William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing. Clifford 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". Biography. 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 scientists, including William Thomson (Lord Kelvin) and James Clerk Maxwell. In 1870, he was part of an expedition to Italy to observe the solar eclipse of 22 December 1870. During  that voyage he 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. Clifford married Lucy Lane on 7 April 1875, with whom he had two children. Clifford enjoyed entertaining children and wrote a collection of fairy stories, "The Little People". Death and legacy. 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. Clifford and his wife are buried in London's Highgate Cemetery, near the graves of George Eliot and Herbert Spencer, just north of the grave of Karl Marx. The academic journal "Advances in Applied Clifford Algebras" publishes on Clifford's legacy in kinematics and abstract algebra. Mathematics. The discovery of non-Euclidean geometry opened  new possibilities in geometry in Clifford's era. The field of intrinsic differential geometry was born, with the concept of curvature broadly applied to space itself as well as to curved lines and surfaces. Clifford was very much impressed by Bernhard Riemann’s 1854 essay "On the hypotheses which lie at the bases of geometry". In 1870, he reported to the Cambridge Philosophical Society on the curved space concepts of Riemann, and included speculation on the bending of space by gravity. Clifford's translation of Riemann's paper was published in "Nature" in 1873. His report at Cambridge, "", was published  in 1876, anticipating Albert Einstein's general relativity by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean metric space. Equidistant curves in elliptic space are now said to be Clifford parallels. Clifford's contemporaries considered him acute and original, witty and warm. He often worked late into the night, which may have hastened his death. He published papers on a range of topics including algebraic forms and projective geometry and the textbook "Elements of Dynamic". His application of graph theory to invariant theory was followed up by William Spottiswoode and Alfred Kempe. Algebras. In 1878, Clifford published  a seminal work, building on Grassmann's extensive algebra. He had succeeded in unifying the quaternions, developed by William Rowan Hamilton, with Grassmann's "outer product" (aka the "exterior product"). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. The versors in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of the inner product and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematician Marcel Riesz. The inner product equips geometric algebra with a  metric, fully incorporating distance and angle relationships for lines, planes, and volumes, while the outer product gives those planes and volumes vector-like properties, including a directional bias. Combining the two brought the operation of division into play. This greatly expanded our qualitative understanding of how objects interact in space. Crucially, it also provided the means for quantitatively calculating the spatial consequences of those interactions. The resulting geometric algebra, as he called it, eventually realized the long sought goal of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space. Moreover, Clifford's algebraic schema  extends to higher dimensions. The algebraic operations have the same symbolic form as they do in 2 or 3-dimensions. The importance of general Clifford algebras has grown over time, while their isomorphism classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions. The realms of real analysis and complex analysis have been expanded through the algebra H of quaternions, thanks to its notion of a three-dimensional sphere embedded in a four-dimensional space. Quaternion versors, which inhabit this 3-sphere, provide a representation of the rotation group SO(3). Clifford noted that Hamilton's  biquaternions were a tensor product formula_1 of known algebras, and proposed instead two other tensor products of H: Clifford argued that the "scalars" taken from the complex numbers C might instead be taken from split-complex numbers D or from the dual numbers N. In terms of tensor products, formula_2 produces split-biquaternions, while formula_3 forms dual quaternions. The algebra of dual quaternions is used to express screw displacement, a common mapping in kinematics. Philosophy. 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 Baruch Spinoza, which Clifford (1878) defined as follows: Regarding Clifford's concept, Sir Frederick Pollock wrote: "Tribal self", on the other hand, 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 anti-spiritual 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. Ethics. In his 1877 essay, "The Ethics of Belief", Clifford argues that  it is immoral to believe things for which one lacks evidence. He describes a ship-owner who planned to send to sea an old and not well built ship full of passengers. The ship-owner had doubts suggested to him that the ship might not be seaworthy: "These doubts preyed upon his mind, and made him unhappy." He considered having the ship refitted even though it would be expensive. At last, "he succeeded in overcoming these melancholy reflections." He watched the ship depart, "with a light heart…and he got his insurance money when she went down in mid-ocean  and told no tales." Clifford argues that the ship-owner was guilty of the deaths of the passengers even though he sincerely believed the ship was sound: ""[H]e had no right to believe on such evidence as was before him"." Moreover, he contends that even in the case where the ship successfully reaches the destination, the decision remains immoral, because the morality of the choice is defined forever once the choice is made, and actual outcome, defined by blind chance, doesn't matter. The ship-owner would be no less guilty: his wrongdoing would never be discovered, but he still  had no right to make that decision given the information available to him at the time. Clifford famously concludes: "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." As such, he is 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. Premonition  of relativity. 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: In his book Elements of Dynamic (1878), he introduced "quasi-harmonic motion in a hyperbola". He wrote an expression for a parametrized unit hyperbola, which other authors later used as a model for relativistic velocity. Elsewhere he states: 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 "" in 1876. In 1910, William Barrett Frankland quoted the "Space-Theory of Matter" in his book on parallelism: "The boldness of this speculation is surely unexcelled in the history of thought. Up to the present, however, it presents the appearance of an Icarian flight." Years later, after general relativity had been advanced by Albert Einstein, various authors noted that Clifford had anticipated Einstein. Hermann Weyl (1923), for instance, mentioned Clifford as one of those who, like Bernhard Riemann, anticipated the geometric ideas of relativity. In 1940, Eric Temple Bell published "The Development of Mathematics", in which  he discusses the prescience of Clifford on relativity: John Archibald Wheeler, during the 1960 International Congress for Logic, Methodology, and Philosophy of Science (CLMPS) at Stanford, introduced his geometrodynamics formulation of general relativity by crediting Clifford as the initiator. In "The Natural Philosophy of Time" (1961), Gerald James Whitrow recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology. Cornelius Lanczos (1970) summarizes Clifford's premonitions: Likewise, Banesh Hoffmann (1973) writes: In 1990, Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight. They conclude that "it was  Clifford, not Riemann, who anticipated some of the conceptual ideas of General Relativity." To explain the lack of recognition of Clifford's prescience, 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 of Clifford and Riemann:[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. William Kingdon Clifford