FOM: Hersh on reproducibility in mathematics & religion

[Martin Davis]

I used religion in my polemic with Reuben Hersh, and since he has
replied, I need to continue the discussion, as I do below. However,
since I know this is a sensitive manner for many, I want to first
emphasize that, although I personally am an atheist, I mean no
disrespect to the religious views that other fom-ers may hold.

> You assert, as if it were common knowledge, that
> rabbis or Catholic theologians have reasoning
> methods by which they obtain reproducible results
> and near unanimous consensus.  This "fact," you
> say, shows that my characterization of mathl by
> its reproducibility of results with respect to
> argument about ideas is false.

The issue under discussion was indeed *reproducibility*. Now in
laboratory science, it is reasonably clear what reproducibility
means. Experimenter A publishes the results obtained in his lab.
Experimenter B working in a different lab "replicates" these results.
That is, B works to duplicate the conditions in A's lab, runs the
corresponding experiments, and (within experimental error) obtains the
same results.

Now, what is it in mathematical practice that remotely compares to
this? Well, as Reuben and I have agreed, there is computation. But
what else? Yes, there are these proofs. But Reuben seems to want to
go to pains to avoid saying that these proofs use correct logical
reasoning, and that is why mathematicians accept them. So in what
sense are proofs "reproducible"? Of course, copy can literally be
"reproduced" using a copying machine or reprinting a file. But surely
Reuben isn't talking about this trivial sense which, in any case, would
apply as well to any nonsense. Proofs can be presented orally in
lectures. They can be rewritten from one presentation to another. But
how do we verify that these are the same proof? Where is the
reproducibility? There is of course the professional *consensus* that
develops; but Reuben has told us specifically that reproducibility is
a requirement additional to consensus. So I must suppose that it is
the consensus about the correctness of a particular proof in addition
to the consensus about the theorem established that Reuben takes to
constitute reproducibility. If I'm wrong about this, then Reueben
should set me straight with a clear explanation of what this
reproducibility consists of. If I'm right, I think the examples I
gave of orthodox Judaism and Catholocism are quite apt.
 
> I must tell you that no one who has any acquailntancae
> with rabbis or theologiansd would make such a
> claim.  As far as rabbis, they  have a common
> method of argument, by referring to scripture and
> commentators, but this does not in the least give them
> reproducibility or consensus.  Among orthodox rabbis, as you can
> easily learn by asking an
> observaant orthodox Jew, there are vast differences
> and bitter quarrels.  If you don't know an
> observant orthodox Jew, just look at the first page
> of most any issue of the Jewish English Weekly Forward.

I promise that as a former member of the faculty of Yeshiva
University, I have known quite a number of orthodox Jews.  One of my
former doctoral students (at NYU) is an orthodox Jew. He is now on
the faculty of Bar-Ilan University in Israel. We had many pleasant
discussions of these matters in his student days, particular about
the position of the many Jews who regard themselves as practicing
their religion without following all of the rules laid down for the
orthodox. I well remember what he said: "There is really no argument
about what it means to be an observant Jew." Vast differences? No
Reuben. Bitter disputes around peripheral matters - sure. But we have
those too.  Just look at fom: Harvey Friedman vs. Lou van den Dries;
Sol Feferman vs. John Steel. 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.
 
> Catholics are a different story.  They have a Pope,
> and when he chooses he can say, "Shut up or get out."
> 
> The disagreement among the American clergy about
> birth control and the Latin American about
>  liberation theology show that until the Pope
> says shut up or get out there is continuing deep
> controversy among Catholic theologians
> 

Again every believing Catholic, anywhere in the world, will give the
same answer as to the correctness of any of the propositions asserted
in the standard catechism. Again, perfect reproducibility.

Reuben, you can't get away from it: we have *correct* proofs. Our
consensus arises from that correctness. While there is doubt about the
correctness of new principles and assumptions (axiom of choice,
Cantor's transfinite) the consensus disappears. The struggle about
incorporating such principles and being convinced that they are
correct is of course a human social process. And you are certainly
right to emphasize that aspect of the matter. But "reproducibility" is
very much beside the point.

Martin