Un Altro Presente: on the historical interpretation of mathematical texts

Abstract

In this paper I discuss different approaches to past mathematical texts. The question I address is: should we stress the continuity of past mathematics with the mathematics practiced today, or should we emphasize its difference, namely what makes it a product of a distant mathematical culture?

Acknowledgements

This is a revised version of a speech I delivered at Caltech on 13 April 2018. I would like to thank the friends at Caltech, most notably Jed Buchwald, Moti Feingold, Tilman Sauer, Noel Swerdlow, Karine Chemla, and two anonymous referees, for useful comments. I should also thank the DocuServe service at Caltech. Ivor Grattan-Guinness’s works cited in this paper, and the many conversations on historiography I entertained with him, shaped many of the ideas here proposed. I dedicate this paper to Ivor’s memory.

Disclosure statement

No potential conflict of interest was reported by the author

Notes

1 Vermeer’s so-called The Art of Painting is characterized by a complex and somewhat esoteric symbolism, endowed according to some commentators with political and/or religious overtones. The identification of this female character with Clio is not certain, but it is accepted by most historians of art. See (Van Gelder 1951; Wheelock 1995, 128–139). According to Sluijter (1998), however, the female figure (maybe Clio) here represents not ‘Historia’ but ‘Fama’.

2 As is well known, it was Max Weber (1864–1920) who made use of the concept of Idealtypus, an ideal type that the sociologist constructs to make sense of the diversity of social phenomena.

3 The contrast between the view of the past of the historian and the view of the mathematician has been discussed at length by Ivor Grattan-Guinness in his classic papers (Grattan-Guinness 2004a, 2004b). It goes without saying that I owe a great deal to my former supervisor. I profited not only from reading his papers, but I still remember the many discussions we entertained on the historiography of mathematics. A recent monograph is (Wardhaugh 2010). Another book, which I found very useful, is (Hodgkin 2005).

4 Intentionality has been the object of debate among philosophers, anthropologists, psychologists and literary critics. A good survey is (Gibbs 1999). I will have more to say about it below.

5 As was the case in 2016 with Andrew Wiles (1953–) for his proof of Fermat’s Last Theorem.

6 This is what Fabio Acerbi (1965–) did with his high-school students at the Liceo Scientifico Magrini in Gemona (Italy), for whom I had the pleasure of lecturing in 2003.

7 ‘Gli autori sono vicine e lontani. Contemporaneamente e paradossalmente vicini e lontani […] Basta scegliere bene entro testi ben conosciuti. Allora non è difficile trovare pagine che appaino singolarmente moderne, tali da essere facilmente integrabili nel dibattito contemporaneo. Ed è poi facile contrapporre ad esse testi dello stesso autore che danno il senso di una irrimediabile distanza o di una vicinanza che era solo apparente’. (Rossi 1999, 28).

8 Robin G. Collingwood (1889–1943) was a British philosopher and practising archaeologist active in the first half of the twentieth century. In his important works on the methodology of history he claimed that in order to interpret a text historically a historian must experience the thoughts of the author. He called this process ‘re-enactment’: the historian ‘must be able to think over again for himself the thought whose expression he is trying to interpret’ (Collingwood 1939, 111–112). This might sound uncontroversial, but for Collingwood a tangible experience of re-enactment in the historian’s mind is what makes historical knowledge possible:

We cannot know how the flowers in the garden of Epicurus smelled, or how the mountain winds felt in Nietzsche’s hair [thus we cannot write history about these]. But we can recreate their thought about their original experience [via the study of available documents, relics, etc.] (Collingwood 1946, 246–247).

Note that Collingwood is talking about an ‘experience’ that allows the historian to live again, so to speak, Epicurus’s and Nietzsche’s mental processes.

9 Clifford Geertz (1926–2006) has been one of the most influential cultural anthropologists active in the second half of the twentieth century. The term ‘thick description’ was adopted by Geertz from the philosopher Gilbert Ryle. Ryle dealt with descriptions of human behavior. Such descriptions could be ‘thin’ when a human behavior is described in its objective and causal dimension (that is, a description is provided about what the behavior consists in and which causes and effects it has). According to Ryle, a ‘thick description’ refers to the context: most notably it refers to who performed the action and their intentions in doing so. For Geertz, the anthropologist has to aim at a thick description: namely, at understanding why people behaved as they did, what they tried to express and whom they addressed. A thick description is always plural, multi-layered, interpretive. It is a semiotic endeavour that goes beyond the factual account: it deals with emotions, webs of beliefs and social relationships. Geertz often underlines the complexity of the cultural context encountered by the anthropologist, who has to focus on microscopic ethnographic happenings and convey a dense account of what made these happenings meaningful for the actors. See (Geertz 1973, 3–30). The historian, it seems to me, rather than encountering the dense, redundant complexity of the actors’ cultural lives, bases his narrative on the lacunose information provided by surviving texts and relics from the past.

10 In Austin’s linguistic theory a locution is what was said or written, illocution is what was meant or intended, and perlocution is what happened as a result of the speech act.

11 A risk of ‘orientalism’ lurks behind this statement, though. See (Chemla and Keller 2017). As Karine Chemla warned me, we should avoid the idea that mathematics from China or India are somehow discontinuous with our present, whereas early modern European mathematics is not.

12 See, for example, (Fischer 1966) on the decline of invariant theory. It is worth noting, however, that this theory, after a decline in the early twentieth century (well documented by Fischer), has come back to life thanks to the work of, among others, David Mumford. Indeed, as I claim in this paper, it is very problematic to state that some part of the mathematics of the past is a dead end. One can never predict the measure in which past mathematics can still be inspirational for contemporary mathematics. I thank an anonymous referee for noting this.

13 A good introduction is Chapter 4 ‘The Reader’ in (Compagnon 2004, 102–122).

14 Charles Leedham-Green, ‘Appendix F, On Newton’s Style and Translating the Principia’, to appear in Newton (to be published). I thank Prof. Leedham-Green for sending me a preliminary version of his translation and notes on Newton’s Principia.

15 This is the version published in the third edition. The second sentence (after ‘For if the focus’) was added in the second (1713) edition. I put in square brackets further additions in the third edition (1726).

16 This corollary has been the object of many studies. A crystal-clear discussion is (Pourciau 1991).

17 On Newton’s admiration for the ancients, see (McGuire and Rattansi 1966; Casini 1981), and the recent (Levitin 2015, 433–446).

18 The indispensable guide to Newton’s mathematics are Tom Whiteside’s (1932–2008) introductions and notes to his edition of Newton’s mathematical papers (Newton 1967–81). Further, what Richard Westfall (1924–96) writes about Newton’s mathematics in his magnificent biography is still important, see (Westfall 1980, 222–232, 377–381, 512–520).

19 Newton refers to the ‘most perfect mechanic of all’ in the Ad lectorem of his Principia. See, (Newton 1999, 381). A magisterial study of Newton’s religious views is (Iliffe 2017).

20 ‘L’équation exprime la nature de la trajectoire cherchée […], dans laquelle équation c est une constante arbitraire pour rendre le tout homogène’.

21 The importance of homogeneity, even in the work of Descartes, was discussed by Leibniz in a manuscript on the mathesis universalis probably composed in 1698–99. He wrote:

Etsi autem Cartesius soleat non raro studio violare legem homogeneorum introductione unitatis, ego tamen ejus rei non magnum usum reperio, et malo cum Vieta, quoad commode licet, etiam in ipsis numeris legem homogeneorum sequi, quia ita sponte naturae nascitur calculus, maximeque id consentit ordini rerum, erroresque etiam facilius evitantur, cum lex homogeneorum inter examina sit calculi Leibniz (1971, 65–66).

I thank David Rabouin for this reference.

22 I owe this reference to Tilman Sauer, who directed the master’s degree thesis by Heitholt (2018).

23 (Weinstock 1982; Weinstock 2000; Arnold 1990). On the debate on Corollary 1, both ancient and contemporary, see (Guicciardini 2015).

24 Symmetry is often invoked by those who study controversies as a feature of correct historical method. Such an ideal is, it should be said, an impossible task. In 1896 Lord Acton exhorted to lofty impartiality: ‘Our Waterloo must be one that satisfies French and English, Germans and Dutch alike’ (Lowenthal 2015, 345).

25 This point has been frequently made. For example, ‘we also want to imagine conversations between ourselves (whose contingent arrangements include general agreements that, e.g. there are no real essences, no God, etc.) and the mighty dead. We want this not simply because it is nice to feel one up on one’s betters, but because we would like to be able to see the history of our race as a long conversational interchange. We want to be able to see it that way in order to assure ourselves that there has been rational progress in the course of recorded history – that we differ from our ancestors on grounds which our ancestors could be led to accept’ (Rorty 1985, 51).