FOM: More on probabilistic proof

[Don Fallis]

At 08:53 AM 9/24/98 -0400, Neil Tennant wrote:
>>Is mathematicians' distaste for
>>inductive evidence due to the epistemic inferiority of inductive
>>evidence or is it due simply to an aesthetic preference on the part of
>>mathematicians?
>
>It's the former. Both scientists' and mathematicians' distaste for
>inductive evidence for *mathematical* claims is due to its epistemic
>inferiority to proper deductive (not: 'probabilistic') PROOF. 

I do not think that this is a foregone conclusion.  In fact, I have argued
elsewhere that some probabilistic methods are epistemically on a par with
conventional mathematical proofs (see Journal of Philosophy, April 1997,
Call # B1 .J7.  If your library is still using the Dewey Decimal System, I
don't know the exact call number, but it should be in the 100s.)  I welcome
comments.

>Don makes out that scientists have a lower epistemic threshold for
>their empirical hypotheses (than mathematicians do for their
>mathematical claims), and thinks that this might help explain why
>scientists would accept probabilistic proofs in mathematics more
>easily than mathematicians would. 

Again, my use of the term "lowered" was tongue-in-cheek.  I think that
scientists have different, not lower, standards of evidence for scientific
claims than mathematicians have for mathematical claims.  Now, with regard
to what standards of evidence scientists have for mathematical claims, you
seem to think that they have the same standards as mathematicians.  Harvey
seems to think that they might have different standards (see point 2 of his
9/22 posting).  I am inclined to agree with Harvey, but I don't have a
large stake in this debate.

>"I am not aware of any such "probability logic" being satisfactorily
>developed in the literature, to the point where the authors of
>so-called probabilistic proofs would be able reliably to claim that
>their proofs could be formalized within that logic, in the way that
>authors of conventional proofs can and do reliably claim that their
>proofs can be formalized within ordinary predicate calculus."

Maybe you are bothered by the fact that I used the term "structure."  Would
it help if I just said that the label "probabilistic" tells us something
about what sort of proof is being offered?

As to your request for a probabilistic logic, I can certainly give you some
references to recent work in this area (see, e.g., Bacchus, F.,
"Representing and reasoning with probabilistic knowledge : a logical
approach to probabilities", Cambridge, Mass. : MIT Press, 1990, Call #
QA273 .B24 1990).  However, I do not know that anyone has proposed anything
that would serve as a general theory of probabilistic proofs.  

In any case, I do not see why this particular point is so pressing.
Mathematicians were proving things, and seemed to have a pretty good idea
of what they were doing, long before Frege showed us how to formalize all
these arguments.  Similarly, I think that it is safe to say that Rabin's
test is a probabilistic proof (and from all accounts a very reliable one)
even in the absence of a general theory of probabilistic proofs.
 
(I doubt that I will have anything more to say about the "structure" of
probabilistic proofs that will be very useful.  I am primarily interested
in investigating the "epistemic status" of probabilistic proofs and we seem
to have veered away from this topic.)

take care,
don

Don Fallis
School of Information Resources & Library Science
University of Arizona