[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 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


----- 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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