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Numerical Methods, Complexity, and Epistemic Hierarchies

Published online by Cambridge University Press:  01 January 2022

Nicolas Fillion  and
Sorin Bangu
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Abstract

Modern mathematical sciences are hard to imagine without appeal to efficient computational algorithms. We address several conceptual problems arising from this interaction by outlining rival but complementary perspectives on mathematical tractability. More specifically, we articulate three alternative characterizations of the complexity hierarchy of mathematical problems that are themselves based on different understandings of computational constraints. These distinctions resolve the tension between epistemic contexts in which exact solutions can be found and the ones in which they cannot; however, contrary to a persistent myth, we conclude that having an exact solution is not generally more epistemologically beneficial than lacking one.

Type
Confirmation Theory
Information
Philosophy of Science , Volume 82 , Issue 5 , December 2015 , pp. 941 - 955
Copyright
Copyright © The Philosophy of Science Association

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Footnotes

†

The current work integrates the two papers we presented at the symposium The Plurality of Numerical Methods in Computer Simulations and Their Philosophical Analysis held at IHPST Paris, November 2011. We thank Julie Jebeile and Anouk Barberousse for organizing the conference. We also thank the audience for this symposium as well as audiences at the WCPA 2014 and the PSA 2014 meetings. In particular, we would like to thank Bob Batterman, Audrey Yap, and Paul Humphreys for very helpful comments.

References

Blum, L. 2004. “Computing over the Reals: Where Turing Meets Newton.” Notices of the American Mathematical Society 51 (9): 1024–37.Google Scholar
Borwein, J., and Crandall, R.. 2013. “Closed Forms: What They Are and Why We Care.” Notices of the American Mathematical Society 60 (1): 50–65.CrossRefGoogle Scholar
Corless, R. M., and Fillion, N.. 2013. A Graduate Introduction to Numerical Methods. New York: Springer.10.1007/978-1-4614-8453-0CrossRefGoogle Scholar
Dahlquist, G., and Bjorck, A.. 1974. Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall. Translated from the 1969 Swedish edition.Google Scholar
Fillion, N., and Corless, R. M.. 2014. “On the Epistemological Analysis of Modeling and Computational Error in the Mathematical Sciences.” Synthese 191:1451–67.CrossRefGoogle Scholar
Humphreys, P. 2004. Extending Ourselves: Computational Science, Empiricism, and Scientific Method. New York: Oxford University Press.CrossRefGoogle Scholar
Matthews, J., and Fink, K.. 1999. Numerical Methods Using MATLAB. 3rd ed. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Trefethen, L. N. 2008. “Numerical Analysis.” In The Princeton Companion to Mathematics, ed. Gowers, T., Barrow-Green, J., and Leader, I.. Princeton, NJ: Princeton University Press.Google Scholar
Turing, A. 1950. “Computing Machinery and Intelligence.” Mind 59 (236): 433–60.Google Scholar
Wang, Q. 1990. “The Global Solution of the n-Body Problem.” Celestial Mechanics and Dynamical Astronomy 50 (1): 73–88.Google Scholar