a novel unification.

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