FOM: Dialogue with Hersh re Silver's "Wagging Dogs"
shipman at savera.com
Thu Dec 17 14:27:00 EST 1998
Wed, 16 Dec 1998 17:26:24 -0700 (MST)
Reuben Hersh <rhersh at math.unm.edu>
shipman at savera.com, Charles Silver <csilver at sophia.smith.edu>
Aristotle is said to have defined "man" as "featherless biped."
Later philosophers pointed out that removing its feathers won't
turn a chicken into a man.
A modern definition might, as you suggest, be in terms of DNA.
Of course that would be backwards, for by knowing what mammal's
DNA to single out requires that we already can tell a man from
a German shepherd.
Going back to your tail-wagging dog, we ought to try a little harder
to answer, how do we recognize a dog when we meet one? Tail-wagging
isn't enough, a dog could be from a tailless breed, or had its
tail chopped off by accident or malice. We might find that we
simply can't explain in a complete and precise manner what is a dog.
Nevertheless, we do know a dog from a dinosaur. This is a question
of tacit knowledge, expounded by Michael Polanyi, and religiously
overlooked and ignored by analytic philosophy.
We know more than we can say.
Since you have been kind enough to resurrect my name from fom'ish
oblivion, let me say what I meant to say when I was trying to say
something about these matters, with reference to mathematics and
To me it seems clear that there are numbers, circles, triangles,
and all sorts of mathematical objects. It also seems clear that
there is only one universe--the physical universe to begin with,
and then the mental and social universes rooted in and growing out
of the physical universe. It seems clear to me that
mathematical objects are not physical, for we do not detect them
with our sense organs or with scientific instruments. It's even
clearer that they are not mental, in the sense of the individual
mind of a single person. But then, observing mathematics in real life,
could see that it was comprised under the heading of the social
At this point one meets the question, how is mathematics distinguished
from other part of the social universe? I concluded that the answer was
not in terms of mathematical subject matter--"math is the science of
number and figure", as Noah Webster would have had it. There is no
to the possible kinds of objects and ideas that mathematics can include.
Neither would I accept a definition in terms of deductive logic, for
deductive logic is important only in the last phase of mathematical
Look under the heading of "Riemann" in my book for evidence that
proof is not the whole story.
I concluded that the distinctive mark of mathematics is that it
virtual unanimity through all its manifold growths and transformations.
Martin Davis argued that this distinction didn't work, because
priests and rabbis also maintain unanimity. As to rabbis, it's simply
false, as you can check any week with the Weekly Forward. As to
the history of the catholic church belies any such unanimity. Look up
under heresies and heretics, of whom Maratin Luther was the most
successful but by no means the only one.
It seems to me, so far as I know, that mathematics is the
only non-physical activity that maintains unanimity. Some of
the physical sciences maintain virtual unanimity, that's
why the definition of mathematics requires both the non-physical
and the unanimous features.
Now, isn't that just like Aristotle's featherless biped?
Even if we never pluck a chicken, we know that featherlessness and
bipedalism are not of the essence in defining a man. This would be
proved, of course, if we come up with a plucked chicken. But even
without physically exhibiting a plucked chicken, we know the
definition is unsatisfactory.
I understand and agree with the feeling that my definition
of mathematics is unsatisfactory. Even if you grant my claim
that any-thing non-physical and maintaining unanimity would
necessarily be a branch of mathematics, that would not abate your
The truth is that there are two kinds of definitions here,
or two uses of definitions. One is classificatory--distinguishing
the definiens from other definienda. The other is essentialist.
Finding those qualities which make the definiens the definiens.
In biological classification, the passage from ancient
to modern is well known. The ancient classifications used the
most obvious traits of an animal, or those traits of use to man,
to make the definition. The comparative study of skeletons showed
which animals are related to which, and which are not; the position
in the scheme of relatedness became the definition.
Saying math does not deal with physical objects does
serve in part to locate it with respect to other disciplines.
How to demarcate it from law, lit crit, theology, economics,
It seems to me its unanimity does the job.
Still granting that just unanimity and non-physicality
do not do complete justice to the essence of mathematics.
Re: wagging dogs
Thu, 17 Dec 1998 10:25:38 -0500
Joe Shipman <shipman at savera.com>
Reuben Hersh <rhersh at math.unm.edu>,
csilver at sophia.smith.edu
That is a very good summary. Would you object if I posted it on the
I agree that your definition of mathematics is classificatory rather
essentialist. However, you are also trafficking in essentialism when
about what mathematicians are "really" doing. This is still all right
classificatory definition (non-physicality and unanimity/consensus) is
incompatible with your essentialist conclusions (mathematics is "really"
The most important disagreement many mathematicians have with you is
account does not allow for an absolute notion of mathematical truth.
important theorems are expressible as sentences in first-order number
and practically all the remaining important theorems are expressible as
sentences in second-order number theory, and almost all mathematicians
that a sentence in first-order number theory has an absolute truth value
independent of anything physical, mental, or social, and most
feel the same way about sentences in second-order number theory. This
stance you reject as "Platonistic".
Charlie argues that unanimity is not enough, that it is possible for
mathematicians to be unanimously WRONG and that your account does not
for this possibility. This is relevant to both the classificatory and
essentialist aspects of your work.
If your classificatory definition is correct, then it is only possible
about a mathematical statement being "true" if the kind of unanimous
recognition you require is never "overturned"; otherwise you do violence
the common meanings attached to the concept "truth". (Some people
that your account did not allow the four-squares theorem to be called
until Lagrange proved it and his proof was generally accepted, but a
unfolding or expansion of the collection of statements deemed "true" is
incompatible with ordinary language, while the possibility of a
"mathematically true" statement having its status changed to "not
as true" or even "false" is.)
If you do not want to say that there is no such thing as mathematical
then you can only deal with the possibility of unanimity being
either denying that this occurs in a serious way or by admitting that it
possible for mathematicians to be unanimously wrong. To eliminate the
of these alternatives, I called for examples, placing the restrictions
the problem have been considered important prior to the announcement of
"proof" (to ensure sufficient scrutiny), that the proof be readable by
person (to ensure a sort of reproducibility), that the "proof" was
accepted for at least 5 years before the consensus changed (again to
that sufficient efforts were made), and that the example occurred in the
Century (because standards of rigor have been high in this century).
examples, both from the theory of simple groups, have been suggested to
privately by f.o.m. subscribers and I am investigating them (one
serious gap in a proof, the other involves a result that actually turned
to be not only unproven but false).
If such examples establish that mathematicians can be unanimously wrong,
poses a serious problem for your essentialist conclusions, because
absolute notion of truth, how is one supposed to be able to say that
wrong? The first person to find that a generally accepted result is
will know that they are wrong even *before* he manages to persuade them
change the consensus, and your account does not seem to allow for this.
you would have difficulty upholding ANY notion of "truth", even a
non-platonic, socially-oriented one, for mathematical statements. (And
would also have to explain what happened when consensus is overturned,
"mistake" could be possible, which would require you to start talking
logic and validity and all that other stuff that Charlie and I and
claim cannot be left out of any "essentialist" account of mathematics.)
Re: wagging dogs
Thu, 17 Dec 1998 10:34:27 -0700 (MST)
Reuben Hersh <rhersh at math.unm.edu>
Joe Shipman <shipman at savera.com>
Reuben Hersh <rhersh at math.math.unm.edu>,
csilver at sophia.smith.edu
Thank you for a courteous, thoughtful message.
I don't mind if you transmit my previous message to the FOM list.
Your concretization and testing of my ideas is interesting. I wish you
well in this investigation. I think G.-C. Rota might be a good person
ask for counterexamples to my views.
One of my unstated assumptions is that the present century is not
the culmination of all centuries with respect to standards of
A favorite example of mine is the Pasch axiom of betweenness in
2-d Euclidean geometry. From the 19th and 20th century vantage
point, Euclid's omission of any axiom of betweenness made a significant
part of his Elements "wrong," in the sense that the proofs were
incomplete. But a more historical view would say that Euclid
was "right", even though his proofs had gaps and he used implicit,
unstated axioms. After all, we don't reject any of his theorems as
"false." He was right in his terms, wrong in ours.
I take it as not unlikely that future centuries will use different
standards of rigor than we. Perhaps stricter, for instance
rejecting our ambiguous use of "exist." Perhaps looser, for
instance accepting as definitive computer proofs that leave us
That would mean that the notion of "proof" would be historically
dependent. If so, we should think of our own notion of proof as
proof a la 20th-21st century. Citing what most mathematicians think
would also be historically conditioned. By what right do we
claim our opinions to be final and eternal?
This doesn't touch the absolute notion of truth. I did not attempt
and would not attempt to disprove that notion, any more than I
would attempt to disprove any other transcendental, absolute belief.
I would put the burden of proof on the other side. Why should we
believe in absolute mathematical truth? Has anyone proved there
is such a thing? The arguments for it are thought to be plausible,
comforting, "obvious" but certainly not rigorous. I think a
scientific attitude is to be skeptical about unseen realities,
especially if belief in them is comforting, perhaps wishful thinking.
If you want to broadcast this letter, go ahead. If not, not.
Thanks for your response. I will send this correspondence to FOM.
Whether or not the 20th Century is a "culmination", the level of rigor
is certainly higher then in previous centuries; it would have been easy
to find examples of unanimous wrongness before 1900 but I felt that
would be less than fair. You raise an interesting point about future
standards of rigor. Would anything we say now is a "gap" in a proof
still be considered so by a future mathematician? In the case where the
claimed result is actually false, obviously so. But if the proof is
merely incomplete the gap may not be considered essential by a future
mathematician. (For a historical example, Cantor's proof of the
well-ordering theorem was incomplete but a modern mathematician who is
well-versed in the Axiom of Choice might fill the "gap" immediately and
regard it as not serious.)
I am informed that one important case of the Feit-Thompson Odd Order
Theorem was overlooked for many years; I need to check this out because
while this is an extremely important theorem the gap may not be serious
enough to count (I think the mathematician who found the gap also filled
it in; if he had not been able to then it clearly would have been
serious enough to count, while if any good finite group theorist would
have been able to quickly fill it in had he found it one could argue
that it was simply an unimportant omission of detail). I am also
informed that a result of Suzuki's about finite simple groups turned out
to be not only unproved but false after standing unchallenged for more
than a decade; this would count as an example if the problem was
important enough, but I can't venture an opinion until I track down the
Euclid's proofs were "wrong" in the sense that he did not make all his
axioms explicit; it is not as easy as one might expect to distinguish
this case from, say, the false 19th-century proofs of FLT which wrongly
assumed unique factorization in number fields. By reinterpreting
Euclid's theorems as only being "about" the structures to which his
implicit axioms applied and saying Euclid never intended to say anything
about nonstandard models of his axioms one can defend him, but there was
definitely a sense in which he was wrong when he chose his axiom set
(some of the axioms he did make explicit are more obvious, and some less
obvious, than the Pasch axiom, and I suspect that if it had been pointed
out to him he would have recognized that it needed to be added).
The reason we have a right to regard some our own notions of proof as
having some eternal validity is that we have managed to formalize them
so that they are themselves mathematical concepts. We can define an
algorithm that outputs all and only the theorems provable in ZFC; this
formalizable notion of proof is NOT, as you have clearly pointed out,
the notion that mathematicians actually use professionally, but it
serves as such a good "classificatory definition" (because nothing
commonly regarded as proved is omitted by this algorithm and any output
of this algorithm would [assuming feasible size] be regarded as proved)
that we have the strong feeling that an "essentialist definition" must
involve something similar.
You are free to reject the notion of "absolute" truth; the question is
whether your account of mathematics allows for ANY kind of "truth",
because it seems that an essential feature of any kind of truth is
incorrigibility. If mathematics is "really" social and mathematicians
are unanimous that something is true for many years, how it is possible
for them to be mistaken unless truth is independent of what they do? It
is a defensible position to say that, for example, the twin prime
conjecture may have no "absolute" truth value, while still maintaining
that theorems of ZFC are "true"; but if mathematics is purely social
then I cannot see how we can ever say any theorem is "true" in a sense
that implies permanence.
-- Joe Shipman
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