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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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

mathematician
and
philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mi ...

philosopher
. Building on the work of
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual wh ...
, he introduced what is now termed
geometric algebra In mathematics, the geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which contains both the scalar (mathematics), ...
, a special case of the
Clifford algebra In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
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 developme ...
,
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, 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 computer hardware , hardware and software. It has sci ...

computing
. Clifford was the first to suggest that
gravitation Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is t ...

gravitation
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 Devon (, archaically known as Devonshire) is a county A county is a geographical region of a country used for administrative or other purposesChambers Dictionary The ''Chambers Dictionary'' (''TCD'') ...
, 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 In public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an organization An organization, or ...
(at age 15) and
Trinity College, Cambridge Trinity College is a constituent college A collegiate university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) education ...
, 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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as num ...

James Clerk Maxwell
. In 1870, he was part of an expedition to Italy to observe the
solar eclipse A solar eclipse occurs when a portion of the Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The r ...
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 University College London, which Trade name, operates as UCL, is a major public university , public research university located in London, United Kingdom. UCL is a Member institutions of the University of London, member institution of the Federa ...
, 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 A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exis ...
. He was also a member of the
London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization An organization, or organisati ...
and the
Metaphysical Society The Metaphysical Society was a famous British debating society Debate is a process that involves formal discourse on a particular topic, often including a moderator and audience. In a debate, argument In logic Logic is an interdiscip ...
. Clifford married
Lucy Lane Lucy Lane is a fictional supporting character in DC Comics DC Comics, Inc. is an American comic book publisher and the flagship unit of DC Entertainment DC Comics, Inc. is an American comic book publisher and the flagship unit of #DC Enter ...
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 An infection is the invasion of an organism's body Tissue (biology), tissues by Pathogen, disease-causing agents, their multiplication, and the reaction of host (biology), host tissues to the in ...

tuberculosis
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 North London is the northern part of London London is the capital city, capital and List of urban areas in the United Kingdom, largest city of England and the United Kingdom. The c ...

Highgate Cemetery
, near the graves of
George Eliot Mary Ann Evans (22 November 1819 – 22 December 1880; alternatively Mary Anne or Marian), known by her pen name A pen name, also called a ''nom de plume'' () or a literary double, is a pseudonym (or, in some cases, a variant form of a real n ...

George Eliot
and
Herbert Spencer Herbert Spencer (27 April 1820 – 8 December 1903) was an English philosopher, biologist A biologist is a professional who has specialized knowledge in the field of biology, understanding the underlying mechanisms that govern the functio ...

Herbert Spencer
, just north of the grave of
Karl Marx Karl Heinrich Marx (; 5 May 1818 – 14 March 1883) was a German philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, M ...

Karl Marx
. The
academic journal An academic or scholarly journal is a periodical publication Periodical literature (also called a periodical publication or simply a periodical) is a category of serial Serial may refer to: Arts, entertainment, and media The presentation of w ...
''
Advances in Applied Clifford Algebras ''Advances in Applied Clifford Algebras'' is a peer-reviewed scientific journal that publishes original research papers and also notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops ...
'' publishes on Clifford's legacy in
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes it ...

kinematics
and
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
.


Mathematics

The discovery of
non-Euclidean geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
opened new possibilities in geometry in Clifford's era. The field of intrinsic
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
was born, with the concept of
curvature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

curvature
broadly applied to
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...

space
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
’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 law, ...
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 natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter an ...
'' 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 physicists of all time. Einstein is known for developing the theory of relativity The theo ...

Albert Einstein
's
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. Equidistant curves in elliptic space are now said to be
Clifford parallel In elliptic geometry Elliptic geometry is an example of a geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of ma ...
s. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
and the textbook ''
Elements of Dynamic ''Elements of Dynamic'' is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positive ...
''. His application of
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
to
invariant theory Invariant theory is a branch of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), ...
was followed up by
William Spottiswoode William H. Spottiswoode HFRSE LLD (11 January 1825 – 27 June 1883) was an English people, English mathematician, physicist and partner in the printing and publishing firm Eyre & Spottiswoode. He was President of the Royal Society from 1878 t ...

William Spottiswoode
and
Alfred Kempe Sir Alfred Bray Kempe Fellow of the Royal Society, FRS (6 July 1849 – 21 April 1922) was a mathematician best known for his work on Linkage (mechanical), linkages and the four colour theorem. Biography Kempe was the son of the Rector of St J ...
.


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 Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion ...
, developed by
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin , name_Latin = Collegium Sanctae et Individuae Trinitatis Reg ...
, with Grassmann's ''
outer product In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
'' (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
versor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of the
inner product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematician
Marcel Riesz Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 1946 * Hungarians, ethnic gro ...

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." . 1976
679 __NOTOC__ Year 679 ( DCLXXIX) was a common year starting on Saturday A common year starting on Saturday is any non-leap year A leap year (also known as an intercalary year or wikt:bissextile, bissextile year) is a calendar year that contain ...
"Letter to
Christian Huygens Christiaan Huygens ( , also , ; la, Hugenius; 14 April 1629 – 8 July 1695), also spelled Huyghens, was a Dutch mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) i ...

Christian Huygens
(8 September 1679)." In ''Philosophical Papers and Letters'' (2nd ed.).
Springer Springer or springers may refer to: Places ;United States * Springer, New Mexico Springer is a town A town is a human settlement In geography Geography (from Greek: , ''geographia'', literally "earth description") is a fi ...
.
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions. The realms of
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

real analysis
and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, which inhabit this 3-sphere, provide a representation of the
rotation group SO(3) In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...
. Clifford noted that Hamilton's
biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions cor ...
s were a
tensor product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

tensor product
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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s C might instead be taken from
split-complex number In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
s D or from the
dual number In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
s N. In terms of tensor products, H \otimes D produces
split-biquaternionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, while H \otimes N forms
dual quaternion In mathematics, the dual quaternions are an 8-dimensional real Algebra over a field, algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using d ...
s. The algebra of dual quaternions is used to express
screw displacement A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation (geometry), translation of a body occurs. Chasles' theorem (kinematics), Chasles' theorem shows that each Euclidea ...
, 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 first principles of being, identity and change, space and time, causality, necessity and possibility. It includes questions about the nature of consciousness and the relationship between ...

metaphysical
conception, suggested to him by his reading of
Baruch Spinoza Baruch (de) Spinoza (; ; ; born Baruch 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 Spanish and Portuguese Jews, Por ...

Baruch Spinoza
, which Clifford (1878) defined as follows: Regarding Clifford's concept,
Sir Frederick Pollock
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 a social Social organisms, including humans, live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchange is voluntary/involuntary. Etymology ...

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 Obscurantism and Obscurationism ( or ) describe the practice of deliberately presenting information in an imprecise, abstruse manner designed to limit further inquiry and understanding. There are two historical and intellectual denotations of ''Ob ...
, 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 Divinity or the divine are things that are either related to, devoted to, or proceeding from a deity A deity or god is a supernatural The supernatural encompasses supposed ...
was still unreconciled with
Darwinism Darwinism is a theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as obse ...
; 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 In the philosophy of mind Philosophy of mind is a branch of that studies the and nature of the and its relationship with the body. The is a paradigmatic issue in philosophy of mind, although a number of other issues are addressed, such as t ...
' influenced
John Hughlings Jackson John Hughlings Jackson, Fellow of the Royal Society, 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, ...

John Hughlings Jackson
's model of the nervous system and, through him, the work of Janet, Freud, Ribot, and Ey.


Ethics

In his 1877 essay, "The Ethics of Belief", Clifford argues that it is immoral to believe things for which one lacks evidence.Clifford, William K. 1877.
The Ethics of Belief
" ''
Contemporary Review ''The Contemporary Review'' is a British biannual, formerly quarterly, magazine A magazine is a periodical publication Periodical literature (also called a periodical publication or simply a periodical) is a category of serial publicat ...
'' 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: "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 American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States ** Americans, citi ...
in his " Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over
evidentialism Evidentialism is a thesis in epistemology Epistemology (; ) is the concerned with . Epistemologists study the nature, origin, and scope of knowledge, epistemic , the of , and various related issues. Epistemology is considered a major sub ...
,
faith Faith, derived from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of ...

faith
, and
overbelief Overbelief (also written as "''over-belief''") is a philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge Knowledge is a familiarity, awareness, or u ...
.


Premonition of relativity

Though Clifford never constructed a full theory of
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
and relativity, there are some remarkable observations he made in print that foreshadowed these modern concepts: In his book
Elements of Dynamic ''Elements of Dynamic'' is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positive ...
(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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...

Karl Pearson
This passage makes reference to
biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions cor ...
s, though Clifford made these into
split-biquaternionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s 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, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. 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, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
had been advanced by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

Albert Einstein
, various authors noted that Clifford had anticipated Einstein.
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens o ...

Hermann Weyl
(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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
, anticipated the geometric ideas of relativity. In 1940,
Eric Temple Bell Eric Temple Bell (7 February 1883 – 21 December 1960) was a Scottish-born mathematician and science fiction writer who lived in the United States for most of his life. He published non-fiction using his given name and fiction as John Taine ...
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 John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist Theoretical physics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and ...
, 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. Sta ...

Stanford
, introduced his
geometrodynamics In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in c ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmet ...
recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology. Cornelius Lanczos (1970) summarizes Clifford's premonitions: :[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: :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.Ruth Farwell, Farwell, Ruth, and Christopher Knee. 1990. ''Studies in History and Philosophy of Science'' 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:
[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.


Selected writings

* 1872. ''On the aims and instruments of scientific thought'', 524–41. * 1876 [1870]. '' On the Space-Theory of Matter''. * 1877. "The Ethics of Belief." ''
Contemporary Review ''The Contemporary Review'' is a British biannual, formerly quarterly, magazine A magazine is a periodical publication Periodical literature (also called a periodical publication or simply a periodical) is a category of serial publicat ...
'' 29:289. * 1878. ''Elements of Dynamic, 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 . * 1881. "Mathematical fragments" (facsimiles). * 1882. ''Mathematical Papers'', edited by Robert Tucker (mathematician), Robert Tucker, with an introduction by Henry John Stephen Smith, 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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...

Karl Pearson
. * 1887. ''Elements of Dynamic'' 2.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.


Quotations

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See also

* Bessel–Clifford function * 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


Citations

*


Further reading

* (The on-line version lacks the article's photographs.) * * * (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. * (See especially Chapter 11) * *


External links

*
William and Lucy Clifford (with pictures)
* * * * Clifford, William Kingdon, William James, and A.J. Burger (Ed.)

* Joe Roone
William Kingdon Clifford
Department of Design and Innovation, the Open University, London. {{DEFAULTSORT:Clifford, William Kingdon 1845 births 1879 deaths 19th-century deaths from tuberculosis 19th-century British philosophers 19th-century English mathematicians English atheists Algebraists British relativity theorists Alumni of Trinity College, Cambridge Fellows of Trinity College, Cambridge Alumni of King's College London Academics of University College London Fellows of the Royal Society Burials at Highgate Cemetery Second Wranglers Panpsychism People from Exeter Tuberculosis deaths in Portugal