# [FOM] Ultrafinitism

Keith Brian Johnson joyfuloctopus at yahoo.com
Mon Feb 9 22:52:45 EST 2009

Jean Paul Van Bendegem wrote, in part:
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(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.

And that seems odd. This kind of paradoxical reasoning is related to the
analysis of vague concepts, so rather than "fuzzy" I would prefer to use the
term "vague".
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But it's unclear to me where you think the problem lies.  Because of the phrasing "I can write down," one might question the truth of (b) for real persons.  One might have just enough time to write down some large number n before dying, but not enough time to write down n.  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).

(Have you already considered such a formulation and rejected it as vague?)

You might, however, intend to call into question something else, like the deductive step from (a) and (b) to (c).  I'm sorry, but it's not clear to me.  What did you have in mind?

Keith Brian Johnson