A sociological question about proof

I've just refereed an expository paper in which the author mentions that
Lambert proved Pi is irrational, that Abel proved that the 5th-degree
equation is not solvable by radicals, that Galois proved that an equation is
solvable by radicals if and only if its Galois group is, and that
Weierstrass proved that the polynomials are dense in C[a,b]. Apart from the
anachronistic formulations of the latter two results (which at the moment I
was too weary even to protest), I suppose I could have pointed out at least
that most of these results turned out to require some emendation later on,
as Hamilton touched up what Abel did, for example. But it did lead me to
wonder: The reason for the "touch-up" was that there was no consensus as to
what constituted a proof at the time. So, is there NOW a permanent
consensus as to what constitutes a proof that was lacking at the time that
Abel and Galois wrote? Of course, anyone can see standards of rigor
evolving over the nineteenth century; but will future mathematicians find it
necessary to go back and touch up the proofs of, say, the ergodic theorem?
My guess is, they will. Something analogous to a need for uniform
convergence, some small subtlety that we all take for granted, will begin to
be questioned, and proofs will have to be rewritten with extra hypotheses to
take account of it. Of course, I have no idea what it will be.

What do others think about this?

Roger Cooke

On this see the entertaining piece "The Nature of Proof?", by Keith Devlin,
Notices of the AMS, November 1992, p. 1065. Among other remarks, it is
suggested there that the issue of what constitutes a valid proof may turn
out to be a matter for the courts. This sends the reader to the article
"Computers, Formal Proofs, and the Law Courts", by Donald MacKenzie, in the
same issue of the Notices.

Joao Filipe Queiro
Universidade de Coimbra
Portugal

On Fri, 31 Jan 2003, Roger Cooke wrote:

(preliminary discourse edited out for brevity)

> So, is there NOW a permanent consensus as to what constitutes a
> proof that was lacking at the time that Abel and Galois wrote?

With respect to Galois, I would suggest Harold M. Edwards' "Galois
Theory", Springer, 1984, ISBN 0-387-90980-X. and the very positive
review of that book by Peter M. Neumann, MAA Monthly (I think),
May 1986, pp 407-411. Edwards believes (as Poisson did not) that
Galois' proof was complete; Neumann is not so sure. Quoting from
Neumann's review:

"(2) The proof of _Lemme III_ is a splendidly controversial matter.
Poisson was unable to understand it and made a note on the manuscript
to say so, but he accepted that the lemma was true by a result of
Lagrange. Galois, incensed, appended "On jugera" to Poisson's note.
Edwards feels that Galois was right and he gives a line of argument
that undoubtedly completes the proof. But to do this he has to read
very much more than is there into what Galois actually wrote, and I
find his justification rather far-fetched. On balance I side with
Poisson: it was up to Galois to be both clear and correct, whereas
what he wrote is far too easily misunderstood."

I do not know if Edwards responded to Neumann's review (which ends
"this is not only a splendid textbook of that subject, but also
an excellent contribution to the study of Galois the mathematician.")

Edwards' many books are always worth the effort, even though I have
yet to complete a single one!

David Derbes [loki@midway.uchicago.edu]

At 07:55 PM 1/31/2003 -0500, Roger Cooke wrote:

<< So, is there NOW a permanent consensus as to what
constitutes a proof ... ? >>

I would say no. See, for example, the following two articles:
A. Jaffee and F. Quinn, ' Theoretical mathematics': Toward a cultural
synthesis of mathematics and theoretical physics, Bull. AMS 29 (1993)
1-13,
and
M. Atiyah et al, Responses to 'Theoretical mathematics': Toward ....,
Bull. AMS 30 (1994) 178-211.

Israel Kleiner
Toronto

I believe that your question is very relevant. Before commenting on your
question I would however like to make a diversion into other sciences.

When I started a more serious study (not research) on the history of
mathematics I, at the same time, also studied the history of science. One
of the first things that struck me in this study, is that what has always
characterized "scientific knowledge", is that it has always been
exaggerated.

We knew what we knew, but we had no idea of how much more "scientific
knowledge" there really was to be known.

Further, since my wife is a medical doctor, I am very much exposed to the
fact, that in medicine, they are usually laughing at what their immediate
predecessors were doing, but sometimes they find that what the
predecessors were laughing at might in fact contain some important, almost
forgotten grain of truth. However, never do t hey question that what they
are doing is at last, the correct way of e.g. treating certain diseases.

So, what about mathematics? First of all, the purpose of a proof is to
convince those that doubt.

(In teaching mathematics it is good to know, that very few students really
doubt the authority of a "good teacher", so what is needed in teaching is
"convincing arguments for students".)

Professional mathematicians are professional doubters, so new mathematical
theorems have to be accompanied by proofs that convince their contemporary
colleagues. To what extent they also convince their successors is somehow
not really relevant.

I have sometimes, in a slightly different form, posed the same question as
you (Roger) do. The way I have formulated is as follows. Suppose that
somebody comes up with an example, of e.g. a set of natural numbers, or
... , which in one stroke shows that say "all modern mathematics is
incorrect", i.e. that we have all been taking something for granted, that
simply was not true. (As far as I am concerned I would not be too
surprised if such an event would take place during my life-time.)

What would happen? My belief is that within, say, ten years "mathematics
would be restored". In a lot of theorems and definitions, additional
assumptions excluding the devastating examples will be inserted.
Nevertheless, it will probably transform mathematics in a way that is at
present completely unpredictable.

Besides this possibility of a "devastating counterexample", there is
another ongoing process, that might transform mathematics in such a way,
that future generations will have the same difficulties when reading our
papers, as we have when reading 19th century mathematics, and that is
computerization. I remember having heard that the late Everett Bishop
believed that mathematics would turn constructive. So far his prophecy has
not yet come true. Nevertheless, with new numerical algorithms and other
mathematical software, more and more exact solutions, or at least,
"arbitrarily good approximations" will be presented. Very likely, some of
these will turn out to be counter examples to proved theorems. This will
lead to a growing uneasiness in trusting old theorems, and might very well
make large parts of our beautiful abstract theories seem very obsolete,
simply because of all the existence theorems that are unable to produce
explicit examples.

Finally, I apologize for discussing my guesses of the future of
mathematics, rather than the past. The only excuse that I have, is that
it is based on historical studies combined with the conviction that our
own time is just a short period in the history of mathematics.

Sten Kaijser

I can provide as an e-attachment a version of that paper to those who request it.

E. Roy Weintraub Professor of Economics Duke University Durham, NC 27708-0097 Phone and Voicemail: 919 660 1838, FAX: 919 684 8974 URL http://www.econ.duke.edu/~erw/erw.homepage.html

> computerization. I remember having heard that the late Everett Bishop

Historical note: That was Errett Bishop (1928-1983). (I have
never known or heard of anyone else having that first name.)

Uppsala cannot possibly be darker or gloomier than Rochester, NY
this afternoon, with dirty old snow in the driveways and a few lazy new
flakes forever in the air.

Ralph A. Raimi

Dear Listmembers,

The question of changing standards of proof over time
is touched upon in many essays in Tymoczko's New Directions
in the Philosophy of Mathematics, revised edition, Princeton,
1998. Some are philosophical and some are historical,
the latter category including my own
"Is Mathematical Truth Time-Dependent" which focuses
on the changing standards of proof in analysis between
the 18th and 19th centuries. Abel's comments on the
lack of rigor in 18th-century arguments as compared
to the standards of Cauchy's Cours d'analyse are a case
in point.
This is a thought-provoking anthology.

Judith V. Grabiner
Flora Sanborn Pitzer Professor of Mathematics
Pitzer College (909) 607-3160

In a message dated Fri, 31 Jan 2003, Roger Cooke <ckrglj@adelphia.net>
wrote:

<< ... will future mathematicians find it necessary to go back
and touch up the proofs of, say, the ergodic theorem? My guess
is, they will. Something analogous to a need for uniform
convergence, some small subtlety that we all take for granted,
will begin to be questioned, and proofs will have to be
rewritten with extra hypotheses to take account of it. Of course,
I have no idea what it will be. >>

I have dusted off my crystal ball and found three candidates for such a
"small subtlety".

1. It is easy to demonstrate that if you select a real number at random from
the interval [0,1], the probability that you have selected a rational number
is zero. I do not know who first demonstrated this, but I believe it was
known to Kolmogorov.

This gives us two different definitions for "probability zero". One is that
an event has probability zero if and only if it is an impossible event. The
other, the one in the preceding paragraph, is that if the sample space has
measure greater than zero (i.e. a continuous or a hybrid distribution) then
the members of a subset of measure zero of the sample space have probability
zero.

So far statisticians are quite comfortable with having these two conflicting
definitions, but I wonder...

2. There is an old paradox about the Cretan who said all Cretans are liars.
Russell made this paradox (or would it be a similar one?) famous as the "set
of all sets not members of themselves" and also as "the [male] barber in the
village who shaves every man in the village who does not shave himself".
John W. Campbell Jr, the long-time editor of the magazine Analog Science
Fiction, gave a real-world example: according to logic an electric doorbell
cannot exist, since the magnet in the bell turns off when it is on and turns
on when it is off.

So far logicians and set theorists have accepted as a theorem (or perhaps
even as an axiom?) that the set of all sets not members of themselves cannot
exist. But suppose someone comes with a new formulation which allows such
sets to exist?

Campbell suggested that his doorbell paradox could be solved by a system of
time-varying logic. I have not been able to find the letter in question,
but I distinctly recall that list-member Alexander Zenkin once posted a
letter on HM describing a system of logic in which such time-varying
propositions could be analyzed. (Professor Zenkin was NOT aware of
Campbell's suggestion).

3. Zeno's "Dichotomy" Paradox can be interpreted as claiming that the limit
of an infinite sequence does not exist. ("Achilles before reaching the end
of the field must reach the half-way point, then the 3/4 point, then the 7/8
point" which in modern notation says that the binary fraction .111111...
does not represent 1.0) Yet mathematicians since at least Madhava (ca. 1400)
have used infinite sequences freely. Therefore there is, buried somewhere
in the foundations of calculus, an axiom that says Zeno is wrong. Can
anyone show that this axiom does not conflict with the other implicit axioms
of calculus?

- James A. Landau

PS My variation on the Probability Zero Paradox: let x be a random variable
from a normal distribution with mean zero and standard deviation 1. What is
the probability that x equals zero? The probability is zero. What is the
probability that x equals one? The probability is also zero. Yet we can
look at any table of the normal density and find that the ordinates for x = 0
and x = 1 are different. Hence P(x=0) = P(x=1) = 0 yet from the ordinates of
the normal density P(x=0) > P(x=1).

At 07:55 PM 1/31/2003 -0500, Roger Cooke wrote:

<< So, is there NOW a permanent consensus as to what constitutes a
proof that was lacking at the time that Abel and Galois wrote? >>

I responded to this question the other day, referring to two articles in
the Bulletin of the AMS which discuss this issue. I omitted to mention a
third, most interesting, article:
W. P. Thurston, On proof and progress in mathematics, Bull. AMS 30 (1994)
161-177.

Israel Kleiner

I have already pointed out in a parallel thread (on "deductive certainty") that the choice of logic underlying the present-day formalist notion of proof is arbitrary. Since formal mathematics has no external anchor in e.g. the empirical world, such arbitrariness can only be resolved through social authority.

Instead of looking at variations in the notion of mathematical proof across time, one can also look at variations in the notion of proof across cultures. While the present-day notion of mathematical proof is at one Platonic extreme in rejecting any role for the empirical in mathematical proof, the Lokayata notion of proof is at the other extreme in rejecting any role for deductive inference in proof.

There are other good sociological reasons why the present-day notion of mathematical proof ultimately rests purely on social authority, so that present-day mathematics is entirely a social construction. My account of these is in an article, "Mathematics and Culture", in _History, Culture, and Truth: Essays Presented to D. P. Chattopadhyaya_, ed. Daya Krishna, and K. Satchidananda Murty, Kalki Prakash, New Delhi, 1999, pp. 179--193. A version of the article is also in _Philosophy of Mathematics Education_, 11, 1999, available online at http://www.ex.ac.uk/~PErnest/pome11/art18.htm .

Further, I have argued (elsewhere) that this social construction has come under pressure from computer technology because of the increasing use of numerical simulation, though the results of such simulation are at present regarded as epistemologically insecure relative to a mathematically proved theorem. From my historical perspective, it seems obvious that it is the current notion of mathematical proof that will have to give way to accommodate this new technology of calculation.

If the present notion of mathematical proof changes, or if the notion of mathematics as proof changes, would the history of mathematics remain the same?

C. K. Raju