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Are there Mathematical Thought Experiments?

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Abstract

With reference to an already existing and relatively widespread use of the expression in question, mathematical “thought experiments” (“TEs”) involve mathematical reasoning in which visualisation plays a relatively more important role. But to ensure an unambiguous and consistent use of the term, certain conditions have to be met: (1) Contrary to what has happened so far in the literature, the distinction between logical-formal thinking and experimental-operational thinking must not be ignored; (2) The separation between the context of discovery and the context of justification is to be rejected, at least in one of the main senses in which it was defended by the logical empiricists and Popper (this excludes any position which, ascribing to mathematical TEs only a heuristic role, regards them an intermediate step to attain more traditional forms of rigour); (3) The distinction between mathematical TEs and formal proofs must be regarded as one of degree, and not as a qualitative one, although this distinction may be used in a de facto way for particular or local purposes.

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Notes

  1. Mach et al. 1906, pp. 197–198; Engl. Transl., p. 144. The literature in recent decades has paid little attention to TEs in mathematics and, with rare exceptions, still less to the comparison between empirical and mathematical TEs. A partial exception is the indirect, though fundamental treatment of TEs in mathematics to be found in Lakatos’s Proofs and Refutations (Lakatos 1963-4), which received relatively more attention (see for example Yuxin 1990; Koetsier 1991; Larvor 1999; Glas 2001a and Glas 2001b; Kühne 2005, pp. 356–366; Sherry 2006; Shaffer 2015; Hertogh 2021. More generally, on mathematical TEs, see above all Witt-Hansen 1976; Mueller 1969; Brown 1999, 2004, 2007, 2011, 2017, and 2021; Glas 2001a and 2001b, Van Bendegem 2003; Buzzoni 2004, 2008, 2011, 2021a and 2021b; Sherry 2006; Starikova 2007; Cohnitz 2008; Starikowa and Giaquinto 2018, Brown 2022, Norton and Parker 2022, Lenhard 2022, Fehige and Vestrucci 2022.

  2. In using the mentioned traditional distinction, however, it is important to emphasise that the expression “applied mathematics” should by no means be understood in the sense that at the beginning there would be pure mathematics, which only later applies to the real world. On the contrary, historically, mathematics first emerged as applied mathematics in the sense that it responded from the outset to demands that came from practical-operational interests. For some further details on the epistemological status of mathematics, see Buzzoni 2011.

  3. Leaving aside (for reasons of space) some differences, I would like to at least mention the broad agreement that exists between the position defended here and Peirce’s considerations on mathematics as diagrammatic reasoning. According to Peirce, in fact, the claim that, as his father Benjamin Pearce had stated, mathematics is “the science which draws necessary conclusions” (Peirce 1895, p. 7), is intimately connected with the importance of graphic or pictorial constructions, or – as Peirce expressed it – with the fact that the mathematician “observes nothing but the diagrams he constructs himself” (Peirce 1895, p. 4). As Peirce noted, in responding to the demands of experience (be they those of the engineer, the insurer, the physicist, or the businessman planning the purchase of land), the mathematician does two things: first, he tries to simplify the facts; and secondly, he tries to formulate another simpler but quite fictitious problem, which “is sufficiently like the question he should answer” to serve as a “substitute” for it. This process, which Peirce calls “skeletonization or diagrammatization of the problem”, makes possible the “tracing the consequences of the hypothesis”, which is the most characteristic aspect of mathematical work (cf. Peirce 1898, pp. 212–213; as is well known, Peirce’s reflections on diagrammatic reasoning have been often taken up in the debate of the last decades on the graphical or diagrammatic character of mathematical proofs (see e.g. Hessler and Mersch (eds.) 2009, Pombo and Gerner (eds) 2010, Panza 2012, Giardino 2013, Meynell 2018, Chapman et al (eds) 2018). In our terminology, the process of “skeletonization” or “diagrammatization” is a process of idealization subordinated to the main purpose of drawing valid consequences within an inferential symbolic space shielded from the real causal interactions present in the natural world.

  4. There is also a sense in which the neopositivist distinction between discovery and justification must be preserved, but this is not what I intend to dwell upon now. For a more detailed treatment of this point, see Buzzoni 2008, above all Ch. 1, Sects. 4–6, and Ch. 3, Sects. 4–6. For more historical details on the discovery/justification distinction, see Schickore and Steinle (eds) 2006, above all Part I and Part II.

  5. Because of this need to distinguish and at the same time connect the conceptual and the concrete-sensible aspects of imagination, the distinction drawn by Stuart 2019 between, on the one hand, imagination as an ability (imagination0), and, on the other hand, two kinds of imagination as different particular manifestations or “uses” of it (imagination1 and imagination2, to which, in my opinion, empirical research could add many more), is important. Stuart sees his distinction as qualitative (or ‘categorical’), and it is in this sense that I accept and use it here. On this point see also Savojardo 2022, who rightly insists on the connection between this distinction, the rejection of the separation between discovery and justification, and the notion of embodied cognition.

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Acknowledgements

This paper has greatly profited from conferences and discussions that took place within the FFIUM Project (Formalism, Formalization, Intuition and Understanding in Mathematics). Many thanks to Gerhard Heinzmann, the principal investigator, who made possible a research stay at the “Archives Henri Poincaré” of the Université de Nancy-Lorraine (15.1.2019-15.2.1919). Ongoing discussions with Gerhard Heinzmann led to a change in my preceding position about mathematical TEs (Buzzoni 2011). But I would like to thank also the entire research group of the FFIUM Project both for conferences held by, and for many helpful discussions with, other members of the Project. Many thanks also to Mike Stuart, who has read an earlier draft of this paper and has made many valuable suggestions for its improvement.

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    Marco Buzzoni

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Buzzoni, M. Are there Mathematical Thought Experiments?. Axiomathes 32 (Suppl 1), 79–94 (2022). https://doi.org/10.1007/s10516-022-09641-3

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