William Clifford | |
|---|---|
.jpg) Likeness, from the frontispiece ofLectures and Essays by the Late William Kingdon Clifford, F.R.S. (1901) | |
| Born | 4 May 1845 (1845-05-04) Exeter, Devon, England |
| Died | 3 March 1879 (1879-03-04) (aged 33) Madeira, Portugal |
| Alma mater | King's College London Trinity College, Cambridge |
| Known for | Clifford algebra Clifford's circle theorems Clifford's theorem Clifford torus Clifford–Klein form Clifford parallel Bessel–Clifford function Dual quaternion Elements of Dynamic |
| Spouse | Lucy Clifford (1875–1879) |
| Scientific career | |
| Institutions | University College London |
| Doctoral students | Arthur Black |
| Signature | |
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William Kingdon Clifford (4 May 1845 – 3 March 1879) was a Britishmathematician andphilosopher. Building on the work ofHermann Grassmann, he introduced what is now termedgeometric algebra. This is a special case of what later became known as theClifford algebra, which was 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 tomathematical physics,[1]geometry,[2] andcomputing.[3] Clifford was the first to suggest thatgravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expressionmind-stuff.
Born inExeter, William Clifford was educated at Doctor Templeton's Academy onBedford Circus and showed great promise at school.[4] He went on toKing's College London (at age 15) andTrinity College, Cambridge, where he was elected fellow in 1868, after beingSecond Wrangler in 1867 and second Smith's prizeman.[5][6] In 1870, he was part of an expedition to Italy to observe thesolar eclipse of 22 December 1870. During that voyage he survived a shipwreck along the Sicilian coast.[7]
In 1871, he was appointed professor of mathematics and mechanics atUniversity College London, and in 1874 became a fellow of theRoyal Society.[5] He was also a member of theLondon Mathematical Society and theMetaphysical Society.
Clifford marriedLucy Lane on 7 April 1875, with whom he had two children.[8] Clifford enjoyed entertaining children and wrote a collection of fairy stories,The Little People.[9]
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 oftuberculosis after a few months, leaving a widow with two children.
Clifford and his wife are buried in London'sHighgate Cemetery, near the graves ofGeorge Eliot andHerbert Spencer, just north of the grave ofKarl Marx.[10]
Theacademic journalAdvances in Applied Clifford Algebras publishes on Clifford's legacy inkinematics andabstract algebra.
"Clifford was above all and before all a geometer."
The discovery ofnon-Euclidean geometry opened new possibilities in geometry in Clifford's era. The field of intrinsicdifferential geometry was born, with the concept ofcurvature broadly applied tospace itself as well as to curved lines and surfaces. Clifford was very much impressed byBernhard Riemann’s 1854 essay "On the hypotheses which lie at the bases of geometry".[11] In 1870, he reported to theCambridge Philosophical Society on the curved space concepts of Riemann, and included speculation on the bending of space by gravity. Clifford's translation[12][13] of Riemann's paper was published inNature in 1873. His report at Cambridge, "On the Space-Theory of Matter", was published in 1876, anticipatingAlbert Einstein'sgeneral relativity by 40 years. Clifford elaboratedelliptic space geometry as anon-Euclideanmetric space. Equidistant curves in elliptic space are now said to beClifford 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 includingalgebraic forms andprojective geometry and the textbookElements of Dynamic. His application ofgraph theory toinvariant theory was followed up byWilliam Spottiswoode andAlfred Kempe.[14]
In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra.[15] He had succeeded in unifying thequaternions, developed byWilliam Rowan Hamilton, with Grassmann'souter product (aka theexterior product). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. Theversors in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of theinner product and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematicianMarcel 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[i] of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space.[16]
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 theirisomorphism classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions.[17]
The realms ofreal analysis andcomplex analysis have been expanded through the algebraH of quaternions, thanks to its notion of athree-dimensional sphere embedded in a four-dimensional space. Quaternionversors, which inhabit this 3-sphere, provide a representation of therotation group SO(3). Clifford noted that Hamilton'sbiquaternions were atensor product of known algebras, and proposed instead two other tensor products ofH: Clifford argued that the "scalars" taken from thecomplex numbersC might instead be taken fromsplit-complex numbersD or from thedual numbersN. In terms of tensor products, producessplit-biquaternions, while formsdual quaternions. The algebra of dual quaternions is used to expressscrew displacement, a common mapping in kinematics.
As a philosopher, Clifford's name is chiefly associated with two phrases of his coining,mind-stuff and thetribal self. The former symbolizes hismetaphysical conception, suggested to him by his reading ofBaruch Spinoza,[5] which Clifford (1878) defined as follows:[19]
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.
— "On the Nature of Things-in-Themselves" (1878)
Regarding Clifford's concept,Sir Frederick Pollock wrote:
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, oratom 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 intoconsciousness, and the rest of the world. This leads to results which would in a loose and popular sense be calledmaterialist. But the theory must, as ametaphysical theory, be reckoned on the idealist side. To speak technically, it is an idealistmonism.[5]
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 towardreligion. 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 favourobscurantism, and to put the claims of sect above those of human society. The alarm was greater, astheology was still unreconciled withDarwinism; and Clifford was regarded as a dangerous champion of the anti-spiritual tendencies then imputed to modern science.[5] There has also been debate on the extent to which Clifford's doctrine of 'concomitance' or 'psychophysical parallelism' influencedJohn Hughlings Jackson's model of the nervous system and, through him, the work of Janet, Freud, Ribot, and Ey.[20]
In his 1877 essay,The Ethics of Belief, Clifford argues that it is immoral to believe things for which one lacks evidence.[21] 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."[21]
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."[ii] 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 with what has come to be known asClifford's principle: "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence."[21]
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 bypragmatist philosopherWilliam James in his "Will to Believe" lecture. Often these two works are read and published together astouchstones for the debate overevidentialism,faith, andoverbelief.
Though Clifford never constructed a full theory ofspacetime andrelativity, there are some remarkable observations he made in print that foreshadowed these modern concepts:In his bookElements of Dynamic (1878), he introduced "quasi-harmonic motion in a hyperbola". He wrote an expression for aparametrized unit hyperbola, which other authors later used as a model for relativistic velocity. Elsewhere he states:[22]
This passage makes reference tobiquaternions, though Clifford made these intosplit-biquaternions as his independent development.The book continues with a chapter "On the bending of space", the substance ofgeneral relativity. Clifford also discussed his views inOn the Space-Theory of Matter in 1876.
In 1910, William Barrett Frankland quoted theSpace-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."[23] Years later, aftergeneral relativity had been advanced byAlbert Einstein, various authors noted that Clifford had anticipated Einstein.Hermann Weyl (1923), for instance, mentioned Clifford as one of those who, likeBernhard Riemann, anticipated the geometric ideas of relativity.[24]
In 1940,Eric Temple Bell publishedThe Development of Mathematics, in which he discusses the prescience of Clifford on relativity:[25]
John Archibald Wheeler, during the 1960 InternationalCongress for Logic, Methodology, and Philosophy of Science (CLMPS) atStanford, introduced hisgeometrodynamics formulation of general relativity by crediting Clifford as the initiator.[26]
InThe Natural Philosophy of Time (1961),Gerald James Whitrow recalls Clifford's prescience, quoting him in order to describe theFriedmann–Lemaître–Robertson–Walker metric in cosmology.[27]
Cornelius Lanczos (1970) summarizes Clifford's premonitions:[28]
Likewise,Banesh Hoffmann (1973) writes:[29]
In 1990,Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight.[30] 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."[30] In 1992, Farwell and Knee continued their study of Clifford and Riemann:[31]
[They] hold that oncetensors 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.
This article incorporates text from a publication now in thepublic domain: Chisholm, Hugh, ed. (1911). "Clifford, William Kingdon".Encyclopædia Britannica. Vol. 6 (11th ed.). Cambridge University Press. p. 506.Lectures on Ten British Mathematicians of the Nineteenth Century.(See especially pages 78–91)
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