[FOM] Are proofs in mathematics based on sufficient evidence?

[Michael Barany]

Monroe,

I would submit that history works much more by penumbras than by
direct arguments (and that this is not a fault, but a reflection of
the material it has to work with and its approach to dealing with it),
but I understand the desire to have things spelled out with more
clarity.  Here's my seat-of-the pants attempt at one possible thread
for the argument.  Hopefully it's at least a little satisfying.

*) There's a famous and possibly apocryphal observation that to enter
Plato's academy one needed to be versed in Euclidean geometry.

*) This observation had great currency in Early Modern Europe (approx
1450-1650, depending on who you ask); it is not hard to find
mathematical texts from this period which refer to rhetoric and
philosophy and, conversely, to find rhetoric and philosophy texts that
invoke Euclid.

*) Geometry thus became one end of a continuum of ways of knowing and
proving, a continuum that included different forms of legal arguments.

*) "Modern science" and the "scientific method" are usually dated to
the mid-1600s, and it is well documented that these drew strongly on
juridical reasoning.  They also (in part because of their connection
to juridical reasoning) became part of the continuum whose end was
Geometry.  (Small elaboration: part of the change that happened in the
mid-1600s was a move from "natural philosophy"---philosophical
theories about the world with minimal connection to observation and
experiment---and "natural history"---roughly, descriptions of the
natural world---to what we now think of as science---roughly, the
connection of theories to the world through experiment.  It was
juridical thinking that helped this transition take place by giving a
model for how what different people see could be used to establish the
"truth" about something.)

*) Over the next 200 years or so, there were many shifts (happening at
different times in different places) in how people saw mathematics in
relation to the other sciences.  For instance, in France they were
initially seen as very different, but by the mid-eighteenth century
descriptions of good mathematical thinking and good scientific
thinking could often be interchangeable.  Then, shortly after 1800,
there was another dramatic shift in France where mathematics and
science were once again torn apart.

*) It is in the context of these shifts that the foundationalists of
the mid-nineteenth century began to develop the standards of rigor we
tend to associate with mathematics today.  They needed, in part, to
take a stand on the math/science question, to say what kind of truth
they were aiming at, and why their method could get that kind of
truth.  The set of things they needed to establish when setting these
standards owes a lot to the historical legacy described earlier, with
juridical reasoning a major player.

Still a rather indirect argument, and there's a lot of circumstantial
linking, but perhaps that makes things a bit more plausible.

Smiles,

Michael

p.s. I'm reminded of how my "Intro to Real Analysis" professor
described mathematical proofs.  Roughly: It's like you're in a trial,
and you have to prove the intermediate value theorem or else you get
put to death.

On Sun, Jul 18, 2010 at 10:20 AM, Monroe Eskew <meskew at math.uci.edu> wrote:
> I would still very much like to see direct argument and evidence for
> the claim that mathematical standards of rigor have been significantly
> shaped by modern-ear juridical standards.  So far I feel we've at most
> seen a "penumbral" justification.
>
> Best,
> Monroe
>