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Clifford, William Kingdon (1845-1879)
"If a man, holding a belief which he was taught in childhood or persuaded of afterwards, keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call in question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it -- the life of that man is one long sin against mankind."

-- W. K. Clifford

William Kingdon Clifford was an English mathematician who also wrote a fair bit on philosophy. Along with Hermann Grassmann, he invented what is now termed geometric algebra, a special case being the Clifford algebras named in his honour, which play a role in contemporary mathematical physics. He was the first to suggest that gravitation might be a manifestation of an underlying geometry. His philosophical writings coined the phrase "mind-stuff".

Born at Exeter, William Clifford showed great promise at school. He went on to King's College London (at age 15) and Trinity College, Cambridge, where he was elected fellow in 1868, after being second wrangler in 1867 and second Smith's prizeman. Being second was a fate he shared with others who became famous mathematicians. e.g., William Thomson, James Clerk Maxwell. In 1870, he was part of an expedition to Italy to observe an eclipse, and survived a shipwreck along the Sicilian coast.

In 1871, he was appointed professor of mathematics and mechanics at University College London, and in 1874 became a fellow of the Royal Society. He was also a member of the London Mathematical Society and the Metaphysical Society.

In 1875, he married Lucy Lane of Barbados. In 1876, Clifford suffered a breakdown, probably brought on by overwork; he taught and administered by day, and wrote by night. A half year holiday in Algeria and Spain allowed him to resume his duties for 18 months, after which he collapsed again. He went to Madeira to recover, but died there of tuberculosis after a few months, leaving a widow with two children. 11 days after his death, Albert Einstein was born, to develop thirty six years later the geometric theory of gravity that Clifford had suggested.

"If he had lived we might have known something." (Said by Isaac Newton of Roger Cotes, but applicable to Clifford.)

Similar to Charles Dodgson, he enjoyed entertaining children, writing a collection of fairy stories, The Little People.

"Clifford was above all and before all a geometer." (H. J. S. Smith). In this he was an innovator against the excessively analytic tendency of Cambridge mathematicians. Influenced by Riemann and Lobachevsky, Clifford studied non-Euclidean geometry. In 1870, he wrote On the space theory of matter, arguing that energy and matter are simply different types of curvature of space. These ideas later played a fundamental role in Albert Einstein's general theory of relativity.

Yet Clifford is now best remembered for his eponymous Clifford algebras, a type of associative algebra that generalizes the complex numbers and William Rowan Hamilton's quaternions. The latter resulted in the octonions (biquaternions), which he employed to study motion in non-Euclidean spaces and on certain surfaces, now known as Klein-Clifford spaces. He showed that spaces of constant curvature could differ in topological structure. He also proved that a Riemann surface is topologically equivalent to a box with holes in it.

His contemporaries considered him a man of extraordinary acuteness and originality, gifted with quickness of thought and speech, a lucid style, wit and poetic fancy, and a social warmth. In his theory of graphs, or geometrical representations of algebraic functions, there are valuable suggestions which have been worked out by others. He was much interested, too, in universal algebra and elliptic functions, his papers "Preliminary Sketch of Biquaternions" (1873) and "On the Canonical Form and Dissection of a Riemann's Surface" (1877) ranking as classics. Another important paper is his "Classification of Loci" (1878). He also published several papers on algebraic forms and projective geometry.

As a philosopher, Clifford's name is chiefly associated with two phrases of his coining, "mind-stuff" and the "tribal self." The former symbolizes his metaphysical conception, suggested to him by his reading of
Spinoza. Sir Frederick Pollock wrote about Clifford as follows:

"Briefly put, the conception is that mind is the one ultimate reality; not mind as we know it in the complex forms of conscious feeling and thought, but the simpler elements out of which thought and feeling are built up. The hypothetical ultimate element of mind, or atom of mind-stuff, precisely corresponds to the hypothetical atom of matter, being the ultimate fact of which the material atom is the phenomenon.

Matter and the sensible universe are the relations between particular organisms, that is, mind organized into consciousness, and the rest of the world. This leads to results which would in a loose and popular sense be called materialist. But the theory must, as a metaphysical theory, be reckoned on the idealist side. To speak technically, it is an idealist monism."

The other phrase, "tribal self," gives the key to Clifford's ethical view, which explains conscience and the moral law by the development in each individual of a "self," which prescribes the conduct conducive to the welfare of the "tribe." Much of Clifford's contemporary prominence was due to his attitude towards religion.

Animated by an intense love of truth and devotion to public duty, he waged war on such ecclesiastical systems as seemed to him to favour obscurantism, and to put the claims of sect above those of human society. The alarm was greater, as theology was still unreconciled with Darwinism; and Clifford was regarded as a dangerous champion of the antispiritual tendencies then imputed to modern science.

He is also well known for arguing that it was immoral to believe things for which one lacks evidence, in his 1879 essay "The Ethics of Belief", which contains the famous principle: "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." As such, he was arguing in direct opposition to religious thinkers which claim faith (i.e. belief in things in spite of the lack of evidence for them) to be virtuous.


"I ... hold that in the physical world nothing else takes place but this variation [of the curvature of space]."

"We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra."

"There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture - that it came to him from outside, and that he did not consciously create it from within."

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The Talk Of Lawrence