The term F distribution is found in Leo A. Aroian, "A study of R. A. Fisher's z distribution and the related F distribution," Ann. Math. Statist. 12, 429-448 (1941).
FACTOR (noun). Fibonacci (1202) used factus ex multiplicatione (Smith vol. 2, page 105).
Factor appears in English in 1673 in Elements of Algebra by John Kersey: "The Quantities given to be multiplied one by the other are called Factors."
FACTOR (verb) appears in English in 1848 in Algebra by J. Ray: "The principal use of factoring, is to shorten the work, and simplify the results of algebraic operations." Factorize (spelled "factorise") is found in 1886 in Algebra by G. Chrystal (OED2).
The term FACTOR ANALYSIS was introduced by Louis L. Thurstone (1887-1955) in 1931 in "Multiple Factor Analysis," Psychological Review, 38, 406-427: "It is the purpose of this paper to describe a more generally applicable method of factor analysis which has no restrictions as regards group factors and which does not restrict the number of general factors that are operative in producing the correlations" (OED2).
FACTOR GROUP. See quotient group.
FACTORIAL. The earlier term faculty was introduced around 1798 by Christian Kramp (1760-1826).
Factorial was coined (in French as factorielle) by Louis François Antoine Arbogast (1759-1803).
Kramp withdrew his term in favor of Arbogast's term. In the Preface, pp. xi-xii, of his "Éléments d'arithmétique universelle," Hansen, Cologne (1808), Kramp remarks:
...je leur avais donné le nom de facultés. Arbogast lui avait substitué la nomination plus nette et plus française de factorielles; j'ai reconnu l'avantage de cette nouvelle dénomination; et en adoptant son idée, je me suis félicité de pouvoir rendre hommage ŕ la mémoire de mon ami. [...I've named them facultes. Arbogast has proposed the denomination factorial, clearer and more French. I've recognised the advantage of this new term, and adopting its philosophy I congratulate myself of paying homage to the memory of my friend.]FEJER KERNEL appears in 1937 in Differential and Integral Calculus, 2nd. ed. by R. Courant: "The expression sm is called the "Fejér kernel", and is of great importance in the more advanced study of Fourier series" [James A. Landau].
FERMAT'S LAST THEOREM. Fermat's General Theorem (referring to this theorem) appears in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].
Fermat's last theorem appears in the title of Gabriel Lamé's "Memoire sur le dernier theoreme de Fermat," C. R. Acad. Sci. Paris, 9, 1839, pp. 45-46. Lamé explained the reason for the term:
De tous les theoremes sur les nombres, enonces par Fermat, un seul reste incompletement demontre. [Of all the theorems on numbers stated by Fermat, just one remains incompletely demonstrated (proved).]In his Rapport sur un memoire de M. Lamé, Cauchy remarks:
L'Academie nous a charges, M. Liouville et moi, de lui rendre compte d'un Memoire de M. Lamé sur le dernier theoreme de Fermat. [The Academy has charged us, Mr Liouville and myself, to review memoir of Mr. Lamé on the last theorem of Fermat.]This citation is from C. R. Acad. Sci. Paris, 9, 1839, pp. 359-363.
Fermat's Undemonstrated Theorem appears in 1845 in Phil. Mag. XXVII. 286 (OED2).
James Joseph Sylvester concluded an 1847 paper as follows: "I venture to flatter myself that as opening out a new field in connexion with Fermat's renowned Last Theorem, and as breaking new ground in the solution of equations of the third degree, these results will be generally allowed to constitute an important and substantial accession to our knowledge of the theory of numbers."
An early use of the phrase "Last Theorem of Fermat" in English appears in "Application to the Last Theorem of Fermat" (1860), in "Report on the Theory of Numbers", part II, art. 61, addressed by Henry J. S. Smith.
The OED2 has this 1865 citation from A Dictionary of Science, Literature, and the Arts, by William T. Brande and Cox: "Another theorem, distinguished as Fermat's last Theorem, has obtained great celebrity on account of the numerous attempts that have been made to demonstrate it."
In May 1816, Carl Friedrich Gauss (1777-1855) wrote a letter to Heinrich Olbers in which he mentioned the theorem. According to an English translation (Singh, p. 105; also an Internet web page), he referred to the theorem as Fermat's Last Theorem. However, in fact Gauss wrote, "Ich gestehe zwar, dass das Fermatsche Theorem als isolierter Satz fuer mich wenig Interesse hat..." (I confess that the Fermat theorem holds little interest for me as an isolated result...)
[Julio González Cabillón and William C. Waterhouse contributed to this entry.]
FERMAT'S LITTLE THEOREM is found in 1913 in Zahlentheorie by Kurt Hensel: "Für jede endliche Gruppe besteht nun ein Fundamentalsatz, welcher der kleine Fermatsche Satz genannt zu werden pflegt, weil ein ganz spezieller Teil desselben zuerst von Fermat bewiesen worden ist." [There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it.]
This citation was provided by Peter Flor to a math history mailing list.
Fermat's "little theorem" is found in English in Irving Kaplansky, "Lucas's Tests for Mersenne Numbers," American Mathematical Monthly, 52 (Apr., 1945) [W. Edwin Clark].
FIBONACCI (as a name for Leonardo of Pisa). There is no evidence that the name Fibonacci was ever used by Leonardo or his contemporaries.
In Baldassarre Boncompagni, Della vita e delle opere di Leonardo Pisano matematico del secolo decimoterzo (1852), Baldassarre Boncompagni listed the writers who used the name "Fibonacci":
John Leslie 1820He also listed writers who explaine "Fibinacci = filio Bonacci"
P. D. Pietro Cossali 1797-99
Giovanni Gabriello Grimaldi 1790-1792
Guillaume Libri 1838-1841
Chasles 1837
Nicollet 1811-1818
S. Ersch & I. G. Gruber 1818 and subsequent years
August de Morgan 1847 (he also used Bonacci)
Flaminio dal Borgo 1765Then his arguments for Fibonacci = de filiis Bonacci follow.
Tiraboschi 1822-1828
Ranieri Tempesti 1787
Giovanni Andres 1808-1817
Grimaldi 1790-1792
Libri 1838-1841
According to Boncompagni, Cossali wrote in Origine, trasporto in Italia, primi progressi in essa dell' algebra (2 vols., Parma 1797-1799) the following, on the last page of volume II:
Nel corso dell'Opera ho chiamato il benemerito Leonardo di Pisa, Leonardo Bonacci, laddove da altri fu detto Leonardo Fibonacci, accozzando la prima sillaba Fi di filius al paterno nome Bonacci. Io ho stimato di volger questo a cognome, come assai volte si č fatto. A taluno sarebbe forse piu\ piacciuto (sic) il dire Leonardo di Bonacci.However, according to Menso Folkerts, Cossali gives on page 1, volume 1, Leonardo's name as "Leonardo Bonacci di Pisa." Later on he only has "Leonardo". In the summary, printed at the beginning of the book, Cossali writes "Leonardo Pisano".
Leonardo Fibonacci appears on pages 2 and 109 in Scritti inediti del P. D. Pietro Cossali. Edited by B. Boncompagni. Roma 1857.
On page 20 of volume two of "Histoire des sciences mathematiques en Italie" (1838) by the historian of mathematics Guillaume Libri (1803-1869) a footnote begins:
Fibonacci est une contraction de filius Bonacci, contraction dont on trouve de nombreux exemples dans la formation des noms des familles toscanes.According to Victor Katz in A History of Mathematics (p. 283), "Leonardo [of Pisa], often known today by the name Fibonacci (son of Bonacci) given to him by Baldassarre Boncompagni, the nineteenth century editor of his works, was born around 1170."
[Most of this entry was taken from a post to the historia matematica mailing list by Heinz Lueneburg.]
The term FIBONACCI SEQUENCE was coined by Edouard Anatole Lucas (1842-1891) (Encyclopaedia Britannica, article: "Leonardo Pisano").
FIBONACCI NUMBER is dated 1890-95 in RHUD2.
FIDUCIAL PROBABILITY and FIDUCIAL DISTRIBUTION first appeared in R. A. Fisher's 1930 paper "Inverse Probability," Proceedings of the Cambridge Philosophical Society, 26, 528-535 (David (2001)).
FIELD (neighborhood). In 1893 in A treatise on the theory of functions J. Harkness and F. Morley used the word field in the sense of an interval or neighborhood:
The function f(x) is said to be continuous at the point c ... if a field (c-h to c+h) can be found such that for all points of this field, |f(x)-f(c)| < epsilon.The term field is not defined therein, suggesting the authors believed it was a common usage.
FIELD (modern definition). The term Zahlkörper (body of numbers) is due to Richard Dedekind (1831-1916) (Kline, page 1146). Dedekind used the term in his lectures of 1858 but the term did not come into general use until the early 1890s. Until then, the expression used was "rationally known quantities," which means either the field of rational numbers or some finite extension of it, depending on the context.
Zahlenkörper appears in Stetigkeit und Irrationale Zahlen (Continuity & Irrational Numbers).
Dedekind used Zahlenkörper in Supplement XI of his 4th edition of Dirichlet's Vorlesungenueber Zahlentheorie, section 160. In a footnote, he explained his choice of terminology, writing that, in earlier lectures (1857-8) he used the term 'rationalen Gebietes' and he says that Kronecker (1882) used the term 'Rationalitaetsbereich'.
Dedekind did not allow for finite fields; for him, the smallest field was the field of rational numbers. According to a post in sci.math by Steve Wildstrom, "Dedekind's 'Koerper' is actually what we would call a division ring rather than a field as it does not require that multiplication be commutative."
Julio González Cabillón believes that Eliakim Hastings Moore (1862-1932) was the first person to use the English word field in its modern sense and the first to allow for a finite field. He coined the expressions "field of order s" and "Galois-field of order s = qn." These expressions appeared in print in December 1893 in the Bulletin of the New York Mathematical Society III. 75. The paper was presented to the Congress of Mathematics at Chicago on Aug. 25, 1893:
Perhaps because of the older mathematical meaning of the English word field, Moore seems to have been very careful in systematically referring to a field of order s and not the shorter term field.3. Galois-field of order s = qn Suppose that we have a system of symbols or marks, µ1, µ2 ... µs, in numbers s, and suppose that these s marks may be combined by the four fundamental operations of algebra ... and that when the marks are so combined the results of these operations are in every case uniquely determined and belong to the system of marks. Such a system of s marks we call a field of order s.
The most familiar instance of such a field, of order s = q = a prime, is the system of q incongruous classes (modulo q) of rational integral numbers a. [...]
It should be remarked further that every field of order s is in fact abstractly considered a Galois-field of order s = qn.
At any event, a decade later Edward V. Huntington wrote:
Closely connected with the theory of groups is the theory of fields, suggested by GALOIS, and due, in concrete form, to DEDEKIND in 1871. The word field is the English equivalent for DEDEKIND's term Körper;. KRONECKER's term Rationalitätsbereich, which is often used as a synonym, had originally a somewhat different meaning. The earliest expositions of the theory from the general or abstract point of view were given independently by WEBER and by Moore, in 1893, WEBER's definition of an abstract field being substantially as follows: [...]The following footnote makes it clear that term field already had the same mathematical meaning at the turn of the century as it does now:The earliest sets of independent postulates for abstract fields were given in 1903 by Professor Dickson and myself; all these sets were the natural extensions of the sets of independent postulates that had already been given for groups.
The most familiar and important example of an infinite field is furnished by the rational numbers, under the operations of ordinary addition and multiplication. In fact, a field may be briefly described as a system in which the rational operations of algebra may all be performed (excluding division by zero). A field may be finite, provided the number of elements (called the order of the field) is a prime or a power of a prime.The quote above is from a paper by Huntington presented to the AMS on December 30, 1904, and received for publication on February 9, 1905.
[Information for this article was contributed by Julio González Cabillón, Heinz Lueneburg, William Tait, and Sam Kutler].
FINITE appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements (OED2).
FINITE CHARACTER. This set-theoretic term - usually applied to "properties" ("collections") of sets - was introduced by John Tukey (1915- ) in Convergence and uniformity in topology, Annals of Math. Studies, No 2, Princeton University Press, 1940, p. 7. Tukey's lemma (a useful equivalent of the axiom of choice) states that every non-empty collection of finite character has a maximal set with respect to inclusion. This result is also known as the "Teichmüller-Tukey lemma" because Oswald Teichmüller (1913-1943) had arrived at it independently in Braucht der Algebriker das Answahlaxiom?, Deutsche Mathematik, vol. 4 (1939), pp. 567-577 [Carlos César de Araújo].
FIRST DERIVATIVE, SECOND DERIVATIVE, etc. Christian Kramp (1760-1826) used the terms premiére dérivée and seconde dérivée (first derivative and second derivative) (Cajori vol. 2, page 67).
However, the DSB implies Joseph Louis Lagrange (1736-1813) introduced these terms in his Théorie des fonctions.
First derivative, used attributively, is found in English in 1850 in The calculus of operations by John Paterson (1801-1883). He also uses the terms second derivative and third derivative, although they are used attributively [University of Michigan Digital Library].
First derivative is found in English in 1881 in A Treatise on Electricity and Magnetism by James Clerk Maxwell: "The first derivatives of a continuous function may be discontinuous" (OED2).
FIXED-POINT (arithmetic) was used in 1955 by R. K. Richards in Arithmetic Operations in Digital Computers [James A. Landau].
FLOATING-POINT is found in 1948 in Math. Tables & Other Aids to Computation III. 318: "Floating-point operation greatly reduces the need for scale factors, but complicates the operations of addition and subtraction" (OED).
The terms FLUXION and FLUENT are associated with Isaac Newton, but Richard Suiseth (also known as Calculator; fl. 1350) used the words fluxus and fluens in his Liber calculationum, according to Carl B. Boyer in The History of the Calculus and its Conceptual Development.
Newton used fluent in 1665 to represent any relationship between variables (Kline, page 340). The word fluxion appears once, perhaps by an oversight, in the Principia (Burton, page 377).
Newton composed the treatise Method of Fluxions in Latin in 1671. [It was first published in 1736, translated into English.] Newton wrote (in translation): "Now those quantities which I consider as gradually and indefinitely increasing, I shall hereafter call fluents, or flowing quantities, and shall represent them by the final letters of the alphabet, v, x, y, and z; ... and the velocities by which every fluent is increased by its generating motion (which I may call fluxions, or simply velocities, or celerities), I shall represent by the same letters pointed...."
FOCUS (of a conic section). Carl Boyer writes in A History of Mathematics, "As the curves are now introduced in textbooks, the foci play a prominent role, yet Apollonius had no names for these points, and he referred to them only indirectly."
The term focus (of an ellipse) was introduced by Johannes Kepler (1571-1630) in a treatment of the conic sections in his Ad Vitellionem paralipomena, quibus Astronomiae pars optica traditur (1604).
Umbilic point is an entry in 1700 in Joseph Moxon's dictionary of mathematics: "Umbilique Points, or the 2 Focus or Centre-Points in an Elipsis."
FOIL (standing for "first, outer, inner, last," a method of multiplying two binomials) is found in Charles P. McKeague, Beginning Algebra: A Text/Workbook Second Edition San Diego: Academic Press (imprint of Harcourt Brace Jovanovich), 1985, ISBN 0-15-505230-6: "Rule: To multiply any two polynomials, multiply each term in the first with each term in the second. There are two wasy we can put this rule to work. FOIL Method."
FOIL is found in 1986 in the third edition of Intermediate Algebra by Charles P. McKeague.
FOIL is also found in 1988 in the third edition of Basic Mathematics for Calculus by Dennis G. Zill, Jacqueline M. Dewar, and Warren S. Wright: "Formula (1), illustrated in Figure 0.10, is sometimes called the FOIL method after the first letter in each of the boldface words." The term is not found in the 1983 second edition of this textbook.
These uses of FOIL were found by James A. Landau. The are almost certainly not the earliest uses; if any readers of this page come across any earlier usages, I would appreciate hearing from you.
FOLIUM OF DESCARTES. According to Eves (page 302), Barrow called this curve la galande.
Roberval, through an error, was led to believe the curve had the form of a jasmine flower, and he gave it the name fleur de jasmin, which was afterwards changed (Smith vol. 2, page 328).
Folium of Descartes was used in 1848 in Differential Calculus (1852) by B. Price (OED2).
The curve is also known as the noeud de ruban.
FOLK THEOREM. This term became very popular in the English literature by the middle of the twentieth century. It is just what the name implies: a result (usually not very deep) which belongs to the "folklore." As far as I know, the first to offer an explicit definition was E. J. McShane in his Retiring Presidential Address ("Maintaining Communication," Amer. Math. Monthly, 1957, 309-317):
One hears of "folk theorems", established (presumably) by some expert, communicated verbally or more likely mentioned in an off-hand way during some conversation with another expert or two, and thereafter unpublishable forevermore because no one would want to publish a "known theorem."[This entry was contributed by Carlos César de Araújo.]
FORMULA appears in the phrase "an algebraic formula" in 1796 in Elements of Mineralogy by Richard Kirwan (OED2).
FOUR-COLOR PROBLEM. The problem itself dates to 1852, when Francis Guthrie, while trying to color the map of counties of England, noticed that four colors sufficed.
The first printed reference is due to Cayley in 1878, "On the colourings of maps.," Proc. Royal Geog. Soc. 1 (1879), 259-261.
Problem of the four colors appears in A. B. Kempe, "On the geographical problem of the four colors," Amer. J. Math., 2 (1879), 193-200.
"Map-Colour Theorem" is the title of a paper by Percy John Heawood (1861-1955) which appeared in the Quarterly Journal of Pure and Applied Mathematics in 1890.
Four-color map theorem is found in R. P. Baker, "The four-color map theorem," American M. S. Bull. (1916).
Four color problem is found in Ph. Franklin, "The four color problem," American J. (1922).
Four colour theorem is found in H. S. M. Coxeter's 1933 revision of Mathematical Recreations and Essays by W. W. Rouse Ball.
Four color conjecture is found in 1969 in F. Harary, Proof Techniques in Graph Theory: "By far the most celebrated problem concerning graphs is the Four Color Conjecture."
FOURIER SERIES appears in Björling, "Fourierska serierna," Öfv. af. Förh. Stockh. (1868).
Fourier's series appears in English in 1872 in Spectrum analysis in its application to terrestrial substances, and the physical constitution of the heavenly bodies by Dr. Heinrich Schellen, translated from a German edition: "Other expansions similar to Fourier's series can be conceived, in which the terms, instead of representing pendulous vibrations, would represent vibrations of any other prescribed form..." [University of Michigan Digital Library].
Fourier series appears in English in 1879 in the article "Function" by Arthur Cayley in the Encyclopaedia Britannica.
FOURIER'S THEOREM is found in English 1834 in Rep. Brit. Assoc. (OED2).
Isaac Todhunter in the third edition of An Elementary Treatise on the Theory of Equations (1875) refers to "a theorem which English writers usually call Fourier's theorem, and which French writers connect with the name of Budan as well as with that of Fourier."
FOURIER TRANSFORM is found in English in 1923 in the Proceedings of the Cambridge Philosophical Society (OED2).
FOURTH DIMENSION. Nicole Oresme (c. 1323-1382) wrote, "I say that it is not necessary to give a fourth dimension" in Quaestiones super geometriam Euclides [James A. Landau].
FRACTAL. According to Franceschetti (p. 357), "In the winter of 1975, while he was preparing the manuscript of his first book, Mandelbrot thought about a name for his shapes. Looking into his son's Latin dictionary, he came across the adjective fractus, from the verb frangere, meaning 'to break.' He decided to name his shapes 'fractals.'"
Fractal appears in 1975 in Les Objets fractals: Forme, hasard, et dimension by Benoit Mandelbrot (1924- ). The title was translated as Fractals: Form, Chance, and Dimension (1977).
Fractal appears (as an adjective in "the idea of the fractal dimension") in the Nov. 1975 Scientific American (OED2).
The OED2 also shows a use of the word by Mandelbrot in Fractals in 1977:
Many important spatial patterns of Nature are either irregular or fragmented to such an extreme degree that..classical geometry..is hardly of any help in describing their form... I hope to show that it is possible in many cases to remedy this absence of geometric representation by using a family of shapes I propose to call fractals - or fractal sets.Johnson (page 155) says the term was coined by Mandelbrot in an article "Intermittent Turbulence and Fractal Dimension" published in 1976.
In The Fractal Geometry of Nature Mandelbrot wrote:
I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible -- and how appropriate for our needs! -- that, in addition to "fragmented" (as infraction or refraction), fractus should also mean "irregular," both meanings being preserved in fragment.According to John Conway, Mandelbrot originally defined fractal to mean "having a possibly fractional dimension." Now it is used most often to describe the self-similarity property that many fractal sets have.The proper pronunciation is frac'tal, the stress being placed as infraction.
The word FRACTION is from the Latin frangere (to break). Some writers called fractions "broken numbers."
In the 12th century Adelard of Bath used minuciae in his Regulae abaci. However in the translation of al-Khowarizmi attributed to Adelard, fractiones is used (Smith vol. 2, page 218).
Johannes Hispalensis in his Liber Algorismi de practica arismetrice used fractiones (Smith vol. 2, page 218).
Fibonacci (1202) generally used fractio.
In English, the word was used by Geoffrey Chaucer (1342-1400) (and spelled "fraccions") about 1391 in A treatise on the Astrolabe (OED2).
Broken number is found in 1542 in Ground of Artes (1575) by Robert Recorde: "A Fraction in deede is a broken number" (OED2).
Fragment is found in English in in 1579 in Stratioticos by Thomas Digges: "The Numerator of the last Fragment to be reduced."
FREQUENCY DISTRIBUTION is found in 1895 in Karl Pearson, Phil. Trans. R. Soc. A. CLXXXVI. 412: "A method is given of expressing any frequency distribution by a series of differences of inverse factorials with arbitrary constants" (OED2).
The term FREQUENTIST (one who believes that the probability of an event should be defined as the limit of its relative frequency in a large number of trials) was used by M. G. Kendall in 1949 in Biometrika XXXVI. 104: "It might be thought that the differences between the frequentists and the non-frequentists (if I may call them such) are largely due to the differences of the domains which they purport to cover" (OED2).
FRUSTUM first appears in English in 1658 in The Garden of Cyrus or the Quincuncial Lozenge, or Net-work Plantations of the Ancients ... Considered by Sir Thomas Brown: "In the parts thereof [plants] we finde..frustums of Archimedes" (OED2).
This word is commonly misspelled as "frustrum" in, for example, Samuel Johnson's abridged 1843 Edition of his dictionary. The word is spelled correctly in the "Frustum" entry and the "Hydrography" entry in the 1857 Mathematical Dictionary and Cyclopedia of Mathematical Science, but it is misspelled in the entry "Altitude of a Frustrum." The word is misspelled in the 1962 Crescent Dictionary of Mathematics and remains misspelled in the 1989 Webster's New World Dictionary of Mathematics, which is a revision of the Crescent dictionary. The word is also misspelled in at least three places in The History of Mathematics: An Introduction (1988) by David M. Burton.
The term FUCHSIAN FUNCTION was coined by Henri Poincaré (1854-1912) (Smith vol. I and Encyclopaedia Britannica, article: "Poincaré"). He used Fuchsian and Kleinean functions for automorphic functions of one complex variable, which he discovered (DSB).
The word FUNCTION first appears in a Latin manuscript "Methodus tangentium inversa, seu de fuctionibus" written by Gottfried Wilhelm Leibniz (1646-1716) in 1673. Leibniz used the word in the non-analytical sense, as a magnitude which performs a special duty. He considered a function in terms of "mathematical job"--the "employee" being just a curve. He apparently conceived of a line doing "something" in a given figura ["aliis linearum in figura data functiones facientium generibus assumtis"]. From the beginning of his manuscript, however, Leibniz demonstrated that he already possessed the idea of function, a term he denominates relatio.
A paper "De linea ex lineis numero infinitis ordinatim..." in the Acta Eruditorum of April 1692, pp. 169-170, signed "O. V. E." but probably written by Leibniz, uses functiones in a sense to denote the various 'offices' which a straight line may fulfil in relation to a curve, viz. its tangent, normal, etc.
In the Acta Eruditorum of July 1694, "Nova Calculi differentialis..." (page 316), Leibniz used the word function almost in its technical sense, defining function as "a part of a straight line which is cut off by straight lines drawn solely by means of a fixed point, and of a point in the curve which is given together with its degree of curvature." The examples given were the ordinate, abscissa, tangent, normal, etc. [Cf. page 150 of Leibniz' "Mathematische Schriften," vol. III, edited by C. I. Gerhardt, Berlin-Halle (Asher-Schmidt), 1849-63.]
In September 1694, Johann Bernoulli wrote in a letter to Leibniz, "quantitatem quomodocunque formatam ex indeterminatis et constantibus," although there is no explicit reference to the Latin term functio. The letter appears in Mathematische Schriften.
On July 5, 1698, Johann Bernoulli, in another letter to Leibniz, for the first time deliberately assigned a specialized use of the term function in the analytical sense, writing "earum [applicatarum] quaecunque functiones per alias applicatas PZ expressae." (Cajori 1919, page 211) [Cf. page 507 of Leibniz' "Mathematische Schriften," vol. III, edited by C. I. Gerhardt, Berlin-Halle (Asher-Schmidt), 1849-63. Also see pages 506-510 and 525-526] At the end of that month, Leibniz replied (p. 526), showing his approval.
Function is found in English in 1779 in Chambers' Cyclopedia: "The term function is used in algebra, for an analytical expression any way compounded of a variable quantity, and of numbers, or constant quantities" (OED2).
(Information for this entry was provided by Julio González Cabillón and the OED2.)
The phrase FUNCTION OF x was introduced by Leibniz (Kline, page 340).
The term FUNCTIONAL CALCULUS was introduced in French by Jacques-Salomon Hadamard (1865-1963) in the preface of his "Leçons sur le calcul des variations" [Lessons on the Calculus of Variations], Paris: Librairie Scientifique A. Hermann et Fils, 1910, p. vii:
Le Calcul des variations n'est autre chose qu'un premier chapitre de la doctrine qu'on nomme aujourd'hui le Calcul Fonctionnel ... [The variational calculus is nothing but a first chapter of the doctrine which one calls today "Functional Calculus"...]For functional, Vito Volterra (1860-1940) used the term "functions of other functions," according to Kramer (p. 550). He used "line function," according to the DSB.
This entry was contributed by Julio González Cabillón.
The term FUNCTIONAL ANALYSIS was introduced by Paul P. Lévy (1886-1971) (Kline, p. 1077; Kramer, p. 550).
FUNCTOR was coined by the German philosopher Rudolf Carnap (1891-1970), who used the word in Logische Syntax der Sprache, published in 1934. For Carnap, a functor was not a kind of mapping, but a function sign - a syntactic entity. In Introduction to Symbolic Logic and its Applications (Dover, 1958) he defined a n-place functor as "any sign whose full expressions (involving n arguments) are not sentences". This definition implies that a functor must stand before its argument-expressions, so that the sign +, for example, is not a functor when used as an infix sign ^ although it has the same logical character as a functor. According to him, the function designated by a functor is the "intension" of that functor, while its "extension" is the "value-distribution" of the function.
As used in category theory, functor was introduced by Samuel Eilenberg and Saunders Mac Lane, borrowing the term from Carnap's Logische Syntax der Sprache. In his Categories for the Working Mathematician (1972) Mac Lane says (pp. 29-30):
Categories, functors, and natural transformations were discovered by Eilenberg-Mac Lane (...) Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: "Category" from Aristotle and Kant, "Functor" from Carnap (...).In A History of Algebraic and Differential Topology 1900-1960 Jean Dieudonné wrote (p. 96):
Perhaps the custom they [S. Eilenberg and S. Mac Lane] had adopted of systematically using notations such as (...) for the various groups they defined in their 1942 paper, suggested to them that they were defining each time a kind of "function" which assigned a commutative group to an arbitrary commutative group (or to a pair of such groups) according to a fixed rule. Perhaps to avoid speaking of the "paradoxical" "set of all commutative groups", they coined the word "functor" for this kind of correspondence; (...)[Carlos César de Araújo]
FUNDAMENTAL EQUATION. This term was used by Leopold Alexander Pars (1896-1985) for the theorem of Lagrange.
The term FUNDAMENTAL FUNCTIONS (meaning eigenfunctions) is due to Poincaré, according to the University of St. Andrews website.
The term FUNDAMENTAL GROUP is found in "The Invariant Theory of the Inversion Group: Geometry Upon a Quadric Surface," Edward Kasner, Transactions of the American Mathematical Society, Vol. 1, No. 4. (Oct., 1900).
FUNDAMENTAL SYSTEM. According to the DSB, Immanuel Lazarus Fuchs (1833-1902) "introduced the term 'fundamental system' to describe n linearly independent solutions of the linear differential equation L(u) = 0."
The term FUNDAMENTAL THEOREM OF ALGEBRA "appears to have been introduced by Gauss" (Smith, 1929, and Burton, page 512).
Fundamental theorem of algebra is found in English in 1891 in a translation by George Lambert Cathcart of the German An introduction to the study of the elements of the differential and integral calculus by Axel Harnack [University of Michigan Historical Math Collection].
FUZZY was coined in 1962 by Lotfi A. Zadeh (1921- ), according to an Internet source. The term appears in 1963 in his paper, "A computational approach to fuzzy quantifiers in natural languages," Computers & Mathematics With Applications 9(1), 149-184.
The OED2 shows this 1964 citation: L. A. Zadeh et al., Memorandum (Rand Corporation) RM-4307-PR 1: "The notion of a 'fuzzy' set..extends the concept of membership in a set to situations in which there are many, possibly a continuum of, grades of membership."
In an interview with Betty Blair, Zadeh said:
I coined the word "fuzzy" because I felt it most accurately described what was going on in the theory. I could have chosen another term that would have been more "respectable" with less pejorative connotations. I had thought about "soft," but that really didn't describe accurately what I had in mind. Nor did "unsharp," "blurred," or "elastic." In the end, I couldn't think of anything more accurate so I settled on "fuzzy".Fuzzy logic appears in 1969 in IEEE Trans. Computers XVIII. 348/2: "In the digital field, pattern recognitions and classification are .. potential users of fuzzy logic" (OED2).
Fuzzy appears in 1976 in Numbers and Games by J. H. Conway: "We say that G and H are confused or that G is fuzzy against H."
Fuzzy mathematics is found in 1963 in the title Journal of Fuzzy Mathematics.
On May 26, 1997, a column by in U. S. News and World Report by John Leo has: "Now 'Deep Thoughts' are available on greeting cards, including one that pokes fun at the fuzzy new math in the schools."
Fuzzy math appears on June 11, 1997, in the Wall Street Journal in the headline "President Clinton's Mandate for Fuzzy Math." In the article, Lynne V. Cheney wrote, "Sometimes called 'whole math' or 'fuzzy math,' this latest project of the nation's colleges of education has some formidable opponents."
In a Presidential debate on Oct. 3, 2000, George W. Bush said, "Look this is the man who's got great numbers. He talks about numbers. I'm beginning to think not only did he invent the Internet, but he invented the calculator. It's fuzzy math."