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Going to the Source(s) of Sources in Mathematicians’ Drafts

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Research in History and Philosophy of Mathematics

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Abstract

The mathematical text in its published form, as we are most used to reading it, is a carefully structured and polished means of communicating results to the scientific community. It is, as Reuben Hersh put it, the ‘front’ of mathematics. In this paper, I propose to look at the ‘back’ of mathematics, at what happens in the privacy of drafts, which can certainly be seen as the mathematician’s laboratory. Considering that these preliminary texts are a part of the mathematical practice—and indeed a crucial one—I will show that they allow us to understand the shaping of mathematics in deep and significant ways. Using a selection of examples, I will focus on questions related to the materiality of mathematical texts, how textual elements and mathematical practices work with each other, the processes of writing in mathematics, and the choices made in writing a text deemed suitable for communication to the scientific community.

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Notes

  1. 1.

    [Medawar(1991)].

  2. 2.

    I will consider as drafts all private research manuscripts, not written to be communicated to another reader. It will not matter, here, whether they are ‘just’ working manuscripts without specific publication intents or part of a process that led to a published work.

  3. 3.

    See, for example, [Lützen(1990), Knobloch(2004), Sauer and Schütz(2020), Bustamante(2022)].

  4. 4.

    [Siegel(1932), Edwards(2010)].

  5. 5.

    [Haffner(2018a), Haffner(2023)].

  6. 6.

    Another example of a new light shed by studying drafts is Clare Moriarty’s recent paper on the draft of Maclaurin’s Treatise on Fluxions and what it reveals of the initial ideological concerns of its author [Moriarty(2022)]. There is a number of recent or ongoing works using archival material akin to drafts, many of which I will cite in this paper—there is also a considerably larger number of available (yet often unexploited) notebooks and drafts that are awaiting exploration. My own project funded by the Émergence(s) program of the Ville de Paris, “Brouillons mathématiques” gathers a small team of historians and philosophers focused on such issues (see http://www.item.ens.fr/brouillons-mathematiques-projet-emergences-2022-2026/).

  7. 7.

    Of course, archives contain many unpublished works, in more or less finished states. As we know, many of them have been published and served greatly in advancing our understanding of the history of mathematics. This is not my focus in this paper. I would rather like to focus, here, on aspects of mathematics in the making, on temporary steps before the completion or final stages of a readable or publishable text.

  8. 8.

    [Haffner(2018a)], [Haffner(2023)].

  9. 9.

    While I do not pretend to possess any sort of expertise on Leibniz’s manuscripts as I might have been able to build for Dedekind’s, the material available is too amazing to be overlooked. My remarks on Leibniz have greatly benefited from my discussions with members of the ERC project PHILIUMM, to whom I am very grateful.

  10. 10.

    See http://eliecartanpapers.ahp-numerique.fr/ and [Haffner(2017a)].

  11. 11.

    On note taking as a private scientific practice, see [Bustamante(2020), Bustamante(2022)].

  12. 12.

    [Struik(1922)].

  13. 13.

    Unless stated otherwise, translations are mine.

  14. 14.

    The interested reader can look directly at Cartan’s notes here: http://eliecartanpapers.ahp-numerique.fr/items/show/37.

  15. 15.

    This is not to say that his reading of his contemporaries’ works never led to important breakthroughs in his own research, as his reading of Weyl’s or of Einstein’s works testify.

  16. 16.

    Notation in Leibniz’s mathematics have been fairly well studied, first in [Cajori(1929)], but also in [Serfati(2005), Knobloch(2010), Trunk(2016), Waszek(2018), Gentil(2021)]. Yet the sheer amount of manuscripts available makes this question a still largely to be explored one.

  17. 17.

    Citations for Leibniz manuscripts are from the so-called Akademie edition [Leibniz(1923–)]. VII designates the seventh series on mathematical writings. 5 means the fifth volume. This precise text is translated and commented in [Leibniz and Child(2008)] (see also [Hofmann(1974)], chap. 13).

  18. 18.

    Child’s translation does not reproduce this note—although both Gerhardt’s original edition and the 2008 edition by the Leibniz-Akademie do.

  19. 19.

    See [Rabouin(2021)].

  20. 20.

    This is a reference to Leibniz’s attempts to denote the coefficients of equations by sequences of numbers in order to make writing determinants easier.

  21. 21.

    See, among others, [Knobloch(2004)], [Trunk(2016), Gentil(2021)] on ambiguous signs, or Arilès Remaki’s rich PhD dissertation [Remaki(2021)].

  22. 22.

    While I will concentrate on computations for mainly incidental reasons—they are the approach favored by the authors I am studying—it should be clear that similar observations can certainly be made concerning diagrammatic experimentation. There are instances in the history of mathematics, of course: Leibniz used diagrams, as [Knobloch(2004)] shows well, and we also find diagrams in Riemann’s, Hurwitz’s, or even Dedekind’s drafts. Mathematician and theoretical computer scientist Viviane Pons, who works on algebraic combinatorics at the Laboratoire de Recherche en Informatique at the Université Paris-Saclay, argues for an experimental approach in mathematics (see https://www.youtube.com/watch?v=3LZiZKgVjaU or https://www.lri.fr/~pons/docs/EAUMP/introduction.pdf), and her own notebooks—which are visible in these links and which she showed me personally—are covered in diagrams, sometimes for dozens of pages.

  23. 23.

    Of course, experimental mathematics is an approach argued for by a number of mathematicians (see, for example, the journal titled and dedicated to Experimental mathematics). For historical sources, see [Echeverria(1992), Echeverria(1996), Goldstein(2008), Goldstein(2011)].

  24. 24.

    See [Remaki(2021), 245], for example.

  25. 25.

    Riemann’s computations can certainly also be interpreted as experimentations, to some extent.

  26. 26.

    This is something that Knobloch also notes in Leibniz’s drafts [Knobloch(2004), 77]. It is certainly not the case of every mathematician.

  27. 27.

    I could not locate a related result in Cartan’s publications (but I could have missed it). The notebook contains a variety of research on manifolds and generalised spaces, some related to these pages, many not. See http://eliecartanpapers.ahp-numerique.fr/items/show/21.

  28. 28.

    http://eliecartanpapers.ahp-numerique.fr/items/show/22.

  29. 29.

    [Dedekind(1888)]. I will refer to it as Zahlen.

  30. 30.

    See [Sieg and Schlimm(2005)] for an analysis of the changes in Dedekind’s concept of number reflected in these manuscripts.

  31. 31.

    These documents were published in [Dugac(1976), 293–308].

  32. 32.

    A lot of the pages of this third draft are notes. In those that are redactions, many are crossed out, and some are written with a pencil. This is not the last version in the sense of the version sent to the printer (which, as far as I know, we do not have).

  33. 33.

    In fact, this type of layout is fairly standard, we find it in some of Leibniz’s texts, in some of Cartan’s, in Borel’s notebooks…

  34. 34.

    The concept of mapping is an important concept in Dedekind’s mathematics, used in many areas of his works, and which is subjected to a number of transformations through its uses in number theory, function theory, set theory…It is not my purpose, here, to analyse the changes of the concept of Abbildung, I only wish to study its definition in the versions of Zahlen.

  35. 35.

    There is little difference between these propositions and the published ones.

  36. 36.

    However, the published version gives a definition of the mapping of a system onto itself that mentions the Abbildung being either similar or not.

  37. 37.

    The blue and purple circles were added by me to locate the use of deutlich and ähnlich, as explained below.

  38. 38.

    See [Haffner(2017b), Haffner(2018b), Haffner(2021)] and hopefully, more to come.

  39. 39.

    See http://archives-bourbaki.ahp-numerique.fr/.

  40. 40.

    These are Leibniz’s first mathematical works, in which we witness the beginning of his method of differences. In these manuscripts, Leibniz only attempt to compute the said sum, but later he tries to generalize his approach to pyramidal numbers. These works are also identified as the origin for Leibniz’s harmonic triangle.

  41. 41.

    The title is not Leibniz’s but the editors’.

  42. 42.

    On (the difficulties of) editing Leibniz, see also [Costa and Pasini(2019)].

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Authors and Affiliations

  1. Institut des textes et manuscrits modernes, ÉNS/CNRS, Paris, France

    Emmylou Haffner

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  1. Emmylou Haffner
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Correspondence to Emmylou Haffner .

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  1. Mathematical, Information & Computer Sciences, Point Loma Nazarene University, San Diego, CA, USA

    Maria Zack

  2. École Normale Supérieure - PSL, Paris, France

    David Waszek

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Haffner, E. (2024). Going to the Source(s) of Sources in Mathematicians’ Drafts. In: Zack, M., Waszek, D. (eds) Research in History and Philosophy of Mathematics. Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-46193-4_6

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