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MODULARITY IN MATHEMATICS

Published online by Cambridge University Press:  21 February 2018

JEREMY AVIGAD
JEREMY AVIGAD*
Affiliation:
Department of Philosophy, Carnegie Mellon University
*
*DEPARTMENT OF PHILOSOPHY CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: [email protected]
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Abstract

In a wide range of fields, the word “modular” is used to describe complex systems that can be decomposed into smaller systems with limited interactions between them. This essay argues that mathematical knowledge can fruitfully be understood as having a modular structure and explores the ways in which modularity in mathematics is epistemically advantageous.

Keywords

00A3000A3503A05modularityphilosophy of mathematicsmathematical practice
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 1 , March 2020 , pp. 47 - 79
Copyright
Copyright © Association for Symbolic Logic 2018 

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References

BIBLIOGRAPHY

Abelson, H. & Sussman, G. J. (1996). Structure and Interpretation of Computer Programs (second edition). Cambridge, MA: MIT Press.Google Scholar
Arana, A. (2008). Logical and semantic purity. Protosociology, 25, 36–48.CrossRefGoogle Scholar
Avigad, J. (2006). Mathematical method and proof. Synthese, 152(1), 105–159.CrossRefGoogle Scholar
Avigad, J. Understanding proofs. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 317–353.CrossRefGoogle Scholar
Avigad, J. (2010). Understanding, formal verification, and the philosophy of mathematics. Journal of the Indian Council of Philosophical Research, 27, 161–197.Google Scholar
Avigad, J. & Harrison, J. (2014). Formally verified mathematics. Communications of the ACM, 57(4), 66–75.CrossRefGoogle Scholar
Avigad, J. & Morris, R. (2014). The concept of “character” in Dirichlet’s theorem on primes in an arithmetic progression. Archive for History of Exact Sciences, 68(3), 265–326.CrossRefGoogle Scholar
Avigad, J. & Morris, R. (2016). Character and object. Review of Symbolic Logic, 9, 480–510.CrossRefGoogle Scholar
Baldwin, C. Y. & Clark, K. B. (1999). Design Rules: The Power of Modularity, Vol. 1. Cambridge, MA: MIT Press.Google Scholar
Barrett, H. C. & Kurzban, R. (2006). Modularity in cognition: Framing the debate. Psychological Review, 113, 628–647.CrossRefGoogle ScholarPubMed
Bhargava, M. (2004). Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations. Annals of Mathematics, 159(1), 217–250.CrossRefGoogle Scholar
Bourbaki, N. (1950). The architecture of mathematics. The American Mathematical Monthly, 57(4), 221–232. Translated from the French by Dresden, Arnold. The original version appeared in F. Le Lionnais ed., Les grands courants de la pensée mathématique, Cahiers du Sud, 1948.CrossRefGoogle Scholar
Buchmann, J. & Vollmer, U. (2007). Binary Quadratic Forms: An Algorithmic Approach. Berlin: Springer.CrossRefGoogle Scholar
Callebaut, W. & Rasskin-Gutman, D. (2005). Modularity: Understanding the Development and Evolution of Natural Complex Systems. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Carruthers, P. (2006). The Architecture of the Mind: Massive Modularity and the Flexibility of Thought. Oxford: Oxford University Press.CrossRefGoogle Scholar
Cox, D. A. (2014). Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication. Hoboken, NJ: Wiley.Google Scholar
de Moura, L., Kong, S., Avigad, J., van Doorn, F., & von Raumer, J. (2015). The Lean theorem prover. In Felty, A. P., and Middeldorp, A., editors. Automated Deduction–CADE-25, 25th International Conference on Automated Deduction, Berlin, Germany, August 1-7, 2015. Cham, Switzerland: Springer International Publishing, pp. 378–388.CrossRefGoogle Scholar
Detlefsen, M. & Arana, A. (2011). Purity of methods. Philosophers’ Imprint, 11(2), 1–20.Google Scholar
Dijkstra, E. W. (1982). On the role of scientific thought. In Dijkstra, E. W., editor. Selected Writings on Computing: A personal Perspective. New York, NY: Springer, pp. 60–66.CrossRefGoogle Scholar
Edwards, H. E. (2001). Riemann’s Zeta Function. Mineola, NY: Dover Publications Inc. Reprint of the 1974 original [Academic Press, New York].Google Scholar
Edwards, H. E. (2005). Essays in Constructive Mathematics. New York: Springer.Google Scholar
Fodor, J. A. (1983). The Modularity of Mind. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Gonthier, G., Asperti, A., Avigad, J., Bertot, Y., Cohen, C., Garillot, F., Roux, S., Mahboubi, A., O’Connor, R., Biha, S. O., Pasca, I., Rideau, L., Solovyev, A., Tassi, E., & Théry, L. (2013). A machine-checked proof of the odd order theorem. InBlazy, S., Paulin-Mohring, C., and Pichardie, D., editors. Interactive Theorem Proving. Lecture Notes in Computer Science, Vol. 7998. Heidelberg: Springer, pp. 163–179.CrossRefGoogle Scholar
Harper, R. (2016). Practical Foundations for Programming Languages (second edition). New York, NY: Cambridge University Press.CrossRefGoogle Scholar
Kitcher, P. (1989). Explanatary unification and the causal structure of the world. In Kitcher, P. and Salmon, W., editors. Scientific Explanation. Minneapolis: University of Minnesota Press, pp. 410–505.Google Scholar
Lemmermeyer, F. (2000). Reciprocity Laws: From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Mancosu, P. Mathematical explanation: Why it matters. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 134–440.CrossRefGoogle Scholar
Mancosu, P. (editor) (2008). The Philosophy of Mathematical Practice. Oxford: Oxford University Press.CrossRefGoogle Scholar
Manders, K. Expressive Means and Mathematical Understanding, manuscript.Google Scholar
Manders, K. The Euclidean diagram. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 80–133.CrossRefGoogle Scholar
Myers, G. J. (1978). Composite/Structured Design. New York: Van Nostrand Reinhold.Google Scholar
Parnas, D. L. (1972). On the criteria to be used in decomposing systems into modules. Communications of the ACM, 15(12), 1053–1058.CrossRefGoogle Scholar
Pitt, D. (2012). Mental representation. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy. Available at https://plato.stanford.edu/archives/spr2017/entries/mental-representation.Google Scholar
Robbins, P. (2009). Modularity of mind. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy. Available at https://plato.stanford.edu/archives/win2017/entries/modularity-mind/.Google Scholar
Robinson, J. A. & Voronkov, A. (editors) (2001). Handbook of Automated Reasoning, Vol. 2. New York: Elsevier, and Cambridge, MA: MIT Press.Google Scholar
Simon, H. A. (1962). The architecture of complexity. Proceedings of the American Philosophical Society, 106(6), 467–482.Google Scholar
Simon, H. A. (1997). Administrative Behavior (fourth edition). New York: Free Press; First edition, New York: Macmillan, 1947.Google Scholar
Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 133–151.CrossRefGoogle Scholar
Struik, D. J. (1969). A Source Book in Mathematics, 1200–1800. Source Books in the History of the Sciences. Cambridge, MA: Harvard University Press.Google Scholar
Tappenden, J. Mathematical concepts and definitions. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 256–275.CrossRefGoogle Scholar
Wirth, N. (1971). Program development by stepwise refinement. Communications of the ACM, 14(4), 221–227.CrossRefGoogle Scholar