Abstract
In his influential book Conceptual Revolutions (1992), Thagard asked whether the question of conceptual change is identical with the question of belief revision. He considered quite seriously the possibility that the answer is positive, for one might argue that they are identical on the ground that “whenever a concept changes, it does so by virtue of changes in the beliefs that employ that concept”.
This chapter was published as Woosuk Park (2010), Belief Revision Vs. Conceptual Change in Mathematics. Studies in Computational Intelligence, 314 (L. Magnani et al. (eds.), Model-based Reasoning in Science and Technology, Boston: Springer), 121–134.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
One might want to find such an application in Rusnock and Thagard (1995, p. 25). However, their focus is on how “to use Darden’s strategies for anomaly resolution to analyze developments in Greek mathematics following the discovery of the incommensurables” (p. 108) rather than the identity or difference between belief revision and conceptual change. I am indebted to one anonymous referee, who drew my attention to Rusnock and Thagard (1995).
- 3.
According to Thagard, we can distinguish between nine degrees of conceptualchange: (1) Adding a new instance; (2) Adding a new weak rule; (3) Adding a new strong rule; (4) Adding a new part-relation; (5) Adding a new kind-relation; (6) Adding a new concept; (7) Collapsing part of a kind-hierarchy; (8) Reorganizing hierarchies by branch jumping; (9) Tree switching (Thagard 1992, p. 35). Thagard claims that among these “(1)–(3) can be interpreted as simple kinds of belief revision”. On the other hand, “(4)–(9) cannot, since they involve conceptual hierarchies” (Ibid., p. 36).
- 4.
The ideas discussed in this section were more extensively presented in Park (2005). As one anonymous referee points out, there is room for further discussion as to whether we can take the principle of informational economy as one preventing large scale changes. For what the principle defends is that informational loss should be avoided if possible while accommodating the new information.
- 5.
The idea of implicit definition has never been clarified completely. We should still wait for a resolution of the famous Frege–Hilbert controversy. As a consequence, for too long we have been lacking a sound understanding of the role and function of definitions in mathematics. Further, as witnessed by Ulrich Majer’s work, we still do not quite understand either the continuity or discontinuity of the traditional and Hilbertian axiomatic method. We even fail to decide whether the problem of implicit definition is merely a side issue, as Majer claims. See Majer (2002).
- 6.
In passing, it is worthwhile to note that Thagard also leaves room for counting branch jumping and tree switching as a kind of belief revision. Anyway, they are “more holistic than piecemeal belief revisions” (emphasis mine) (28, p. 36, emphasis mine).
- 7.
Mancosu refers back to Giuseppe Biancani (1566–1624) for this belief: “According to Biancani, the objects of mathematics are quantities abstracted from sensible matter. Arithmetic and geometry, which together constitute pure mathematics respectively deal with discrete and continuous magnitude” Mancosu (1996, p. 16a) and see also, Park (2009).
- 8.
Here we seem to face a problem of how to find a way out from what I would call “the Hilbertian dilemma”: Are we to emphasize the novelty of Hilbert’s axiomatic method or the continuity of the history of axiomatic method? If we grasp the second horn, as Majer does, we might be giving up the hope for understanding the so-called second birth of mathematics in the 19th and 20th centuries. If we grasp the first horn, as logical positivists and Friedman do, we should make clear how the method of implicit definition truly works in scientific as well as mathematical practice.
References
Alchorron, C., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction functions and their associated revision functions. The Journal of Symbolic Logic, 50, 510–530.
Dedekind, R. (1967) Letter to Keferstein. English translation by Stefan Bauer-Mengelberg and Hao Wang in van Heijenoort, pp. 98–103.
Dunmore, C. (1992). Meta-level revolutions in mathematics. In D. Gillies (Ed.), revolutions in mathematics (pp. 209–225). Oxford: Oxford University Press.
Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (2003). Many-dimensional modal logics: Theory and applications. Amsterdam: Elsevier (134 W. Park).
Gärdenfors, P. (1988). Knowledge in flux: Modeling the dynamics of Epistemic States. Cambridge: MIT Press.
Gärdenfors, P., & Rott, H. (1995). Belief revision. In D. M. Gabbay et al. (Eds.), Handbook of logic in artificial intelligence and logic programming. Epistemic and temporal reasoning, vol. 4, (pp. 35–132). Oxford: Clarendon Press.
Gillies, D. (Ed.). (1992). Revolutions in mathematics. Oxford: Oxford University Press.
Hilbert, D. (1996). ¨Uber den Zahlbegriff. Jahresbericht der Deutschen Mathematiker—Vereinigung 8, 180–184, English translation in Ewald 1089–1096.
Kline, M. (1972), Mathematical Thought from Ancient to Modern Times, Oxford: Oxford University Press.
Majer, U. (2002). Hilbert’s program to axiomatize physics (in analogy to geometry) and its impact on Schlick, Carnap and other members of the Vienna Circle. In M. Heidelberger & F. Stadler (Eds.), History of philosophy and science (pp. 213–224). Boston: Kluwer.
Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford: Oxford University Press.
Nagel, E. (1979). Teleology revisited and other essays in the philosophy and history of science. New York: Columbia University Press.
Park, W. (2005). Belief revision in Baduk. Journal of Baduk Studies, 2(2), 1–11.
Park, W. (2009). The status of scientiae mediae in the history of mathematics. Korean Journal of Logic, 12(2), 141–170.
Pycior, H. M. (1997). Symbols, impossible numbers, and geometric entanglements: British algebra through the commentaries on Newton’s universal arithmetick. Cambridge: Cambridge University Press.
Rott, H. (2000). Two dogmas of belief revision. Journal of Philosophy, 97, 503–522.
Rott, H. (2001). Change, choice and inference: A study of belief revision and nonmonotonic reasoning. Oxford: Clarendon Press.
Rusnock, P., & Thagard, P. (1995). Strategies for conceptual change: Ratio and proportion in classical greek mathematics. Studies in History and Philosophy of Science, 26, 107–131.
Sieg, W., & Schlimm, D. (2005). Dedekind’s analysis of number: systems and axioms. Synthese, 147, 121–170.
Thagard, P. (1992). Conceptual revolutions. Princeton: Princeton University Press.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Park, W. (2017). Belief Revision Versus Conceptual Change in Mathematics. In: Abduction in Context. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-319-48956-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-48956-8_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-48955-1
Online ISBN: 978-3-319-48956-8
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)