Volume 7 · Number 2 · Pages 141–149
A Defense of Strict Finitism

Jean Paul Van Bendegem

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Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evaluation of arguments and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in (the foundations of) mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem (why is mathematics so successful in its applications?). Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account.

Key words: mathematics, finite, largest number, infinite, limits, budget constraints


Van Bendegem J. P. (2012) A defense of strict finitism. Constructivist Foundations 7(2): 141–149. http://constructivist.info/7/2/141

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Alberts G. & Blauwendraat H. (eds.) (2000) Uitbeelden in de wiskunde. CWI, Amsterdam. ▸︎ Google︎ Scholar
Alberts G. (2000) Twee geesten van de wiskunde: Biografie van David van Dantzig. CWI, Amsterdam. ▸︎ Google︎ Scholar
Baker A. (1984) A concise introduction to the theory of numbers. Cambridge University Press, Cambridge. ▸︎ Google︎ Scholar
Dummett M. (1978) Wang’s paradox. In: Dummett M., Truth and other enigmas. Duckworth, London: 248–268. ▸︎ Google︎ Scholar
Epstein R. L. & Carnielli W. A. (2000) Computability. Computable functions, logic, and the foundations of mathematics with computability: A timeline. Second Edition. Wadsworth, London. ▸︎ Google︎ Scholar
Field H. (1980) Science without numbers: A defence of nominalism. Princeton University Press, Princeton. ▸︎ Google︎ Scholar
Gödel K. (1971) On formally undecidable propositions of Principia mathematica and related systems I. In: van Heijenoort J. (ed.) From Frege to Gödel. Harvard University Press, Cambridge, MA: 592–617. Originally published in German in 1931. ▸︎ Google︎ Scholar
Groenink A. (1993) Uism and short sighted models. Ph.D. Thesis, University of Utrecht. ▸︎ Google︎ Scholar
Isles D. (1992) What evidence is there that 2^65536 is a natural number? Notre Dame Journal of Formal Logic 33: 465–480. ▸︎ Google︎ Scholar
Isles D. (1994) A finite analog to the Löwenheim–Skolem theorem. Studia Logica 53: 503–532. ▸︎ Google︎ Scholar
Lavine S. (1994) Understanding the infinite. Harvard University Press, Cambridge, MA. ▸︎ Google︎ Scholar
Leng M. (2010) Mathematics and reality. Oxford University Press, Oxford. ▸︎ Google︎ Scholar
Marion M. (1998) Wittgenstein, finitism, and the foundations of mathematics. Clarendon Press, Oxford. ▸︎ Google︎ Scholar
Mawby J. (2005) Strict finitism as a foundation of mathematics. Ph.D. Thesis, Glasgow University. ▸︎ Google︎ Scholar
Mycielski J. (1981) Analysis without actual infinity. Journal of Symbolic Logic 4: 625–633. ▸︎ Google︎ Scholar
Nelson E. (1986) Predicative arithmetic. Princeton University Press, Princeton. ▸︎ Google︎ Scholar
Parikh R. (1971) Existence and feasibility in arithmetic. Journal of Symbolic Logic 36: 494–508. ▸︎ Google︎ Scholar
Presburger M. (1929) Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Comptes Rendus du I congrés de Mathématiciens des Pays Slaves: 92–101. ▸︎ Google︎ Scholar
Priest G. (1994) Is arithmetic consistent? Mind 103: 337–349. ▸︎ Google︎ Scholar
Priest G. (1994) What could the least inconsistent number be? Logique et Analyse 37: 3–12. ▸︎ Google︎ Scholar
Priest G. (1997) Inconsistent models of arithmetic I: Finite models. Journal of Philosophical Logic 26: 223–235. ▸︎ Google︎ Scholar
Priest G. (2000) Inconsistent models of arithmetic II: The general case. Journal of Symbolic Logic 65: 1519–1529. ▸︎ Google︎ Scholar
Rodych V. (2000) Wittgenstein’s anti-modal finitism. Logique et Analyse 43: 301–333. ▸︎ Google︎ Scholar
Rotman B. (1988) Toward a semiotics of mathematics. Semiotica 72: 1–35. ▸︎ Google︎ Scholar
Rotman B. (1993) Ad infinitum. The ghost in Turing’s machine. Taking god out of mathematics and putting the body back in. Stanford University Press, Stanford. ▸︎ Google︎ Scholar
Sorensen R. (2006) Vagueness. In: Zalta E. N. (ed.) The Stanford encyclopedia of philosophy. Fall 2008 Edition. http://plato.stanford.edu/archives/fall2008/entries/vagueness/
Styrman A. (2009) Finitist critique on transfinity: An investigation of infinity, collection theory and continuum. Master’s Thesis, University of Helsinki. ▸︎ Google︎ Scholar
Van Bendegem J. P. (1994) Strict finitism as a viable alternative in the foundations of mathematics. Logique et Analyse 37: 23–40. ▸︎ Google︎ Scholar
Van Bendegem J. P. (1995) In defence of discrete space and time. Logique et Analyse 38: 127–150. ▸︎ Google︎ Scholar
Van Bendegem J. P. (1999) Why the largest number imaginable is still a finite number. Logique et Analyse 42: 107–126. ▸︎ Google︎ Scholar
Van Bendegem J. P. (2000) How to tell the continuous from the discrete. In: Beets F. & Gillet E. (eds.) Logique en perspective. Mélanges offerts à Paul Gochet. Ousia, Brussels: 501–511. ▸︎ Google︎ Scholar
Van Bendegem J. P. (2002) Finitism in geometry. In: Zalta E. N. (ed.) The Stanford encyclopedia of philosophy. Spring 2002 Edition. http://plato.stanford.edu/entries/geometry-finitism/
Van Bendegem J. P. (2002) Inconsistencies in the history of mathematics: The case of infinitesimals. In: Meheus J. (ed.) Inconsistency in science. Kluwer, Dordrecht: 43–57. ▸︎ Google︎ Scholar
Van Bendegem J. P. (2003) Classical arithmetic is quite unnatural. Logic and Logical Philosophy 11: 231–249. ▸︎ Google︎ Scholar
Van Bendegem J. P. (2010) Een verdediging van het strikt finitisme. Algemeen Nederlands Tijdschrift voor Wijsbegeerte 102(3): 164–183. ▸︎ Google︎ Scholar
Van Dalen D. (1978) Filosofische grondslagen van de wiskunde. Van Gorcum, Assen. ▸︎ Google︎ Scholar
Van Dantzig D. (1956) Is 101010 a finite number? Dialectica 9: 273–278. ▸︎ Google︎ Scholar
Welti E. (1987) Die Philosophie des strikten Finitismus. Entwicklungstheoretische und mathematische Untersuchungen über Unendlichkeitsbegriffe in Ideengeschichte und heutiger Mathematik. Peter Lang, Bern. ▸︎ Google︎ Scholar
Wittgenstein L. (1984) Bemerkungen über die Grundlagen der Mathematik. Edited by Anscombe G. E. M., Rhees R. & von Wright G. H. Suhrkamp, Frankfurt. Originally published in 1956. English translation: Wittgenstein L. (1956) Remarks on the foundations of mathematics. Edited by von Wright G. H., Rhees R. & Anscombe G. E. M., translated by Anscombe G. E. M. Basil Blackwell, Oxford. ▸︎ Google︎ Scholar
Wright C. (1980) Wittgenstein on the foundations of mathematics. Harvard University Press, Cambridge MA. ▸︎ Google︎ Scholar
Wright C. (1982) Strict finitism. Synthese 51: 203–282. ▸︎ Google︎ Scholar
Yessenin-Volpin A. (1970) The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics. In: Kino A., Myhill J. & Vesley R. (eds.) Intuitionism and proof theory. North-Holland, Amsterdam: 3–45. ▸︎ Google︎ Scholar

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