William Kingdon Clifford
   HOME

TheInfoList



OR:

William Kingdon Clifford (4 May 18453 March 1879) was a British
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, mathematical structure, structure, space, Mathematica ...
and
philosopher Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, 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 with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
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 is the development of mathematics, 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 de ...
,
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, and
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and the development of both computer hardware, hardware and softw ...
. Clifford was the first to suggest that
gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression ''mind-stuff''.


Biography

Born in
Exeter Exeter ( ) is a City status in the United Kingdom, cathedral city and the county town of Devon in South West England. It is situated on the River Exe, approximately northeast of Plymouth and southwest of Bristol. In Roman Britain, Exeter w ...
, William Clifford was educated at Doctor Templeton's Academy on Bedford Circus and 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 university, public research university in London, England. King's was established by royal charter in 1829 under the patronage of George IV of the United Kingdom, King George IV ...
(at age 15) and
Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, 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 ...
, where he was elected fellow in 1868, after being
Second Wrangler At the University of Cambridge in England, a "Wrangler" is a student who gains first-class honours in the Mathematical Tripos competition. The highest-scoring student is the Senior Wrangler, the second highest is the Second Wrangler, and so on ...
in 1867 and second Smith's prizeman. 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 Earth, totally or partially. Such an alignment occurs approximately every six months, during the eclipse season i ...
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 (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...
, 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 society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
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), also known colloquially as the "white death", or historically as consumption, is a contagious disease usually caused by ''Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can al ...
after a few months, leaving a widow with two children. Clifford and his wife are buried in London's Highgate Cemetery, near the graves of George Eliot and
Herbert Spencer Herbert Spencer (27 April 1820 – 8 December 1903) was an English polymath active as a philosopher, psychologist, biologist, sociologist, and anthropologist. Spencer originated the expression "survival of the fittest", which he coined in '' ...
, just north of the grave of
Karl Marx Karl Marx (; 5 May 1818 – 14 March 1883) was a German philosopher, political theorist, economist, journalist, and revolutionary socialist. He is best-known for the 1848 pamphlet '' The Communist Manifesto'' (written with Friedrich Engels) ...
. The
academic journal An academic journal (or scholarly journal or scientific journal) is a periodical publication in which Scholarly method, scholarship relating to a particular academic discipline is published. They serve as permanent and transparent forums for the ...
'' Advances in Applied Clifford Algebras'' publishes on Clifford's legacy in
kinematics In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with s ...
and
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
.


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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
was born, with the concept of
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
broadly applied to
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
itself as well as to curved lines and surfaces. Clifford was very much impressed by
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
’s 1854 essay "On the hypotheses which lie at the bases of geometry". 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 of Riemann's paper was published in ''
Nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
'' in 1873. His report at Cambridge, " On the Space-Theory of Matter", was published in 1876, anticipating
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
's
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. 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 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 (''p ...
and the textbook '' Elements of Dynamic''. His application of
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
to invariant theory was followed up by William Spottiswoode and Alfred Kempe.


Algebras

In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra. He had succeeded in unifying the quaternions, developed by
William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
, with Grassmann's ''
outer product In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions ''n'' and ''m'', the ...
'' (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, ofte ...
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 "Letter to Christian Huygens (8 September 1679)." In ''Philosophical Papers and Letters'' (2nd ed.).
Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...
.
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 or morphism 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 the ...
classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions. The realms of
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
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 a ...
. Clifford noted that Hamilton's biquaternions were a
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
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 for ...
s C might instead be taken from
split-complex number In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...
s D or from the dual numbers N. In terms of tensor products, H \otimes D produces
split-biquaternion In mathematics, a split-biquaternion is a hypercomplex number of the form : q = w + x\mathrm + y\mathrm + z\mathrm , where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each co ...
s, 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 examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the conceptual framework of h ...
conception, suggested to him by his reading of Baruch Spinoza, 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 a range of social system, social-cultural systems, including designated religious behaviour, behaviors and practices, morals, beliefs, worldviews, religious text, texts, sanctified places, prophecies, ethics in religion, ethics, or ...
. 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, obscurantism or obscurationism is the Anti-intellectualism, anti-intellectual practice of deliberately presenting information in an wikt:abstruse, abstruse and imprecise manner that limits further inquiry and understanding of a subj ...
, and to put the claims of sect above those of human society. The alarm was greater, as
theology Theology is the study of religious belief from a Religion, religious perspective, with a focus on the nature of divinity. It is taught as an Discipline (academia), academic discipline, typically in universities and seminaries. It occupies itse ...
was still unreconciled with
Darwinism ''Darwinism'' is a term used to describe a theory of biological evolution developed by the English naturalist Charles Darwin (1809–1882) and others. The theory states that all species of organisms arise and develop through the natural sel ...
; 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'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 and psychologist. The first educator to offer a psychology course in the United States, he is considered to be one of the leading thinkers of the late 19th c ...
in his " Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over evidentialism,
faith Faith is confidence or trust in a person, thing, or concept. In the context of religion, faith is " belief in God or in the doctrines or teachings of religion". According to the Merriam-Webster's Dictionary, faith has multiple definitions, inc ...
, and overbelief.


Premonition of relativity

Though Clifford never constructed a full theory of
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
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 biostatistician and mathematician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university ...
This passage makes reference to biquaternions, though Clifford made these into
split-biquaternion In mathematics, a split-biquaternion is a hypercomplex number of the form : q = w + x\mathrm + y\mathrm + z\mathrm , where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each co ...
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, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. 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 as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
had been advanced by
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
, 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, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
(1923), for instance, mentioned Clifford as one of those who, like
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
, 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 John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr to e ...
, during the 1960 International Congress for Logic, Methodology, and Philosophy of Science (CLMPS) at Stanford, introduced his geometrodynamics formulation of general relativity by crediting Clifford as the initiator. In ''The Natural Philosophy of Time'' (1961), Gerald James Whitrow recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology. Cornelius Lanczos (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 (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'' 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:
heyhold 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" ( facsimiles). * 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 biostatistician and mathematician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university ...
. * 1887. ''Elements of Dynamic'' 2.Clifford, William K. 1996 887 "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 publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...
.
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

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


See also

* Bessel–Clifford function * Clifford's principle * Clifford analysis * Clifford gates * Clifford bundle * Clifford module * Clifford number * Motor * Rotor * Simplex *
Split-biquaternion In mathematics, a split-biquaternion is a hypercomplex number of the form : q = w + x\mathrm + y\mathrm + z\mathrm , where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each co ...
* 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 Scientists from Exeter Tuberculosis deaths in Portugal British epistemologists