FOM: Further dialogue with Hersh on Certainty and Truth
Joe Shipman
shipman at savera.com
Wed Dec 23 08:49:28 EST 1998
Subject: Re: FOM: certainty
Date: Sun, 20 Dec 1998 15:15:18 -0700 (MST)
From: Reuben Hersh <rhersh at math.unm.edu>
To: Vladimir Sazonov <sazonov at logic.botik.ru>
CC: Joe Shipman <shipman at savera.com>, Charles Silver
<csilver at sophia.smith.edu>
I take it you declare disagreement with me on two counts: one, my
statement that mathematics is not mental; two, my underestimating
in your opinion the role of deductive proof.
You cite Lobatchewsky as proof that mathematics is located in
the individual mind of individual mathematicians. For no one
he knew understood his new geometry, he worked it out all alone.
The social world of mathematics includes, not only presently
active mathematicians, but also the accumulated archive of books and
articles, and even the word of mouth tradition inherited from
mathematicians of the past. Did Lobatchewsky, or could anybody,
invent non-Euclidean from scratch, all out of his head? Certainly
not. The very name non-Euclidean indicates that it is a continuation,
an outgrowth, of Euclidean geometry. Not, of course, in the sense of
continuing with Euclid's postulates and axioms. But rather, by
changing axiom #5, it then pursues the same questions and analogous
theorems as in Euclid. Lobatchevsky had a university math educuation,
that is to say, he was socialized into the mathematical culture of his
time. His work on geometry was a natural continuation of the geometry
he had been taught. If you doubt this, remember that Gauss and Bolyai
independently arrived at the same thing.
The mathematical culture throws up questions and problems,
and often several mathematicians seize on the same problem,
either knowing or not knowing of each
other. When I say math is not mental. I mean it is not part of
the private, un-transmittable inner world of individuals. It is
social, in that it is part of a shared, transmitted body of
knowledge and skills. Any piece of nonmaterial culture--law,
religion, political ideology, folklore, literature, music, etc.
is realized by individual consciousness of those participating in it.
But its existence and features are not dependent on any individual's
consciousness. If you and I die tomorrow, War and Peace and Crime and
Punishment will survive. And so will Euclidean and non-Euclidean
geometry. They survive because they are not limited to any individual
consciousness, they are art of a social consciousness.
What about the role of deductive proof, which you think I
underestimate?
I can explain this by rejecting your stataement that until
a conjecture has a deductive proof, it isn't mathematics.
I would agree if you said, it isn't part of proved mathematics.
Is the Riemann conjecture part of mathematics? It hasn't been
proved, so you evidently claim it isn't. What is it part of, then?
Music? Nuclear physics? Bourgeois ideology? Don't be silly.
I suppose you believe that Fermat's last theorem has been proved.
Before Wiles cleaned it up, it wasn't part of mathematics. Surely
all the partial results that had accumulated over the centuries, in
unsuccessful attacks on FLT, were mathematics, for they did prove
something or other. But the conjecture or problem that they were
struggling to prove, that wasn't mathematics?????
To me, mathematics includes all the mathematical work that
mathematicians do. Making conjectures, gathering evidence
for or against conjectures, coming up with defective proofs,
finally making a complete proof. You say none of that was
mathematics, until the final stage is reached.
You say all mathematics is striving for deductive proof. May
I point out that this notion is only 100 years old, or so.
Newton didn't do mathematics. Fourier didn't do mathematics.
What about Oliver Heaviside? When criticized for not having
rigorously justified his heuristic method of solving differential
equations, he answered, something like, "Does my ignorance of
the details of the process of digestion prevent me from enjoying my
dinner?"
Applied mathematics today overwhelmingly depends on computer
solutions of differential equations. Solutions thgat are
rigorously justified only in simple special cases,
which are not at all the cases that the applied mathematician is
interested in.
So you say applied mathematics isn't mathematics?
>From the applied viewpoint, a rigorous justification of
an algorithm is a plus, it adds some confidence to the
calculations with that algorithm. If the algorithm is
one that has been used dozens of times or hundreds of
times, all over the world, the extra confidence from the
rigorous proof is not decisive.
You could say, when you say mathematics, you mean pure
mathematics. I revert to the earlier discussion of
Fermat's last theorem.
After all, this disagreement is largely one of emphasis.
I'm in favor of rigorous proof, in fact I never published
a research paper that wasn't entirely rigorous.
But in judging the role of rigorous proofs, I take into account the
overall picture of what mathematicians do.
Reuben Hersh
From: Joe Shipman <shipman at savera.com>
To: Reuben Hersh <rhersh at math.unm.edu>
CC: Vladimir Sazonov <sazonov at logic.botik.ru>, Charles
Silver <csilver at sophia.smith.edu>
Reuben --
This is a good response to Sazonov. My earlier remarks about the role
of
truth are not thereby refuted, because my view of what mathematicians
are
really doing (finding out the truth about mathematical statements, by
various means, of which formal provability is merely the most reliable)
allows for the other types of mathematical activity you describe. My
view
requires an explanation of the connection between these activities and
abstract mathematical truth; one such explanation would be that the
Universe follows mathematical laws, and by investigating it we obtain
information about these laws, and the mathematics necessary to express
these laws is thus in some sense established (Putnam's thesis, which
Feferman uses to distinguish between mathematics necessary for science
and
mathematics which is purely a creation of the human mind). This was
usefully discussed on the FOM forum earlier this year.
-- Joe Shipman
Subject: truth
Date: Mon, 21 Dec 1998 10:54:12 -0700 (MST)
From: Reuben Hersh <rhersh at math.unm.edu>
To: Vladimir Sazonov <sazonov at logic.botik.ru>, Joe Shipman
<shipman at savera.com>,
Charles Silver <csilver at sophia.smith.edu>
Shipman and Silver say tht mathematicians want truth, and my view of the
nature of math has no place for truth.
I kind of see what you mean.
How does this work out for me personally? I certainly believe that
correct statements in mathematics are true. What is "correct"? Well,
correctly proved theorems are true. But many other statements are also
true. For instance, all those statements which eventually some time in
the future will
be proved. They are already true, even if we have only a suspicion or a
hope of their truth, or even no inkling of it. Any purely finitary
statement is true, or else its negation is true; (I say purely finitary
to avoid the intuitionist briar patch.) A simple counting argument
shows
that most of such true finitary statements will never be proved
(assuming
that the life span of the human race and also of its computing machines
is
finite.) So truth includes more than proved.
These statements are trivially obvious to most mathematicians,
whether Platonists, formalists, or agnostics. How do they stand up in
view of my social constructivist stance?
Sentences about social constructions can be true or false (not both.)
Social constructions aren't always imaginary fantasies, they can exist
in
hard reality. As I explained in my book, an eviction notice from your
apartment or a draft notice from your draft board or an overdrawn notice
from your bank are real as real can be, and you'd better treat them as
such. Yet they aren't physically real, even though they have a piece of
paper as their representation. Tearing up the draft notice piece of
paper
and the eviction notice piece of paper won't keep you out of the army
and
in your apartment. Much less are they mere thoughts in your mind, which
you can nullify by simply changing the subject of your thoughts. No,
they
are hard reality, very hard, and you had better recognize them as such.
The reality of those nasty ntices is social. They are part of the
social system, of which you are also a part, and their reality is more
compelling than the reality of a pebble in the street.
What does this have to do with mathematics? Since you must agree that
it's true that you've been served notice to evict within seven days,
then you are putting a truth valuation on something which exists, not
physically or mentally, only socially. Social reality can and does have
true or false valuation.
Another exmple of such is the Pythagorean theorem, a true fact about
certain social entities, namely, right triangles and the lengths
of their sides.
Now you protest. Your eviction notice is purely contingent. The nature
of the world does not require your eviction from your apartment. The
Pythagorean theorem, on the other hand, is a necessary truth, as soon as
we start talking about the lengths of the sides of right triangles the
theorem is there, even before we discover it. Necessity versus
contingency--what a polar oppposition! How can I identify such
different things?
Evidently there are different kinds of social constructions. Some are
contingent, some are necessary. Our state of mathematical knowledge
implies theorems and facts that we do not yet know, even that we will
never know. Eviction notices are contingent social constructions,
mathematics is largely a necessary social construction.
We must acknowledge an element of contingency in mathematics. If Cantor
had died in the cradle, would we have set theory? Maybe yes, maybe no.
Contingent. The mathematical questions we ask ourselves and each other
seem to be partly inevitable and partly accidental. Once the question
is
asked, then we expect the answer to be necessary (even if the answer is
that there is no answer.)
Reuben Hersh
From: Joe Shipman <shipman at savera.com>
To: Reuben Hersh <rhersh at math.unm.edu>
CC: Vladimir Sazonov <sazonov at logic.botik.ru>, Charles
Silver <csilver at sophia.smith.edu>
Reuben --
I still don't understand, from your response below, what you think is
going on
when mathematicians are unanimously wrong. Do you recognize the
possibility
that there could be universal consent that a theorem has been proved,
such
that this consensus is nonetheless mistaken and capable of being
reversed?
Does it make sense to say that "they thought the theorem was true when
in fact
it was false"? If you reject any notion of mathematical truth that is
independent of mathematicians, I don't know how you explain what is
going on.
-- Joe
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