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 (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.

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.

On April 7, 1875, Clifford married Lucy Lane. 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, leaving a widow with two children. 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.

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 Cemetery just north of the grave of Karl Marx, and near the graves of George Eliot and Herbert Spencer.

"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 and biquaternions. Clifford employed these 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 published several papers on algebraic forms and projective geometry and the textbook Elements of Dynamic.

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.

Though Clifford never constructed a full theory of spacetime and relativity, there are some remarkable observations he made in print that foreshadowed these modern concepts:

The geometry of rotors and motors ... forms the basis of the whole modern theory of the relative rest (Static) and the relative motion (Kinematic and Kinetic) of invariable systems.

Common Sense of the Exact Sciences (1885)

This passage makes reference to biquaternions, though Clifford made these into split - biquaternions as his independent development. The book continues with a chapter "On the bending of space", the substance of general relativity. Clifford also discussed his views in On the Space - Theory of Matter in 1876.

Clifford's anticipation of Einstein has been frequently noted:

In 1923 Hermann Weyl mentioned Clifford as one of those who, like Bernhard Riemann, anticipated the geometric ideas of relativity.

In 1940 Eric Temple Bell published his The Development of Mathematics. There on pages 359 and 360 he discusses the prescience of Clifford on relativity:

Bolder even than Riemann, Clifford confessed his belief (1870) that matter is only a manifestation of curvature in a space - time manifold. This embryonic divination has been acclaimed as an anticipation of Einstein’s (1915 – 16) relativistic theory of the gravitational field. The actual theory, however, bears but slight resemblance to Clifford’s rather detailed creed. As a rule, those mathematical prophets who never descend to particulars make the top scores. Almost anyone can hit the side of a barn at forty yards with a charge of buckshot.

Also in 1960, at Stanford University for the International Congress for Logic, Methodology, and Philosophy of Science, John Archibald Wheeler introduced his geometrodynamics formulation of general relativity by crediting Clifford as the initiator.

In his The Natural Philosophy of Time (1961, 1980) Gerald James Whitrow recalls Clifford's prescience by quoting him to describe the Friedmann - Lemaitre - Robertson - Walker metric in cosmology.

In 1970 Cornelius Lanczos summarizes Clifford's premonitions this way:

[He] with great ingenuity foresaw in a qualitative fashion that physical matter might be conceived as a curved ripple on a generally flat plane. Many of his ingenious hunches were later realized in Einstein's gravitational theory. Such speculations were automatically premature and could not lead to anything constructive without an intermediate link which demanded the extension of 3 - dimensional geometry to the inclusion of time. The theory of curved spaces had to be preceded by the realization that space and time form a single four - dimensional entity.

In 1990 Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight. They conclude "it was Clifford, not Riemann, who anticipated some of the conceptual ideas of General Relativity". To explain the backward attitude to Clifford, they point out that he was an expert in metric geometry, and "metric geometry was too challenging to orthodox epistemology to be pursued." In 1992 Farwell and Knee continued their study with "The Geometric Challenge of Riemann and Clifford" They "hold that once tensors had been used in the theory of general relativity, the framework existed in which a geometrical perspective in physics could be developed and allowed the challenging geometrical conceptions of Riemann and Clifford to be rediscovered."