[FOM] Ultrafinitism
Walter Read
read at csufresno.edu
Tue Feb 10 13:35:53 EST 2009
It seems to me that the response to Jean Paul Van Bendegem's comment on ultrafinitism
has been going down a less interesting path, concentrating on parses of "writing down a
numeral". I think that the more interesting issue is distinguishing "any numeral" from "all
numerals", i.e., unbounded from infinite or, in older terminology, "potential" from "actual"
infinities. Since Cantor - more precisely, since a generation or so after Cantor - these
concerns have largely faded away, but at one time they engaged the best minds of the era.
-Walt
Walt Read
Computer Science, MS ST 109
CSU, Fresno
Fresno, CA 93740
Email: read at csufresno.edu
Tel: 559 278 4307
559 278 4373 (dept)
Fax: 559 278 4197
http://www.csufresno.edu/csci/
----- Original Message -----
From: Alex Blum <blumal at mail.biu.ac.il>
Date: Tuesday, February 10, 2009 8:10 am
Subject: [FOM] Ultrafinitism
To: Foundations of Mathematics <fom at cs.nyu.edu>
> Jean Paul Van Bendegem presents a putative counterexample to a
> generalization of mathematical induction. He writes, in part:
>
> "(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."
>
>
> Keith Brian Johnson questions (b), for, he writes: "One might have
> just
> enough time to write down some large number n before dying, but not
> enough time to write down n[+1]. Or one might run out of paper (or
> the
> amount of material in the universe might limit how many numbers
> could
> actually be written down). Or one might be limited, when
> conceiving of
> numbers, by his own brain's finitude. So, as a practical matter,
> (b)
> might be false. Naturally, I would think the argument should be
> so formulated as to render such practical considerations
> irrelevant,
> e.g., with an "in principle" inserted: for all n, if I can in
> principle
> write down n, then I can in principle write down n+1. I.e., if a
> hypothetical being unconstrained by spacetime limitations or mental
> finitude could conceive of n, then that being could conceive of
> n+1.
> (Whether such a being *would* conceive of n+1 is unimportant; what
> matters is that there is no mathematical reason why he couldn't.)
> Similarly, it's clearly false that I personally physically can
> write
> down all numerals, but "I can, in principle, write down all
> numerals,"
> where "in principle" is so construed as to leave me unconstrained
> by
> spacetime limitations or mental finitude, doesn't seem similarly
> false
> (unless one picks on the notion on writing down numerals as
> necessarily
> physical, in which case I would replace my writing down of numerals
> by that hypothetical being's conception of numbers)."
> ...
>
> The properties of numbers in mathematical induction hold of numbers
> irrespective of how they are named. Since a number may be named
> in a
> notation which could never be completed, (b), even if true,
> need not
> be true, and thus the predicate 'I can write down the numeral' is
> inappropiate for mathematical induction.
>
>
> Alex Blum
>
>
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