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Structural Semiotics as an Ontology of Mathematics

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Handbook of the History and Philosophy of Mathematical Practice

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Abstract

This paper introduces a minimalist framework for structuralist and post-structuralist semiotics based on the work of de Saussure, Lévi Strauss, and Derrida. It then applies this framework to mathematical case studies. Finally, it explains what it means to consider semiotics as an ontology of mathematics and provides examples of this approach.

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Notes

  1. 1.

    Wittgenstein’s discussion of mathematical language(s) (1976, 1981) can be read as an attempt to tease layers of language that we tend to mix together around mathematics; however, Wittgenstein never considered himself, as far as I know, to be doing “semiotics.”

  2. 2.

    Within an appropriate ontological framework, these two examples might arguably be reducible to semiotic values, but I am not interested in pursuing such a reductive trajectory here.

  3. 3.

    In contemporary computational settings, such binary commutation tests are usually replaced by a statistical analyses of a concrete textual corpus (see Gastaldi 2021).

  4. 4.

    This is a take on Lévi-Strauss’s so called canonic formula (Lévi-Strauss 1955, 442). Since Lévi-Strauss himself was quite cavalier about it (Sperber 1985, 65–66), I allow myself to improvise around the original, rather than attempt to capture the original nuance.

  5. 5.

    Of course, this may change when this practice is considered within richer contexts, such as real numbers or temperature measurement, but a change of context often changes sign relations, as we will see below.

  6. 6.

    I owe my exposure to this system to its independent development by Dirk Schlimm in his unpublished “A decimal place-value system without zero” (2012), which lists earlier publications of this recurrent idea.

  7. 7.

    We have noted that the semiotic values of signs involve not only intra-linguistic commutations (e.g., the relation between 0 and other digits), but also cross-linguistic ones (e.g., the relation between different systems of number representation). This is an important aspect of mathematical signs. Historically, we can witness it in the case of negative numbers. In one context, they evolved through binomials: subtraction of two terms. Some of the earliest acknowledgments of negative numbers are in the form “0 less a number,” and the rules for applying arithmetical operations to these numbers are analogous to the rules for applying operations to combinations of the form “number less number” where the result is positive (Wagner 2017, ch. 2; La Nave and Mazur 2002). However, the articulation of these new negative numbers also depended on their cross-linguistic translation to debts. Neither kind of relation determined negative numbers alone. Similarly, in the case of complex numbers, sets of rules (again, by analogy to other binomials and their roots) were usually not enough for mathematicians to endorse them as meaningful. Geometric representations also played a part, even before the emergence of the modern complex plane (Wagner 2010).

  8. 8.

    Note that I insist on positive numbers, so that the use of 0 by itself, which implies that it is a numerical value, is not available as evidence in our discussion. This is, I admit, an artificial move, as will become apparent in the next section.

  9. 9.

    As my students occasionally did in the context of discussions of real number representation and naïve infinite set theory at the early stages of an introductory logic and set theory course.

  10. 10.

    At any rate, the analysis above is only meant to show how the theory works, not to make proper historical claims. The latter may be found in analyses such as Rotman (1987) and Herreman (2001).

  11. 11.

    Some scholars consider post-structuralism as a reaction against structuralism. I tend, like others (e.g., Maniglier 2006) to see it as a continuation and acknowledge that some of the principles most strongly identified with post-structuralism are already there in earlier structuralist thought. What changed is, I think, mostly the emphasis and goals, not the basic toolkit.

  12. 12.

    Note, however, Wittgenstein’s statement: “‘But you will surely admit that there is a difference between pain behaviour with pain and pain-behaviour without pain.’ – Admit it? What greater difference could there be?” (Wittgenstein 2009, §304). Wittgenstein does not deny the private experience, he only objects to its object-ification.

  13. 13.

    It might sound more sensible to characterize mathematical signs in terms of operations rather than commutations, but, as we noted above, an operation is something that allows commuting the array of signs that forms the operation and the array of signs that forms the result of the operation. Moreover, this is not the only relevant kind of commutation, as we saw in the examples above.

  14. 14.

    Exemplary analysis in non-European contexts, which relate the semiotic-material and conceptual aspects of mathematics include Chemla (2018, 2020) and Keller (2015).

  15. 15.

    This convergence between Latour and semiotics is no accident – Latour’s thought emerged, at least in part, in the context of French structuralist and post-structuralist semiotics.

  16. 16.

    Isabelle Stengers (2010–2011) can be read as (but should not be reduced to) someone who incorporates Latour’s and Cartwright’s approaches, dwelling on the various weaknesses of scientific chains of reference without thereby dismissing them.

  17. 17.

    Karen Barad (2007) goes farthest in attempting to design a full-fledged feminist post-structuralist ontology that insists on the inseparability of materiality and semiotics, and on the fact that materiality is given objectively only in semiotic co-articulations.

  18. 18.

    In terms of Netz’s paper, the scope of “semiotic” here covers both his “structural” and “material,” so this is not the problem. But the commitment to a semiotic ontology for mathematics might still invoke other kinds of objections.

  19. 19.

    In its original form (postulate V of the Euclid’s Elements) or in any of its modern forms, e.g.: for any straight line and any point outside it, a unique parallel to this line passes through this point.

  20. 20.

    There are other arguments for claiming that arcs of great circle on a sphere are not straight line segments – for example, the fact that two such segments may intersect in two points. But the fact is that the question was never even raised (if it had been raised, some creative tinkering might have resolved it). The above argument did not need to arise, because considering an arc as a straight line, as far as I can tell, was simply out of the question.

  21. 21.

    This process spanned much of the twentieth century and continues, for example, in the co-articulation of computer verified proofs and contemporary mathematical practice, as discussed in Avigad and Harrison (2014).

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Authors and Affiliations

  1. ETH, Zurich, Switzerland

    Roi Wagner

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  1. Roi Wagner
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Correspondence to Roi Wagner .

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Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. ETH Zurich, Zurich, Switzerland

    Juan Luis Gastaldi

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Wagner, R. (2023). Structural Semiotics as an Ontology of Mathematics. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_131-1

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  • DOI: https://doi.org/10.1007/978-3-030-19071-2_131-1

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  • Print ISBN: 978-3-030-19071-2

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