[FOM] Ultrafinitism

[Walter Read]

   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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