by Leo Corry
1. Some Recent Debates
a tremendous influence on mathematical research and teaching, especially
in the "pure" branches, all over the world. This influence surely counts
as one of the main factors
behind the apparently robust status of the belief in the eternal character
of mathematical knowledge over this period of time. This belief is far
from having disappeared from the mathematical scene (and perhaps this is
not without justification), but at the same time, interesting debates have
arisen around it. In the present section, I mention some of the most recent,
dissenting views.
In 1980, the mathematician Morris Kline published a provocative book
entitled _Mathematics. The Loss of Certainty_ (Kline 1980). The main thesis
of the book is that the received view of mathematics, according to which
this discipline "is regarded as the acme of exact reasoning, a body of
truths in itself, and the truth about the design of nature", is plainly
false! Kline presented in his book a historical account of the rise of
mathematics to the unparalleled heights of "prestige, respect and glory"
that were accorded to it from ancient times and well into the nineteenth
century. This development, however, was followed by what Kline sees as a
total debacle in which all certainty about the truth of mathematics and,
especially, concerning the question of which approach to the foundations
of mathematics is correct and secure, was lost. In Kline's words:
It is now apparent that the concept of a universally accepted,
infallible body of reasoning - the majestic mathematics of
1800 and the pride of man - is a grand illusion. Uncertainty
and doubt concerning the future of mathematics have replaced
the certainties and complacency of the past. The disagreement
about the foundations of the "most certain" science are both
surprising and, to put it mildly, disconcerting. The present
state of mathematics is a mockery of the hitherto deep-rooted
and widely reputed truth and logical perfection of mathematics.
(Kline 1980, 6)
"The Age of Reason", Kline concluded in the introductory chapter of his
book, "is gone."
The details of Kline's arguments and the question whether or not they
lead to his sweeping, appalling, conclusions will not concern us here. The
interested reader can consult the book and judge this by herself. There is
no doubt, however, that Kline's claims, whether well taken or not, whether
supported by sound historical evidence or not, were rather uncommon at the
time of their publication, especially coming from a prominent mathematician
like himself.
Kline's book did not immediately give raise to any kind of open controversy.
If one has to judge according to published reactions, then the conclusion
is that the book was largely ignored by mathematicians. The few published
reviews of this book suggest that the mathematical community may have even
been hostile to the kind of arguments put forward by Kline. At the same
time, however, one can see in retrospect that Kline pointed to a direction
that was soon to be followed by others, who would undertake a reexamination
of the character of eternal truth commonly attributed to mathematical
knowledge.
Substantial evidence that such a reexamination was under way appeared in
1985 in a collection of articles edited by Thomas Tymoczko, under the name
_New Directions in the Philosophy of Mathematics_. The articles in this
collection, written by mathematicians, as well as by philosophers and
historians, put forward a philosophy of mathematics that Tymoczko calls
"quasi-empiricist", and that is opposed to the view that had dominated
discourse in this domain, at least since the 1920s. Tymoczko called this
formerly dominant view "foundationalism", and he characterized it as the
search for the true foundations of what is assumed, in the first place,
to be a system of certain, unchanging, knowledge. This view includes, of
course, three main schools of philosophy of mathematics in the present
century, namely, logicism, formalism and intuitionism.
The quasi-empiricist works that Tymoczko collected in his volume share a
common interest in the processes of production, communication and change
of mathematical knowledge, rather than focusing on the finished, and
allegedly definitive, versions of it. Also, they coincide in stressing
that the nature of mathematics can only be elucidated when this science
is considered to be an organic, lively body of knowledge, and that the
analysis of its foundations is only a very partial perspective of this
more general task. They stressed that mathematical knowledge arises as
part of a social process in which elements of uncertainty, such as plain
mistakes, empirical considerations, heuristic factors, and even tastes
and fads, may play some role. This view does not necessarily imply a
relativistic account of mathematics, but it does dispute, perhaps from
a perspective somewhat different from the one suggested by Kline, some
basic, accepted beliefs concerning the nature of mathematical knowledge
as a body of eternal, unshakable truth.
Tymoczko's collection included recent articles, as well as less recent
ones, such as those by Imre Lakatos, who began publishing his
idiosyncratic work on the philosophy of mathematics in the late sixties.
But one main, direct, motivation behind Tymoczko's publication was the
recent rise to prominence of new kinds of proofs that departed from the
classical Euclidean paradigm, a long-dominant one, on which the classical
view of the eternal nature of mathematical truth was based. Among those
new kinds of proofs, especial attention was accorded to "computer-assisted
proofs", (e.g. to the four color problem), but also to "very long proofs"
(e.g. to the simple, finite groups classification theorem), and to proofs
that established that a theorem was true with an "extremely high
probability", rather than with absolute, Euclidean or deductive, certainty
(e.g. Rabin 1976 on the distribution of prime numbers). This is not the
place to describe all these kinds of proofs and the philosophical questions
they raise. The point here is simply to make clear that some actual
mathematical developments raised pressing questions and posed new challenges
that somehow clashed with the received conception of mathematics as a body
of eternal truths. These questions were addressed mainly by philosophers
and historians of mathematics, and under the influence of their works some
observers went so far as to pronounce the classical conception of proof
officially dead (Horgan 1993). The reactions of most working mathematicians
to these developments were at this stage either indifferent or hostile to
the conclusions that some non-mathematicians were deriving from their
second-hand knowledge of them (Thurston 1994).
A noteworthy event in the debate on the eternal character of mathematical
truths took place quite recently, when some mathematicians - in fact, some
very prominent mathematicians - came forward with their own proposals to
change the accepted canons of mathematical publishing. By doing so, they
anticipated that a broader spectrum of what constitutes the actual process
of mathematical research and knowledge will become public and will be
shared by the mathematical community at large. This process will affect
the conception of mathematics as a body of eternal truths, and it will
contribute, so these mathematicians hope, to the enhanced development of
their discipline.
A by now well-known manifestation of this trend was an interchange
published over the pages of the _Bulletin of the American Mathematical
Society_ in 1993 and 1994. It started with an article by Arthur Jaffe
and Frank Quinn, both distinguished mathematicians. As a matter of fact,
Jaffe, a mathematical physicist from Harvard, is presently President of
the AMS. Their article bears the title "Theoretical Mathematics: Toward
a Cultural Synthesis of Mathematics and Theoretical Physics". According
to the authors, recent events in the development of mathematics and
physics dictate the need for a redefinition of the relations between
the two sciences. In particular, they claimed, there has recently been
an intense activity in physics that has yielded many new insights into
pure mathematical fields. Some of these results were eventually taken
over by mathematicians and reworked according to their professional
tastes, but originally they were produced by the physicists without
themselves abiding by the standards set by the mathematical community
for their own works. Jaffe and Quinn had in mind, among others, the
recent work of Edward Witten in string theory, and they thought that
mathematicians should encourage the production of works similar to this
one. In their view, without an active initiative to do so, the current
professional mores would hinder such contributions and thus cut a vital
source for inspiration and insight for mathematics.
Faced with such a situation, the article suggests the need to adopt in
mathematics a division of labor accepted in physics throughout the present
century, namely, that between theoretical and experimental physics. How
is this division translatable into mathematics? Jaffe and Quinn compared
the initial stages of mathematical discovery, involving speculation,
intuition and convention, to the work of the theoretical physicists. Like
experiment in physics, rigorous mathematical proof is introduced only
later in order to correct, refine and validate the results and insights
obtained in the earlier stage. Thus, while admitting that the terms
"theoretical" and "experimental" mathematics may be somewhat confusing
at first, Jaffe and Quinn suggested the following prescription for a
healthy, future development of mathematics:
The mathematical community has evolved strict standards
of proof and norms that discourage speculation. These
are protective mechanisms that guard against the more
destructive consequences of speculation; they embody the
collective mathematical experience that the disadvantages
outweigh the advantages. On the other hand, we have seen
that speculation, if properly undertaken, can be
profoundly beneficial. Perhaps a more conscious and
controlled approach that would also allow us to reap the
benefits but avoid the dangers is possible. The need to
find a constructive response to the new influences from
theoretical physics presents us with both an important
test case and an opportunity.
Mathematicians should be more receptive to theoretical
material but with safeguards and a strict honesty. The
safeguards we propose are not new; they are essentially
the traditional practices associated with conjectures.
However a better appreciation of their function and
significance is necessary, and they should be applied
more widely and more uniformly. Collectively, our proposals
could be regarded as measures to ensure 'truth in
advertising,' [e.g.:] "Theoretical work should be
explicitly acknowledged as theoretical and incomplete;
in particular, a major share of the credit for the final
result must be reserved for the rigorous work that
validates it." (Jaffe and Quinn 1993, 10)
It was clear to the editors of the _Bulletin_ that a proposal of this kind
would not pass in silence. Even before publication they asked several
leading mathematicians to write their opinions and reactions, to be
published in a forthcoming issue of the journal (Atiyah et al. 1994;
Jaffe & Quinn 1994). The published reactions ranged from a total rejection
(e.g., by Saunders Mac Lane), to a criticism of the general, authoritative,
tone adopted by the authors when suggesting new standards for publication
(e.g., by Michael Atiyah, Armand Borel, and Benoit Mandelbrot), to a
general agreement in principle but disagreement in the details of the
proposal (by William Thurston and Albert Schwartz), to a disagreement in
principle but agreement with some of the details (by Ren/e Thom).
But the debate remained open and one may expect, if only for the
prominence of the mathematicians involved, that the issues raised by it
will not be forgotten very soon. As a matter if fact, on February 12, 1996,
a colloquium was held at Boston University, on "Proof and Progress in
Mathematics", which was basically a follow up of this interchange. Jaffe
and Mac Lane were again among the discussing parties, together with other
mathematicians, such as the late Gian-Carlo Rota, from MIT, and the Harvard
mathematician Barry Mazur. New issues were raised in this meeting, which
in retrospect seem inevitable. Such is the case, e.g., of the role of
electronic communications among mathematicians, Internet, etc. The
pervasiveness of these new media raises the need to redefine some well-
established concepts pertaining the mathematical profession: publishing,
definitive versions, authorship of ideas and results, etc.
Parallel to the Jaffe-Quinn proposal for reconsidering the accepted norms
of publication in mathematics, the role of rigor in proof and - implicitly
at the very least - of the eternal character of mathematical truth, I want
to mention here an additional, similar, debate involving prominent
mathematicians. This one was sparkled by Doron Zeilberger, from Temple
University, in an article bearing the provocative name of "Theorems for a
Price: Tomorrow's Semi-Rigorous Mathematical Culture." Based, among others,
on the innovations implied by his own important mathematical contributions,
Zeilberger attempted in this article to attack a conception of mathematics,
which, although still dominant today, is in his view actually obsolete and
bound to be changed by a new mathematical culture. Today's conception was
characterized by Zeilberger as follows:
The most fundamental precept of the mathematical faith
is _thou shalt prove everything rigorously_. While the
practitioners of mathematics differ on their views of
what constitutes a rigorous proof, and there are
fundamentalists who insist on even a more rigorous rigor
than the one practiced by the mainstream, the belief in
this principle could be taken as the _defining property_
of _mathematician_. (Zeilberger 1994, 11. Italics in
the original)
This conception, promised Zeilberger, will soon be preserved only by a
small sect of fringe mathematicians, that, in spite of the deep changes
expected, will choose to keep abiding by the now orthodox conception. In
order to support his claim and make explicit whom he refers to with this
description, he cites the 1993 article by Jaffe and Quinn. The reader
thus understands that Zeilberger is going to present a truly radical
proposal for the future of mathematics.
Zeilberger makes his point by referring to the so-called algorithmic
proof theory of hypergeometric identities, a field to which he made
significant contributions. This theory considers identities involving
certain functions, and proves or refutes them by means of algorithms
that reduce any given identity of this kind to an auxiliary one,
involving only specific polynomials. Today the theory can be successfully
applied to a wide range of known identities, but, as Zeilberger explains,
it is natural to expect that in the future one might construct examples
of identities, whose reduction using the known algorithms in any computer
will involve prohibitive amounts of running time or of memory. Performing
the algorithms in this case would lead to absolute certainty concerning
the truth or falsity of the identities, but the price (in dollars) one
would have to pay for doing so would be enormous. On the other hand, it
is possible to apply a different kind of algorithms from which we will be
able to answer the same question, not with full certainty, but with a very
high probability and for free, or for a very low price in terms of computer
resources.
I already mentioned above "probabilistic" proofs, namely, arguments that
assign a very high probability to statements of the kind "Theorem X is
true." Michael Rabin had published one such argument in 1976 concerning
the statement that a certain number is prime. Rabin devised an algorithm,
each iteration of which raises the probability in question. Thus, the
idea of a probabilistic proof is not a new one. Still, Rabin did never
claim that this should become a mainstream way of supporting mathematical
truth. Moreover, it seems quite clear that Rabin would be very much
pleased to have a deductive arguments to prove, in the classical and (by
implication) conclusive way, what his probabilistic proof seemed only to
support with a very high likelihood. Zeilberger, on the contrary, is
explicitly arguing for the adoption of these kinds of proofs as the
standard, mainstream vindication of the truth of a mathematical statement.
Zeilberger invokes two arguments to support his position: First, he says,
it is likely that few new, non-trivial, results might be proved through
deductive arguments. Second: the price of the latter will become
increasingly high. It is pertinent to quote here Zeilberger himself:
As wider classes of identities, and perhaps even other
kinds of classes of theorems, become routinely provable,
we might witness many results for which we would know
how to find a proof (or refutation); but we would be
unable or unwilling to pay for finding such proofs, since
"almost certainty" can be bought so much cheaper. I can
envision an abstract of a paper, c.2100, that reads,
"We show in a certain precise sense that the Goldbach
conjecture is true with probability larger than 0.99999
and that its complete truth could be determined with a
budget of $ 10 billion." (...)
As absolute truth becomes more and more expensive, we
would sooner or later come to grips with the fact that
few non-trivial results could be known with old-fashioned
certainty. Most likely we will wind up abandoning the
task of keeping track of price altogether and complete
the metamorphosis to nonrigorous mathematics.
(Zeilberger 1994, 14)
A reply to Zeilberger's article was published very soon by George E.
Andrews. Andrews is himself a mathematician of no less merits than
Zeilberger (in fact the two have collaborated on many occasions). It
is instructive to read Andrews in order to realize that, in spite of
the strong arguments put forward, and in spite the verve expressed,
by Zeilberger, Jaffe, Quinn, and others, the idea of eternal truth
in mathematics will not disappear so soon, if only for reasons that
touch to the sociology of the profession, but certainly also for
reasons deeper than that. In a formulation that might warmly be adopted
by many colleagues Andrews disputed Zeilberger's position with the
following words:
Through the summer of 1993 I was desperately clinging to
the belief that mathematics was immune from the giddy
relativism that has pretty well destroyed a number of
disciplines in the university. Then came the October
_Scientific American_ and John Horgan's article, "The
death of proof" [Horgan 1993]. The theme of this article
is that computers have changed the world of mathematics
forever, in the process making proof an anachronism.
Oh well, all my friends said, Horgan is a non-mathematician
who got in way over his head. Apart from his irritating
comments and obvious slanting of the material, "The death
of proof" actually contains interesting descriptions of a
number of important mathematics projects. Indeed, as
W. Thurston has said [Thurston 1994] "A more appropriate
title would have been 'The Life of Proof'."
Then came [Zeilberger's article] (...) Unlike Horgan,
Zeilberger is a first-rate mathematician. Thus one expects
that his futurology is based on firm ground. So what is his
evidence for this _paradigm shift_? It was at this point
that my irritation turned to horror. (Andrews 1994, 16)
Andrews described in some technical detail why, in his view, Zeilberger
specific arguments do not support his claim. At the same time Andrews
stressed an important component of mathematical knowledge that in his
view Zeilberger's perspective failed to stress: the role of insight.
Perhaps it is also relevant to cite here a further reaction to
Zeilberger's and Andrews's articles, that appeared under the title of
"Making Sense of Experimental Mathematics" (Borwein et al. 1996). This
article attempts to put the whole debate raised by mathematicians such
as Jaffe, Quinn and Zeilberger, in a broader context, and to find a
common ground that might be accepted by a larger portion of the
mathematical community. The article put forward some arguments which
are interesting in themselves, but that is not the point I want to
stress here. What I find worth of special attention is the fact that
one of its authors, Jonathan Borwein, works at the "Center for
Experimental and Constructive Mathematics", Simon Fraser University.
This is only one of this kind of institutions active today in many
universities around the world. Thus, while the debate on new ways to
legitimize mathematical truth is still an open one, new institutions
are being built which already promote work based on the new principles.
One should not be surprised, then, to realize that Zeilberger's vision
of "theorems for a price" might become reality, much sooner than he has
envisaged, though not literally in the sense described in his article:
it is not unlikely that financial support for "Centers for Experimental
Mathematics" around the world might soon surpass the one allocated for
more traditionally-oriented departments, thus dictating, for a price,
what kinds of theorems are going to be proved and in which direction
mathematics is going to advance in the foreseeable future. Institutional
factors have been decisive throughout history in shaping the course of
development of mathematical ideas (through education, grants,
appointments, promotion, etc.), and given the present state of academic
research, such consideration will only become increasingly important.
At any rate, it is not the aim of this article to elucidate the future
course of mathematical research into one of the directions suggested by
the mathematicians mentioned in the foregoing pages. The aim of this
section is just to indicate an interesting turn that the idea of eternal
truth in mathematics has undergone over the last ten years. This makes
more perspicuous the relevance of the historical analysis that was
presented in the preceding sections. If this article had been written in
the early eighties it could have started with the following words:
"Mathematics is the scientific discipline in which the idea of eternal
truths is most deeply entrenched. In fact, unlike other sciences,
twentieth-century developments have only strengthened this historically
conditioned tendency." However, in view of the development mentioned,
I must erase the second sentence of the quotation, and instead start as
follows: "Mathematics is the scientific discipline in which the idea of
eternal truths has historically been most deeply entrenched, although
recent developments have modified this to a certain degree, in a
direction whose actual significance is still to be definitely evaluated."
2. Summary and Concluding Remarks
The foregoing sections discussed the views of some leading
twentieth-century mathematicians concerning the status of truth in their
discipline. Neither Bourbaki's nor Hilbert's views in this context is
monolithic, yet, in general they share the belief in the eternal character
of mathematical truth which has basically been unchallenged throughout
history, and still remains so. The interesting debates and nuances that
this issue raises pertain to the ways of achieving these truths.
In Bourbaki's conception, the conjunction of the structural, the axiomatic
and the reflexive images of mathematical knowledge together produce an
image of mathematics that, besides leading to the discovery of new eternal
mathematical truths in a unprecedentedly effective way, bears itself the
character of eternal truth: Bourbaki's image of mathematics is bound to
remain unchanged as well as the truths to which discovery it leads.
Hilbert's views, on which Bourbaki's are supposedly based, were much
more multifarious. The eternal truths of mathematics are, in his view,
attained in complex ways. The axiomatic method was seen as a very useful,
but in no way infallible, tool leading to such truths. It helps mainly
in the identification of precisely those places where falsity or
contradiction has entered into mathematical reasoning, but even a mindful
and able use of the method leaves much room for error, uncertainty,
innovation and need for change. In his published works on physics, for
example, Hilbert's axiomatic treatment of theories (e.g., radiation
theory or general relativity) suggests an air of definiteness, but in his
lectures he put forward a somewhat more tentative approach. And certainly,
Hilbert did not think that the axiomatic method, or any other mathematical
idea for that matter, can lead to a definitive scheme for organizing
science.
My discussion of Hilbert, Bourbaki, and eternal truths in mathematics,
also helps clarifying, I believe, the background to the recent debates
mentioned in \par. 4 above. These debates are clearly debates about the
images of mathematical knowledge. They attempt to establish the
disciplinary boundaries of legitimate mathematical knowledge. They do not
in general question the eternal character of the truths that currently
exist in, and that must be added to, the body of mathematical knowledge.
Rather they question whether or not, by accepting new forms of
legitimation, new 'truths' are going to be accepted which perhaps will
bear an essentially different, and therefore undesired, character. On the
one side of the debate are those who claim that departing from the
established model of proof, basically as embodied in Bourbaki's textbooks,
will be detrimental for the future of mathematics as we know it today,
precisely because it will cast serious doubt on the character of the kind
of truth involved in it. The other party involved in this debate does not
seem to wish to change the character of mathematical truth as such.
Rather, their claim is that introducing additional, legitimate models of
proofs will not threat mathematics as a science of certain knowledge, and
at the same time it will significantly enlarge its scope.
It therefore seems that Bourbaki's ambition of establishing once and for
all the images of mathematics according to which mathematical research
will have to proceed in the future is being questioned today in directions,
and with an intensity, that not even Bourbaki's critics could have
envisaged in the past.
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[end of part 3/3]
Leo Corry
Cohn Institute for History and Philosophy Science and Ideas
Tel-Aviv University, Ramat Aviv, 69978, ISRAEL
corry@post.tau.ac.il