Earliest Known Uses of Some of the Words of Mathematics (M)

Maclaurin's theorem also appears in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson: "A particular case of this formula is commonly called Maclaurin's theorem, because it was first made generally known by that writer. It had been given previously, however, by Stirling, another Scotch mathematician; and therefore, if a particular case of Taylor's general theorem should be named after any other mathematician, this ought to be called Stirling's theorem." Thomson subsequently uses the term Stirling's theorem throughout the book.

McLaurin's formula appears in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science.

Maclaurin's series is found in 1902 in the Encyclopaedia Britannica in the article "Infinitesimal Calculus":

This result is usually called Maclaurin's series, having been given in his Fluxions (1742). It had, however, been previously published by Stirling in his Meth. Diff. (1717); but neither Stirling nor Maclaurin laid any claim to the theorem as being original, both referring it to Taylor.

MAGIC SQUARE is found in the title Des quassez ou tables magiques by Frenicle de Bessy (1605-1675).

The first citation in the OED2 is in 1704 in Lexicon technicum, or an universal English dictionary of arts and sciences by John Harris.

Benjamin Franklin used the term in his autobiography:

This latter station was the more agreeable to me, as I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so unentertaining that I was induc'd to amuse myself with making magic squares or circles, or any thing to avoid weariness; and I conceiv'd my becoming a member would enlarge my power of doing good.

The term MANDELBROT SET was coined by Adrien Douady, according to an Internet web page.

MANIFOLD was apparently introduced as Mannigfaltigkeit by Bernhard Riemann (1826-1866).

MANTISSA is a late Latin term of Etruscan origin, originally meaning an addition, a makeweight, or something of minor value, and was written mantisa. In the 16th century it came to be written mantissa and to mean appendix (Smith vol. 2, page 514).

Numerous sources, including Smith (vol. 2, page 524), Boyer (page 345), the Century Dictionary (1889-97), and Webster's New International Dictionary (1909), claim that mantissa was introduced by Henry Briggs (1561-1631) in 1624 in Arithmetica logarithmica. However, this information apparently is incorrect. Johannes Tropfke in his "Geschichte der Elementar-Mathematik, vol. 2, 3rd edition 1933, says "Das Fachwort Mantisse hatte Briggs noch nicht" (p. 252). [Christoph J. Scriba]

According to Cajori (1919, page 152), the word mantissa was first used by John Wallis in 1693:

Ejusque partes decimales abscissas, appendicem voco, sive mantissam.

Mantissa was also used by Leonhard Euler in 1748:

Constat ergo logarithmus quisque ex numero integro et fractione decimali et ille numerus integer vocari solet characteristica, fractio decimalis autem mantissa. (The logarithm consists of an integral part, called the characteristic, and a decimal fraction, called the mantissa.)

Gauss suggested using the word for the fractional part of all decimals: "Si fractio communis in decimalem convertitur, seriem figurarum decimalium ... fractionis mantissam vocamus ..." (Smith vol. 2, page 514).

MAPPING is found in "On the Metric Geometry of the Plane N-Line," F. Morley, Transactions of the American Mathematical Society, Vol. 1, No. 2. (Apr., 1900).

MARKOV CHAIN. The phrase "les châines de Markoff" is found in 1930 by Kaucky [James A. Landau].

MARKOV PROCESS occurs in Kosaku Yosida and Shizuo Kakutani, "Markoff process with an enumerable infinite number of possible states," Jap. J. Math. 16 (1939).

The term also appears in Shizuo Kakutani, "Some results in the operator-theoretical treatment of the Markoff process," Proc. Imp. Acad. Jap. 15 (1939).

The term MARRIAGE THEOREM was introduced by Hermann Weyl (1885-1955) in "Almost periodic invariant vector sets in a metric vector space", Amer. J. Math. 71 (1949), 178-205, according to Konrad Jacobs in Measure and Integral, Academic Press, 1978. The theorem is also called "Hall's theorem" or "Hall's marriage theorem" since it was first proved by Philip Hall in 1935 [Carlos César de Araújo].

MATH is dated ca. 1878 in MWCD10.

The phrase "Math: books" is found in the writings of Isaac Newton, but apparently the colon indicates this is an abbreviation [James A. Landau, Axel Harvey].

MATHS. The first citation in the OED2 is 1911: "The Answers to Maths. Ques. were given us all this morning." This citation is from the collected letters of Wilfred Edward Salter Owen, published in 1967. The next OED2 citation is from Wireless World in 1917: "Extremely 'rusty' in 'maths'." Apparently there is not a period in this use of the word.

MATHEMATICAL EXPECTATION was used by DeMorgan in 1838 in An Essay on Probabilities (1841) 97: "The balance is the average required, and is known by the name of mathematical expectation" (OED2).

See also expectation.

The term MATHEMATICAL INDUCTION was introduced by Augustus de Morgan (1806-1871) in 1838 in the article Induction (Mathematics) which he wrote for the Penny Cyclopedia. De Morgan had suggested the name successive induction in the same article and only used the term mathematical induction incidentally. The expression complete induction attained popularity in Germany after Dedekind used it in a paper of 1887 (Burton, page 440; Boyer, page 404).

See also complete induction.

MATHEMATICAL LOGIC occurs in 1858 in On Syllogisms by A. DeMorgan (OED2).

According to the University of St. Andrews website, Ernst Schröder (1841-1902) "seems to be the first to use the term mathematical logic."

MATHEMATICAL RIGOR. Leonhard Euler used a term in 1755 in Institutiones calculi differentialis which is rendered "mathematical rigor" in an English translation.

MATHEMATICIAN is first found in Higden's Polychronicon, translated 1432-50. (The word is spelled "mathematicions.") (OED2).

MATHEMATICS. Pythagoras is said to have coined the words philosophy for "love of wisdom" and mathematics for "that which is learned."

Mathematics is found in English in 1581 in Positions, wherein those primitive circumstances be examined, which are necessarie for the training up of children by Richard Mulcaster. (The word is spelled "mathematikes.") (OED2)

The term MATRIX was coined in 1850 by James Joseph Sylvester (1814-1897):

[...] For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding of pth order.

Kline (page 804) says, "The word matrix was first used by Sylvester when in fact he wished to refer to a rectangular array of numbers and could not use the word determinant, though he was at that time concerned only with the determinants that could be formed from the elements of the rectangular arry."

Katz (1993) says: "The English word matrix meant 'the place from which something else originates.' Sylvester himself made no use of the term at the time. It was his friend Cayley who put the terminology to use in papers of 1855 and 1858."

In 1851 Sylvester informally uses the term matrix as follows:

Form the rectangular matrix consisting of n rows and (n+1) columns

[matrix]

Then all the n+1 determinants that can be formed by rejecting any one column at pleasure out of this matrix are identically zero.

[Randy K. Schwartz and Julio González Cabillón]

MATROID. In a effort to axiomatize the notion of "independence" that arises in graph theory and in vector spaces theory, Hassler Whitney coined the term "matroid" and introduced it in his fundamental paper On the abstract properties of linear independence, Amer. J. Math. 57 (1935) 509-533. The choice of the name arose because he took as an initial model the finite sets of linearly independent column vectors of a matrix over a field. In his paper Whitney gave several equivalent characterizations of a matroid, but the general idea is that of a finite set endowed with a "independence structure" (just as a topological space is a set endowed with a "closeness structure"). Extensions to infinite sets and additional contributions were made by Saunders Mac Lane (1936), R. Rado (1942), W. T. Tutte (1961) and many others. [Carlos César de Araújo]

The term MAXIMUM LIKELIHOOD was introduced by Sir Ronald Aylmer Fisher in his paper "On the Mathematical Foundations of Theoretical Statistics," in Philosophical Transactions of the Royal Society, April 19, 1922. In this paper he made clear for the first time the distinction between the mathematical properties of "likelihoods" and "probabilities" (DSB).

The solution of the problems of calculating from a sample the parameters of the hypothetical population, which we have put forward in the method of maximum likelihood, consists, then, simply of choosing such values of these parameters as have the maximum likelihood. Formally, therefore, it resembles the calculation of the mode of an inverse frequency distribution. This resemblance is quite superficial: if the scale of measurement of the hypothetical quantity be altered, the mode must change its position, and can be brought to have any value, by an appropriate change of scale; but the optimum, as the position of maximum likelihood may be called, is entirely unchanged by any such transformation. Likelihood also differs from probability in that it is not a differential element, and is incapable of being integrated: it is assigned to a particular point of the range of variation, not to a particular element of it.

MEAN ERROR. The 1845 Encyclopedia Metropolitana has "mean risk of error" (OED2).

In 1878, Petrie, in Jrnl. Anthrop. Inst. wrote, "the mean error being 7 inches on 130 feet" (OED2).

In 1894 in Phil. Trans. Roy. Soc, Karl Pearson has "error of mean square" as an alternate term for "standard-deviation" (OED2).

In Higher Mathematics for Students of Chemistry and Physics (1912), J. W. Mellor writes:

In Germany, the favourite method is to employ the mean error, which is defined as the error whose square is the mean of the squares of all the errors, or the "error which, if it alone were assumed in all the observations indifferently, would give the same sum of the squares of the errors as that which actually exists." ...

The mean error must not be confused with the "mean of the errors," or, as it is sometimes called, the average error, another standard of comparison defined as the mean of all the errors regardless of sign.

MEANS. According to Smith (vol. 2, page 483), "The terms 'means,' 'antecedent,' and 'consequent' are due to the Latin translators of Euclid."

The term MEAN SQUARE DEVIATION (apparently meaning variance) appears in a paper published by Sir Ronald Aylmer Fisher in 1920 [James A. Landau].

The term MEAN VALUE THEOREM is found in "A Simple Proof of the Fundamental Cauchy-Goursat Theorem," Eliakim Hastings Moore, Transactions of the American Mathematical Society, Vol. 1, No. 4. (Oct., 1900).

In Differential and Integral Calculus by Virgil Snyder and J. L. Hutchinson (1902) the theorem is called "the theorem of mean value."

In Advanced Calculus by Edwin Bidwell Wilson (1912), the theorem is called the "theorem of the mean."

The term MEASURABLE FUNCTION was used by Arnaud Denjoy (1884-1974) (Kramer, p. 648).

An early use of the term is N. Lusin, "Sur les propriétés des fonctions mesurables," Comptes Rendua Acad. Sci. Paris, 154 (1912).

MEASURE. The following articles feature some uses of the term measure.

Giuseppe Vitali, Sul problema della misura dei gruppi di punti di una retta Bologna: Tip. Gamberini e Parmeggiani (1905).

"On Non-Measurable Sets of Points, with an Example," Edward B. Van Vleck, Transactions of the American Mathematical Society, Vol. 9, No. 2 (Apr., 1908): "Lebesgue's theory of integration is based on the notion of the measure of a set of points, a notion introduced by BOREL and subsequently refined by LEBESGUE himself."

Nikolai Luzin, "Sur les propriétês des fonctions mesurables," Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 154 (1912).

Waclaw Sierpinski, "Sur quelques problèmes qui impliquent des fonctions non-mesurables," Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 164 (1917).

Henri Lebesgue, "Remarques sur les théories de la mesure et de l'intégration," Annales Scientifiques de l'Ecole Normale Supérieure (3) 35, pp 191-250 (1918) [James A. Landau].

Émile Borel (1871-1956), who created the theory of the measure of sets of points, wrote: "La définition de la mesure des ensembles linéaires bien définis m'est entièrement due" (The definition of the measure of well defined linear sets, is entirely due to me) [Udai Venedem].

MECHANICAL QUADRATURE is found in F. G. Mehler, "Bemerkungen zur Theorie der mechanischen Quadraturen," J. Reine angew. Math 63 (1864) [James A. Landau].

MEDIAN (in statistics) was used by Francis Galton in Report of the British Association for the Advancement of Science in 1881: "The Median, in height, weight, or any other attribute, is the value which is exceeded by one-half of an infinitely large group, and which the other half fall short of" (OED2).

MEDIAN (of a triangle) appears in the Encyclopaedia Britannica of 1883 (OED2).

MEDIATE is found in Dorothy Wrinch, "On Mediate Cardinals," American Journal of Mathematics 45 (1923) [James A. Landau].

MERSENNE NUMBER is found in É. Lucas, Récréations Mathématiques, tome II, Note II, "Sur les nombres de Fermat et de Mersenne" (1883).

Mersenne's number is found in English in 1892 in Messenger of Math. XXI. 40: "The riddle as to how Mersenne’s numbers were discovered remains unsolved" (OED2).

Mersenne number is found in English in the 1911 Encyclopaedia Britannica: "Similar difficulties are encountered when we examine Mersenne's numbers, which are those of the form 2^p - 1, with p a prime; the known cases for which a Mersenne number is prime correspond to p = 2, 3, 5, 7, 13, 17, 19, 31, 61" (OED2).

Mersenne prime is found in English in D. H. Lehmer (Editor & Reviewer), "A New Mersenne Prime," Note 138, MTAC 6 (1952).

MESSENGER PROBLEM. In 1930, Karl Menger (1902-1985) mentioned the messenger problem, referring to the problem of finding the shortest Hamiltonian path, according to an Internet web page.

METABELIAN GROUP appears in William Benjamin Fite, "On Metabelian Groups," Transactions of the American Mathematical Society 3 (July, 1902): "We define a Metabelian Group as a group whose group of cogredient isomorphisms is abelian."

The term METAMATHEMATICS goes back to the 1870s where it was used as a pejorative (intending to put it in the same light as metaphysics) in discussions of non-Euclidean geometries.

In the 1890 Funk & Wagnalls Dictionary the word is defined as "The philosophy or metaphysics of mathematics."

It was first used in its modern sense by David Hilbert (1862-1943) in a 1922 lecture. [Michael Detlefsen, Carlos César de Araújo]

METHOD OF EXHAUSTION. The Flemish Jesuit mathematician Gregorius a Sancto Vincentio (or Gregory St. Vincent) (1584-1667) was "probably the first to use the word exhaurire in a geometrical sense" (Cajori 1919). The term method of exhaustion arose from this word.

Vincentio used the term in 1647, according to A Concise History of Mathematics by Dirk J. Struik, third edition.

The term METHOD OF LEAST SQUARES was coined by Adrien Marie Legendre (1752-1833), appearing in Sur la Méthode des moindres quarrés [On the method of least squares], the title of an appendix to Nouvelles méthodes pour la détermination des orbites des comètes (1805). The appendix is dated March 6, 1805 [James A. Landau].

The term METRIC SPACE is due to Felix Hausdorff (1869-1942). The term metrischer raum is found in Grundzüge der Mengenlehre (1914).

METRIC SYSTEM. The metric system is explained in Noah Webster's 1806 dictionary under the heading "New French Weights and Measures."

The term French system was used in 1821 by John Quincy Adams: "The French system..is in design the greatest invention of human ingenuity since that of printing" (OED2).

The term French decimal system is used in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science.

Metric system is found in English in the "Metric Weights and Measures Act, 1864."

Gram is found in English in Aug. 1797 in Nicholson's Journal where it is spelled "gramme." Kilogram and liter are found in English in Aug. 1797 in Journal of Natural Philosophy. Kilometer, milliliter, millimeter, and milligram are found in English in Noah Webster's 1806 A Compendious Dictionary of the English Language, although kilometer is spelled "chiliometer." Metric ton is dated ca. 1890 in MWCD10.

MILLER-RABIN TEST is found in H. W. Lenstra, Jr. "Primality testing," Number theory and computers, Studyweek, Math. Cent. Amsterdam 1980, and in Louis Monier, "Evaluation and comparison of two efficient probabilistic primality testing algorithms," Theor. Comput. Sci., 12 (1980).

Related terms are found in H. W. Lenstra, Jr., "Miller's primality test," Inf. Process. Lett. 8 (1979) and Tore Herlestam, "A note on Rabin's probabilistic primality test," BIT, Nord. Tidskr. Informationsbehandling 20 (1980).

MILLIARD. Gulielmus Budaeus (1467-1540) used the term in his De Asse et Partibus eius Libri V. In the Paris edition of 1532, the following appears: "hoc est denas myriadu myriadas, quod vno verbo nostrates abaci studiosi Milliartu appellat, quasi millionu millione" (Smith vol. 2, page 85).

MILLION. According to Smith (vol. 2, page 81), Maximus Planudes (c. 1340) "seems to have been among the first of the mathematicians to use the word."

Ghaligai wrote that Maestro Paulo da Pisa "La settima dice numero di milione" (read the seventh order as millions). Smith (vol. 2, page 81) writes that this Paulo may have been Paolo Dagomari (b. 1281; d., 1365 or 1374).

Million was used in English in 1362 in Piers Plowman by William Langland (c. 1334-c.1400): "Coueyte not his goodes / For millions of moneye."

According to Smith (vol. 2, page 81), the word first appeared in a printed work in the Treviso arithmetic of 1478. According to Johnson (page 157), it first appeared in print in 1494 in Summa de Arithmetica, by Luca Paciola (1445-1514).

Million appears in the King James Bible: "And they blessed Rebekah, and said unto her, Thou art our sister, be thou the mother of thousands of millions, and let thy seed possess the gate of those which hate them" (Gen. 24: 60). The word was also used by Shakespeare a number of times.

To avoid confusion, mathematicians tended to use "thousand thousand" into the sixteenth century.

The term MINOR was apparently coined by James Joseph Sylvester, who wrote in Philos. Mag. Nov. 1850:

Now conceive any one line and any one column to be struck out, we get ... a square, one term less in breadth and depth than the original square; and by varying in every possible manner the selection of the line and column excluded, we obtain, supposing the original square to consist of n lines and n columns, n² such minor squares, each of which will represent what I term a First Minor Determinant relative to the principal or complete determinant. Now suppose two lines and two columns struck out from the original square ... These constitute what I term a system of Second Minor Determinants; and ... we can form a system of rth minor determinants by the exclusion of r lines and r columns.

MINUEND is an abbreviation of the Latin numerus minuendus (number to be diminished), which was used by Johannes Hispalensis (c. 1140) (Smith vol. 2, page 96).

In English, minuend was used in 1706 by William Jones in Synopsis palmariorum matheseos, or a new introduction to the mathematics (OED2).

MINUS. See plus.

MINUS SIGN. Negative sign appears in 1668 in T. Brancker, Introd. Algebra: "The Sign for Subtraction is - i.e. Minus, or the Negative Sign.

Minus sign appears in a 1909 Webster dictionary.

MIXED NUMBER appears in English in 1542 in The Ground of Artes by Robert Recorde: "mixt numbers (that is whole numbers with fractions)" (OED2).

MÖBIUS STRIP appears in 1904 in E. R. Hedrick, translation of Goursat's Course in Mathematical Analysis (as "Möbius' strip) (OED2).

MODE was coined by Karl Pearson (1857-1936). In 1895 he wrote in the Philosophical Transactions of the Royal Society, "I have found it convenient to use the term mode for the abscissa corresponding to the ordinate of maximum frequency. Thus the "mean," the "mode," and the "median" have all distinct characters."

The term MODULAR ARITHMETIC was coined by Gauss, according to an Internet website.

The term is dated 1959, in English, in MWCD10.

MODULAR CURVE appears in 1883 in the title The Modular Curves of an Uneven Order by H. J. S. Smith (OED2).

The term MODULAR EQUATION was introduced by Jacobi [Encyclopaedia Britannica (1902), article "Infinitesimal Calculus"; Smith (1906)].

The OED2 shows a use of the term in 1845 by DeMorgan in Encyclopaedia Metropolitana.

MODULAR FORM occurs in the heading "Definite Modular Forms" in "Definite Forms in a Finite Field," Leonard Eugene Dickson, Transactions of the American Mathematical Society, Vol. 10, No. 1. (Jan., 1909).

MODULAR FUNCTION. Christoph Gudermann (1798-1852) called elliptical functions "Modularfunctionen" (DSB).

Modular function is found in English in 1894 in Theory of Functions by Forsyth (OED2).

MODULO appears in English in 1897 in Bull. Amer. Math. Soc. III. 381: "Congruences irreducible modulo p (p = prime)" (OED2).

Carlos César de Araújo notes that besides its use as a technical term, modulo is being widely used by mathematicians in a related charming sense as a slang expression. He provides these examples:

• "The following proof is self-contained modulo the standard material on operators and inner-product spaces."
• "He called them continuous functionals. It was clear that modulo unimportant differences these two classes of functionals were equivalent."
• "Turing's work showed that, modulo a universal Turing machine, hardware and software are interchangeable."

In all these examples, "modulo" can be replaced by "taking for granted" or else "except for". He has not found any explicit discussion of this usage in print, modulo a short passage in Gregory Chaitin's homepage, where he says:

One could almost imagine a journal of experimental number theory. For example, there are papers published by number theorists which are, mathematicians say, "modulo the Riemann hypothesis." That is to say, they're taking the Riemann hypothesis as an axiom, but instead of calling it a new axiom they're calling it a hypothesis.

MODULUS (in logarithms) was used by Roger Cotes (1682-1716) in 1722 in Harmonia Mensurarum: Pro diversa magnitudine quantitatis assumptae M, quae adeo vocetur systematis Modulus. Cotes also coined the term ratio modularis (modular ratio) in this work.

Modulus (a coefficient that expresses the degree to which a body possesses a particular property) appears in the 1738 edition of The Doctrine of Chances: or, a Method of Calculating the Probability of Events in Play by Abraham De Moivre (1667-1754) [James A. Landau].

Modulus (in number theory) was introduced by Gauss in 1801 in Disquisitiones arithmeticae:

Si numerus a numerorum b, c differentiam metitur, b et c secundum a congrui dicuntur, sin minus, incongrui; ipsum a modulum appelamus. Uterque numerorum b, c priori in casu alterius residuum, in posteriori vero nonresiduum vocatur. [If a number a measure the difference between two numbers b and c, b and c are said to be congruent with respect to a, if not, incongruent; a is called the modulus, and each of the numbers b and c the residue of the other in the first case, the non-residue in the latter case.]

Modulus (of a complex number) was introduced by Augustin-Louis Cauchy (1789-1857) in 1821.

The term MODULUS OF TRANSFORMATION was used in 1882 by George M. Minchin in Uniplanar Kinematics of Solids and Fluids: "It will be convenient to speak of this quantity K as a modulus of transformation" (OED2).

The term MONOGENIC (for a function having a single derivative at a point) was introduced by Augustin-Louis Cauchy (1789-1857).

MONOMIAL appears in English in a 1706 dictionary.

MONOTONIC is found in 1901 in Ann. Math. II: "It follows that f(s) is a monotonic function that actually decreases in parts of the interval..." (OED2).

It is also found in W. F. Osgood, "On the Existence of a Minimum of the Integral...," Transactions of the American Mathematical Society, 2 (Apr., 1901). The term is probably considerably older.

MONTE CARLO. The method as well as the name for it were apparently first suggested by John von Neumann and Stanislaw M. Ulam. In an unpublished manuscript, "The Origin of the Monte Carlo Method," dated Apr. 12, 1983, Ulam wrote that the method came to him while playing solitaire during an illness in 1946, and that what seems to be the first written account of the method was given by von Neumann in a letter to Robert Richtmyer of Los Alamos in early 1947.

Monte Carlo method occurs in the title "The Monte Carlo Method" by Nicholas Metropolis in the Journal of the American Statistical Association 44 (1949).

Monte Carlo method also appears in 1949 in Math. Tables & Other Aids to Computation III: "This method of solution of problems in mathematical physics by sampling techniques based on random walk models constitutes what is known as the 'Monte Carlo' method. The method as well as the name for it were apparently first suggested by John von Neumann and S. M. Ulam" (OED2).

MOORE SPACE. This name was introduced by F. Burton Jones in Concerning normal and completely normal spaces (Bull. Amer. Math. Soc. 43 (1937) 671-677, p.675) for a topological space satisfying "Axiom 0 and parts 1, 2, and 3 of Axiom 1 of R. L. Moore’s Foundations of Point Set Theory" (Amer. Math. Soc. Coll. Publ. 13, NY, 1932). It was in that paper (p. 676) that Jones stated for the first time the famous normal Moore space conjecture: "Is every normal Moore space M metric [metrizable]?" Despite considerable effort spent in seeking a solution, the question was "settled" only in 1970, when Tall and Silver (by using a Cohen model) showed its undecidability from traditional set theory. [Carlos César de Araújo]

The term MORAL EXPECTATION was used by Daniel Bernoulli.

MULTIPLY was used in English as a verb ("multiply by two") about 1391 by Chaucer in A Treatise on the Astrolabe (OED2).

MULTIPLICATION was used by Chaucer in a non-mathematical sense about 1384 and in a mathematical sense in 1390 by John Gower in Confessio amantis III 89 (OED2).

MULTIPLICATION TABLE. Table of multiplication appears in 1594 in Exercises (1636) by Blundevil: "Before I teach you the true order of multiplying, I thinke it good to set you downe a Table of Multiplication" (OED2).

Multiplication table appears in 1674 in Arithmetic by Samuel Jeake: "To learn by heart the Table commonly called Multiplication Table" (OED2).

MULTIPLICATIVE IDENTITY and MULTIPLICATIVE INVERSE are found in 1953 in First Course in Abstract Algebra by Richard E. Johnson [James A. Landau].

MULTIVARIATE is found in J. Wishart, "The generalized product moment distribution in samples from a normal multivariate population," Biometrika 20A, 32 (1928) [James A. Landau].