FOM: Fallis on probability proofs

[Don Fallis]

At 12:23 PM 9/23/98 -0400, Neil Tennant wrote:
>Don Fallis wrote
>
>>scientists (unlike mathematicians) have long since lowered their
>>standards and commonly accept inductive evidence for scientific
>>claims.
>
>What were the scientists' standards before they were thus "lowered"?

My statement was intended to be a bit tongue-in-cheek.  There is a common
presumption (among many philosophers and mathematicians) that deductive
evidence is epistemically superior to inductive evidence.  I do not happen
to agree with this presumption.  In particular, as I argue in my JPhil
paper, I do not think that mathematicians would be lowering their epistemic
standards by appealing to a probabilistic proof to establish the truth of a
mathematical claim.

I was just trying to point out that looking at how scientists treat
probabilistic proofs is unlikely to be very helpful in our investigation of
the epistemic status of probabilistic proofs.  Scientists commonly appeal
to inductive evidence for the truth of scientific claims.  As a result,
their acceptance of probabilistic proofs is not very surprising.

Mathematicians, however, try not to appeal to inductive evidence for the
truth of mathematical claims (at least they do if they want to get
published in the "Annals of Mathematics").  So here we have an opportunity
to compare the epistemic virtues of deductive evidence and inductive
evidence.  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?

In any case, to answer your question, Aristotle for one thought that the
truth of claims about the physical world could be established purely by
ratiocination (and indeed that this was the best way to establish such
claims).  It was not until Francis Bacon that anyone insisted on empirical
confirmation.

>How can the label give information as to the *structure* of the proof?
>After all, if I were to say of a conventional proof that it was a
>"truth-value proof", this would carry no information at all as to its
>*structure*. All I would know is that if the premisses of the proof
>have truth-value T, then so does its conclusion. That leaves open
>every possibility as to its deductive structure (i.e. the patterning
>of steps of inference within the proof).

Of course, from the fact that some labels (such as "truth value proof") do
not convey information about the structure of a proof, I don't think that
you can infer that no labels convey such information.

>An alternative picture of probabilistic proof might be that its
>conclusion doesn't explicitly register the probability of S. Instead,
>its conclusion is S itself, but the steps within the proof are made in
>such a way that, though they do not guarantee truth-transmission, they
>nevertheless guarantee that the probability-value of 1 for each of the
>premisses does not degrade below 1-2^(-n) (for some suitably large n).

A deductive proof of a proposition about probabilities is, of course, still
a deductive proof (and not a probabilistic proof) - and might very well get
published in the "Annals of Mathematics."  

Your alternative picture, however, seems to capture the meaning of the term
"probabilistic proof" (at least as it has been used on FOM).  In any case,
let us adopt this for the moment as the definition.  When a mathematician
labels a proof as "probabilistic," she conveys to the reader that the proof
is one that meets this definition (and that does not meet the definition of
a deductive proof).  Now, if we were in the fortunate position of having a
probabilistic logic that could formalize all probabilistic proofs, then the
label "probabilistic" might be even more informative that it now is.
However, this does not mean that the label does not currently convey some
information about the proof. 

In fact, if the label were not somewhat informative (and we were not in
fairly close agreement about what information it conveys), then we could
not even be having a meaningful discussion about the epistemic status of
such proofs.  (I'm going to go out on a limb and presume that we are having
such a discussion.) 

take care,
don

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