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William Kingdon Clifford (4 May 18453 March 1879) was an English
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
and
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
. Building on the work of Hermann Grassmann, he introduced what is now termed
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
, a special case of the
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
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 Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, and
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, ...
. Clifford was the first to suggest that
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression ''mind-stuff''.

# Biography

Born at
Exeter Exeter () is a city in Devon, South West England. It is situated on the River Exe, approximately northeast of Plymouth and southwest of Bristol. In Roman Britain, Exeter was established as the base of Legio II Augusta under the personal comm ...
, William Clifford showed great promise at school. He went on to
King's College London King's College London (informally King's or KCL) is a public research university located in London, England. King's was established by royal charter in 1829 under the patronage of King George IV and the Duke of Wellington. In 1836, King's ...
(at age 15) and
Trinity College, Cambridge Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge or Oxford. ...
, 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 James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
. In 1870, he was part of an expedition to Italy to observe the
solar eclipse A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby obscuring the view of the Sun from a small part of the Earth, totally or partially. Such an alignment occurs during an eclipse season, approximately every six mon ...
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 , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = ...
, and in 1874 became a fellow of the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
. He was also a member of the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical ...
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 Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, ...
after a few months, leaving a widow with two children. Clifford and his wife are buried in London's
Highgate Cemetery Highgate Cemetery is a place of burial in north London, England. There are approximately 170,000 people buried in around 53,000 graves across the West and East Cemeteries. Highgate Cemetery is notable both for some of the people buried there as ...
, near the graves of George Eliot and
Herbert Spencer Herbert Spencer (27 April 1820 – 8 December 1903) was an English philosopher, psychologist, biologist, anthropologist, and sociologist famous for his hypothesis of social Darwinism. Spencer originated the expression " survival of the f ...
, just north of the grave of
Karl Marx Karl Heinrich Marx (; 5 May 1818 – 14 March 1883) was a German philosopher, economist, historian, sociologist, political theorist, journalist, critic of political economy, and socialist revolutionary. His best-known titles are the 1848 ...
. The
academic journal An academic journal or scholarly journal is a periodical publication in which scholarship relating to a particular academic discipline is published. Academic journals serve as permanent and transparent forums for the presentation, scrutiny, and ...
'' Advances in Applied Clifford Algebras'' publishes on Clifford's legacy in
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
and
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
.

# Mathematics

The discovery of non-Euclidean geometry opened new possibilities in geometry in Clifford's era. The field of intrinsic
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
was born, with the concept of
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
itself as well as to curved lines and surfaces. Clifford was very much impressed by
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
’s 1854 essay "On the hypotheses which lie at the bases of geometry". In 1870, he reported to the
Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of l ...
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 Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
'' in 1873. His report at Cambridge, " On the Space-Theory of Matter", was published in 1876, anticipating
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
. 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 In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
to invariant theory was followed up by William Spottiswoode and
Alfred Kempe Sir Alfred Bray Kempe FRS (6 July 1849 – 21 April 1922) was a mathematician best known for his work on linkages and the four colour theorem. Biography Kempe was the son of the Rector of St James's Church, Piccadilly, the Rev. John Edward ...
.

## Algebras

In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra. He had succeeded in unifying the
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
, developed by
William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
, with Grassmann's ''
outer product In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of nu ...
'' (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 In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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"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." Leibniz, Gottfried. 1976
679 __NOTOC__ Year 679 ( DCLXXIX) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. The denomination 679 for this year has been used since the early medieval period, when the Anno Domini calendar ...
"Letter to
Christian Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...
(8 September 1679)." In ''Philosophical Papers and Letters'' (2nd ed.). Springer.
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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions. The realms of real analysis and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
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) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is ...
. Clifford noted that Hamilton's biquaternions were a
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
$H \otimes C$ of known algebras, and proposed instead two other tensor products of H: Clifford argued that the "scalars" taken from the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s C might instead be taken from
split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s D or from the dual numbers N. In terms of tensor products, $H \otimes D$ produces split-biquaternions, while $H \otimes N$ 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 Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
conception, suggested to him by his reading of
Baruch Spinoza Baruch (de) Spinoza (born Bento de Espinosa; later as an author and a correspondent ''Benedictus de Spinoza'', anglicized to ''Benedict de Spinoza''; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Portuguese-Jewish origin, ...
, 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 Religion is usually defined as a social- cultural system of designated behaviors and practices, morals, beliefs, worldviews, texts, sanctified places, prophecies, ethics, or organizations, that generally relates humanity to supernatur ...
. 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 In philosophy, the terms obscurantism and obscurationism describe the anti-intellectual practices of deliberately presenting information in an abstruse and imprecise manner that limits further inquiry and understanding of a subject. There are two ...
, and to put the claims of sect above those of human society. The alarm was greater, as
theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing th ...
was still unreconciled with
Darwinism Darwinism is a theory of biological evolution developed by the English naturalist Charles Darwin (1809–1882) and others, stating that all species of organisms arise and develop through the natural selection of small, inherited variations tha ...
; 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 John Hughlings Jackson, FRS (4 April 1835 – 7 October 1911) was an English neurologist. He is best known for his research on epilepsy. Biography He was born at Providence Green, Green Hammerton, near Harrogate, Yorkshire, the youngest s ...
'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.Clifford, William K. 1877.
The Ethics of Belief
" '' Contemporary Review'' 29:289.
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: "'' had no right to believe on such evidence as was before him''."The italics are in the original. 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." 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 William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States. James is considered to be a leading thinker of the lat ...
in his " Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over evidentialism,
faith Faith, derived from Latin ''fides'' and Old French ''feid'', is confidence or trust in a person, thing, or In the context of religion, one can define faith as "belief in God or in the doctrines or teachings of religion". Religious people ofte ...
, and overbelief.

# Premonition of relativity

Though Clifford never constructed a full theory of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
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: :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.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 Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
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 General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Clifford also discussed his views in '' 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." Years later, after
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
, various authors noted that Clifford had anticipated Einstein.
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
(1923), for instance, mentioned Clifford as one of those who, like
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
, 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: :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 Congress for Logic, Methodology, and Philosophy of Science (CLMPS) at
Stanford Stanford University, officially Leland Stanford Junior University, is a Private university, private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. S ...
, introduced his
geometrodynamics In theoretical physics, geometrodynamics is an attempt to describe spacetime and associated phenomena completely in terms of geometry. Technically, its goal is to unify the fundamental forces and reformulate general relativity as a configurati ...
formulation of general relativity by crediting Clifford as the initiator. In ''The Natural Philosophy of Time'' (1961),
Gerald James Whitrow Gerald James Whitrow (9 June 1912 – 2 June 2000) was a British mathematician, cosmologist and science historian. Biography Whitrow was born on 9 June 1912 at Kimmeridge in Dorset, the elder son of William and Emily (née Watkins) Whitrow. ...
recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology.
Cornelius Lanczos __NOTOC__ Cornelius (Cornel) Lanczos ( hu, Lánczos Kornél, ; born as Kornél Lőwy, until 1906: ''Löwy (Lőwy) Kornél''; February 2, 1893 – June 25, 1974) was a Hungarian-American and later Hungarian-Irish mathematician and physicist. Acco ...
(1970) summarizes Clifford's premonitions: : ewith 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 Banesh Hoffmann (6 September 1906 – 5 August 1986) was a British mathematician and physicist known for his association with Albert Einstein. Life Banesh Hoffmann was born in Richmond, Surrey, on 6 September 1906. He studied mathematics and ...
(1973) writes: :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. Farwell, Ruth, and Christopher Knee. 1990. ''
Studies in History and Philosophy of Science ''Studies in History and Philosophy of Science'' is a series of three peer-reviewed academic journals published by Elsevier. It was established in 1970 as a single journal, and was split into two sections–''Studies in History and Philosophy of S ...
'' 21:91–121.
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:
hey Hey or Hey! may refer to: Music * Hey (band), a Polish rock band Albums * ''Hey'' (Andreas Bourani album) or the title song (see below), 2014 * ''Hey!'' (Julio Iglesias album) or the title song, 1980 * ''Hey!'' (Jullie album) or the title ...
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.

# Selected writings

* 1872. ''On the aims and instruments of scientific thought'', 524–41. * 1876 870 '' On the Space-Theory of Matter''. * 1877. "The Ethics of Belief." '' Contemporary Review'' 29:289. * 1878. '' Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies''. **Book I: "Translations" **Book II: "Rotations" **Book III: "Strains" * 1878. "Applications of Grassmann's Extensive Algebra." '' American Journal of Mathematics'' 1(4):353. * 1879: ''Seeing and Thinking''—includes four popular science lectures: **"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. * 1881. "Mathematical fragments" (
facsimile A facsimile (from Latin ''fac simile'', "to make alike") is a copy or reproduction of an old book, manuscript, map, art print, or other item of historical value that is as true to the original source as possible. It differs from other forms of ...
s). * 1882. ''Mathematical Papers'', edited by Robert Tucker, with an introduction by Henry J. S. Smith. * 1885. ''The Common Sense of the Exact Sciences'', completed by
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university st ...
. * 1887. ''Elements of Dynamic'' 2.Clifford, William K. 1996
887 __NOTOC__ Year 887 ( DCCCLXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. Events By place Europe * November 17 – East Frankish magnates revolt against the inept emperor ...
"Elements of Dynamic" 2. In ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', edited by W. B. Ewald. Oxford.
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print book ...
.
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''"

# Quotations

---- ---- ---- ---- ----

* Bessel–Clifford function * Clifford's principle * Clifford analysis *
Clifford gates In quantum computing and quantum information theory, the Clifford gates are the elements of the Clifford group, a set of mathematical transformations which normalize the ''n''-qubit Pauli group, i.e., map tensor products of Pauli matrices to ten ...
*
Clifford bundle In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any ( pseudo) Riemannian ...
*
Clifford module In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra ''C'' is a central simple algebra over some field extension ''L'' of the field ''K'' over which the quadratic form ''Q'' defining ''C'' is de ...
* Clifford number * Motor * Rotor *
Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
* Split-biquaternion *
Will to Believe Doctrine "The Will to Believe" is a lecture by William James, first published in 1896, which defends, in certain cases, the adoption of a belief without prior evidence of its truth. In particular, James is concerned in this lecture about defending the rati ...

*