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| {{Infobox scientist
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| |name = William Clifford
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| |image = Clifford William Kingdon.jpg
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| |image_size = 300px
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| |caption = William Kingdon Clifford (1845–1879)
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| |birth_date = {{birth-date|4 May 1845}}
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| |birth_place = [[Exeter]], [[Devon]], [[England]]
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| |death_date = {{death-date|3 March 1879 }} (aged 33)
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| |death_place = [[Madeira]], [[Portugal]]
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| |residence = [[England]]
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| |citizenship =
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| |nationality = [[England|English]]
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| |ethnicity =
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| |field = [[Mathematics]]<br>[[Philosophy]]
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| |work_institutions = [[University College London]]
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| |alma_mater = [[King's College London]]<br>[[Trinity College, Cambridge]]
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| |doctoral_advisor =
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| |doctoral_students = [[Arthur Black (mathematician)|Arthur Black]]
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| |known_for = [[Clifford algebra]]<br>[[Clifford's theorem on special divisors|Clifford's theorem]]<br>[[Clifford–Klein form]]<br>[[Clifford parallel]]<br>[[Bessel–Clifford function]]<br>[[Dual quaternion]]<br>''[[Elements of Dynamic]]''
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| |author_abbrev_bot =
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| |author_abbrev_zoo =
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| |influences = [[Georg Friedrich Bernhard Riemann]]<br>[[Nikolai Ivanovich Lobachevsky]]
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| |influenced =
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| |prizes =
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| |religion = [[Anglican]] turned [[atheist]]<ref>http://www.massey.ac.nz/~wwifs/mathnews/NZMS86/news86a.shtml</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'', August 20, 2001, Pg. 19.</ref>
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| |footnotes = He was married to the novelist [[Lucy Clifford]].
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| |signature =
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| }}
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| '''William Kingdon Clifford''' [[Fellow of the Royal Society|FRS]] (4 May 1845 – 3 March 1879) was an [[England|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]],<ref>{{cite book|last=Doran|first=Chris|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=Grassmann's Legacy in From Past to Future: Graßmann's Work in Context, Petsche, Hans-Joachim, Lewis, Albert C., Liesen, Jörg, Russ, Steve (ed)|year=2011|publisher=Springer|location=Basel, Germany|isbn=978-3-0346-0404-8|pages=243–260|url=http://dx.doi.org/10.1007/978-3-0346-0405-5_22}}</ref> and [[computing]].<ref>{{cite book|last=Dorst|first=Leo|title=Geometric Algebra for Computer Scientists|year=2009|isbn=9780123749420|pages=664|publisher=Morgan Kaufman|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".
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| ==Biography==
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| Born at [[Exeter, England|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 (University of Cambridge)|wrangler]] in 1867 and second Smith's prizeman.<ref>{{acad|id=CLFT863WK|name=Clifford, William Kingdon}}</ref> (Being second was a fate he shared with others who became famous mathematicians: for example, [[William Thomson, 1st Baron Kelvin|William Thomson]] (Lord Kelvin), or [[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.<ref>{{Cite book | last = Chisholm | first = M. | title = Such Silver Currents | year = 2002 | publisher = The Lutterworth Press | location = Cambridge | page = 26 | isbn = 0-7188-3017-2}}</ref>
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| 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]].
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| On April 7, 1875, Clifford married [[Lucy Clifford|Lucy Lane]].<ref>{{Cite book | last = Stephen | first = 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 | edition = | page = 20 | url = http://www.openlibrary.org/details/lecturesessays01clifiala }}</ref> 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.
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| Similar to [[Charles Lutwidge Dodgson]], he enjoyed entertaining children, writing a collection of fairy stories, ''The Little People''.
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| 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]].
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| == Mathematician ==
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| "Clifford was above all and before all a geometer." ([[Henry John Stephen Smith|H. J. S. Smith]]). | |
| The discovery of [[non-Euclidean geometry]] opened new possibilities in geometry in Clifford's era. The field of [[differential geometry]] was born, where [[curvature]] applied to [[space]] as well as curves 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]] (1854, 1867) [http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html On the hypotheses which lie at the bases of geometry], [[Habilitationsschrift]] and posthumous publication, translated by Clifford, link from 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>W.K. Clifford (1873) "On the hypotheses which lie at the bases of geometry", [[Nature (journal)|Nature]] 8:14 to 17, 36, 37; also Paper #9 in ''Mathematical Papers'' (1882), page 55, synopsis pp 70,1</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.
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| In 1878 he published his seminal work 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|url=http://www.jstor.org/stable/10.2307/2369379}}</ref> He had succeeded in unifying the [[quaternions]], developed by [[William Rowan Hamilton]], and Grassmann's outer product (also known as the [[exterior product]]). He did this by defining a geometric product, composed of the sum of the [[inner product]] and the outer product. The former equips geometric algebra with a metric, fully incorporating distance and angle relationships for lines, planes, and volumes. The latter gives those planes and volumes vector-like properties including a directional bias. The resulting geometric algebra, as he called it, realizes the long sought goal 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=http://geocalc.clas.asu.edu/html/Evolution.html}}</ref> Moreover, it is fully extensible to any number of higher dimensions. The algebraic operations have the same fundamental nature as they do in 2 or 3-dimensions. The importance of geometric algebra - and of other Clifford algebras - has grown enormously over time. Variations have been shown to be [[isomorphic]] to [[complex numbers]], [[vector algebra]], [[matrices]], [[tensor algebra]], [[differential geometry]], and other mathematical systems, as well as quaternions. They also touch on many aspects of [[group theory]].<ref>{{cite journal|last=Dechant|first=Pierre-Phillipe|title=A Clifford algebraic framework for Coxeter group theoretic computations|journal=ArXiv e-print|date=Jul, 2012|url=http://adsabs.harvard.edu/abs/2012arXiv1207.5005D}}</ref>
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| [[File:William Kingdon Clifford by John Collier.jpg|right|thumb|<center>Clifford by [[John Collier (artist)|John Collier]]]]
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| Clifford's contemporaries considered him acute and original, witty and warm. He was also very productive working late into the night, which may have led to his early and unfortunate 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|author1=Norman L. Biggs|author2=Edward Keith Lloyd|author3=Robin James Wilson|title=Graph Theory: 1736-1936|url=http://books.google.com/books?id=XqYTk0sXmpoC&pg=PA67|accessdate=30 July 2013|year=1976|publisher=Oxford University Press|isbn=978-0-19-853916-2|page=67}}</ref>
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| == Evolution of Clifford Algebras ==
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| The [[academic journal]] ''[[Advances in Applied Clifford Algebras]]'' has provided an outlet for the inheritors of Clifford’s legacy in [[kinematics]] and [[abstract algebra]]. When he was alive the realms of [[real analysis]] and [[complex analysis]] had been expanded by the algebra '''H''' of [[quaternion]]s, where there is a [[three-dimensional sphere]]. Quaternion [[versor]]s that inhabit this 3-sphere provide a representation of the [[rotation group SO(3)]] . Clifford elaborated [[elliptic geometry#Elliptic space|elliptic space geometry]] as a [[non-Euclidean geometry|non-Euclidean]] [[metric space]]. Equidistant curves in elliptic space are said to be [[Clifford parallel]]s.
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| Clifford noted that Hamiltion’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.
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| ==Philosopher==
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| [[File:Clifford William Kingdon desk.jpg|desk|right|thumb|William Kingdon Clifford]]
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| 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, 3rd Baronet|Sir Frederick Pollock]] wrote about Clifford as follows:
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| <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]]."</blockquote>
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| Clifford himself defined "mind-stuff" as follows (1878, "On the Nature of Things-in-Themselves", ''Mind'', Vol. 3, No. 9, pp. 57–67):
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| <blockquote>"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."</blockquote>
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| 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.<ref>Berrios G E (2000) Body and Mind: C K Clifford. ''History of Psychiatry'' 11: 311-338</ref>
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| Arguing that it was immoral to believe things for which one lacks evidence, his 1877 essay "The Ethics of Belief" 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 [[Pragmatism|pragmatist]] philosopher [[William James]] in his [[Will to Believe Doctrine|"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]].
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| ==Premonition of relativity==
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| 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:
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| 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,
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| :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>''Common Sense of the Exact Sciences'' (1885), page 214 (page 193 of the Dover reprint), 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>
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| This passage makes reference to [[biquaternion]]s, though Clifford made these into [[split-biquaternion]]s as his independent development.
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| 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.
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| In 1910 William Barrett Frankland quoted the ''Space-Theory of Matter'' in his book on parallelism.<ref>William Barrett Frankland (1910) ''Theories of Parallelism'', pp 48,9, [[Cambridge University Press]]</ref> He wrote:
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| :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.
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| Years later, after [[general relativity]] had been advanced by [[Albert Einstein]], various authors noted that Clifford had anticipated Einstein:
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| In 1923 [[Hermann Weyl]] mentioned Clifford<ref>''Raum Zeit Materie'', page 101, [[Springer-Verlag]], Berlin</ref> as one of those who, like [[Bernhard Riemann]], anticipated the geometric ideas of relativity.
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| 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:
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| :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.
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| 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.<ref>J. Wheeler (1960) "Curved empty space as the building material of the physical world: an assessment", in Ernest Nagel (1962) ''Logic, Methodology, and Philosophy of Science'', Stanford University Press</ref>
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| In his ''The Natural Philosophy of Time'' (1961, 1980) [[Gerald James Whitrow]] recalls Clifford's prescience by quoting him to describe the [[Friedmann–Lemaître–Robertson–Walker metric]] in cosmology (1st ed pp 246,7; 2nd ed p 291).
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| In 1970 [[Cornelius Lanczos]] summarizes Clifford's premonitions this way:
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| :[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.<ref>[[Cornelius Lanczos]] (1970) ''Space through the Ages: The evolution of geometrical ideas from Pythagoras to Hilbert and Einstein'', page 222, [[Academic Press]]</ref>
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| In 1973 [[Banesh Hoffmann]] wrote:
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| :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.<ref>[[Banesh Hoffmann]] (1973) "Relativity" in ''Dictionary of the History of Ideas'' 4:80, [[Charles Scribner's Sons]]</ref>
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| 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." <ref>Farwell & Knee (1990)''Studies in History and Philosophy of Science'' 21:91–121</ref>
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| In 1992 Farwell and Knee continued their study with "The Geometric Challenge of Riemann and Clifford"<ref>Farwell & Knee (1992) in ''1830–1930: A Century of Geometry'', pages 98 to 106, Lecture Notes in Physics #402, Springer-Verlag ISBN 3-540-55408-4</ref>
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| 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."
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| == Selected writings ==
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| Most of his work was published posthumously.
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| *1877. [http://www.uta.edu/philosophy/faculty/burgess-jackson/Clifford.pdf "The Ethics of Belief"], ''Contemporary Review''.
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| *1878. [http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=04370002 Elements of Dynamic], London: MacMillan & Co; on-line presentation by [[Cornell University]] ''Historical Mathematical Monographs''.
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| *1879. [http://books.google.com/books?id=Tdrry7p7DeMC&printsec=frontcover&dq=william+kingdon+clifford&as_brr=1#PPP9,M1 ''Seeing and Thinking''], popular science lectures.
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| *1879. [http://www.openlibrary.org/details/lecturesessaysby02clifrich ''Lectures and Essays''], with an introduction by [[Sir Frederick Pollock, 3rd Baronet|Sir Frederick Pollock]].
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| *1882. ''Mathematical Papers'' (at [http://books.google.com/books?id=LzXW1Lj62AMC ''Google Books'']; at [http://www.ams.org/bookstore-getitem/item=chel/210.h ''American Mathematical Society'']), edited by R Tucker, with an introduction by [[Henry John Stephen Smith|Henry J. S. Smith]].
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| *1885. [http://books.google.com/books?id=kAUAAAAAQAAJ&printsec=frontcover&dq=exact+sciences&as_brr=1#PPR3,M1 ''The Common Sense of the Exact Sciences'']. Completed by [[Karl Pearson]].
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| *1887. ''Elements of Dynamic'', vol. 2, in Ewald, William B., ed., 1996. ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols. Oxford University Press.
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| *1872. ''On the aims and instruments of scientific thought'', 524-41.
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| *1876. ''[[s:On the Space-Theory of Matter|On the Space-Theory of Matter]]''.
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| ==Quotations==
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| [[File:Clifford grave 1986.jpg|thumb|Marker for W. K. Clifford and his wife in Highgate Cemetery (''ca''. 1986)]]
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| *"I ... hold that in the physical world nothing else takes place but this variation [of the curvature of space]." ''Mathematical Papers''.
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| *"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")
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| *"It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." ''The Ethics of Belief'' (1879)
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| *"I was not, and was conceived. I loved and did a little work. I am not and grieve not." - ''epitaph''.
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| *"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." - ''Contemporary Review'' (1877)
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| == See also ==
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| * [[Clifford–Klein form]]
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| * [[Will to Believe Doctrine]]
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| ==Notes==
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| {{Reflist}}
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| == References ==
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| *{{Cite EB1911|wstitle=Clifford, William Kingdon}}
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| == Further reading ==
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| * {{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/ }} (The on-line version lacks the article's photographs.)
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| * {{cite book | last = Chisholm | first = M. | coauthors = | title = Such Silver Currents - The Story of William and Lucy Clifford, 1845-1929 | publisher = The Lutterworth Press | year = 2002 | location = Cambridge, UK | isbn = 0-7188-3017-2}}
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| * Ruth Farwell & Christopher Knee (1990) "The End of the Absolute: a nineteenth century contribution to General Relativity", ''Studies in History and Philosophy of Science'' 21: 91–121.
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| * {{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 = http://books.google.com/?id=qWQSAAAAIAAJ&printsec=frontcover&dq=Lectures+on+Ten+British+Mathematicians+of+the+Nineteenth+Century#PPA78,M1 }} (See especially pages 78 – 91)
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| *Madigan, Timothy J. (2010). ''''W.K. Clifford and "The Ethics of Belief"'' Cambridge Scholars Press, Cambridge, UK 978-1847-18503-7.
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| * {{cite book | last = Penrose | first = Roger | title = The Road to Reality: A Complete Guide to the Laws of the Universe | publisher = Alfred A. Knopf | year = 2004 | location = }} (See especially Chapter 11)
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| * {{Cite book | last = Stephen | first = 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 | edition = | url = http://www.archive.org/details/lecturesessays01clifiala }}
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| * {{Cite book | last = Stephen | first = 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 | edition = | url = http://www.archive.org/details/lecturesessaysby02clifrich }}
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| == External links ==
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| {{wikisource author}}
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| {{Wikiquote}}
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| {{commons category}}
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| * [http://www.williamandlucyclifford.com/ William and Lucy Clifford (with pictures)]
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| * "''[http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Clifford.html William Kingdon Clifford]''". School of Mathematics and Statistics, University of St Andrews, Scotland.
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| * Clifford, William Kingdon, William James, and A.J. Burger (Ed.), ''[http://ajburger.homestead.com/ethics.html The Ethics of Belief]''".
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| * [http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=10610&pt=William%20Kingdon%20Clifford Clifford's gravesite]
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| * "''[http://www.1911encyclopedia.org/William_Kingdon_Clifford William Kingdon Clifford]''". 1911 Encyclopædia Britannica.
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| * Joe Rooney [http://oro.open.ac.uk/8455/01/chapter4(020507).pdf William Kingdon Clifford], Department of Design and Innovation, the Open University, London.
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| {{philosophy of religion}}
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| {{Authority control|VIAF=51772037}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Clifford, William Kingdon
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = British mathematician and philosopher
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| | DATE OF BIRTH = 4 May 1845
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| | PLACE OF BIRTH = [[Exeter]], [[Devon]], [[England]]
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| | DATE OF DEATH = 3 March 1879
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| | PLACE OF DEATH = [[Madeira]], [[Portugal]]
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| }}
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| {{DEFAULTSORT:Clifford, William Kingdon}}
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| [[Category:1845 births]]
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| [[Category:1879 deaths]]
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| [[Category:Deaths from tuberculosis]]
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| [[Category:19th-century philosophers]]
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| [[Category:19th-century English mathematicians]]
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| [[Category:English atheists]]
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| [[Category:Algebraists]]
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| [[Category:Relativists]]
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| [[Category:Alumni of Trinity College, Cambridge]]
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| [[Category:Fellows of Trinity College, Cambridge]]
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| [[Category:Alumni of King's College London]]
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| [[Category:Academics of University College London]]
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| [[Category:Fellows of the Royal Society]]
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| [[Category:Burials at Highgate Cemetery]]
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| [[Category:Second Wranglers]]
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| [[Category:People from Exeter]]
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