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On Abducing the Axioms of Mathematics

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Abduction in Cognition and Action

Part of the book series: Studies in Applied Philosophy, Epistemology and Rational Ethics ((SAPERE,volume 59))

Abstract

How do we discover and justify axioms of mathematics? In view of the long history of the axiomatic method, it is rather embarrassing that we are still lacking a standard answer to this simple question. Since the axiom of choice is arguably one of the most frequently discussed famous axioms throughout the history of mathematics, Thomas Forster’s recent identification of the axiom as an inference to the best explanation (IBE) provides us with a nice point of departure. I will argue that, by separating sharply between abduction and IBE, we can give a convincing account of both the discovery and the justification of the axioms of mathematics.

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Notes

  1. 1.

    Irvine cites Steiner [59], Parsons [42], and Lakatos [25] as examples of such an interpretation. Also, he points out that this interpretation is indeed correct for other logicists such as Frege [7, p. 2f] and Hempel [13].

  2. 2.

    Patton cites Irvine [21], Godwyn and Irvine [10], Hager [12], and Gandon [9] as examples.

  3. 3.

    Even in quoting from Russell’s letter to Jourdain (Letter of March 15, [53]; reprinted in Grattan-Guiness [11], 107), she is again following Moore’s lead (Patton [43], p. 117; Moore [35], p. 122).

  4. 4.

    Please refer to my more extensive discussion of Zermelo in Chap. 5 of Park [40].

  5. 5.

    I also discussed this problem in Park [41]. I am delighted by Raftopoulos’s reminder that Hintikka [18, 19] already argued for the distinction between abduction and IBE (Raftopoulos [51], 262–263). In fact, Woods also discuss the problems related to abduction and IBE in his writings. For example, he discusses them in connection with the problem of abductive premiss-searches (Woods [62], p. 2, [63], p. 136). Woods’ most recent attempt to prove the impossibility of identifying abduction with IBE is discussed in Park [41].

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Authors and Affiliations

  1. Korea Advanced Institute of Science and Technology, Daejeon, South Korea

    Woosuk Park

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  1. Philosophy and Science Education, University at Buffalo, Amherst, NY, USA

    John R. Shook

  2. Faculty of Educational Sciences, University of Helsinki, Helsinki, Finland

    Sami Paavola

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Park, W. (2021). On Abducing the Axioms of Mathematics. In: Shook, J.R., Paavola, S. (eds) Abduction in Cognition and Action. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-030-61773-8_8

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