[FOM] PoV on Ultrafinitism - Mathematical Induction Trickiness

Dennis E. Hamilton dennis.hamilton at acm.org
Tue Feb 10 14:54:37 EST 2009

My first reaction to the quandary over mathematical induction in this note
is to suggest that (c) is better replaced by the simple observation that I
can write down the progression of numerals as long as I want without any
repetitions (taking into consideration the other properties of the numbers
under successor).

On looking deeper into the appeal to mathematical induction, I note that (b)
is not proven but is being asserted -- it is an empirical claim, rather
different than the usual practice of applying the second step in the
principle of mathematical induction.

However, even accepting (b), I believe the proper inference is 

    (c') I can write down any numeral.

There's nothing in the usual statement of the principle of mathematical
induction (say, for Peano Arithmetic) that says anything about predicates on
the numbers as a totality, and in any case IcanWriteDownUniqueNumeralFor(x)
for x a number, is not such a predicate.

 - Dennis 

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
jean paul van bendegem
Sent: Sunday, February 08, 2009 10:41
To: Bill Taylor; Foundations of Mathematics
Subject: Re: [FOM] PoV on Ultrafinitism

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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