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William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an Englishmarker mathematician and philosopher. Along with Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics and geometry. He was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression "mind-stuff".

Early life and education

Born at Exetermarker, William Clifford showed great promise at school. He went on to King's College Londonmarker (at age 15) and Trinity College, Cambridgemarker. At the latter, he was second wrangler in 1867 and second Smith's prizeman.He was elected fellow in 1868. Being second was a fate he shared with others who became famous mathematicians. e.g., William Thomson (Lord Kelvin), 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.

Career in academia

In 1871, he was appointed professor of mathematics and mechanics at University College Londonmarker, 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 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 the island of Madeira to recover, but died there of tuberculosis after a few months. Eleven days later, Albert Einstein was born, who would go on to develop the geometric theory of gravity that Clifford had suggested nine years earlier.


"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 complex quaternions (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. On Clifford algebras, quaternions, and their role in contemporary mathematical physics, see Penrose (2004).

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."

Clifford himself defined "mind-stuff" as follows (1878, "On the Nature of Things-in-Themselves," Mind, Vol. 3, No. 9, pp. 57-67):
"That element of which, as we have seen, even the simplest feeling is a complex, I shall call Mind-stuff.
A moving molecule of inorganic matter does not possess mind or consciousness ; but it possesses a small piece of mind-stuff.
When molecules are so combined together as to form the film on the under side of a jelly-fish, the elements of mind-stuff which go along with them are so combined as to form the faint beginnings of Sentience.
When the molecules are so combined as to form the brain and nervous system of a vertebrate, the corresponding elements of mind-stuff are so combined as to form some kind of consciousness ; that is to say, changes in the complex which take place at the same time get so linked together that the repetition of one implies the repetition of the other.
When matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition."

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 toward religion. Animated by an intense love of his conception 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. There has also been debate on the extent to which Clifford’s doctrine of ‘concomitance’ or ‘psychophysical parallelism’ influenced John Hughlings Jackson’s model of the nervous system and through him the work of Janet, Freud, Ribot, and Ey.

For arguing that it was immoral to believe things for which one lacks evidence, in his 1877 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 for whom "blind faith" (i.e. belief in things in spite of the lack of evidence for them) was a virtue. This paper was famously attacked by pragmatist philosopher William James in his "Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over evidentialism, faith, and overbelief.

Selected writings

Most of his work was published posthumously.

Personal life

On April 7, 1875, Clifford married Lucy Lane and they had two children. Similar to Charles Dodgson, he enjoyed entertaining children, writing a collection of fairy stories, The Little People. Clifford and his wife are buried in London's Highgate Cemeterymarker just north of the grave of Karl Marx, and near the graves of George Eliot and Herbert Spencer.


Marker for W.
Clifford and his wife in Highgate Cemetery (ca. 1986)

  • "I ... hold that in the physical world nothing else takes place but this variation [of the curvature of space]." Mathematical Papers.
  • "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." (From a lecture to the Royal Institution titled "Some of the conditions of mental development")
  • "It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." The Ethics of Belief (1879)
  • "I was not, and was conceived. I loved and did a little work. I am not and grieve not." - epitaph.

See also


  1. Berrios G E (2000) Body and Mind: C K Clifford. History of Psychiatry 11: 311-338

Further reading

  • (The on-line version lacks the article's photographs.)

  • (See especially pages 78 – 91)

  • (See especially Chapter 11)

External links

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