Abstract
The goal of this paper is to answer the following question: Does it make sense to speak of thought experiments not only in physics, but also in mathematics, to refer to an authentic type of activity? One may hesitate because mathematics as such is the exercise of reasoning par excellence, an activity where experience does not seem to play an important role. After reviewing some results of the research on thought experiments in the natural sciences, we turn our attention to experiments and thought experiments in mathematics, especially in fundamental mathematics, and investigate what thought experiments can teach us there. If we accept the principle that mathematical practice can sometimes be best described by a pragmatist approach where the concept of experience in mathematics is considered from a new perspective, thought experiments can in some cases be a useful instrument different from both ‘mathematical experiments’ and ‘formal’ proofs: they are mathematical experiments using deviant methods.
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Notes
Nevertheless, according to Imre Lakatos (1976, 11) ‘Thought experiment was the most ancient pattern of mathematical proof.’ Lakatos, as Ernst Mach before him, contrasted informal thought experiments in number theory and geometry—which start from observations – with formal proofs. Here, thought experiments are also linked to informal proofs or to the starting point of mathematics from experience.
With respect to the mode of comprehension, this point was actually already put forth in arguments against the logicisation of mathematical proofs: according to Poincaré, a proof in its logical form is conceptually insufficient for understanding the extensive character of the result with respect to the premises. For experimental mathematicians, ‘formal’ proofs are sometimes incomprehensible due to their excessively abstract character. Poincaré wishes to supplement the standard ‘formalism’ of proof by an architecture (cf. Detlefsen 1992), while the experimental mathematicians would supplement it by a calculation.
According to a tradition spanning from Peirce to the Erlangen School of Lorenzen/Lorenz, the pragmatic approach can be characterised by a set of at least five theses:
(1) It is a method, not a system.
(2) Abandoning every absolute empirical criterion or absolute rationale, it takes as its basis observations and statements of common sense.
(3) It admits a convergence between ontology and the theory of knowledge and considers fruitfulness as an important criterion for the acceptability of propositions.
(4) It rejects a separation in principle between the context of justification and the context of invention as well as between the context of justification and the context of understanding.
(5) It admits a correlation between knowledge and value.
By forming this picture, my aim is neither to articulate a particular historical position held by various thinkers, nor do I intend to present the features shared by such positions.
The function sin(x) and the infinite product have exactly the same roots and have the same value at x = 0 and so “Euler asserts that they describe the same function. Euler is correct that they describe the same function, but these reasons are insufficient to guarantee it. For example, the function \(e^{X} \frac{{sinx}}{x}\) also has the same roots and the same value at x = 0, but is a different function.” (Sandifer 2003, 1).
References:
Ayoub R (1974) Euler and the Zeta Function. Am Math Mon 81:1067–1086
Borwein J and Bailey D (2004) Mathematics by Experiment: plausible reasoning in the 21st Century, Natick MA: AK Peters
Brown JR (1986) Thought experiments since the scientific revolution. Int Stud Philos Sci 1:1–15
Buzzoni M (2008) Thought experiment in the natural sciences. An operational and reflective-transcendental conception, Würzburg: Königshausen & Neumann
Buzzoni M (2011) Kant und das Gedankenexperiment. Über eine kantische Theorie der Gedankenexperimente in den Naturwissenschaften und in der Philosophie. Deut Z Philos 59:93–107
Buzzoni M (2021) Mathematical vs empirical thought experiments. In: Calosi C, Graziani P, Pietrini D and Tarozzi G (eds), Experience, abstraction, and the scientific of the World. Festschrift for Vincenzo Fano on his 60th Birthday, Franco Angeli Editore, Milan, pp 215–228
Detlefsen M (1992) Poincaré against the Logicicians. Synthese 90:349–378
Engel P (2011) Philosophical thought experiments: In or out of the armchair? In: Ierodiakonou K and Roux S (eds), Thought experiments in methodological and historical contexts, Leiden–Boston, pp 145–163
Goffi, J-Y and Roux S (2018) A dialectical account of thought experiments. In Brown JR, Fehige Y and Stuart M (eds.). A Routledge companion to thought experiment, London–Boston: Routledge, pp 439–453
Gonseth F (1945–55) La géométrie et le problème de l'espace, vols. I–VI, Neuchâtel: Griffon
Goodman N (1969) Languages of Art. An approach to a theory of symbols. Oxford University Press, London
Heinzmann G (ed) (1986), Poincaré, Russell, Zermelo et Peano. Textes de la discussion (1906–1912) sur les fondements des mathématiques: des antinomies à la prédicativité, Paris: Blanchard
Heinzmann G (2013) L’intuition épistémique. Une approche pragmatique du contexte de justification en mathématiques et en philosophie, Paris: Vrin, collection Mathesis
Kuhn TS (1977) The essential tension. Selected Studies in scientific tradition and change, Chicago–London: University of Chicago Press
Lakatos I (1976) Proofs and refutation – the logic of mathematical discovery. Cambridge University Press, London
Lenhard J (2018) Thought experiments and simulation experiments. Exploring hypothetical worlds. In: Brown JR, Fehige Y, and Stuart M (eds). A Routledge companion to thought experiment, London–Boston: Routledge, pp 484–497
Lorenz K (2010) Logic, language and method. On polarities in human experience, Berlin: De Gruyter
Marconi D (2017) Philosophical thought experiments: the case for Engel. Philos Sci 21(3):111–124
Norton JD (1996) Are thought experiments just what you thought? Can J Philos 26:333–366
Norton JD (2004) Why thought experiments do not transcend empiricism. In: Hitchcock C (ed) Contemporary debates in the philosophy of science. Blackwell, Oxford, pp 44–66
Peirce CS (1933–1958) Collected papers (edited by Hartshorne Ch and Weiss P), Volume I–VI, Cambridge, MA: Belknap Press
Poincaré H (1898) On the foundations of geometry. The Monist IX: 1–43
Poincaré H (1902) La Science et l’hypothèse, Paris: Flammarion. English translation Science and Hypothesis: the complete text, New York–London: Bloomsbury, 2017
Reiss J (2016) Thought experiments in economics and the role of coherent explanations. Stud Metodol 36:113–130
Sellars W (1956) Empiricism and the philosophy of mind, Cambridge, MA–London: Harvard University Press (1997)
Sandifer Ed (2003) How Euler did it,http://eulerarchive.maa.org/hedi/HEDI-2004-03.pdf
Sundholm G (2008) Proofs as acts and proofs as objects: some questions for dag Prawitz. Theoria 64(2–3):187–216
Sundholm G (2012) ‘Inference versus consequence’ revisited: inference, consequence, conditional, implication. Synthese 187:943–956
Acknowledgements
I would like to thank my Nancy colleagues Jean-Pierre Ferrier, Philippe Lombard, and Andrew Arana for their advice and numerous discussions of this article first presented in the FFIUM seminar (French–German ANR/DFG project), where the mathematical example of the Basel problem was presented by Phillipe Lombard, and my proposal in general analysed by Jean-Pierre Ferrier, to whom I am indebted for several other analyses. I am especially grateful to Marco Buzzoni (Macerata), who sparked my interest in this topic, directed me to relevant literature, and shared with me many ideas during our discussions in Macerata, in Nancy and during our distance discussion at the FFIUM seminar. I am grateful to Marco Panza (Chapman (Orange)/Paris-Sorbonne) for his reformulation of the hierarchy in chapter 2.2 and I finally thank Anna Pilatova for her linguistic corrections of an earlier version of this paper, that will be published in the proceedings of the XVIth CDLMPST, hold in Prague. The present version contains substantial improvements to the older Prague paper.
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Heinzmann, G. Mathematical Understanding by Thought Experiments. Axiomathes 32 (Suppl 3), 871–886 (2022). https://doi.org/10.1007/s10516-022-09640-4
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DOI: https://doi.org/10.1007/s10516-022-09640-4