He acknowledges that our concept of proof has been influenced by the history of
mathematics, since "mathematics is a subject done by real people, with our own interests,
our personal perspectives, our own ways of working, our pressures of life, our strengths and
our shortcomings," but maintains that it is not completely determined by it (p. 134).
Jaffe then poses seven controversial questions, among which the following four have the most
direct bearing on proof.
Is mathematics a science? Jaffe concludes that mathematics
is indeed a science, but that there is a qualitative difference between the nature of
verification in mathematics and in physics, with mathematical proof having the "highest
degree of certainty possible to man" (p. 135).
What is mathematical proof?
Jaffe points out that little mathematics is written in the form of a traditional proof
(though mathematicians agree on what that form is), and asks whether the relaxation of
mathematical standards may not endanger the reliability of mathematical results.
Is mathematics proof necessary? Physicists routinely use mathematical
results that are still conjectures and often find they "yield the most accurate comparisons
between theory and experimental measurement," but in Jaffe's view this does not constitute
proof. "Scientific hypotheses come and go," but mathematics is different from
science because it lasts an eternity."
Can proof survive the interaction with
physics? The working style of physicists, unburdened as it is by the details of proof,
appeals to some mathematicians and may have a negative influence on the mathematics literature,
but Jaffe does not believe it is a serious threat to mathematics.
In the next paper, Saunders MacLane defends the necessity of mathematical proof against assaults from both physicists and educators.
He adheres to a strict notion of proof as "following the logical canons of proof."
Having said that, he also recognizes that "real proof is not simply a formalized document,
but a sequence of ideas and insights." He then takes issue with a comment by Atiyah
(1994) to the effect that mathematics, in the face of new developments, may need to rethink
its high standards of proof and make room for "a more buccaneering style." The term
"buccaneering" provided MacLane with the opportunity to append a nautical ballad
about the style of Captain Kidd, a pirate of dubious repute, adding that "we do not need
such styles in mathematics."
Jaako Hintikka's article, "A Revolution in the Foundations of Mathematics?," is a rather technical paper, sometimes confusing, that requires a fair knowledge of mathematical logic.
It begins with a discussion of first-order logic and its shortcomings. It then goes on to
present the traditional two-tier model of the foundations of mathematics, the first tier
consisting of a semantically complete first-order logic and the second consisting of set
theory and/or higher-order logic with no hope for completeness. Claiming that this generally
accepted picture of mathematics is really wrong, Hintikka elaborates upon a new perspective
on the foundations of mathematics in which set theoretical assumptions are no longer the core
of mathematical thinking. His new perspective suggests that mathematical thinking
be "thought of as combinatorial, dealing with the structures of particular objects,
especially with questions as to which configurations of individuals are possible and
impossible." (p. 167).
Gian-Carlo Rota contributed two papers to this collection.
In the first, "The Phenomenology
of Mathematical Beauty," he claims that artists like to stress the technical aspects of
their art, while mathematicians tend to stress the aesthetic aspects of their work. While there
are courses offered on art appreciation, however, no one has ever heard of courses on
"mathematical beauty appreciation." The rest of the essay is a discussion of the way in
which the word "beautiful" is applied to mathematics in general and to theorems and
proofs in particular. Mathematics educators will note that in Rota's opinion, teachers'
attempts to arouse interest in mathematics on the basis of beauty are bound to fail, mainly
because "appreciation of mathematical beauty requires thorough familiarity with mathematics,
and such familiarity is arrived at the cost of time, effort, exercise and Sitzfleisch." (p. 177).
His second paper is on the phenomenology of mathematical proof.
He first explores the
concept of proof by showing by means of examples that the accepted description of mathematical
proof is unrealistic; his idea of what "realistic" means is much influenced by
Husserl's rules for a realistic description. He goes on to discuss proof by verification,
arguing that verifications, though formally correct, are not satisfactory, because they are
restricted to showing that a theorem is true. Such proofs have little value, in his view, as
they cannot be turned into a proof technique suitable for other theorems.
Rota next asks
"Is all verification proof?," citing the four-colour conjecture. He points out that
mathematicians are still not satisfied, even though the conjecture has been verified with
computers, because they would like to have reasons why and because "they are on the lookout
for an argument that will make all computer programs obsolete." (p. 186).
From here he
moves to the question of "Theorems or proofs?," asserting that there are two schools
of thought, one that claims that mathematics is primarily composed of theorems and that it
is irrelevant how these are proved, whereas the second claims that mathematics is made out of
proofs with theorems merely serving as "arbitrary stoppers" in its development.
He then
discusses what he calls "pretending" in mathematics. Rota calls mathematical proof a
form of "pretending," because the actual presentation of a proof using the pedantic
grammar of an axiomatic method causes mathematicians to pretend that mathematics is
the axiomatic method. This pretense conceals the real point of a proof, which lies in the
understanding it provides and the possibilities it opens up for further work.
Rota then asks
"Are there definitive proofs?," and shows that mathematicians keep busy discovering
new and simpler proofs to replace old and lengthy ones, as their understanding of the reasons
behind the theorems increases. In the final section titled "Secret Life of Mathematics,"
Rota arrives at the conclusion that there is a need for a serious discussion of notions such
as "future possibility," "understanding" and "evidence" in connection with
proof.
The last paper in this collection is on the growth of conjecture as a formal tool in mathematics.
Barry Mazur notes that it is only recently that "the art of conjecturing has achieved a
formidable, and quite formal, prominence in the mathematical landscape." He defines the
term "architectural conjectures" as precise statements that are expected to prove true.
Such conjectures are very important to mathematics, because they point to research directions
in a concrete way. Mazur cautions that conjectures might well become an impediment to progress
if, as David Kazhdan remarked during the symposium, they come to resemble the notorious
"five-year plans" of the former Soviet Union.
The discussion then moves to the
relationship between conjecture and the axiomatic method, to the idea that conjecture is
motivated by analogy, to conjectures stemming from research programs, and to very specific
conjectures, such as the Ur-Poincaré and the Weil conjectures.
This issue is a wonderful addition to the literature on proof. It offers many mathematical
arguments for the mathematician, many philosophical arguments for the philosopher and many
arguments relevant to contemporary debate in mathematics education. Mathematics educators
will appreciate the thoughts of active mathematicians and the beautiful style in which most
of the articles are written.