FOM: Hersh on reproducibility in mathematics & religion

[Martin Davis]

At 09:26 AM 1/15/98 -0700, Reuben Hersh wrote:

>As I take it, you say that (1) rabbis and theologians have as much
>consensus as mathematicians, and (2) it makes no senbse to talk
>about reproducible results in math.
>
>By reproducible results in math, I mean that different people
>tackling the same problem, possibly by very different methods,
>get the same result.  For instance, they say there are 100 proofs
>of the Pythagorean theorem.  Epstein's book on pde has six different
>methods to solve the Dirichlet problem.  Quadratic reciprocity I
>understand has many proofs, including 6 by Gauss.  This doesn't
>happen in any other humanistic study.  Therefore, I say, it
>characterizes math among the humanistic studies.  Humanistic
>because it studies human non-material creations, as do
>literature, philosophy, history etc.  Another way to say the
>same thing is that math  is both scientific and humanistic,
>or both a science and a humanity.

OK. I'm truly glad to have that explained. I really had no idea in what
sense you found "reproducibility" in mathematics. What you refer to is
important and significant. It is called "consistency". We won't find one
proof demonstrating quadratic reciprocity, and another showing that it's
false. In fact it was precisely because consistency was felt to be
threatened by the contradictions in set-theoretic reasoning that such great
mathematicians as Hilbert and Weyl were pushed to foundational
investigations. I find the relationship to reproducibility in laboratory
science rather far-fetched.

>"Correct" is fine with me.  I have no objection to your saying
>mathematical reasoning is correct.  But how do you know it's
>correct?  Did it come with a seal from the hand of Jove or Jehovah,
>or dragged down by Moses from Sinai?  I don't think so.  Can you
>use mathematical reasoning to prove that mathematical reasoning
>is "correct?"  That doesn't sound right.  

In fact Hilbert's program attempted to do something very much like that.
Although it did not succeed, it led directly to G\"odel's work, and it left
behind a vigorous and important branch of logic: proof theory. The answer to
your question is complex. It is certainly in part what you say. But it is
also in part because the rules of correct logical reasoning have been
codified (by Frege), and much of mathematics can be obtained using those
rules on simple and unobjectionable axioms. Attempting to get even better
answers is very much what FOM as a field is about.

>Now what about rabbis and theologians?  You have the nerve to
>tell me that rabbis have a consensus not to eat bacon!

This takes nerve? And why do you misquote me again? Here's what I said:

"But ask an orthodox Jew whether it is
permitted to eat bacon or to use electricity on Saturday, and you
will get full unanimity. The answer to your query will be the same
anywhere in the world. Full reproducibility."
 
You reply:

>Of course anyone who is a rabbi accepts what is required to
>be a rabbi.  

I didn't mention rabbis. I referred to orthodox Jews. Is what I said wrong?

Reuben, with all respect I think that we've come as far as we can, and had
best agree to diasagree. I certainly have pushed the religion example as far
as I want to. It was only meant to demonstrate that there is a lot more to
mathematical belief than mere consensus. By mentioning consistency (whatever
you choose to call it), and agreeing that mathematical reasoning is correct,
our difference is considerably narrowed. I do wish this aspect had been more
prominent in your book.

One last comment: I do not believe at all that mathematics requires or can
generally be expected to attain "absolute certainty". I think sometimes we
do have it and it is very nice. But, especially as one meddles with the
higher transfinite, one must expect to be satisfied with far less. G\"odel
had some interesting comments along these lines.

Be well, Martin