FOM: Hersh on reproducibility in mathematics & religion

[Reuben Hersh]

 On Sun, 11 Jan 1998, Martin Davis wrote:

> 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
> 
Prof.  Simpson:  I apologize for using such a long quote.  It seemed
better than trying to summarize it all.

Dear Martin:  I apologize for forgetting about your service at
Yeshiva.

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.

"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.  It's reproducibility
of proofs and calculations that convinces us they are correct, 
and result in reasoned and voluntary consensus.

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

Of course anyone who is a rabbi accepts what is required to
be a rabbi.  Ditto for a catholic theologian.  That is not
a matter of voluntary consensus, it's a matter of following the
rules (which are accepted voluntarily when one chooses
to adopt the particular religion.)  The issue of consensus 
arises when there is a quesion or a difference of opinion within
the group.  Mathematicians with great consistency atttain consensus
about the correctness of mathematical results  (not about taste or
value or importance or usefulness.)

Religious groups don't consistently attain such consensus.  They may tolerate
differences, or they may expel heretics, or they may split.  
How many different Christian churches are listed in your local
phone book?  In Spain in the 15th century they had little
ceremonies called auto da fe.  Consensus?

I don't have a copy of What is Mathematics, Really? close to hand.  If 
you can put your hand on your copy, I suggest you turn to the section on 
David Hume and read his quoted words about mathematical certainty.

All the best,

Reuben