FOM: Hersh on consensus and reproducibility

[Martin Davis]

At 03:18 PM 1/3/98 -0700, Reuben Hersh wrote:

>	TURNING FROM PROF. TRAGESSER TO MY FRIEND M. DAVIS,
>	DOES MY CHARACTERIZATION OF MATH ADMIT CATHOLIC OR
>	ORTHODOX JEWISH THEOLOGY?
>
>	I DON'T THINK SO.  I AM AFRAID I FAILED TO STRESS THATOF
>	THE TWO CRITERIA FOR A SCIEN CE, REPRODUCIBILITY AND CONSENSUS,
>	IT IS REPRODUCIBILITY THAT COJES FIRST.  CONSENSDUS IS ONLY
>	A CONSEQUENCE (IN SCIENCE ) OF   RPRODUCIBILITY.  CONSENSUS
>	AT A NAZI PARTY RALLY IN 1034 IS NOT BASED ON REPRODUCIBIITY,
>	AND DOES NOT MAKE NAZISM A SCIENCE.
>
>	IF SOMEONE MAKES A CLIAM IN CHEMISTRY, SAY, IT MAY HAPPEN
>	THAT NIS RESULT IS IN PRINCIPLE IMPOSSIBLE TO REPEAT, OR THAT
>	OTHERS CAN REPEAT THE EXPERIMENT, BUT DO NOT OBTAIN HIS CLAIMED
>	RESULTS.  THEN THE CLAIM IS N O ACCEPTED.  (THIS OBSERVATION
>	WAS USERD BY KARL POPPER TO SOLVE HIS "DEMARACATION PROBLEM"
>	BETWEEN SCVIENCE AND NON-SCIENCE.  ON THE OTHER HAND, IF HIS
>	RESULT IS SUCCESSFULLY REPEATED BY SEVERAL ACKNOWLEDGED EXPERTS.
>	IT IS ACCEPTED INTO THE BODY OF CHEMISTRY.
>
>	THE PATTERN IN MATH IS VERY SIMILAR, EXCEPT THAT CALCULATIONS
>	AND ACCEPTED PATTERNS OF REASONING TAKE THE PLACE OF EXPERIMENT.
>
>	NOW, LET'S TAKE THE HOLY TRINITY, OR THE IMMACULATE CONCEPTION ,
>	OR THE ILNFALLIBILITY OF THE POPE, FOR INSTANCE.  WHAT 
>	EXPERIMENT, WHAT CALCULATION CAN MARTIN LUTHER (FOR INSTANCE)
>	MAKE TO EITHER CONFIRM OR DISCONFORM ANY ITEM OF CATHOLIC DOGMA?
>

What is it in mathematics that my old friend (and co-author of two
Scientific American articles) Reuben Hersh takes to be "reproducible"? He
says "calculations and accepted patterns of reasoning". Calculations are
surely reproducible; but of course they inevitably play an important but
strictly subsidiary role in establishing mathematical truth. There's the
wonderful example of a real-valued function studied by Littlewood (in fact
defined as the difference of two function each asymptotic to the number of
primes <x) found to always be positive by calculation: Littlewood showed
that in fact the function changes its sign infinitely often but that the
first value for which it is negative is some monstrous number likely greater
than the number of atoms in the universe. That leaves "accepted" patterns of
reasoning.

Now "accepted" and "by consensus" clearly mean the same thing. So Reuben's
version of  "reproducible" is not a condition additional to "consensus" but
just more of the same. Anyhow, how are these patterns reproducible? We can
write the proofs in our textbooks or articles and present them in lectures.
Trained mathematicians will nod their heads and agree. THIS FORM OF
REPRODUCIBILITY IS JUST AS AVAILABLE TO CATHOLIC THEOLOGIANS. They have
their "accepted patterns of reasoning". If Reuben can name Luther, I can
name Brouwer. (The comparison with religion is not new; cf. Berkeley.)

Of course, the crucial point about the "patterns of reasoning" used in
mathematics is that they are (or at the boundary of what is clearly
understood, meant to be) CORRECT. We know that Lagrange's theorem (that
every positive integer is the sum of four squares) is TRUE because we
possess a CORRECT PROOF (in fact many) of it. It does have computational and
empirical consequences. Thus, we can assert with confidence that even when
the dinosaurs roamed the planet, any assemblage of (say) pebbles was capable
of being rearranged into 4 separate assemblages, each arranged to form a
square. To deny this on the grounds that the theorem involves numbers, and
thus has no validity outside of a social consensus is as inappropriate as to
deny the existence of those dinosaurs since no people were there to see them.

Mathematical truth is easy to understand for universal statements (Pi-1 in
the lingo). For statements involving higher order abstractions there are
real conceptual difficulties. An expert like Sol Feferman can doubt whether
some of these have any sense at all let alone a truth value. It is the
problematic character of mathematical truth at a (shifting) boundary of what
is understood that provides the problems for workers in FOM. It is their
scientific conquests that provide the developing consensus that Reuben sees
as just out there, while heaping scorn (in his "What is Mathematics Really")
on those he refers to as foundationists.

Martin