Abstract
Defenders of the enhanced indispensability argument argue that the most effective route to platonism is via the explanatory role of mathematical posits in science. Various compelling cases of mathematical explanation in science have been proposed, but a satisfactory general philosophical account of such explanations is lacking. In this paper, I lay out the framework for such an account based on the notion of “the mathematical stance.” This is developed by analogy with Dennett’s well-known concept of “the intentional stance.” Roughly, adopting the mathematical stance towards a particular physical phenomenon involves treating it as an abstract mathematical structure for the purposes of prediction and explanation. Interestingly, Dennett himself frequently draws analogies between his intentional stance towards beliefs and desires and scientists’ stance towards centers of gravity. I explore the theoretical role played by centers of gravity within science and discuss how an indispensabilist platonist ought to categorize the ontological status of this type of posit. I conclude with some thoughts on how an approach based on the mathematical stance might be developed into a more general philosophical account of the application of mathematics in science.
Similar content being viewed by others
Notes
Bueno & Colyvan [2011, p. 347].
ibid.
For further elaboration of this problem see Nguyen & Frigg [2017]. Their proposed solution involves the notion of a “structure generating description,” which uses physical properties to carve up a given system into objects and relations from which the resulting structure can be abstracted. However it is not clear that this solves the Assumed Structure Problem rather than simply pushing the question back to what makes one structure generating description the uniquely correct one.
See Baker [2009].
See e.g. Leng [2010, Chapter 6].
In the sense that a line drawn vertically down from the center of gravity does not meet any part of the base.
Baker [2017].
Because any water added will immediately be above the height of the center of gravity of the glass.
This is not the only reason. One problem with the honeycomb example is that the mathematics concerns a 2-dimensional optimization problem whereas actual bees are facing a 3-dimensional optimization problem. Another problem is that there are alternative, non-optimization-based explanations for the hexagonal shape that are based just on the physical behavior of liquid wax as it forms into cells. For more details, see Raz [2013].
Baker [2005, p. 228].
Lange [2018]. Lange actually couches the example in terms of center of mass, not center of gravity, but the core point remains the same.
Dennett [1992].
For further discussion of how (and where) to draw the boundary between abstract and concrete, see Rosen [2017], especially the section on the non-spatiality criterion.
One of the interesting (and complicating) factors here is that centers of gravity tend to play a role in scientific explanations that is more token-based than type-based. This is unlike more canonical mathematical explanations. Numbers may play an explanatory role for one physical phenomenon (e.g. cicada periods), and those very same numbers may also play an explanatory role in a quite distinct physical phenomenon (e.g. bicycle gear ratios). In the case of centers of COG’s, by contrast, each different stability explanation is likely to involve a distinct COG.
Dennett [1988, p. 496].
ibid.
Dennett [2009, p. 349].
Dennett [1991, p. 27].
Dennett [2009, p. 340].
Different predictive and explanatory goals may also yield different levels of structural stance. The physical stance is at a lower level than any structural stance (and I leave open whether the physical stance should itself be considered structural, since this hinges on more general issues concerning structural realism in the philosophy of science).
Perhaps these two categories of case can be brought closer together by thinking of the first example as also (in a sense) involving the violation of an idealizing assumption. The assumption in this case is that by representing each bridge as a link in the graph, each link thereby represents a viable crossing.
Dennett [2009, p. 341].
ibid.
Dennett [1988, p. 497].
Dennett [1987, p. 72].
e.g. “Belief is a perfectly objective phenomenon.” (Dennett [1987, p. 15])
Dummett [1978, p. xxxviii].
See e.g. Shapiro [1997].
Dennett [1987, p. 72].
op. cit., pp. 72–3.
Demeter [2009, p. 62].
Thus while (for example) recent work on mathematical explanation within mathematics has drawn attention to the potential explanatory value of proofs, the target of the explanation is a result in pure mathematics, and thus it only has ‘mathematics-independent’ predictive or explanatory value when it is itself applied.
Bibliography
Baker, A. (2005). Are There Genuine Mathematical Explanations of Physical Phenomena? Mind, 114, 223–238
Baker, A. (2009). Mathematical Explanation in Science. British Journal for the Philosophy of Science, 60, 611–633
Baker, A. (2017). Mathematics and Explanatory Generality. Philosophia Mathematica, 25, 194–209
Bueno, O., & Colyvan, M. (2011). An Inferential Conception of the Application of Mathematics. Nous, 45, 345–374
Demeter, T. (2009). Two Kinds of Mental Realism. Journal for General Philosophy of Science, 40, 59–71
Dennett, D. (1969). Content and Consciousness. London: Routledge & Kegan Paul
Dennett, D. (1971). Intentional Systems. Journal of Philosophy, 68, 87–106
Dennett, D. (1987). The Intentional Stance. Cambridge, Mass: MIT Press
Dennett, D. (1988). Précis of The Intentional Stance. The Behavioral and Brain Sciences, 11, 495–546
Dennett, D. (1991). Real Patterns. Journal of Philosophy, 88, 27–51
Dennett, D. (1992). “The Self as a Center of Narrative Gravity. In F. Kessel, P. Cole, & D. Johnson (Eds.), ” Self and Consciousness: Multiple Perspectives. Erlbaum: Hillsdale, NJ
Dennett, D. (1996) Kinds of Minds. New York: Basic Books
Dennett, D. (2009). “Intentional Systems Theory. In B. McLaughlin, et al. (Ed.), ” The Oxford Handbook of Philosophy of Mind (pp. 339–350). Oxford: Oxford University Press
Dummett, M. (1978). Truth and Other Enigmas. Cambridge, MA: Harvard University Press
Joyce, R. (2013). Psychological Fictionalism and the Threat of Fictionalist Suicide. The Monist, 96, 517–538
Lange, M. (2018). “How the Explanations of Natural Laws Make Some Reducible Physical Properties Natural and Explanatorily Powerful. In W. Ott, & L. Patton (Eds.), ” Laws of Nature. Oxford: Oxford University Press
Leng, M. (2010). Mathematics and Reality. Oxford: Oxford University Press
Nguyen, J., & Frigg, R. (2017). “Mathematics is Not the Only Language in the Book of Nature,” Synthese
Pincock, C. (2004). A Revealing Flaw in Colyvan’s Indispensability Argument. Philosophy of Science, 71, 61–79
Pincock, C. (2007). A Role for Mathematics in the Physical Sciences. Nous, 41, 253–275
Raz, J. (2013). On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb. Philosophia Mathematica, 21, 351–360
Rosen, G. (2017). “Abstract Objects,” Stanford Encyclopedia of Philosophy, online at https://plato.stanford.edu/entries/abstract-objects/
Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article belongs to the Topical Collection: Explanatory and Heuristic Power of Mathematics.
Rights and permissions
About this article
Cite this article
Baker, A. The mathematical stance. Synthese 200, 53 (2022). https://doi.org/10.1007/s11229-022-03458-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11229-022-03458-8