[FOM] A formalism for Ultrafinitism
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Mon May 24 22:36:26 EDT 2004
Jean-Paul van Bendegem details some recent work relevant to the
problem of formalising an appropriate form of finitism. The most
recent issue of MIND (vol 113.450, April 2004) contains two articles
showing how relevant the issue is.
Both articles deliberately aim to oppose finitism with respect to
Decision Theory, although the first ('Vexing Expectations' by Harris
Nover and Alan Hajek) first produces a new paradox ('The Pasadena
Paradox', a development of the St Petersburg Paradox) which shows
there are severe difficulties with infinitary versions of this. The
authors go on to consider two finitary ways to avoid their paradox,
'restrict decision theory to finite space state spaces', and
'restrict decision theory to bounded utility functions', and find
neither plausible. But they end without a solution, saying merely
'If [the paradox] is going to be cured, some other kind of medicine
will be required' (p248).
Their arguments against finitism in this area, moreover, are nowhere
near as tight as those in the second article ('Bayesianism, Infinite
Decisions, and Binding' by Frank Arntzenius, Adam Elga and John
Hawthorne), which tries to show there is a way out from several other
paradoxes in infinitary Decision Theory. However, there are some
clear fallacies in some of this paper's reasoning, it seems to me,
which again leaves the matter wide open.
A one place, for instance, the second authors have Donald Trump
obtaining ten roubles on an infinite number of occasions, on
condition that he burns each time the lowest numbered of his then
total pile of roubles, the roubles being labelled with distinct
natural numbers, and successive piles being obtained ever more
quickly so that all the burning has been done in the open interval up
to 1 pm on a certain day. They claim (p253) that Trump has no
roubles at that time, so that although each gift looked a good deal,
the totality of them is not. But from the case of Thomson's Lamp we
know that nothing is determinable about the limit point - and one
cannot presume normal causal continuity, if one is thinking of an
infinite number of burnings.
Some of the burnt roubles might be resurrected at 1 pm, for instance,
which is a quite appropriate idea in the context of the authors'
extensive discussions of God, Satan and Eternal Life. Here, though,
they forget about the latter supposed possibility in connection with
Satan trying to put Eve into a bind, when faced with an infinite
number of pieces of apple. Satan says (p262) that if Eve takes only
a finite number of pieces she will not be removed from the Garden of
Eden, but if she takes an infinite number she will be - and so
seemingly puzzles her about how many she should take. But clearly,
under the same set of inifinitary assumtions, she should take a piece
every half an hour, for instance, since if she lives forever there
will be no time left to be put out of the Garden. Again, one cannot
presume that things are normal, so that Eve has only a finite life,
in a context where she is said to be making an infinite number of
decisions, and that raises very pointedly the major problem with a
Decision Theory which allows such things.
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater
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