[FOM] Ultrafinitism
Alex Blum
blumal at mail.biu.ac.il
Tue Feb 10 10:36:00 EST 2009
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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