1 Thought Experiments in Mathematics

The first books on thought experiments (TEs) appeared in 1991 (Brown 1991 and Horowitz & Massey (eds.) 1991). Since then, the fast-growing literature on the subject has generated a large number of different views (for a survey see above all Fehige and Brown 2010, and Stuart, Fehige & Brown (eds.) 2018). However, this debate has dealt relatively little with the problem of mathematical TEs, and has devoted even less attention to the comparison between TEs in mathematics and in the empirical sciences, although an indirect, though fundamental treatment of TEs in mathematics can be found in Lakatos 1963-4.Footnote 1

This neglect of mathematics might have something to do with the complexity of the topic, since many fundamental and difficult questions arise in connection with TEs in mathematics. It is already sufficiently difficult to define TEs and assess their epistemological and methodological status without introducing issues concerning the nature of mathematics. But the most important reason lies, in our opinion, in the fact that the similarity between these two types of thought experiments, with rare exceptions, was usually assumed without discussion, starting from Mach himself (Mach 1906, pp. 197–198; Engl. Transl., p. 144). For this rason the question of what, if anything, distinguishes mathematical TEs from empirical ones remains an important question in today’s debate, and the papers of this issue can be considered a first step in the direction of bridging this gap.

2 The Papers of this Issue

In light of the problem just pointed out in the philosophy of TEs in mathematics, James Robert Brown’s essay “Rigour and Thought Experiments: Burgess and Norton” makes an important contribution, both in its pars destruens and in its pars construens, toward grasping the distinctive feature of thought experiments in mathemtics. As is well known, Brown argues for a largely Platonic view, according to which we have a certain (albeit fallible) cognitive capacity that allows us to grasp truth both in the abstract realm of mathematics and - at least with respect to certain aspects of it - in that of empirical reality. From here, he discusses John Burgess’ views on the nature of mathematical rigour and John Norton’s views on the nature of thought experiments. In both cases, in order to evaluate them, it is necessary to reconstruct the starting point, a mathematical proof or a thought experiment respectively. Against Burgess, Brown adduces examples from picture proofs, which, although irreducible to formal proofs, are authentic and reliable proofs. Against Norton, he adduces some examples in which reconstruction appears irrelevant, since the initial TEs are absolutely convincing in themselves, even if there is no logically reconstructed version of them. The pars construens follows directly from the pars destruens: mathematical proofs need not be linked to formal proofs, and TEs are not necessarily inferences from empirically grounded premises. The realm of abstract entities is so varied in its content that it is difficult to ascribe it to a single type of proof. The concept of proof subsumes under itself not only traditionally understood proofs, but also those that make use of diagrams, intuitions, statistical arguments, etc.

The paper by John D. Norton and M.W. Parker (“An Infinite Lottery Paradox”) ends with an open conclusion that leads us, albeit implicitly, to a point of view opposite to Brown’s, since it examines a case in which the conflict of opposing intuitions does not seem resolvable on the basis of a criterion that is itself based on intuitions. The authors discuss a mathematical TE concerning a fair and infinite lottery, in which an imaginary machine selects fairly from a countable infinity of possible outcomes, numbered 1, 2, 3, … This TE was used by de Finetti to show that complete additivity does not hold when the elementary cases each have probability zero, while Benci, Horsten and Wenmackers used it to argue for infinitesimal probabilities. The two authors of this paper have in the past defended opposing views concerning this TE. Norton argued that the infinite lottery TE speaks not only against a countably additive probability measure, but even against a finitely additive notion of chance. M.W. Parker, on the contrary, developed a different analysis, in terms of a comparative chance relation that restores finite additivity (in a comparative form) and makes every outcome set more probable than any of its proper subsets. Against the background of this opposition, the paper examines both the basic assumptions of the two accounts and the objections that can be raised against one and the other. As already said, the paper ends with an open conclusion, which ultimately depends on the adoption of intuitions consistent with the Euclidean axiom that the whole is greater than the part or, alternatively, with Cantor’s criterion of one-to-one correspondence.

In “Mathematical Understanding through Thought Experiments”, Gerhard Heinzmann explores the question of whether it makes sense to talk about TEs not only in physics, but also in mathematics, where there is no direct recourse to experience. After an overview of some research on thought experiments in the natural sciences, the author turns his attention to “mathematical experiments” and “mathematical thought experiments,“ particularly in fundamental mathematics. Just as the TE in science should, at least in principle, be realizable, i.e., translatable into real experiments, so too should TEs in mathematics be translatable, at least in principle, into mathematical experiments anchored in the standard grammar and logical and semantic norms of the time, which are the pragmatic basis of their rigour. From this point of view, thought experiments, as distinct from both ‘mathematical experiments’ and ‘formal’ proofs, are mathematical experiments with deviant methods. In a mathematical thought experiment, an idea that does not fit well into the established language seems promising enough to be included in the family, even if we do not know exactly how to integrate it into the accepted formalism or how to formalise it clearly to allow for an acceptable implementation of that formalism. From this point of view, a mathematician using a thought experiment is aware that he is deviating from a norm.

Both James Robert Brown and Gerhard Heinzmann argue that one of the possible purposes of mathematical TEs is to provide us with simple mathematical understanding. Marco Buzzoni, in “Are there Mathematical Thought Experiments?” argues that this view tacitly accepts the separation between the context of discovery and the context of justification (a distinction strongly questioned by Brown’s analysis and explicitly rejected by Heinzmann). In the pars construens of his article, Buzzoni agrees with Brown on the fundamental importance of visualization for mathematical TEs - importance that is also in agreement with relevant uses of the expression in the literature - but only under certain conditions, the most important of which are the following: (1) contrary to the literature so far, the distinction between logical-formal thinking and experimental-operational thinking should not be ignored, but explicitly drawn; (2) the separation between discovery and justification (at least in one of the main senses) is to be rejected along with all the corollaries that descend from it (including the claim that mathematical TEs only have a heuristic role, as well as the possibility of “pure” mathematical understanding that is separated from any reasons in favor of the mathematical proposition in question); (3) The distinction between mathematical TEs and formal proofs must be regarded as one of degree, not kind, although this distinction may be used in a de facto way for particular or local purposes.

Johannes Lenhard (“Proof, Semiotics, and the Computer. On the Relevance and Limitation of Thought Experiment in Mathematics”) also believes that TEs are not reducible to heuristics, and this very fact explains why they are so important and powerful in mathematics. This is the first of the two main claims in his paper. The main argument for this claim is based on a semiotic analysis of how mathematics works with signs. Based on that analysis, formal symbols do not eliminate TEs (replacing them with something rigorous), but rather provide a new stage for them. Like the investigation around empirical reality, the formal world requires exploration that may hold surprising discoveries. In this sense, thought experiments are not exclusive to the empirical sciences, but also take place in mathematics. The second claim in Lenhard’s article concerns an aporia that highlights the limitations of TEs. The aporia arises when mathematical arguments become so complex that they are no longer completely accessible or surveyable, causing one of the main conditions of scientific thinking to fail. To illustrate this problem, Lenhard draws on the work of Vladimir Voevodsky (1966–2017), Fields medalist in 2002), who argued that even the purest branches of mathematics cannot entirely avoid a certain “inaccessibility” of proofs, which often results in an impasse in consensus even among the specialists in a field. Checking proofs by computer is certainly a viable way out, but it too has some limitations that are difficult to overcome.

Valeria Giardino, in “Experimenting with triangles”, inquires about the existence of TEs in mathematics and their relationship to other types of experiments. The main purpose of the article is to offer a framework for making order in the conceptual space between experimentation and proof. The author draws a distinction between thought experiments, real experiments, quasi-experiments (experiments that are performed to look for connections and possible deductive paths, close to inferential views of TEs such as Norton’s) and experimental proofs in pure mathematics. This framework highlights the importance of visualizations in mathematics, but distinguishes between a kind of visualization that is more “free” and related to imagination and internalized constraints - as in TEs - and another kind of visualization that relies on diagrams and more generally on systems of representations and their constituent constraints. Although in TEs mathematicians can legitimately play with their imaginations and “see” new possibilities, the internalized constraints they use must be carefully checked, and the natural way to do this would be to start with a TE and then find a way to define an actual experiment or “quasi-experiment” that would confirm the intuition obtained in the TE. Regarding the comparison between TEs in mathematics and the empirical sciences, Giardino uses her framework to help define an analogy with what happens in the reality of the laboratory, since even in the natural sciences there are both thought experiments and real-world experiments.

Finally, Yiftach Fehige and Andrea Vestrucci, in “On Thought Experiments, Theology, and Mathematical Platonism”, compare TEs in mathematics with TEs in theology, starting from the question whether mathematical Platonism can be a ‘facilitator’ in clarifying the relationship between mathematics and theology. On the one hand, TEs seem to play a comparable role in theology and mathematics; on the other hand, however, there exists an important asymmetry in ontological commitments. A realist commitment plays a much more constitutive role in theology than in mathematical practice. This asymmetry is placed by the authors at the basis of a comparison between mathematics and theology in terms of Platonism. It is true that both mathematical objects and God are inaccessible to the senses, have no causes, do not occupy a position in space and time, and exist independently of the mind; but unlike mathematical objects, the Christian God intervenes in the real world, is the cause of everything that exists, has become man, and has thus occupied a position in space and time. In the above comparison, therefore, only independence from the human mind remains, which however does not help much, considering the existence of even nominalist interpretations of mathematics. The aforementioned asymmetry thus ends up questioning the function of Platonism as a “facilitator” for clarifying the relationship between mathematics and theology.