Computation and Indispensability

Jeffrey Ketland



This article provides a computational example of a mathematical explanation within science, concerning computational equivalence of programs. In addition, it outlines the logical structure of the reasoning involved in explanations in applied mathematics. It concludes with a challenge that the nominalist provide a nominalistic explanation for the computational equivalence of certain programs.


applicability of mathematics; computation; indispensability

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Arntzenius, Frank, and Cian Dorr, 2012, “Calculus as geometry”, Chapter 8 of F. Arntzenius (ed.), Space, Time, and Stuff, Oxford: Oxford University Press. DOI:

Baker, Alan, 2005, “Are there genuine mathematical explanations of physical phenomena?”, Mind 114: 223–238. DOI:

Burgess, J.P., and Gideon Rosen, 1997, A Subject with No Object. Strategies for Nominalistic Interpretation of Mathematics, Oxford: Clarendon Press. DOI:

Church, Alonzo, 1932, “A set of postulates for the foundations of logic”, Annals of Mathematics 33: 346–366.

Church, Alonzo, 1933, “A set of postulates for the foundations of logic” (second paper), Annals of Mathematics 34: 839–864.

Church, Alonzo, 1958, “Letter to Goodman (1 Dec 1958)”. Reprinted in [Goodman 1972].

Field, Hartry, 1980, Science Without Numbers: A Defence of Nominalism, Princeton, N.J.: Princeton University Press.

Frege, Gottlob, 1891, “Function and concept”. English translation of [Frege, 1891] “Funktion und Begriff” (der Jenaischen Gesellschaft für Medizin und Naturwissenschaft) in P. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege, Oxford: Blackwell, 1980.

Feynman, Richard, 1965, The Character of the Physical Law, London: Penguin Books, 1992. Page reference to the 1992 edition. DOI:

Goodman, Nelson, 1972, Problems and Projects, New York: The Bobbs-Merrill Company Inc.

Kleene, S.C., and J.B. Rosser, 1935, “The inconsistency of certain formal logics”, Annals of

Mathematics 36: 630–637. DOI:

Klement, Kevin, 2003, “Russell’s 1903–1905 anticipation of the lambda calculus”, History and Philosophy of Logic 24 (1): 15–37. DOI:

Mancosu, Paolo, 2018, “Explanation in mathematics”, Stanford Encyclopedia of Philosophy. URL:".

Putnam, Hilary, 1971, Philosophy of Logic. Reprinted in [Putnam 1979]. Page reference to [Putnam, 1979].

Putnam, Hilary, 1975, “What is mathematical truth?”, in [Putnam, 1979]. Page reference to [Putnam, 1979].

Putnam, Hilary, 1979, Mathematics, Matter and Method. Philosophical Papers, Vol. 1, Cambridge University Press, Cambridge. Second edition. DOI:

Quine, W.V., 1948, “On what there is”, Revue of Metaphysics 2: 21–38.

Quine, W.V., 1973, “The limits of knowledge”, Radio Talk. Published in [Quine, 1976], The Ways of Paradox. Revised enlarged second edition. Harvard University Press, Cambridge, Ma.

Quine, W.V., 1995, From Stimulus to Science, Harvard University Press, Cambridge, Ma.

Quine, W.V., and Nelson Goodman, 1947, “Steps towards a constructive nominalism”, Journal of Symbolic Logic 12: 105–122. Reprinted in [Goodman, 1972].

Russell, Bertrand, 1903–05, The Collected Papers of Bertrand Russell, Vol. 4, Foundations of Logic 1903–05, edited by Alasdair Urquhart, London: Routledge, 1994.

Simpson, S., 2010, Subsystems of Second-Order Arithmetic, 2nd edition, Cambridge: Cambridge University Press. DOI:

Synge, J.L., and B.A. Griffith, 1959, Principles of Mechanics, Third edition, McGraw-Hill Book Company Inc., New York.

Wetzel, L., 2009, Types and Tokens: On Abstract Objects, Cambridge, Ma.: MIT Press. DOI:

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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