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Philosophia Mathematica 2005 13(2):174-186; doi:10.1093/philmat/nki021
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Philosophia Mathematica (III), Vol. 13 No. 2 © Oxford University Press, 2005, all rights reserved

Computational Structuralism{dagger}

Volker Halbach* and Leon Horsten**

*New College Oxford OX1 3BN, England volker.halbach{at}philosophy.oxford.ac.uk
**Center for Logic and Philosophy of Science, Institute of Philosophy, University of Leuven Kardinaal Mercierplein 2, B-3000 Leuven, Belgium Leon.Horsten{at}hiw.kuleuven.ac.be

According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by (nonstandard) models that do not instantiate this structure. Proponents of structuralism have put forward various accounts of how we succeed in fixing one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On this account, the intended models of arithmetic are the notation systems with recursive operations on them satisfying the Peano axioms.

[A]m Anfang [...] ist das Zeichen.

(Hilbert [1935], p. 163)


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