[FOM] Ultrafinitism

[Alex Blum]

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