[FOM] A formalism for Ultrafinitism

[jean paul van bendegem]

In response to the message of Bill Taylor, a few remarks concerning
ultrafinitism:

* something that comes real close to your proposal has been done many years
ago by Jan Mycielski: "Analysis without Actual Infinity". Journal of
Symbolic Logic, vol. 46, number 3, 1981, pp. 625-633.

* recently, Graham Priest and Chris Mortensen have done similar things
although their inspiration comes from resp. paraconsistent and relevant
logic:
Graham Priest: "Inconsistent Arithmetics", in: Vincent F. Hendricks and
Jacek Malinowski, "Trends in Logic, 50 Years of Studia Logica", Kluwer
Academic, Dordrecht, 2003, pp. 273-299
Chris Mortensen: : "Inconsistent Mathematics. Mathematics and Its
Applications", vol. 312. Dordrecht: Kluwer Academic Publishers, 1995.

* using their work I have done a few attempts myself in that direction:
"Strict Finitism as a Viable Alternative in the Foundations of Mathematics".
Logique et Analyse, vol. 37, 145, 1994 (date of publication: 1996), pp.
23-40.
"Why the largest number imaginable is still a finite number". Logique et
Analyse, vol. 41, 161-162-163, 1998 (date of publication: 2001), pp.
107-126.

* and there is from the historical point of view, the nice study of Ernst
Welti, "Die Philosophie des strikten Finitismus. Entwicklungstheoretische
und mathematische Untersuchungen über Unendlichkeitsbegriffe in
Ideengeschichte und heutiger Mathematik", Bern, Peter Lang, 1987
(unfortunately not translated).

So, I am tempted to say that that the idea of ultrafinitism is a good idea,
at least worthwhile to have a look at.

Jean Paul

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Jean Paul Van Bendegem
Centrum voor Logica en Wetenschapsfilosofie
Vrije Universiteit Brussel
Pleinlaan 2
B-1050 Brussel
Belgium