Abstract
In the early 20th century, the image of mathematics as a pure discipline, fully autonomous from anything else, and in fact hardly comparable with ‘the sciences’, was well established within and without the maths community. We shall here consider three different settings or periods and look at different meanings and implications of purity as a value: the rise of purism around 1800–1850; the search for balance and unity [Einheitlichkeit] in Klein’s Göttingen, 1890–1914; the extremist modernism of the 1920s and 30s and some of its versions of purity.
This is an expanded version of the talk given at MC2, London, De Morgan House, 15 Sept 2013. I thank participants for their comments, David Rowe for making available a piece from Hilbert’s Nachlass, and two unknown referees for their challenging remarks.
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Notes
- 1.
For the case of Italy, see e.g. Brigaglia’s contribution to Goldstein et al. (2006, esp. p. 431, 447–448).
- 2.
A contrary example is Plücker, who had been educated in France and whose research oscillated between geometry and experimental physics; and it may be relevant that Klein was his assistant.
- 3.
Schröder (1873, 1–2). Gauss argued that knowledge of space must be gained from experience, while arithmetic is purely a priori (letter to Bessel in 1830, Ferreirós 1999, 15; see Ferreirós 2006). Kronecker (1887, 253) began his well known paper on the number concept referring to these ideas of Gauss, and insisted that “arithmetizing” algebra and analysis required the elimination of notions of continuity and irrational magnitudes, which had been motivated by the “applications to geometry and mechanics”. For Dedekind, see Ferreirós (1999, 243–44, 247); the topic is also related to the idea of axiom and in particular to the “Cantor-Dedekind” axiom of continuity of the line.
- 4.
On this topic see Boniface’s contribution to Goldstein et al. (2006, 328–329, 335); and the contribution by Petri and Schappacher, pp. 363–364.
- 5.
See Goldstein et al. (2006). The examples given in the previous paragraph are representative, all of them coming from the North of Germany—with the only apparent exception of Schröder (born in Mannheim, he worked at Baden-Baden at the time of his Lehrbuch); however, Schröder studied at Heildelberg when maths and physics at this university were strongly influenced by the powerful Königsberg school (O. Hesse, Kirchhoff, etc.) and he went to Königsberg himself for two years.
- 6.
This is from a paper published in Jahresbericht der DMV, 1905. See also the interesting remarks of the short-lived Eisenstein, in Wussing (1969, 270); or see Ferreirós (1999, 10 and 28).
- 7.
See Dedekind’s Werke, vol. 2, 54–55.
- 8.
Gauss to Schumacher, September 1, 1850; in Werke X.1; quoted in Ferreirós (2006, 216) (my translation, checked with Dunnington’s).
- 9.
See Schubring (2005, pp. 483–486), which offers a good summary of the rise of pure maths in Germany, by an expert in the topic.
- 10.
The reader should know that at this time there were no universities in France, they had been abolished in the wake of the Revolution.
- 11.
One might add that e.g. Weierstrass and Dedekind, two of the most relevant promoters of arithmetisation and pure mathematics after 1850, showed little or no trace of positivism in their views and ideals (in contrast to, say, physicists Kirchhoff and Helmholtz).
- 12.
- 13.
- 14.
- 15.
The original text is perhaps stronger, Jean Paul Richter: Hesperus oder 45 Hundposttage (1795), 13 Hundspottag: “sie ehren [some people] in der reinen Mathesis und in reiner Weibertugend [virtue of wives] nur beider Verwandlung in unreine für Fabriken und Armeen, in der erhabnen Astronomie nur die Verwandlung der Sonnen in Schrittzähler und Wegweiser für Pfefferflotten, und im erhabensten magister legens nur den anködernden Bierkranz für arme Universitäten.”
- 16.
See Ferreirós (2006, 219–220). I tend to think that Gauss moved away from his youthful emphasis on purism, to insist more and more on scientific method (to judge from his library, over the years he developed a philosophical interest in empiricism), and of course he was happy to contribute to electricity, geomagnetism, etc. Perhaps he was like the real Archimedes, after all!
- 17.
See Bottazzini (2001), ‘From Paris to Berlin’, which to a large extent draws on Italian testimonies.
- 18.
Gispert’s chapter in Goldstein et al. (1996, 401).
- 19.
Quoted by Gispert, op. cit., 401. Darboux created the Bulletin des sciences mathématiques, in part at least, to revert this situation.
- 20.
The sequence was: (1) introduction to the theory of analytic functions, (2) elliptic functions, (3) Abelian functions, and (4) Calculus of variations or applications of elliptic functions. The puristic orientation should be obvious. Kummer taught analytic geometry, mechanics, the theory of surfaces, and number theory, also with great numbers of students (as many as 250).
- 21.
1874 lectures, Hettner transcription, quoted in Ferreirós (2006, 211).
- 22.
See the interview by Kuhn for the SHQP project, which can be found in http://www.aip.org/history/ohilist/4562.html.
- 23.
“Weierstrass is in the first place a logician; he proceeds slowly, systematically, step by step. Where he works, he tries to attain the final form.” (op. cit., 246) But Klein, like Poincaré, emphasized that mathematics will “never be completed” by logical deduction, that intuition is necessary and indispensable; indeed he praised much more the new ideas introduced by Riemann on the basis of geometric intuition and physical ideas.
- 24.
In his Speech before the Göttingen Maths Club on the occassion of his 60th birthday (Hilbert's Nachlass, Cod. Ms. 741; I thank David Rowe for making a transcription available), translations mine.
- 25.
It is well known that Kronecker and Weierstrass, although good friends early on, ended up having opposite viewpoints and strained relations around 1880.
- 26.
Hermite was a purist in very good relations to the Berlin people, but one should not take for granted that he shared the values of the Berlin school of Weierstrass; for an account of the peculiarities of his views, see Goldstein (2011). I regard him as ‘pure’ mostly because of the orientation of his work, but readers should not fancy that he shared the views or attitudes of a Dedekind.
- 27.
In 1902, Klein and Hilbert maneuvered to create a third professorship in pure maths, which placed Göttingen above any other German university; it went to Minkowski, and eventually to Landau (even though Landau had a typical Berlin orientation, see above)—further showing the breadth of the Göttingen spirit of unification.
- 28.
In the 1890s, Klein developed many initiatives to improve teacher education, such as establishing summer schools or cooperating with the teachers’ association. His lectures on Elementary Mathematics from an advanced viewpoint, published in three volumes, became a model textbook. See e.g. Schubring, ‘Felix Klein’, in F. Furinghetti and L. Giacardi (eds), History of ICMI, http://www.icmihistory.unito.it/portrait/klein.php.
- 29.
- 30.
As witnessed by Cantor, Hilbert, Hausdorff, Weyl, etc., but also Poincaré, Brouwer outside Germany.
- 31.
Wigner’s perspective of mathematics (1960), which exemplifies the impact of the 20th century image in its formalistic version, amounts to a severe misrepresentation of mathematical knowledge, of its historical evolution, and its relations to the sciences.
- 32.
Something which prominent physicist C.F. von Weizsäcker tried to promote in 1979.
- 33.
The SA or Sturmabteilung, also known as Brownshirts, was the paramilitary wing of the Nazi party, crucial in Hitler’s rise to power. [Its university members were responsible, among other things, for the boycott to E. Landau in 1933. Bieberbach commented: “The instinct of the Göttingen students was that Landau was a type who handled things in an un-German manner”; quoted in Siegmund-Schultze (2009, 73)].
- 34.
By which I mean work on abstract structures which subsume the previous concrete “objects”, axiomatic approaches based on set theory, characterization of systems “up to isomorphism”.
- 35.
So cheap that they were done merely with chalk, or with paper and pencil—not so anymore, with the increasing use of computers…
- 36.
It is quite different, if one focuses on a more narrowly defined topic, such as purity of methods in relation to mathematical proofs and mathematical topics; see the relevant papers in Mancosu (2008).
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Ferreirós, J. (2016). Purity as a Value in the German-Speaking Area. In: Larvor, B. (eds) Mathematical Cultures. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28582-5_13
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