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Are Euclid’s Diagrams ‘Representations’? On an Argument by Ken Manders

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Part of the Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques book series (ACSHPM)

Abstract

In his well-known paper on Euclid’s geometry, Ken Manders sketches an argument against conceiving the diagrams of the Elements in ‘semantic’ terms, that is, against treating them as representations—resting his case on Euclid’s striking use of ‘impossible’ diagrams in some proofs by contradiction. This paper spells out, clarifies and assesses Manders’s argument, showing that it only succeeds against a particular semantic view of diagrams and can be evaded by adopting others, but arguing that Manders nevertheless makes a compelling case that semantic analyses ought to be relegated to a secondary role for the study of mathematical practices.

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Notes

  1. 1.

    ‘Artifacts in a practice that gives us a grip on life are sometimes thought of in semantic terms—say, as representing something in life. There is, of course, an age-old debate on how geometrical diagrams are to be treated in this regard.’ (Manders, 2008, 84).

  2. 2.

    Manders (2008, 84); my emphasis.

  3. 3.

    Manders (2008, 84).

  4. 4.

    See Heath (1908, II: 12) = Vitrac (1990, I: 399–400) = Heiberg (1883, I: 176–177).

  5. 5.

    A few elements of context about this choice are in order. As recently shown by Ken Saito, Byzantine manuscripts of Euclid—which, save for isolated fragments, are the oldest we have—typically display diagrams that differ significantly from those of modern editions, including critical ones. (For an introduction, see Saito and Sidoli, 2012, Saito, 2009, 817–825 or Saito, 2012—the latter discussing the very Proposition III.5 taken as an example here. For a fuller overview of diagrams in manuscript sources of the first books of Euclid’s Elements, see Saito, 2006, 2011.) The difference is of particular interest in the case of reductio proofs, as the manuscript diagrams are often much more blatantly ‘wrong’ or ‘impossible’ than those of modern editions. This is why I have chosen, in this paper, to reproduce the diagrams from ‘Codex B’, a 888 C.E. manuscript, which, though removed from Euclid himself by almost 1200 years, is one of the oldest still extant. (The letters standardly used to refer to manuscripts of the Elements go back to Heiberg’s authoritative nineteenth-century critical edition of the Greek text; for a list, see Heiberg, 1883, I: V–X or Saito, 2006, 95–96.)

  6. 6.

    Manders (2008, 85–86).

  7. 7.

    For a survey of the role of Euclid’s diagrams in his proofs, see for instance Netz (1999, 175–182).

  8. 8.

    Heath (1908, I: 241–242) = Vitrac (1990, I: 194–195) = Heiberg (1883, I: 10–13).

  9. 9.

    In general, Manders calls ‘co-exact’ those properties that can be read off from diagrams in Euclid; as for Panza, the fact that diagrams of Euclidean geometry allow attributing some of their properties to the corresponding geometrical objects is what he calls their ‘local role’, and those properties that geometrical objects are taken to inherit from their diagrammatic representations are what he calls ‘diagrammatic attributes’ (see Panza 2012, in part. 72–82).

  10. 10.

    Trans. from Heath (1908, II: 12), where (for consistency with the Codex B diagram) I have replaced Heath’s Roman letters with Heiberg’s Greek letters. See also Vitrac (1990, I: 400) = Heiberg (1883, I: 177).

  11. 11.

    Manders (2008, 85); KM’s emphasis. The claims ‘in force’ within a reductio context refer to the hypotheses under which one arrives at a contradiction; the terminology here comes from an analogy with natural deduction, in which inferences are relative to a context defined by the undischarged assumptions under which it is made.

  12. 12.

    Heath (1908, II: 8–9) = Vitrac (1990, I: 394–395) = Heiberg (1883, I: 168–171).

  13. 13.

    Heath (1908, II: 23–24) = Vitrac (1990, I: 412–413) = Heiberg (1883, I: 192–195).

  14. 14.

    As Rabouin (2015, 115–118, 126–131) shows, the kinds of distortions that reductio proofs require easily produce incorrect results in other situations: some form of selective control over diagram distortions is clearly going on; see Manders (2008, 109–118) for further discussion.

  15. 15.

    Manders (2008, 86).

  16. 16.

    Manders (1996, 391). (This quotes comes, not from his most famous 2008 paper, but from a previous publication on the topic; his view on this did not change, however.)

  17. 17.

    See, in particular, the collective volume Allwein and Barwise (1996).

  18. 18.

    The original version of Mumma’s system did not define a formal semantics for its diagrams, but this is possible and is done in Mumma (2019).

  19. 19.

    As a matter of fact, the propositions from Euclid’s arithmetical books also contain diagrams of sorts, which represent by way of lines the numbers discussed in the text; but, in contrast to the geometrical case, these diagrams do not play much of a role in proofs. See, e.g., Mueller (1981, 67).

  20. 20.

    Heath (1908, II: 323–324) = Vitrac (1990, II: 328) = Heiberg (1883, II: 234–237). In modern terms, if two integers A and B are such that there are no smaller integers C and D such that \(\frac {C}{D} = \frac {A}{B}\), then A and B are relatively prime.

  21. 21.

    Manders (2008, 86).

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Acknowledgements

Without David Rabouin, who initially drew my attention to this issue and discussed it with me on numerous occasions, this paper would not exist. I would also like to thank Marco Panza, Jeremy Avigad, Ken Manders, and Dirk Schlimm for useful discussions.

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Waszek, D. (2022). Are Euclid’s Diagrams ‘Representations’? On an Argument by Ken Manders. In: Zack, M., Schlimm, 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/10.1007/978-3-030-95201-3_7

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