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Wigner’s Puzzle on Applicability of Mathematics: On What Table to Assemble It?


Attempts at solving what has been labeled as Eugene Wigner’s puzzle of applicability of mathematics are still far from arriving at an acceptable solution. The accounts developed to explain the “miracle” of applied mathematics vary in nature, foundation, and solution, from denying the existence of a genuine problem to designing structural theories based on mathematical formalism. Despite this variation, all investigations treated the problem in a unitary way with respect to the target, pointing to one or two ‘why’ or ‘how’ questions to be answered. In this paper, I argue that two analyses, a semantic analysis ab initio and a metatheoretical analysis starting from the types of unreasonableness involved in this problem, will establish the interdisciplinary character of the problem and reveal many more targets, which may be addressed with different methodologies. In order to address objectively the philosophical problem of applicability of mathematics, a foundational revision of the problem is needed.

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Fig. 1


  1. 1.

    In a methodological sense.

  2. 2.

    See (Bangu, 2012, pp. 133–139) for a well-organized presentation of these solutions.

  3. 3.

    It is important to note that in this context of a philosophical puzzle, a ‘solution’ actually means the deductive chain or argument linking a set of concepts, hypotheses, and principles to the “unexplained” facts, even if those hypotheses and principles are not entirely tested and established or those concepts are not entirely tested for adequacy. With this definition, a solution is in fact a rational method available for making the unreasonable reasonable, but it needs a further, deeper analysis of the elements linked.

  4. 4.

    See, for instance Ginammi (2011), Bangu (2016), Maudlin (2016) and McDonnell (2017), to name just a few of the most relevant.

  5. 5.

    In the interpretation of Islami (2017), this particularity turns, for Wigner, the metaphysical question into an epistemological one which Wigner actually solves for the specific case of theoretical physics by invoking the division between events, laws of nature, and invariance principles. As concerns the quality of ‘unreasonable’ granted by a physicist rather than philosopher, Ferreirós (2017) argues that the main reason for such qualification is even Wigner’s formalist views on mathematics and on its relation to theoretical physics.

  6. 6.

    I do not take these formulations of Wigner’s facts as a generalization, nor as interpretation, but rather as a conceptual unification with the philosophical framework of the problem. The acronyms in the list are mine.

  7. 7.

    The distinction between semantic and descriptive applicability has already been made in the literature and is justified through the fact that the semantic applicability has more of a logical nature; however, this is not the logic of a descriptive language, but rather the logic of the application. These features are precisely represented in the Fregean account of semantic applicability. For a succinct presentation of this account in relation with the philosophical problem of applicability, see (Steiner 1998, pp. 13–23).

  8. 8.

    For instance Wigner (1960), Hamming (1980) and Sarukkai (2005).

  9. 9.

    We refer here to ‘empirical context’ as a target of application in the sense of the standard model of application of Bueno and Colyvan (2011) that is characterized through a structural idealized description facilitating the application.

  10. 10.

    A language with both mathematical and physical terms.

  11. 11.

    For an insightful analysis of the application of mathematics in biology, see Islami and Longo (2017).

  12. 12.

    The methodological success also has an empirical nature, as its ultimate indicators rely on the observational base consisting of the recorded individual empirical successes.

  13. 13.

    In ordinary language, the term success seems to be used preponderantly with the accidental (lucky) component, being attributed to a human endeavor that by its nature is subject to incertitude, subjectivism, imperfect preparation, and contingency, and it is exactly this incertitude that justifies the emphasis put on the result of the “successful” endeavor. Of course, the accidental/non-accidental distinction is related to the empirical/methodological one.

  14. 14.

    In a sense that involve both scientific and mathematical practice.

  15. 15.

    Idealization as a determinant procedure that makes application of mathematics possible is an idea that has been stressed by most of the philosophers dealing with applied mathematics, so the ‘fudging’ solution cannot be attributed to a certain author. This solution is named as such in Bangu (2012).

  16. 16.

    We will sketch a formal definition of the unreasonableness in Sect. 5.

  17. 17.

    My emphasis.

  18. 18.

    Either in sense of having both mathematical and physical terms, or of a combination of common and scientific language.

  19. 19.

    This view accommodates well with what we call the inseparability thesis as a solution to Wigner’s puzzle through dissolving. The main premise of this solution is that the mathematical content cannot be separated within a physical theory and so mathematics is not something that can be added externally to the theory as a whole. In this way, the mere concept of application of mathematics in physics is in contradiction with this internally holistic view of physical theories. Proponents of this thesis are Boniolo and Budinich (2005, pp. 78–86).

  20. 20.

    Any analysis of the applied mathematics practice can account for the failures (or inaccurate results) as a result of inaccurate representation through either inadequate mappings or idealizations. Moreover, failures are the main argument for the proponents of a solution of dissolving Wigner’s puzzle (claiming that the number of failures, not the number of successes, is significant). However, if we stay with the concept of a general (perpetual) applicability instead of a partial one, their argument loses power, and the accurate results stands as a premise for this kind of unreasonableness.

  21. 21.

    In sense of non-normative.

  22. 22.

    My emphasis.

  23. 23.

    Take as a label for this general connection the famous Euler’s formula.

  24. 24.

    Two excellent succinct histories of this discovery and its background are presented in Bangu (2012, pp. 91–95) and Steiner (1998, pp. 90–94).

  25. 25.

    Steiner qualifies such an analogy as Pythagorean, arguing with several examples on the ground that “[T]he use of Pythagorean analogies is typical of twentieth-century physical thinking for two main reasons: the rich development of modern mathematics and the lack of any alternatives” (Steiner 1998, pp. 76–94).

  26. 26.

    In the philosophical literature dedicated to the subject of new particle discovery and also of the role of mathematics in natural sciences, such discoveries are usually called predictions (or e-predictions). Although the distinction prediction (on a par with explanation)/discovery certainly has significance with respect to the epistemology of scientific practice and its results, we must limit the discussion to what concerns the unreasonableness in Wigner’s empirical hypothesis. This is why we will stay with the term discovery, especially for catching the differences between the various aspects of facts a to e. Note also that prediction—among others—is also one of the functions/goals of a mathematical model, which is the main concern of facts d and e.

  27. 27.

    A detailed analysis of and comparison between the two types, on epistemological grounds, is made by Bangu (2012, pp. 97–108).

  28. 28.

    Here the success is the empirical confirmation of the discovered entity.

  29. 29.

    This assumption may be challenged by recent works in the field of theory of explanation that employ mathematical explanations and abstract explanations; see, for instance, Pincock (2015).

  30. 30.

    Note that the lack of epistemic justification that characterizes this type of unreasonableness has nothing to do with the general concept of discovery, which assumes non-justification as almost a necessity. The circumstances of a discovery (walking, taking a bath, dreaming, guessing, etc.) need not bear any epistemic value for the discovery; otherwise, it would instead be an inference. That is, in the case of the mathematically driven discovery, the observation of the formalism is not such a circumstance, since it has the epistemic value of the equal status.

  31. 31.

    Of course, the personification is metaphoric.

  32. 32.

    The two questions are not equivalent, but share the same subject and are related to the same issue.

  33. 33.

    For a detailed history of the discovery of the Higgs boson as a study case for the problem of the relation between experimental data, simulation and theoretical mathematical modeling, see Morrison (2015, pp. 287–316).

  34. 34.

    Such an identity relation is actually a criterion of identity; that is, without such a relation, one of the relata would not be what it is. For instance, the relation of set membership is an identity relation between the set and any of its elements, since without it, that set would not be what it is.

  35. 35.

    From conceptual and theoretical frameworks, to local theories and even to well-established universal theories (the case of revolutions).

  36. 36.

    See Field (1980), Hellman (1989), Balaguer (1996), and Azzouni (2004, 2006).

  37. 37.

    See Boniolo and Budinich (2005).

  38. 38.

    See footnotes 2 and 3.

  39. 39.

    It is not the place here to make a review of these (partial) results, but they definitely deserve a further paper. We would just mention as significant the research on perceptual mathematics (Teissier 2005; Ye 2009; Mujumdar and Singh 2016), the constructivist views of the act of mathematical modeling and the constructivist representations of reality (Hennig 2010), the dissolution of abstract structures and the ‘blurring problem’ in ontic structuralism (French and Ladyman 2003, 2011; Cao 2003), the works of Batterman (2002, 2009) on the asymptotic modeling and non-idealization modeling, and the dynamical account of the inferential conception of applied mathematics (Räz and Sauer 2015). They are just pathfinders toward a non-classical (in sense of dealing with the unreasonable per our (I, F) scheme) account of Wigner’s problem from which solutions can be developed.

  40. 40.

    Dissolving means that there is not a genuine problem with a well determined sense, and triviality means that the problem is trivially resolvable, having the solution visible from a subtle perspective.

  41. 41.

    In sense of sentential conjunction.

  42. 42.

    Even if posed, it was not qualified as a puzzle, and effectiveness of science was not seen as unreasonable (although mathematized science is included). Or, at least, we accepted it with no such long debate and intellectual effort as in the case of mathematics.


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Correspondence to Cătălin Bărboianu.

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Bărboianu, C. Wigner’s Puzzle on Applicability of Mathematics: On What Table to Assemble It?. Axiomathes 30, 423–452 (2020).

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  • Applicability of mathematics
  • Unreasonableness
  • Wigner’s puzzle
  • Applied mathematics