FOM: unsurveyability
M. Randall Holmes
M.R.Holmes at dpmms.cam.ac.uk
Mon Oct 5 06:09:46 EDT 1998
(J. Kennedy writes:)
Wittgensteinian skepticism challenges the part of Holmes' position stated
above. Wittgenstein's idea being (I think) that proofs that are too big to
grasp all at once DO present a challenge to the indubitability of
mathematics. He uses in support of this the example that there is no way to
know that the rule we use to multiply numbers is stable under the passage
from small to large numbers. Kripke's "Wittgenstein: On Rules and Private
Languages" gives an exposition of this. (Please forgive if this is not
exactly the right title.) It is a (to me) elusive and difficult argument
that seems at first glance implausible in the extreme. Does anyone have a
clear description of Wittgenstein's postion? I am not sure I understand
Wittgenstein's argument very well.
(Holmes writes:)
More profound thinkers than Wittgenstein have recognized this as a
problem. It is discussed by Descartes, for example.
I'm not familiar with the exact argument of Wittgenstein, but I am
familiar with contexts in which such scruples can arise. For example,
if one is an ultrafinitist, one may flatly deny that proofs of
sufficiently large finite size whose existence can be deduced formally
(though they are in practice completely unsurveyable) even exist, much
less prove anything.
One may also deny the validity of induction principles; the typical
ultrafinitist probably does accept that given natural numbers can be
multiplied, but he may doubt that exponentiation can be applied to
even rather small and familiar numbers (2^1000 might be doubted to
exist). Formal systems can be devised which have the defects that
ultrafinitists fear (or hope?) that arithmetic displays; though
models of such systems are always actually infinite, due to the
model-builder's misguided respect for consistency :-)
Humor aside, such systems are very interesting objects of study. I
think that such study is useful, becaue we probably should have some
notion of how to make sense of mathematics if the universe really is
finite; there is an elegant account of what we are doing if we grant
ourselves even a countable infinity (and so can use the usual
semantics of first-order logic), but one ought to take bad
possibilities into account.
The argument in question is, of course, a challenge to the validity of
mathematics; it is also a challenge to the validity of reason in
general (which is how Descartes saw it). Descartes answered it by
appealing to the essential trustworthiness of God. I would answer it
by saying that if it holds water we cannot demonstrate anything in any
field (and appeals to social consensus do no good against this
argument; a society is just as susceptible to the spell of Descartes's
demon as an individual -- in fact, more so, because societies are less
intelligent than individuals ;-) ).
Sincerely, M. Randall Holmes
holmes at math.idbsu.edu or mrh29 at dpmms.cam.ac.uk
http://diamond.idbsu.edu/~holmes
Boise State University and the University of Cambridge
must be held harmless for any silly thing I may say.
"And God posted an angel with a flaming sword at the gates
of Cantor's paradise, that the slow-witted
and the deliberately obtuse might not glimpse
the wonders therein." (Holmes, with apologies to Hilbert)
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