[FOM] A formalism for Ultrafinitism

[Hartley Slater]

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