Finitism versus Infinitism: No "truths'' are lost

Jan Heylen

Abstract


No "truths" of infinitism with respect to natural numbers (i.e. there are infinitely many natural numbers) are lost, says Van Bendegem, who defends strict finitism with respect to natural numbers (i.e. there are only finitely many natural numbers). In this paper it is shown that one can use Van Bendegem's own strategy to similarly conclude that no "truths" of finitism are lost to the infinitist. It will be further argued that, as a consequence, the debate between finitist and infinitist is not about the truth beyond the "truths'' or that the "truths" are not adequate representations of the ontological commitments of the two frameworks.


References


Boolos, G. S., Burgess, J. P. and Jeffrey, R. C. (2003), Computability and Logic, fourth edn, Cambridge University Press, New York.

Carnap, R. (1950), ‘Empiricism, semantics, and ontology’, Revue Internationale

de Philosophie 4(11), 20–40.

Chalmers, D. J. (2009), Ontological anti-realism, in D. J. Chalmers, D. Manley

and R. Wasserman, eds, ‘Metametaphysics: New Essays on the Foundations

of Ontology’, Oxford University Press.

Gödel, K. (1986), Collected Works, Volume I, Publications 1929–1936, Oxford University Press, chapter Zum intuitionistische aussagenkalkül, pp. 286–295.

Horsten, L. (1998), ‘In defense of epistemic arithmetic’, Synthese 116(1), 1–25.

Horsten, L. (2010), ‘Apofatisch finitisme?’, Algemeen Nederlands Tijdschrift voor Wijsbegeerte 102(3), 184–187.

Quine,W. V. (1948), ‘On what there is’, Review of Metaphysics 2(1), 21–38.

Shapiro, S. (1985), Epistemic and intuitionistic arithmetic, in S. Shapiro, ed.,

‘Intensional Mathematics’, Vol. 113 of Studies in Logic and the Foundations of

Mathematics, North-Holland, pp. 11–46.

Van Bendegem, J. P. (1999), ‘Why the largest number imaginable is still a finite

number’, Logique et Analyse 42(165-166), 107–126.

Van Bendegem, J. P. (2003), ‘Classical arithmetic is quite unnatural’, Logic and

Logical Philosophy 11(n/a), 231–249.

Van Bendegem, J. P. (2010), ‘Een verdediging van het strikt finitisme’, Algemeen Nederlands Tijdschrift voor Wijsbegeerte 102(3), 164–183.

Van Bendegem, J. P. (2012), ‘A defense of strict finitism’, Constructivist Foundations 7(2), 141–149.


Refbacks

  • There are currently no refbacks.