[FOM] Epistemology and the Philosophy of Mathematics (was Mathematical Experiments)

[Don Fallis]

Hi,

Bill Taylor ( http://www.cs.nyu.edu/pipermail/fom/2003-June/006941.html
) wrote:
> Proof doesn?t have all that much significance to epistemic matters.
>
> Proof is for *hygiene*, not discovery.
>
> [The possible presence of mistakes is] not of great concern to the
philosophy OF MATH.

I thought that your original essay (
http://www.cs.nyu.edu/pipermail/fom/2003-June/006864.html ) was a good
presentation of the standard view about mathematical evidence. 
However, the comments above seem to me to be somewhat non-standard.

(a) The importance of deductive proof in mathematics certainly goes
beyond just the epistemic role of establishing that mathematical claims
are true.  (After all, mathematicians look for shorter and more elegant
proofs of theorems that have already been proved.)  However, most
mathematicians (and most philosophers of mathematics) seem to be
committed to deductive proof being *epistemically* better than
probabilistic proof.

Note: My own (non-standard) contention is that deductive proof may not
really be epistemically better than (at least some examples of)
probabilistic proof.  Once we note that neither probabilistic proof nor
deductive proof allows us to know for sure that a mathematical claim is
true, it is not immediately clear exactly why deductive proof is
superior.

(b) The epistemology of mathematics is typically considered to be part
of the philosophy of mathematics.  In addition, the epistemology of
mathematics is not just about how we come to believe mathematical
claims, but about how we are *justified* in believing them.  And,
whether our beliefs could be mistaken (and how likely it is that they
are mistaken) is critical to this issue of justification.

Bill Taylor wrote:
> True, probabilistic proof and regular (published) proof(2) are both
> forms
> of *evidence*; but evidence of quite different types.  The former is
> merely evidence of a certain *claim*, perhaps quite strong evidence;
> whereas the latter is evidence of both the claim *and* the existence
> of another thing - the implied proof-1 object.    Significantly
> different!

I beg to differ.  At least in the case of primality testing, they are
*not* significantly different in this particular respect.  A
probabilistic primality test is also evidence of both the claim (that a
number is prime) *and* the existence of a proof-1 (that the number is
prime).  If a number is prime, then it follows (trivially) that there
is a proof-1 of this fact.  Thus, since a probabilistic primality test
provides strong evidence that a number is prime, it also provides
strong evidence that there is a proof-1 of this fact.

At least for finite combinatorial facts, evidence of the truth of a
mathematical claim is also evidence of the existence of a proof-1 of
the claim.  Thus, deductive proof cannot be epistemically better than
probabilistic proof simply because it provides evidence of the
existence of a proof-1.

See page 181 and pages 184-185 of
http://links.jstor.org/sici?sici=0022-362X%28199704%2994%3A4%3C165%3ATESOPP%3E2.0.CO%3B2-B
for a more detailed discussion of this issue.

take care,
don

Don Fallis
School of Information Resources
University of Arizona
http://www.sir.arizona.edu/faculty/fallis.html