[FOM] PoV on Ultrafinitism

jean paul van bendegem jpvbende at vub.ac.be
Sun Feb 8 13:40:51 EST 2009

As someone interested in ultrafinitism or strict finitism (the label I 
prefer), I would rather not "promote" the connection between strict finitism 
and fuzzy mathematics. There are alternative approaches relying, e.g., on 
paraconsistent logic or similar/related logical systems.

I do understand perfectly the idea that the idea of  "the natural numbers" 
is a crystalline abstract jewel of ours, although I would not count myself 
as someone to whom it was once crystal clear (but that is perhaps I am 
mostly involved with philosophy and not maths). My doubts are fueled by 
reasonings of the following kind:

(a) I can write down the numeral 0 (or 1, does not matter),
(b) for all n, if I can write down n, I can write down n+1 (or the successor 
of n),
hence, by mathematical induction,
(c) I can write down all numerals.

And that seems odd. This kind of paradoxical reasoning is related to the 
analysis of vague concepts, so rather than "fuzzy" I would prefer to use the 
term "vague".

Jean Paul Van Bendegem 

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