Decline,Then Recovery: An Overview of Activity in the History of Mathematics during the Twentieth Century

Scriba C.J. Dauben J. W. (eds). Writing the history of mathematics: Its historical development (Basel, 2002). It is important to distinguish the history of mathematics from mathematical history. The latter is a branch of applied mathematics, in which the mathematics involved could be modern or even new: For example, some kind of statistics used to assess prosopographically. say, priests in eighteenth-century Mexico, or methods such as seriation for retrieving data from incomplete sources such as archaeological remains (see, for example, Rashevsky N. , Looking at history through mathematics (Cambridge, Mass., 1968)). While a small field, mathematical history, sometimes known as “cliometrics”, has been practised long enough to have its own history (see ref. 23). A related subject is computing history, where software methods are developed specially for historical needs; the Association for Computing and History furthers the cause internationally.

Schulze F. (ed.), B. G. Teubner: Geschichte der Firma (Leipzig, 1911), esp. pp. 266–374 on mathematics.

Burkhardt H. F. K. L. , “Entwicklungen nach oscillirenden Functionen und Integration der Differentialgleichungen der mathematischen Physik”, Jahresbericht der Deutschen Mathematiker-Vereinigung, x/2 (1901–8), xii + 1804 pp.; the snippet occupies pp. 819–1354 of Part 2, Section 1 of the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen.

Gebhardt M. , Die Geschichte der Mathematik im mathematischen Unterrichte der höheren Schulen Deutschlands (Leipzig and Berlin, 1912); and Schröder J. , Die neuzitliche Entwicklung des mathematischen Unterrichts an der höheren Mädchenschulen Deutschlands (Leipzig and Berlin, 1913).

See especially Lorey W. , Das Studium der Mathematik an den deutschen Universitäten seit Anfang des 19. Jahrhunderts (Leipzig and Berlin, 1916).

This Commission deserves a full-length study: Meanwhile, see Goray D. (eds), One hundred years of L'enseignement mathématique: Moments of mathematics education in the twentieth century (Geneva, 2003), especially the articles by F. Furinghetti and G. Schubring on pp. 19–66; Schubring G. , “The cross-cultural ‘transmission’ of concepts — The First International Mathematics Commission reform around 1900” (Bielefeld University, 1988: Institute for Mathematics Education, occasional paper, no. 92); and Grattan-Guinness I. , “European mathematical education in the 1900s and 1910s: Some published and unpublished surveys”, in Ausejo E. Hormigon M. (eds), Messengers of mathematics: European mathematical journals (1800–1946) (Madrid, 1993), 117–30.

Klein F. , Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (2 parts, Berlin, 1926–27; repr. New York, n.d.); idem, Elementare Mathematik vom höheren Standpunkt aus. esp. 3rd edn (3 vols, Berlin, 1925–28); English transl. of vols i–ii: Elementary mathematics from an advanced standpoint (2 vols, New York, 1932; repr. New York, 1939).

Cantor M. (ed.), Vorlesungen über die Geschichte der Mathematik, iv (Leipzig, 1908); Günther's article is on pp. 1–36 and von Braunmühl's on pp. 403–50.

This manuscript, a palimpsest, came into private hands seemingly around 1920, and re-emerged only at auction in the late 1990s. Its anonymous purchaser is financing the cleaning of the manuscript at the Walters Art Gallery, Baltimore, and the transcription of the texts (an extremely difficult task, as the mathematical ones are the originals). Exhibitions of the folios and an edition are planned for 2006, the centenary of Heiberg's identification.

Siegmund-Schulze R. , Mathematische Berichterstattung in Hitlerdeutschland: Der Niedergang des “Jahrbuch über die Fortschritte der Mathematik” (Göttingen, 1993).

See already Higgins T. J. , “Biographies and collected works of mathematicians”, American mathematical monthly, li (1944), 433–45, and addenda in lvi (1949), 310–12.

See especially the ten essays comprising the second parts of Gauss , Werke, vols x–xi (Leipzig, 1922–33; repr. Hildesheim, 1973), 1,300 pages in all. Some essays had appeared earlier as papers with the Göttingen Academy.

Eneström G. , Verzeichniss der Schriften Leonhard Eulers (Leipzig, 1910–13). See also ref. 36.

Stäckel P. , “Entwurf einer Einteilung der sämtlichen Werke Leonhard Eulers”, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, liv (1909), 261–88. A somewhat revised version appeared in Jahresbericht der Deutschen Mathematiker-Vereinigung, xix (1910), part 2, 104–16, 128–42.

Rudio F. (President of the Euler Commission), “Vorwort zu der Gesamtausgabe der Werke Leonhard Eulers”, in Euler , Opera omnia, ser. 1, vol. i (Leipzig, 1911), pp. ix–lxi, see pp. xxxiii–xxxiv.

On the history of the relationship between the two encyclopaedias, see Gispert H. , “Les débuts de l'histoire des mathématiques sur les scènes internationales et le cas de l'entreprise encyclopédique de Felix Klein et Jules Molk”, Historia mathematica, xxvi (1999), 344–60, and further historical items cited there. In 1995 the published parts of the Encyclopédie were photoreprinted with new editorial material by Gispert; a collective history of it is in preparation. An example of an incomplete article, important because not obvious, is Vilfredo Pareto's account of mathematical economics, which happened to end with a sentence on the last line of p. 640 (= the 20th signature) of its Part.

The philosophers published their papers in the Revue de métaphysique et de morale from 1914 to 1916; a report by A. Reymond was provided in L'enseignement mathématique, (1) xvi (1914), 370–8. The educators quickly published their proceedings in this same volume on pp. 167–226, 245–534 (a part comprising national reports on teaching the calculus); the meeting is appraised by Kahane J.-P. Nabonnand P. in Goray (eds), op. cit. (ref. 6), 167–78, 229–49.

Reymond , op. cit. (ref. 17), 370–1.

Loria G. , Guida allo studio della storia delle mathematiche, 1st edn (Milan, 1916). He published a second edition in 1946.

The notes are found principally in the five editions of the Peanists' compendia of Formulario di matematica (Turin, 1895–1908). One member of the group was the historian Giovanni Vacca (1872–1953), who at the International Congress of Mathematicians in Paris in 1900 drew Couturat's attention to the logic manuscripts of Leibniz.

Jourdain P. E. B. , Selected essays on the history of set theory and logics (1906–1918), ed. by Grattan-Guinness I. (Bologna, 1991).

With the third edition (1896) Rouse Ball changed the title of his book to Mathematical recreations past and present. The posthumous editions (11th-13th) were edited by Coxeter H. S. M. ,.

For collections of reprints of historical articles largely from Biometrika, see Kendall M. G. (ed.), Studies in the history of statistics and probability (2 vols: Vol. i ed. with Pearson E. S. , London, 1970; vol. ii with Plackett R. L. , London, 1977). Pearson's own historical lectures were published posthumously thanks to his son: The history of statistics in the 17th and 18th centuries …, ed. by Pearson E. S. (London and High Wycombe, 1978). Some of his other work belongs to the history of mathematical history mentioned in ref. 1.

Watts P. , “Ship” and “Shipbuilding”, in Encyclopaedia Britannica, 11th edn (London, 1911), xxiv, 860–922, 922–81.

Parshall Karen Hunger Rowe David E. , The emergence of the American mathematical research community, 1876–1900: J. J. Sylvester, Felix Klein, and E. H. Moore ([Providence, R.I.], 1994).

McCoy R. E. , Open Court: A centennial bibliography 1887–1987 (La Salle, Ill., 1987); and Henderson H. , Catalyst for controversy: Paul Carus of Open Court (Carbondale, Ill. 1993). The archives of the Company, held at Southern Illinois University at Carbondale, are amazingly rich.

Mikami Y. , The development of mathematics in China and Japan (Leipzig, 1913; repr. New York, 1974); and Mikami Y. Smith D. E. , A history of Japanese mathematics (Chicago, 1914).

Sinaceur H. , “Structure et concept dans l'épistemologie mathématique de Jean Cavaillès”, Revue d'histoire des sciences, xl (1987), 5–30 (see also pp. 117–29).

van Berkel K. , Dijksterhuis: Een biografie (Amsterdam, 1996).

Tropfke J. , Geschichte der Elementarmathematik, i, 4th edn, ed. by Vogel K. Reich K. Gericke H. (New York and Berlin, 1979).

Høyrup J. , “Changing trends in the historiography of Mesopotamian mathematics”, History of science, xxxiv(1991), 1–32.

I owe this information to the late John Fauvel.

On the role of Sarton for the history of science in general see Isis, lxxv (1984), 6–62.

Reid C. , The search for E. T. Bell, also known as John Taine (Washington, D.C., 1993).

Hofmann J. E. , Ausgewählte Schriften zur Geschichte der Mathematik, ed. by Scriba C. J. (2 vols, Hildesheim, 1990).

For a listing of the contents of the three series of Euler's publications, see Speiser A. , “Einleitung der sämtlichen Werke Leonhard Eulers”, Commentarii mathematici Helvetici, xx (1948), 288–318. A summary history of the edition to date is provided in Burckhardt J. J. , “Die Euler-Kommission der Schweizerischen Gesellschaft — Ein Beitrag zur Editionsgeschichte”, in Burckhardt (eds), Leonhard Euler: Beiträge zu Leben und Werk (Basel, 1983), 501–10, followed by his massive secondary bibliography on Euler (pp. 511–52) and preceded by Biermann K.-R. , “Aus der Vorgeschichte der Euler-Ausgabe 1783–1907” (pp. 489–500).

Rostand F. , Souci d'exactitude et scrupules des mathématiciens (Paris, 1960), and Sur la clarté des démonstrations mathématiques (Paris, 1962).

The Bourbakists as mathematicians are themselves the subject of historical study, by Liliane Beaulieu (Paris).

Yushkevich A. P. , Istoriya mathematiki v Rossii do 1917 goda (Moscow, 1968). A few more (partial) histories of mathematics in a country have been produced since; for example, Finland, Sweden, Poland, Hungary and the USA (ref. 65).

Contrast the five-volume Polnoe sobranie sochinenii (Lobachevsky, 1946–51; Chebyshev, 1944–51) with the two-volume editions of Oeuvres (respectively 1883–86, 1899–1907).

Gottwald S. (eds), Lexikon bedeutender Mathematiker (Leipzig, 1990).

Struik D. , “The historiography of mathematics from Proklos [sic] to Cantor”, Schriftenreihe NTM, xvii (1980), 1–22.

Truesdell's main writings are to be found in the Euler Opera omnia, ser. 2, vols xi–xiii (Zurich, 1954–60).

May K. O. , Bibliography and research manual in the history of mathematics (Toronto, 1973).

Dauben J. W. (ed.), The history of mathematics from Antiquity to the present: A selective bibliography (New York, 1985); and Lewis A. C. (ed.), The history of mathematics: A selective bibliography (Providence, R.I., 2000), CD-ROM.

For a bibliography of Italian work, see Barbieri F. Pepe L. , “Bibiliografia Italiana di storia delle mathematiche 1961–1990”, Bollettino di storia delle scienze mathematiche, xii (1992), 1–181.

[ Guerraggio A. (ed.)], La matematica Italiana tra le due guerre mondiale (Bologna, 1987).

Available from the editors, the publication series carries the title “Quaderni del Centro Studi della Matematica Mediovale”. The most up to date bibliography for Arrighi is given by C. Simonetti in his Festschrift: Franci R. (eds), Itinera matematica (Siena, 1996), 375–425.

Kolmogorov A. N. Yushkevich A. P. (eds), Mathematics of the 19th century (3 vols, Basel, 1992–98), 2nd. edn of vol. i (2001); original edn, Matematika XIX veka (Moscow, 1978–87). The earlier trio is Yushkevich (ed.), Istoriya matematiki s drevneisnikh vremen do nachala XIX veka (Moscow, 1970–72).

For an overview see Gerdes P. , “On mathematics in the history of sub-Saharan Africa”, Historia mathematica, xxi (1994), 345–76.

See, for example, Ascher M. , Mathematics elsewhere (Princeton, 2002).

See, for example, Hogendijk J. P. , “Two editions of Ibn al-Haytham's Completion of the conics”, Historia mathematica, xxix (2002), 247–65.

Grattan-Guinness I. (chief ed.), History of the mathematical sciences (Delhi, 2003).

For example Fourier's Joseph Théorie analytique de la chaleur (Paris, 1822) appeared in Japanese in 1993 and in Chinese in 1994 — Shortly after a Spanish translation of 1992.

Hoe J. , “The Jade Mirror of the Four Unknowns — Some reflections”, New Zealand mathematical chronicle, vii (1978), 125–56.

For a review of editions old, new and pending, see Giusti E. Pepe L. (eds), Edizioni critiche e storia della matematica (Pisa, 1986).

Knobloch E. , “Leibniz und die Herausgabe seines wissenschaftlichen Nachlasses”, Akademie der Wissenschaften zu Berlin: Jahrbuch, 1988 (publ. 1989), 475–83.

These volumes were to have been edited by Eric Aiton (1920–91), the editor of the volume on cosmic physics and the only Briton among the various editors. It would be nice if the second series was finished by 2007, a century after the start of the edition.

Mathúna D. Ó. , The Bernoulli project: Historic origins, development of mathematical works and the evolution of the Bernoulli edition (Basel, 2000).

Winter E. , Bernard Bolzano (Stuttgart, 1969).

Bernkopf M. , book review, Isis, lxii (1971), 532.

History of modern mathematics, i–ii, ed. by Rowe D. E. McCleary J. (New York, 1989); iii, ed. by Knobloch E. Rowe D. E. (New York, 1994).

Biermann K.-R. , Die Mathematik und ihre Dozenten an der Berliner Universität 1810–1933 (Berlin (DDR); 1st edn 1973, 2nd edn 1988).

Gispert H. , La France mathématique: La Société Mathématique de France (1870–1914) (Paris, 1992).

See especially Parshall K. Rowe D. E. , The emergence of the American mathematical community (Providence, 1994). Note also Duren P. (ed.), A century of mathematics in America (3 vols, Providence, R.I., 1988–89), which launched the American Mathematical Society book series mentioned in Section 9; and Shell-Gellasch A. (ed.), History of undergraduate mathematics in America (West Point, 2002).

Two pioneering sources are Otte M. Jahnke H. (eds), Epistemological and social problems of the sciences in the early nineteenth century (Dordrecht, 1980); and Mehrtens H. Bos H. Schneider I. (eds), Social history of nineteenth century mathematics (Basel, 1981). More recent successors include C. Goldstein and others (eds), L'Europe mathématique: Mathematical Europe (Paris, 1996); and Bottazzini U. Dahan A. (eds), Changing images in mathematics: From the French Revolution to the new millennium (London, 2001). An early deployment of postmodernism in our subject was made in Mehrtens H. , Moderne Sprache Mathematik (Frankfurt/Main, 1990), in the context of foundational studies in mathematics around 1900.

Notable recent contributions include Ingrao B. Israel G. , The invisible hand: Economic equilibrium in the history of science (Cambridge, Mass., 1990); and Mirowski P. (ed.), Natural images in economic thought (Cambridge, 1994).

Recent notable works include Grave P. Radelet-De Benvenuto E. (eds), Entre mécanique et architecture / Between mechanics and architecture (Basel, 1995); Field J. V. , The invention of infinity: Mathematics and art in the Renaissance (Oxford, 1997); and Gouk P. , Music, science and natural magic in seventeenth-century England (New Haven and London, 1999).

On Christianity, Chase G. B. Jongsma C. (eds), Bibliography of Christianity and mathematics: 1910–1983 (Sioux Center, Iowa Dordt College Press, 1983: Available on www.messiah.edu/acdept/-depthome/mathsci.acms/index.html) lists publications on a wide range of themes, including philosophical, logical and educational issues, but rather few specifically on history. In particular, it confirms the lack of literature (as of 1983) on the period after 1750, which is explored in Grattan-Guinness I. , “Christianity and mathematics: Kinds of link, and the rare occurrences after 1750”, Physis, n.s., xxviii (2001), 467–500. The converse relationship is tentatively explored in I. Grattan-Guinness, “Manifestations of mathematics in and around the Christianities: Some examples and issues”, Historia scientarum, (2) xi (2001), 48–85; also in Revista Brasiliera de historia da matematica, i (2001), 21–56.

See, for example, Grinstein L. S. Campbell P. J. (eds), Women of mathematics: A biobibliographical sourcebook (New York, 1987).

Hankins T. L. , Sir William Rowan Hamilton (Baltimore, 1980); Hodges A. , Alan Turing: The enigma (London, 1983); Gillispie C. C. (main author), Pierre Simon Laplace: A life in exact science (Princeton, 1997); and Dauben J. W. , Georg Cantor (Cambridge, Mass., 1979).

Dieudonné J. (ed.), Abrégé d'histoire des mathématiques (2 vols, Paris, 1978). The great difference between the mathematics profession in the two centuries is also passed over. But several articles are excellent.

On the historiography of mechanics see Grattan-Guinness I. , “The varieties of mechanics by 1800”, Historia mathematica, xvii (1990), 313–38.

A recent pioneering volume on military mathematics across the two World Wars is Booß-Bavnbek B. Høyrup J. (eds), Mathematics and war (Basel, 2003).

See especially Wallis P. J. Wallis R. , Biobibliography of British mathematics and its applications, Part 2: 1701–1760 (Newcastle-upon-Tyne, 1986).

For a non-specialist presentation of these changes of interpretation, together with references to the main recent historical literature, see Grattan-Guinness I. , “Numbers, magnitudes, ratios and proportions in Euclid's Elements: How did he handle them?”, Historia mathematica, xxiii (1996), 355–75 (with printing correction, xxiv (1997), 213).

Høyrup J. , Lengths, widths, surfaces: A portrait of old Babylonian algebra and its kin (New York, 2002).

Gillies D. (ed.), Revolutions in mathematics (Oxford, 1992).

This issue is developed in some detail in Grattan-Guinness I. , “History or heritage? An important distinction in mathematics and for mathematics education”, American mathematical monthly, cxi (2004), 1–12; and “The mathematics of the past: Distinguishing its history from our heritage”, Historia mathematica, to appear.

Grattan-Guinness I. , “History of mathematics”, in Dorling A. (ed.), Use of mathematics literature (London, 1977), 60–77.

The same point about selective teaching from published “Vorlesungen” applies not only to Cantor but also possibly to von Braunmühl's large history of trigonometry (Section 3). The publisher for both authors was Teubner, who regularly published substantial “Vorlesungen” on mathematical subjects; hence there was some tradition in that form of title.

See, for example, Fauvel J. (ed.), History in the mathematics classroom: The IREM papers (Leicester, 1990).

See, for example, Calinger R. (ed.), Vita mathematica: Historical research and integration with teaching (Washington, D.C., 1996).

I am indebted for this information to Kirsti Andersen.

Fauvel J. van Maanen J. (eds), History in mathematics education: The ICMI study (Dordrecht, 2000).

Toeplitz's treatment of the calculus was published posthumously as Die Entwicklung der Infinitesimalrechnung (Darmstadt, 1949); English translation, The calculus: A genetic approach (Chicago, 1963).

Sarton G. , The study of the history of mathematics (Cambridge, Mass., 1936; repr. New York, 1957), 4.

A selection of examples, in connection with the predicament of the British Society for the History of Mathematics, is given in Grattan-Guinness I. , “A residual category: Some reflections on the history of mathematics and its status”, Mathematical intelligencer, xv/4 (1993), 4–6.

Grattan-Guinness I. (ed.), History in mathematics education (Paris, 1987). This old tome is not mentioned in the recent volume of the same title (ref. 85); such is history. A moderate amount of historical material, mostly for informative use, appears in Grinstein L. S. Lipsey S. I. (eds), Encyclopedia of mathematics education (New York and London, 2001).

Grattan-Guinness I. , “Does History of Science treat of the history of science? The case of mathematics”, History of science, xxvii (1990), 149–73.

Demidov S. S. Folkerts M. (eds), “History and the historiography of mathematics”, Archives internationales d'histoire des sciences, xlii (1992), 5–144.

See ref. 1.

The critical bibliographies in Isis for the period would provide many basic data, using the appropriate parts of the cumulative version for 1913–65 edited by Magda Whitrow (5 vols, London, 1971–82). Also useful are the lists of various kinds furnished in Sarton G. , Horus: A guide to the history of science (New York, 1952); and in Loria , op. cit. (ref. 19).

Dauben J. W. , “Historia mathematicae: Journals of the history of mathematics”, in Beretta M. others (eds), Journals and the history of science (Florence, 1998), 1–30; and “Historia mathematica: 25 years/context and content”, Historia mathematica, xxvi (1999), 1–28.

Mohan M. , “Useful web links on history of mathematical sciences”, Ganita-Bharati, xxv (2003), 29–44.

This rainbow metaphor formed the sub-title of my own general history, The Fontana history of the mathematical sciences (London, 1997), which also trades as The Norton history of the mathematical sciences (New York, 1998). I also rejected the normal chronological balance mentioned in Section 10: 1800 is reached only halfway through.