References analysis of the Wikipedia article "History of the function concept" in the English version

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"...although we do not find in [the mathematicians of Ancient Greece] the idea of functional dependence distinguished in explicit form as a comparatively independent object of study, nevertheless one cannot help noticing the large stock of functional correspondences they studied." (Medvedev 1991, pp. 29–30) Medvedev, Fyodor A. (1991). Scenes from the History of Real Functions. Birkhauser. ISBN 9780817625726.

Medvedev 1991, p. 47. Medvedev, Fyodor A. (1991). Scenes from the History of Real Functions. Birkhauser. ISBN 9780817625726.

Lobachevsky 1951, p. 43 as quoted in Medvedev 1991, p. 58. Lobachevsky, Nikolai (1951). Works. Moscow-Leningrad.{{cite book}}: CS1 maint: location missing publisher (link) Medvedev, Fyodor A. (1991). Scenes from the History of Real Functions. Birkhauser. ISBN 9780817625726.

Dirichlet 1889, p. 135 as quoted in Medvedev 1991, pp. 60–61. Dirichlet, G. P. Lejeune (1889). Gesammelte Werke, Bd. I. Berlin. ISBN 9780828402255.{{cite book}}: CS1 maint: location missing publisher (link) Medvedev, Fyodor A. (1991). Scenes from the History of Real Functions. Birkhauser. ISBN 9780817625726.

See Medvedev 1991, pp. 55–70 for further discussion. Medvedev, Fyodor A. (1991). Scenes from the History of Real Functions. Birkhauser. ISBN 9780817625726.

Frege cautions that the function will have "argument places" where the argument should be placed as distinct from other places where the same sign might appear. But he does not go deeper into how to signify these positions and Russell 1903 observes this. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

"In Mathematics, my chief obligations, as is indeed evident, are to Georg Cantor and Professor Peano. If I had become acquainted sooner with the work of Professor Frege, I should have owed a great deal to him, but as it is I arrived independently at many results which he had already established", Russell 1903, p. viii. He also highlights Boole's 1854 Laws of Thought and Ernst Schröder's three volumes of "non-Peanesque methods" 1890, 1891, and 1895 cf Russell 1903, p. 10 Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

Russell 1903, p. 505. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

Russell 1903, pp. 5–6. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

Russell 1903, p. 7. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

Russell 1903, p. 19. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

Russell 1903, p. 99. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

Russell 1903, pp. 523–529. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press.

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Kleiner, Israel (2009). "Evolution of the Function Concept: A Brief Survey". In Marlow Anderson; Victor Katz; Robin Wilson (eds.). Who Gave You the Epsilon?: And Other Tales of Mathematical History. MAA. pp. 14–26. ISBN 978-0-88385-569-0.

Eves dates Leibniz's first use to the year 1694 and also similarly relates the usage to "as a term to denote any quantity connected with a curve, such as the coordinates of a point on the curve, the slope of the curve, and so on" (Eves 1990, p. 234). Eves, Howard (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover. ISBN 0-486-69609-X.

N. Bourbaki (18 September 2003). Elements of Mathematics Functions of a Real Variable: Elementary Theory. Springer Science & Business Media. pp. 154–. ISBN 978-3-540-65340-0.

Eves 1990, p. 234. Eves, Howard (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover. ISBN 0-486-69609-X.

Eves 1990, p. 235. Eves, Howard (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover. ISBN 0-486-69609-X.

Eves 1990, p. 235 Eves, Howard (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover. ISBN 0-486-69609-X.

Fourier 1822. Fourier, Joseph (1822). Théorie analytique de la chaleur. Paris: Firmin Didot Père et Fils.

For example: "A general function f(x) is a sequence of values or ordinates, each of which is arbitrary...It is by no means assumed that these ordinates are subject to any general law; they may follow one another in a completely arbitrary manner, and each of them is defined as if it were a unique quantity." (Fourier 1822, p. 552) Fourier, Joseph (1822). Théorie analytique de la chaleur. Paris: Firmin Didot Père et Fils.

Über die Darstellung ganz willkürlicher Funktionen durch Sinus- und Cosinusreihen," 1837 (Dirichlet 1889, pp. 135–160). Dirichlet, G. P. Lejeune (1889). Gesammelte Werke, Bd. I. Berlin. ISBN 9780828402255.{{cite book}}: CS1 maint: location missing publisher (link)

Eves asserts that Dirichlet "arrived at the following formulation: "[The notion of] a variable is a symbol that represents any one of a set of numbers; if two variables x and y are so related that whenever a value is assigned to x there is automatically assigned, by some rule or correspondence, a value to y, then we say y is a (single-valued) function of x. The variable x . . . is called the independent variable and the variable y is called the dependent variable. The permissible values that x may assume constitute the domain of definition of the function, and the values taken on by y constitute the range of values of the function . . . it stresses the basic idea of a relationship between two sets of numbers" Eves 1990, p. 235 Eves, Howard (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover. ISBN 0-486-69609-X.

Lakatos, Imre (1976). Worrall, John; Zahar, Elie (eds.). Proofs and Refutations. Cambridge: Cambridge University Press. p. 151. ISBN 0-521-29038-4. Published posthumously.

Gardiner, A. (1982). Understanding infinity, the mathematics of infinite processes. Courier Dover Publications. p. 275. ISBN 0-486-42538-X.

De Morgan 1847, p. 1. De Morgan, Augustus (1847). Formal Logic, or The Calculus of Inference, Necessary and Probable. Walton and Marberly.

Boole 1848 in Grattan-Guinness & Bornet 1997, pp. 1, 2 Grattan-Guinness, Ivor; Bornet, Gérard (1997). George Boole: Selected Manuscripts on Logic and its Philosophy. Springer-Verlag. ISBN 3-7643-5456-9.

Boole 1848 in Grattan-Guinness & Bornet 1997, p. 6 Grattan-Guinness, Ivor; Bornet, Gérard (1997). George Boole: Selected Manuscripts on Logic and its Philosophy. Springer-Verlag. ISBN 3-7643-5456-9.

Boole circa 1849 Elementary Treatise on Logic not mathematical including philosophy of mathematical reasoning in Grattan-Guinness & Bornet 1997, p. 40 Grattan-Guinness, Ivor; Bornet, Gérard (1997). George Boole: Selected Manuscripts on Logic and its Philosophy. Springer-Verlag. ISBN 3-7643-5456-9.

Eves 1990, p. 222. Eves, Howard (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover. ISBN 0-486-69609-X.

Some of this criticism is intense: see the introduction by Willard Quine preceding Russell 1908a Mathematical logic as based on the theory of types in van Heijenoort 1967, p. 151. See also in von Neumann 1925 the introduction to his Axiomatization of Set Theory in van Heijenoort 1967, p. 395 ——; Russell, Bertrand (1967) [1908a]. "Russell (1908a) Mathematical logic as based on the theory of types". ibid. pp. 150–182. With commentary by Willard Quine. van Heijenoort, Jean (1976) [1967]. From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931 (3rd printing ed.). Harvard University Press. ISBN 0-674-32449-8.

——; Frege, Gottlob (1967) [1879]. "Frege (1879) Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought". ibid. pp. 1–82.

——; von Neumann, John (1967) [1925]. "von Neumann (1925) An axiomatization of set theory". ibid. pp. 393–413.

van Heijenoort, Jean (1976) [1967]. From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931 (3rd printing ed.). Harvard University Press. ISBN 0-674-32449-8.

——; Frege, Gottlob (1967) [1879]. "Frege (1879) Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought". ibid. pp. 1–82.

Boole 1854, p. 86. Boole, George (1854). An Investigation into the Laws of Thought on which are founded the Laws of Thought and Probabilities. Walton and Marberly.
cf Boole 1854, pp. 31–34. Boole discusses this "special law" with its two algebraic roots x = 0 or 1, on page 37. Boole, George (1854). An Investigation into the Laws of Thought on which are founded the Laws of Thought and Probabilities. Walton and Marberly.
Although he gives others credit, cf Venn 1881, p. 6 Venn, John (1881). Symbolic Logic. Macmillan.
Venn 1881, pp. 86–87. Venn, John (1881). Symbolic Logic. Macmillan.
cf van Heijenoort's introduction to Peano 1889 in van Heijenoort 1967. For most of his logical symbolism and notions of propositions Peano credits "many writers, especially Boole". In footnote 1 he credits Boole 1847, 1848, 1854, Schröder 1877, Peirce 1880, Jevons 1883, MacColl 1877, 1878, 1878a, 1880; cf van Heijenoort 1967, p. 86). ——; Peano, Giuseppe (1967) [1889]. "Peano (1889) The principles of arithmetic, presented by a new method". ibid. pp. 83–97. With commentary by van Heijenoort. van Heijenoort, Jean (1976) [1967]. From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931 (3rd printing ed.). Harvard University Press. ISBN 0-674-32449-8.

doi.org

Katz, Victor; Barton, Bill (October 2007). "Stages in the History of Algebra with Implications for Teaching". Educational Studies in Mathematics. 66 (2): 192. doi:10.1007/s10649-006-9023-7. S2CID 120363574.

Edwards 2007, p. 47. Edwards, Harold M. (2007). "Euler's definition of the derivative". Bulletin of the American Mathematical Society. 44 (4): 575–580. doi:10.1090/s0273-0979-07-01174-3. MR 2338366.

Luzin 1998, p. 263. Translation by Abe Shenitzer of an article by Luzin that appeared (in the 1930s) in the first edition of The Great Soviet Encyclopedia Luzin, N. (1998). "Function: Part II". The American Mathematical Monthly. 105 (3): 263–270. doi:10.2307/2589085. JSTOR 2589085.

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