View source for William Kingdon Clifford

{{Short description|British mathematician and philosopher (1845–1879)}}
{{other people||William Clifford (disambiguation)}}
{{Use dmy dates|date=October 2022}}
{{Use British English|date=March 2017}}
{{Infobox scientist
| name              = William Clifford
| image             = Clifford William Kingdon.jpg
| caption           = William Kingdon Clifford (1845–1879)
| honorific_suffix  = {{post-nominals|country=GBR|size=100%|FRS}}
| birth_date        = {{birth date text|4 May 1845}}
| birth_place       = [[Exeter]], Devon, England
| death_date        = {{death date text|3 March 1879 }} (aged 33)
| death_place       = [[Madeira]], Portugal
| citizenship       =
| alma_mater        = [[King's College London]]<br>[[Trinity College, Cambridge]]
| doctoral_advisor  = 
| doctoral_students = [[Arthur Black (mathematician)|Arthur Black]]
| known_for         = [[Clifford algebra]]<br>[[Clifford's circle theorems]]<br>[[Clifford's theorem on special divisors|Clifford's theorem]]<br>[[Clifford torus]]<br>[[Clifford–Klein form]]<br>[[Clifford parallel]]<br>[[Bessel–Clifford function]]<br>[[Dual quaternion]]<br>''[[Elements of Dynamic]]''
| author_abbrev_bot = 
| author_abbrev_zoo = 
| signature         = 
| spouse            = [[Lucy Clifford]] (1875–1879)
| ethnicity         = 
| field             = [[Mathematics]]<br>[[Philosophy]]
| work_institutions = [[University College London]]
| prizes            = <!--|religion          = [[Anglican]] turned [[atheist]]<ref>{{cite web|url=http://www.massey.ac.nz/~wwifs/mathnews/NZMS86/news86a.shtml|title=New Zealand Mathematical Societu Newsletter Number 86, December 2002|work=massey.ac.nz}}</ref><ref>"I once wrote a book about the Victorian crisis of faith and entitled it, borrowing from a poem of Hardy's, God's Funeral. I included Carlyle, [...] as well as the out-and-out atheists such as W K Clifford [...]." A N Wilson, 'Browning's faith kept the snake wriggling underfoot', ''Daily Telegraph'', 20 August 2001, p. 19.</ref>--> 
}}
'''William Kingdon Clifford''' (4 May 1845{{snd}}3 March 1879) was a British [[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]],<ref>{{cite book|last1=Doran|first1=Chris|last2=Lasenby|first2=Anthony|title=Geometric Algebra for Physicists|year=2007|publisher=Cambridge University Press|location=Cambridge, England|isbn=9780521715959|pages=592|url=http://www.cambridge.org/us/academic/subjects/physics/theoretical-physics-and-mathematical-physics/geometric-algebra-physicists}}</ref> [[geometry]],<ref>{{cite book|last=Hestenes|first=David|title=From Past to Future: Graßmann's Work in Context |year=2011|publisher=Springer|location=Basel, Germany|isbn=978-3-0346-0404-8|pages=243–260|doi=10.1007/978-3-0346-0405-5_22|chapter=Grassmann's legacy|chapter-url=https://davidhestenes.net/geocalc/pdf/GrassmannsLegacy4.pdf}}</ref> and [[computing]].<ref>{{cite book|last=Dorst|first=Leo|title=Geometric Algebra for Computer Scientists|year=2009|isbn=9780123749420|pages=664|publisher=[[Morgan Kaufmann]]|location=Amsterdam|url=http://www.geometricalgebra.net/}}</ref> 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 in [[Exeter, England|Exeter]], William Clifford was educated at Doctor Templeton's Academy on [[Bedford Circus, Exeter|Bedford Circus]] and showed great promise at school.<ref >{{Cite book
  |title=Exeter Past
  |first=Hazel  |last=Harvey
  |isbn=1-86077-006-1
  |year=1996
  |publisher=Phillimore
  |pages=64–66
}}</ref> He went on to [[King's College London]] (at age 15) and [[Trinity College, Cambridge]], where he was elected fellow in 1868, after being [[Wrangler (University of Cambridge)|Second Wrangler]] in 1867 and second Smith's prizeman.{{sfn|Chisholm|1911|p=506}}<ref>{{acad|id=CLFT863WK|name=Clifford, William Kingdon}}</ref> In 1870, he was part of an expedition to Italy to observe the [[solar eclipse of December 22, 1870|solar eclipse]] of 22 December 1870. During that voyage he survived a shipwreck along the Sicilian coast.<ref>{{Cite book | last = Chisholm | first = M. | title = Such Silver Currents | year = 2002 | publisher = The Lutterworth Press | location = Cambridge | page = 26 | isbn = 978-0-7188-3017-5}}</ref>

In 1871, he was appointed professor of mathematics and mechanics at [[University College London]], and in 1874 became a fellow of the [[Royal Society]].{{sfn|Chisholm|1911|p=506}}  He was also a member of the [[London Mathematical Society]] and the [[Metaphysical Society]].

Clifford married [[Lucy Clifford|Lucy Lane]] on 7 April 1875, with whom he had two children.<ref>{{Cite book | last1 = Stephen | first1 = Leslie | last2 = Pollock | first2 = Frederick | title = Lectures and Essays by the Late William Kingdon Clifford, F.R.S | place = New York | publisher = Macmillan and Company | year = 1901 | volume = 1 | page = 20 | url = http://www.openlibrary.org/details/lecturesessays01clifiala | access-date = 8 March 2008 | archive-date = 3 March 2008 | archive-url = https://web.archive.org/web/20080303102243/http://www.openlibrary.org/details/lecturesessays01clifiala | url-status = dead }}</ref> Clifford enjoyed entertaining children and wrote a collection of fairy stories, ''The Little People''.<ref>{{cite book|last=Eves|first=Howard W.|title=In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes|publisher=Prindle, Weber and Schmidt|year=1969|volume=3&ndash;4|pages=91&ndash;92}}</ref>

===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 ==
{{Blockquote|text="Clifford was above all and before all a geometer."|author=[[Henry John Stephen Smith]]{{sfn|Chisholm|1911|p=506}}|title=|source=}}
[[File:Clifford-2-2-vol 1-3.jpg|thumb|259x259px|Title page of Volume 1 (1878) containing books I-III of Clifford's ''"[[Elements of Dynamic]]"'']]
[[File:Clifford-2.jpg|left|thumb|223x223px|Volumes 1 (1878) and 2 (1887) containing books I-IV of Clifford's "[[Elements of Dynamic]]"]]
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".<ref>[[Bernhard Riemann|Riemann, Bernhard]]. 1867 [1854]. "[http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html On the hypotheses which lie at the bases of geometry]" ([[Habilitation]]sschrift), translated by W. K. Clifford. – via School of Mathematics, [[Trinity College Dublin]].</ref> 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<ref>Clifford, William K. 1873. "On the hypotheses which lie at the bases of geometry." [[Nature (journal)|''Nature'']] 8:14–17, 36–37.</ref><ref>Clifford, William K. 1882. "Paper #9." P. 55–71 in ''Mathematical Papers.''</ref> of Riemann's paper was published in ''[[Nature (journal)|Nature]]'' in 1873. His report at Cambridge, "[[s:On the Space-Theory of Matter|On the Space-Theory of Matter]]", was published in 1876, anticipating [[Albert Einstein]]'s [[general relativity]] by 40 years. Clifford elaborated [[elliptic geometry#Elliptic space|elliptic space geometry]] as a [[non-Euclidean geometry|non-Euclidean]] [[metric space]]. Equidistant curves in elliptic space are now said to be [[Clifford parallel]]s.
[[File:William Kingdon Clifford by John Collier.jpg|right|thumb|{{center|Clifford by [[John Collier (Pre-Raphaelite painter)|John Collier]]}}]]

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 form]]s 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]].<ref>{{cite book|last1=Biggs|first1= Norman L.|first2= Edward Keith|last2=Lloyd|first3= Robin James|last3=Wilson|at=[https://books.google.com/books?id=XqYTk0sXmpoC&pg=PA67 p. 67]|title=Graph Theory: 1736-1936|title-link=Graph Theory, 1736–1936|publisher=Oxford University Press|year=1976|isbn=978-0-19-853916-2}}</ref>

===Algebras===
In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra.<ref>{{cite journal|last=Clifford|first=William|title=Applications of Grassmann's extensive algebra|journal=American Journal of Mathematics|year=1878|volume=1|issue=4|pages=350–358|jstor=2369379|doi=10.2307/2369379}}</ref> He had succeeded in unifying the [[quaternions]], developed by [[William Rowan Hamilton]], with Grassmann's ''[[outer product]]'' (aka the [[Exterior algebra|''exterior product'']]). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. The [[versor]]s 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<ref group="lower-roman">"I believe that, so far as geometry is concerned, we need still another analysis which is distinctly geometrical or linear and which will express situation directly as algebra expresses magnitude directly."

[[Gottfried Wilhelm Leibniz|Leibniz, Gottfried]]. 1976 [1679]. "Letter to [[Christiaan Huygens|Christian Huygens]] (8 September 1679)." In ''Philosophical Papers and Letters'' (2nd ed.). [[Springer Publishing|Springer]].</ref> of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space.<ref>{{cite web|last=Hestenes|first=David|title=On the Evolution of Geometric Algebra and Geometric Calculus|url=https://davidhestenes.net/geocalc/html/Evolution.html}}</ref>

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.<ref>{{cite journal|last=Dechant|first=Pierre-Philippe|title=A Clifford algebraic framework for Coxeter group theoretic computations|journal=Advances in Applied Clifford Algebras|date=March 2014|volume=14|issue=1|pages=89–108|doi=10.1007/s00006-013-0422-4|bibcode=2012arXiv1207.5005D|arxiv=1207.5005|s2cid=54035515}}</ref>

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 [[versor]]s, which inhabit this 3-sphere, provide a representation of the [[rotation group SO(3)]]. Clifford noted that Hamilton's [[biquaternion]]s were a [[tensor product#Tensor product of algebras|tensor product]] <math>H \otimes C</math> of known algebras, and proposed instead two other tensor products of '''H''': Clifford argued that the "scalars" taken from the [[complex number]]s '''C''' might instead be taken from [[split-complex number]]s '''D''' or from the [[dual number]]s '''N'''. In terms of tensor products, <math>H \otimes D</math> produces [[split-biquaternion]]s, while <math>H \otimes N</math> forms [[dual quaternion]]s. The algebra of dual quaternions is used to express [[screw theory#Homography|screw displacement]], a common mapping in kinematics.

[[File:Clifford William Kingdon desk.jpg|thumb|William Kingdon Clifford (1901), as shown on the [[Book frontispiece|frontispiece]] of ''Lectures and Essays'', vol. 2.<ref>Frontispiece of ''Lectures and Essays by the Late William Kingdon Clifford, F.R.S.'', vol 2.</ref>|alt=Clifford (1901), as shown on the frontispiece of Lectures and Essays, vol. 2.]]

==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 [[Metaphysics|metaphysical]] conception, suggested to him by his reading of [[Baruch Spinoza]],{{sfn|Chisholm|1911|p=506}} which Clifford (1878) defined as follows:<ref>Clifford, William K. 1878. "On the Nature of Things-in-Themselves." [[Mind (journal)|''Mind'']] 3(9):57–67. {{doi|10.1093/mind/os-3.9.57}}. {{JSTOR|2246617}}.</ref>

{{Blockquote|text=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.|author=|title=|source="On the Nature of Things-in-Themselves" (1878)}}

Regarding Clifford's concept, [[Sir Frederick Pollock, 3rd Baronet|Sir Frederick Pollock]] wrote:

{{blockquote|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 [[materialism|materialist]]. But the theory must, as a [[metaphysics|metaphysical]] theory, be reckoned on the idealist side. To speak technically, it is an idealist [[monism]].{{sfn|Chisholm|1911|p=506}} }}

''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.{{sfn|Chisholm|1911|p=506}} There has also been debate on the extent to which Clifford's doctrine of '[[Concomitance (doctrine)|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.<ref>Clifford, C. K., and [[G. E. Berrios]]. 2000. "Body and Mind." ''[[History of Psychiatry (journal)|History of Psychiatry]]'' 11(43):311–38. {{doi|10.1177/0957154x0001104305}}. {{PMID|11640231}}.</ref>

===Ethics===
[[File:Highgate Cemetery - East - William Kingdon Clifford 01.jpg|thumb|The grave in Highgate Cemetery - East - of William Kingdon Clifford, just north of the grave of [[Karl Marx]].]]
In his 1877 essay, ''The Ethics of Belief'', Clifford argues that it is immoral to believe things for which one lacks evidence.<ref name=":0">Clifford, William K. 1877. "[http://www.uta.edu/philosophy/faculty/burgess-jackson/Clifford.pdf The Ethics of Belief] {{Webarchive|url=https://web.archive.org/web/20140123160039/http://www.uta.edu/philosophy/faculty/burgess-jackson/Clifford.pdf |date=23 January 2014 }}." ''[[The Contemporary Review|Contemporary Review]]'' 29:289.</ref> 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."<ref name=":0" />

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''."<ref group="lower-roman">The italics are in the original.</ref> 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 as [[Clifford's principle]]: "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence."<ref name=":0" />

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 [[Pragmatism|pragmatist]] philosopher [[William James]] in his "[[The Will to Believe|Will to Believe]]" lecture. Often these two works are read and published together as [[Touchstone (metaphor)|touchstones]] for the debate over [[evidentialism]], [[faith]], and [[overbelief]].

==Premonition of relativity==
Though Clifford never constructed a full theory of [[spacetime]] and [[Special relativity|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 [[unit hyperbola#Parametrization|parametrized unit hyperbola]], which other authors later used as a model for relativistic velocity. Elsewhere he states:<ref>Clifford, William K. 1885. ''[[iarchive:commonsenseexac01clifgoog|Common Sense of the Exact Sciences]]''. London: [[Kegan Paul, Trench and Company|Kegan Paul, Trench and Co.]] p. 214.</ref>

: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.<ref group="lower-roman">This passage is immediately followed by a section on "The bending of space." However, according to the preface (p.vii), this section was written by [[Karl Pearson]]</ref>

This passage makes reference to [[biquaternion]]s, though Clifford made these into [[split-biquaternion]]s 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 ''[[s:On the Space-Theory of Matter|On the Space-Theory of Matter]]'' 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."<ref>Frankland, William Barrett. 1910. ''Theories of Parallelism''. Cambridge: [[Cambridge University Press]]. pp. 48–49.</ref> 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.<ref>[[Hermann Weyl|Weyl, Hermann]]. 1923. ''Raum Zeit Materie''. Berlin: [[Springer-Verlag]]. p. 101</ref>

In 1940, [[Eric Temple Bell]] published ''The Development of Mathematics'', in which he discusses the prescience of Clifford on relativity:<ref>[[Eric Temple Bell|Bell, Eric Temple]]. 1940. ''The Development of Mathematics''. pp. 359–60.</ref>

: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.

[[John Archibald Wheeler]], during the 1960 International [[International Union of History and Philosophy of Science#DLMPST|Congress for Logic, Methodology, and Philosophy of Science]] (CLMPS) at [[Stanford University|Stanford]], introduced his [[geometrodynamics]] formulation of general relativity by crediting Clifford as the initiator.<ref>[[John Archibald Wheeler|Wheeler, John Archibald]]. 1962 [1960]. "Curved empty space as the building material of the physical world: an assessment." In ''Logic, Methodology, and Philosophy of Science'', edited by [[Ernest Nagel|E. Nagel]]. [[Stanford University Press]].</ref>

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.<ref>[[Gerald James Whitrow|Whitrow, Gerald James]]. 1961. ''The Natural Philosophy of Time'' (1st ed.). pp. 246–47.—1980 [1961]. ''The Natural Philosophy of Time'' (2nd ed.). pp. 291.</ref>

[[Cornelius Lanczos]] (1970) summarizes Clifford's premonitions:<ref>[[Cornelius Lanczos|Lanczos, Cornelius]]. 1970. ''Space Through the Ages: The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein''. [[Academic Press]]. p. 222.</ref>

:[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.

Likewise, [[Banesh Hoffmann]] (1973) writes:<ref>[[Banesh Hoffmann|Hoffmann, Banesh]]. 1973. "Relativity." ''Dictionary of the History of Ideas'' 4:80. [[Charles Scribner's Sons]].</ref>

:Riemann, and more specifically Clifford, conjectured that forces and matter might be local irregularities in the curvature of space, and in this they were strikingly prophetic, though for their pains they were dismissed at the time as visionaries.

In 1990, [[Ruth Farwell]] and Christopher Knee examined the record on acknowledgement of Clifford's foresight.<ref name=":1">[[Ruth Farwell|Farwell, Ruth]], and Christopher Knee. 1990. ''[[Studies in History and Philosophy of Science]]'' 21:91–121.</ref> 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."<ref name=":1" /> In 1992, Farwell and Knee continued their study of Clifford and Riemann:<ref>[[Ruth Farwell|Farwell, Ruth]], and Christopher Knee. 1992. "The Geometric Challenge of Riemann and Clifford." Pp. 98–106 in ''1830–1930: A Century of Geometry'', edited by L. Boi, D. Flament, and J. Salanskis. ''[[Lecture Notes in Physics]]'' 402. [[Springer Berlin Heidelberg]]. {{isbn|978-3-540-47058-8|}}. {{doi|10.1007/3-540-55408-4_56}}.</ref><blockquote>[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.</blockquote>

== Selected writings ==
* 1872. ''On the aims and instruments of scientific thought'', 524–41.
* 1876 [1870]. ''[[s:On the Space-Theory of Matter|On the Space-Theory of Matter]]''.<ref>Clifford, William K. 1876 [1870]. "[[wikisource:On the Space-Theory of Matter|On the Space-Theory of Matter]]." [[Mathematical Proceedings of the Cambridge Philosophical Society|''Proceedings of the Cambridge Philosophical Society'']] 2:157–58. {{OCLC|6084206}}. {{OL|20550270M}}. {{Internet Archive|id=proceedingscamb06socigoog|name=proceedingscamb06socigoog|page=175}}</ref><ref>Clifford, William K. 2007 [1870]. "On the Space-Theory of Matter." P. 71 in ''Beyond Geometry: Classic Papers from Riemann to Einstein'', edited by P. Pesic. Mineola: [[Dover Publications]]. {{Bibcode|2007bgcp.book...71K}}.</ref>
* 1877. "The Ethics of Belief." ''[[The Contemporary Review|Contemporary Review]]'' 29:289.<ref name=":0" /><ref>Clifford, William K. 1886 [1877]. "[http://ajburger.homestead.com/files/book.htm#foot2return The Ethics of Belief]" (full text). ''Lectures and Essays'' (2nd ed.), edited by [[Leslie Stephen|L. Stephen]] and F. Pollock. [[Macmillan Publishers|Macmillan and Co]]. – via A. J. Burger (2008).</ref>
* 1878. ''[[Elements of Dynamic|Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies]]''.<ref>Clifford, William K. 1878. [[iarchive:elementsofdynami01clifiala/|''Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies'' I, II, & III]]. London: [[Macmillan Publishers|MacMillan and Co.]] – via Internet Archive.</ref>
**Book I: "Translations"
**Book II:  "Rotations"
**Book III: "Strains"
* 1878. "Applications of Grassmann's Extensive Algebra." ''[[American Journal of Mathematics]]'' 1(4):353.<ref>Clifford, William K. 1878. "Applications of Grassmann's Extensive Algebra." ''[[American Journal of Mathematics]]'' 1(4):353.  {{doi|10.2307/2369379}}.</ref>
* 1879: ''Seeing and Thinking''<ref>Clifford, William K. 1879. ''[[iarchive:seeingandthinki00goog/page/n12/mode/2up|Seeing and Thinking]]''. London: [[Macmillan Publishers|Macmillan and Co.]]</ref>—includes four popular science lectures:{{sfn|Chisholm|1911|p=506}}
**"The Eye and the Brain"
**"The Eye and Seeing"
**"The Brain and Thinking"
**"Of Boundaries in General"
* 1879. ''Lectures and Essays'' I & II, with an introduction by [[Sir Frederick Pollock, 3rd Baronet|Sir Frederick Pollock]].<ref>Clifford, William K. 1901 [1879]. ''[[iarchive:lecturesessays01clifiala/page/n10/mode/2up|Lectures and Essays]]'' I (3rd ed.), edited by [[Leslie Stephen|L. Stephen]] and [[Sir Frederick Pollock, 3rd Baronet|F. Pollock]]. New York: [[Macmillan and Co.|The Macmillan Company]].</ref>
* 1881. "Mathematical fragments" ([[facsimile]]s).<ref>Clifford, William K. 1881. "[http://www.babordnum.fr/files/original/0b6d9b68b9348a5b95fff70afb205468.pdf Mathematical Fragments]" (facsimile).  London: [[Macmillan Publishers|Macmillan Company]].

Located at [[University of Bordeaux]]. Science and Technology Library. FR 14652.</ref>
* 1882. ''Mathematical Papers'', edited by [[Robert Tucker (mathematician)|Robert Tucker]], with an introduction by [[Henry John Stephen Smith|Henry J. S. Smith]].<ref>Clifford, William K. 1882. [[iarchive:mathematicalpap00smitgoog/page/n9|''Mathematical Papers'']], edited by [[Robert Tucker (mathematician)|R. Tucker]], introduction by [[Henry John Stephen Smith|H. J. S. Smith]]. London: [[Macmillan Publishers|MacMillan and Co]]. – via [[Internet Archive]].</ref>
* 1885. ''The Common Sense of the Exact Sciences'', completed by [[Karl Pearson]].<ref>Clifford, William K. 1885. ''[[iarchive:commonsenseexac01clifgoog/page/n10/mode/2up|The Common Sense of the Exact Sciences]]'', completed by [[Karl Pearson|K. Pearson]]. London: [[Kegan Paul, Trench and Company|Kegan, Paul, Trench, and Co.]]</ref>{{sfn|Chisholm|1911|p=506}}
* 1887. ''Elements of Dynamic'' 2.<ref>Clifford, William K. 1996 [1887]. "Elements of Dynamic" 2. In ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', edited by [[William B. Ewald Jr.|W. B. Ewald]]. Oxford. [[Oxford University Press]].</ref>
<gallery>
File:Clifford-1.jpg|1885 copy of "''The Common Sense of the Exact Sciences''"
File:Clifford-1-2.jpg|Title page of an 1885 copy of "''The Common Sense of the Exact Sciences''"
File:Clifford-1-3.jpg|Table of contents page for an 1885 copy of "''The Common Sense of the Exact Sciences''"
File:Clifford-1-4.jpg|First page of an 1885 copy of "''The Common Sense of the Exact Sciences''"
</gallery>

==Quotations==
{{Copy to Wikiquote}}{{Blockquote|text="I…hold that in the physical world nothing else takes place but this variation [of the curvature of space]."|author=|title=''Mathematical Papers'' (1882)|source=}}
----{{Blockquote|text="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."|author=|title="Some of the conditions of mental development" (1882), lecture to the [[Royal Institution]]|source=}}
----{{Blockquote|text="It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence."|author=|title=''The Ethics of Belief'' (1879) [1877]|source=}}
----{{Blockquote|text="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."|author=|title=''The Ethics of Belief'' (1879) [1877]|source=}}
----{{Blockquote|text="I was not, and was conceived. I loved and did a little work. I am not and grieve not."|author=|title=Epitaph|source=}}
----

== See also ==
* [[Bessel–Clifford function]]
* [[Clifford's principle]]
* [[Clifford analysis]]
* [[Clifford gates]]
* [[Clifford bundle]]
* [[Clifford module]]
* [[Multivector|Clifford number]]
* [[Motor variable|Motor]]
* [[Rotor (mathematics)|Rotor]]
* [[Simplex]]
* [[Split-biquaternion]]
* [[Will to Believe Doctrine]]

==References==

=== Notes ===
{{Reflist|group=lower-roman}}

=== Citations ===
{{Reflist|30em}}

*{{EB1911|wstitle=Clifford, William Kingdon |volume=6 |page=506}}

== Further reading ==

* {{cite journal | last = Chisholm | first = M. | title = William Kingdon Clifford (1845-1879) and his wife Lucy (1846-1929) | journal = [[Advances in Applied Clifford Algebras]] | year = 1997 | volume = 7S | pages = 27–41 | url = http://www.clifford-algebras.org/ | access-date = 22 April 2007 | archive-date = 3 August 2020 | archive-url = https://web.archive.org/web/20200803211720/http://clifford-algebras.org/ | url-status = dead }} (The on-line version lacks the article's photographs.)
* {{cite book | last = Chisholm | first = M. | title = Such Silver Currents - The Story of William and Lucy Clifford, 1845-1929 | publisher = The Lutterworth Press | year = 2002 | location = Cambridge, UK | isbn = 978-0-7188-3017-5}}
* {{cite journal | last1 = Farwell | first1 = Ruth | last2 = Knee | first2 = Christopher | year = 1990 | title = The End of the Absolute: a nineteenth century contribution to General Relativity | journal = Studies in History and Philosophy of Science | volume = 21 | issue = 1 | pages = 91–121 | doi=10.1016/0039-3681(90)90016-2| bibcode = 1990SHPSA..21...91F }}
* {{cite book | last = Macfarlane | first = Alexander | title = Lectures on Ten British Mathematicians of the Nineteenth Century | year = 1916 | publisher = John Wiley and Sons | location = New York | url = https://archive.org/details/lecturesontenbr00macfgoog | quote = Lectures on Ten British Mathematicians of the Nineteenth Century. }} (See especially pages 78–91)
*Madigan, Timothy J. (2010). ''W.K. Clifford and "The Ethics of Belief'' Cambridge Scholars Press, Cambridge, UK 978-1847-18503-7.
* {{cite book | last = Penrose | first = Roger | title = The Road to Reality: A Complete Guide to the Laws of the Universe | url = https://archive.org/details/roadtorealitycom00penr_0 | url-access = registration | publisher = Alfred A. Knopf | year = 2004 | isbn = 9780679454434 }} (See especially Chapter 11)
* {{Cite book | last1 = Stephen | first1 = Leslie | last2 = Pollock | first2 = Frederick | title = Lectures and Essays by the Late William Kingdon Clifford, F.R.S | place = New York | publisher = Macmillan and Company | year = 1879 | volume = 1 | url = https://archive.org/details/lecturesessays01clifiala }}
* {{Cite book | last1 = Stephen | first1 = Leslie | last2 = Pollock | first2 = Frederick | title = Lectures and Essays by the Late William Kingdon Clifford, F.R.S | place = New York | publisher = Macmillan and Company | year = 1879 | volume = 2 | url = https://archive.org/details/lecturesessaysby02clifrich }}

== External links ==
{{sister project links|d=Q323228|s=Author:William Kingdon Clifford|n=no|b=no|v=no|voy=no|m=no|mw=no|species=no|c=Category:William Kingdon Clifford|wikt=no}}
*{{Gutenberg author|id=45742}}
* [http://www.williamandlucyclifford.com/ William and Lucy Clifford (with pictures)]
* {{MacTutor|id=Clifford}}
* {{Internet Archive author |sname=William Kingdon Clifford |sopt=t}}
* {{Librivox author |id=2893}}
* Clifford, William Kingdon, William James, and A.J. Burger (Ed.), [http://ajburger.homestead.com/ethics.html ''The Ethics of Belief''].
* Joe Rooney [http://oro.open.ac.uk/8455/01/chapter4(020507).pdf William Kingdon Clifford], Department of Design and Innovation, the Open University, London.

{{philosophy of religion}}

{{Authority control}}

{{DEFAULTSORT:Clifford, William Kingdon}}
[[Category:1845 births]]
[[Category:1879 deaths]]
[[Category:19th-century deaths from tuberculosis]]
[[Category:19th-century British philosophers]]
[[Category:19th-century English mathematicians]]
[[Category:English atheists]]
[[Category:Algebraists]]
[[Category:British relativity theorists]]
[[Category:Alumni of Trinity College, Cambridge]]
[[Category:Fellows of Trinity College, Cambridge]]
[[Category:Alumni of King's College London]]
[[Category:Academics of University College London]]
[[Category:Fellows of the Royal Society]]
[[Category:Burials at Highgate Cemetery]]
[[Category:Second Wranglers]]
[[Category:Panpsychism]]
[[Category:Scientists from Exeter]]
[[Category:Tuberculosis deaths in Portugal]]
[[Category:British epistemologists]]