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% review of "18 Unconventional Essays on the Nature of Mathematics"
% reviewed by Edward Nelson, nelson@math.princeton.edu

{\it 18 Unconventional Essays on the Nature of Mathematics}. Edited
by Reuben Hersh. Springer, New York, 2006, xxi + 326 pp., ISBN
0-387-25717-9, \$49.95.

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{\it Reviewed by} Edward Nelson
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Although several of the essays in this collection are by
sociologists, none addresses the central question of the sociology
of mathematics: why is it that mathematicians are such nice people?
We are no respecters of persons (in that curious phrase that means
we do respect persons but pay little attention to the trappings of
age, position, or prestige), we take equal delight in fierce
competition and collaborative effort, and we are quick to say ``I
was wrong.'' Perhaps some of us know an exception that proves the
rule, but by and large I speak sooth, especially when one compares
mathematicians to our colleagues in the humanities.


But alas!~one major qualification is still needed. The bias against
women in our field is yet with us, equally pernicious when it is
unwitting. To take just one example, the wide-ranging and all-too-brief
commentary by Mary Beth Ruskai on the decline of science [1]
raises a number of issues quite relevant to the
collection under review. Its inclusion would have enriched this
all-male book.


How does one explain that we are so lovable? Is there something in
the nature of mathematics that attracts gentle souls? Possibly, but
another explanation is more convincing. We are singularly blessed in
that the worth of a mathematical work is judged largely by whether
the {\it proof\/} is correct, and this is something on which we all
agree (eventually), despite the fact that we may have divergent
views on the nature of mathematics, as the volume under review amply
shows. This is a singular fact. In art, projection of personality
may prevail; in the humanities, the power of position may prevail;
in science, the prevailing fad may prevent the publication
even of excellent work---but we are
extraordinarily fortunate that in our field none of this matters.


So what is a proof? Like obscenity, we all know it when we see it,
but it is hard to define---unless one is a formalist. In my opinion,
the gap between rigorous argument and formal proof in the sense of
mathematical logic is one that will close. In the lifetime of most
of my readers, it will be common for a referee to submit the paper
to a computer program to verify the correctness of the proof, thus
freeing the referee to evaluate the work in terms of originality,
depth, and importance.


The nature of proof is one of the themes of the essays in this book.
But the typesetting of the book is a disgrace. Here are a few
verbatim examples, though they will give my spelling checker fits.
(This is the spelling checker that suggested replacing {\it
finitist\/} by {\it dentist}, and {\it Fock space\/} by---but the
{\it Monthly\/} is a family magazine.)
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\qquad the derivative of $f\ {\rm o}\ g$ is $f\!'\ {\rm o}\ g\ ^
{\scriptscriptstyle *}\ g'$ \quad [p.~39]


\qquad as in Figure 1 \quad [p.~57; there is no figure]


\qquad miracle-as R\'enyi had Hippocrates say-that \quad [p.~78]


\qquad Hausdorff1 space \quad [p.~105]


\qquad `the necessary residue of' the extinction of the ego' \quad
[p.~107]


\qquad in mathema tics was held to he the \quad [p.~112]


\qquad in terms of the amalgamations al' thinking/scribbling \quad
[p.~121]


\qquad ofprevious \quad [p.~227]


\medskip


I could go on, but these are more than enough to show the careless
and uncaring treatment of this book by the publisher. Is this what
is meant by ``value added'' by which the publisher justifies a
charge of \$49.95 for a paperback? It is true that all of these
passages, except the one on p.~121, are understandable, but that is
not the point. When my brother John was taking freshman French, the
professor corrected a student's mistake and the student said, ``But
a Frenchman would have understood me.'' The professor replied,
``Yes, and dogs understand each other by sniffing one another's
behinds.'' The responsibility for such messes is ours. We have the
right and the obligation to require of publishers that they not
mangle manuscripts submitted to them, that they look at the book at
least to catch glaring errors before offering it for sale, that they
have some skill at mathematical typesetting, and even that the
manuscript be read by someone with sufficient literacy to catch
errors such as {\it affect} for {\it effect} [p.~159].


This book is not, and is not intended to be, a source book such as
the very valuable selection [2] by Benacerraf and Putnam.
Rather the essays are
chosen to be diverse and provocative. The chief pleasure in reading
such a book is mentally arguing with the essayists. Gentle Reader,
double your pleasure, double your fun: argue with the present
reviewer as well.


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{\it R\'enyi.} The first essay is a delightful Socratic dialogue,
perhaps no more unhistorical than those of Plato, on the nature of
mathematics. It ends with an impassioned speech by Socrates on the
virtue of using the mathematical method in philosophy---in strong
contrast to Rota's views (see below).


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{\it Celluci.} In this introduction to his book on philosophy and
mathematics, the author presents 13 points of the ``dominant view''
of the philosophy of mathematics, points that he says he will
counter. Most of these points of the dominant view are quite
sensible, and one wishes they were expressed in greater detail,
especially when he quotes Dummett.


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{\it Thurston}. This is a humane, balanced, and deeply personal
account of the author's experience and views of mathematics. I found
it well worth pondering.


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{\it Aberdein}. This essay contains an interesting discussion of the
computer-assisted proof of the four-color theorem by Appel and
Haken, if one ignores the jargon about patterns of argument.


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{\it Rav}. The author says, ``It is an intellectual scandal that
some philosophers of mathematics can still discuss whether whole
numbers exist or not,'' thus dismissing one of the central problems
of the field. As do several of the other essayists, he emphasizes
the importance of a solid orientation towards the practice of
mathematics. In short, he offers us a descriptive rather than a
normative philosophy of mathematics, based in his account on
``evolutionary epistemology.'' But is a descriptive account what is
needed? Are we so sure of the essential correctness of current
mathematical practice that no critical study of it is required? A
descriptive philosophy of law written in the nineteen-thirties might
well have included a description of lynch law without comment.


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{\it Brian Rotman.} Once I led a junior seminar on the foundations of
mathematics. This was a mistake: the students did not have enough
experience of mathematics to appreciate the foundational problems.
But we had fun; the twelve students read in the literature and made
presentations on Platonic realism, constructivism, and formalism. In
the last meeting I asked the students to guess what my position on
foundations was. Ten thought I was a Platonist and two thought I was
a constructivist; not one guessed the awful truth. 
At least I knew I
was innocent of any charge of proselytizing.


Rotman's essay achieves the astonishing feat of making me wish to
leap to the defense of Platonism against his attacks. This is
because it contains passages such as ``Frege's anti-psychologism and
his obsession with eternal truth correspond to his complete
acceptance of the two poles of the subjective/objective opposition -
an opposition which is the {\it sine qua non\/} of
nineteenth-century realism.'' Or again, ``Whether one sees realism
as a mathematical adjunct of capitalism or a theistic wish for
eternity, the semiotic point is the same\dots .''


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{\it MacKenzie.} The frightening possibility is raised that the
question of what is a proof may, and almost did, reach the law
courts.


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{\it Stanway.} The most interesting passage here is the description
[p.~152] of Hardy and Littlewood's highly civilized four axioms for
successful collaboration. When I was a graduate student, I heard a
lecture by Littlewood on the art of work. The audience was
mesmerized. Apart from his emphasis on the need for vacations, what
I chiefly remember is his advice to finish an evening's work in
mid-thought to provide an entry point in the morning.


Stanway discusses the effects of digital technology, but sensibly
concludes, ``There are good reasons to believe, however, that
despite changes in patterns of collaboration, doing mathematics in
the twenty-first century, will not be too unlike doing mathematics
in the twentieth century.''


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{\it N\'u\~nez.} The author's thesis is that ``by finding out that
real numbers `really move,' we can see that even the most abstract,
precise, and useful concepts human beings have ever created are
ultimately {\it embodied.}'' But he provides little evidence that
this thesis is true or even interesting.


I once had an experience that at first sight might be seen as
supporting the author's argument. One morning I was preparing for a
graduate class in dynamics and worked out the formula for the flow
generated by the Lie product of two vector fields. This was already
in the literature, in a paper by Helgason and possibly elsewhere,
but it was harder to search the literature in those days. A simple
calculation produced the answer:
$$\lim_{n\to\infty}\left(U\left(\sqrt{t\over n}\right)
V\left(\sqrt{t\over n}\right) U\left(\sqrt{t\over n}\right)^{-1}
V\left(\sqrt{t\over n}\right)^{-1}\right)^n$$ where $U$ and $V$ are
the flows produced by the two vector fields. As I stared at this
formula---do something, do something else, undo the first, undo the
second, repeat many times---something about it seemed familiar. And
then I felt the similarity in the muscles of my arms: drive, steer,
reverse drive, reverse steer, repeatedly. I went to Woolworth's and
bought a toy car to illustrate the formula for the class, for which
I received a sitting ovation. (But when I wrote this up in lecture
notes I got the illustration all wrong, as my brother Jim pointed
out: I drew a wagon rather than a car.) This was fun, but does it
say anything of the slightest significance about mathematics? Are we
to believe that Sophus Lie in his seminal investigations was
expressing his embodied experience of parking a car?


The article is illustrated by some funny pictures of people
gesturing while lecturing about mathematics, with their eyes blocked
out by black rectangles. I was reminded of my most memorable
encounter with gesture at a mathematics lecture. Will Feller was a
great showman in addition to being a deep and subtle mathematician.
Once at a colloquium talk he gave the audience {\it roared\/} with
laughter. I won't attempt to say what was so funny because it
wasn't; only Feller's showmanship made it so. On another occasion in
a seminar Feller was discussing an intricate combinatorial problem
about random walks. It was quite hard to follow. (But the purpose of
a mathematical lecture is not to convey technical knowledge but to
entertain and impart a feeling for what is important and exciting. A
class attempts to do both, which is one reason that teaching is so
difficult and rewarding.) At one point, Feller said, ``Now follow
the trajectory until the first time it goes {\it down},''
accompanied by a dramatic gesture {\it up\/} with the chalk.


N\'u\~nez objects to the notion of continuity as used in
mathematics, wishing to replace it by ``natural continuity,'' which
seems to mean something like ``continuous with a locally rectifiable
graph and crossing each horizontal line in a locally monotonic
fashion.'' One gets the impression that he dislikes
mathematics---certainly he dislikes mathematics as mathematicians
practice it.


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{\it Gowers}. The author expounds a non-extreme version of formalism
in an engaging way. By ``non-extreme'' I mean, for example, that he
wants to say that there is a fact of the matter as to whether the
decimal expansion of $\pi$ contains a string of a million sevens. But
his feeling is less strong about the twin primes conjecture, and he
points out that the first question is a ``there exists'' problem
while the second is a ``for all there exists'' problem. Since the
argument that the natural numbers form a model for arithmetic
depends on the cogency of there being a fact of the matter as to
whether an {\it arbitrary\/} closed formula of arithmetic is true,
it would be quite interesting to hear the author's views on his
reasons for believing arithmetic to be consistent (if he does).


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{\it Azzouni.} The author gives a witty and instructive discussion
of what he calls, in a happy phrase, ``the benign fixation of
mathematical practice.''


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{\it Rota.} The Italian word {\it geniale\/} means possessed of
genius. Certainly GianCarlo Rota was genial in both the Italian and
the English meanings of the word. I wish he were still with us to
counter my comments on his essay.


His main concern is with philosophy rather than mathematics. It is
an angry attack on what he calls ``mathematicizers of philosophy.''
And the attack is angry: {\it snobbish symbol dropping,
preposterous, bewitched, enslaved, absurd pretense, unable or
afraid, slavish and superficial imitation, damage to philosophy,
dictatorial regime, resorted to the ruse, derelict in their duties,
outrageous proposition, failed mathematicians, today's impoverished
philosophy, catastrophic misunderstanding\/} are some of the terms
he uses. (But yes, GianCarlo {\it was\/} genial.)


In the course of this jeremiad he makes some strange comments about
mathematics. The strangest is this: ``No mathematician will ever
dream of attacking a substantial mathematical problem without first
becoming acquainted with the {\it history\/} of the problem.'' I
have been doing mathematics for sixty years and I never encountered
this idea before. None of my teachers ever said anything similar,
and I certainly never gave any of my students such misguided advice.
The way to begin work on a substantial problem is with a fresh idea;
it is not very important in the beginning whether it be right or
flawed, since the important thing is to begin. Many young people
hesitate to begin research feeling that they do not {\it know\/}
enough, but much study is a weariness of the flesh. It is a mistake
to become indoctrinated with the methods of the past, which by
definition were insufficient for the substantial problem at hand.


Another is this: ``Suppose you are given two formal presentations of
the same mathematical theory. The definitions of the first
presentation are the theorems of the second, and vice versa. \dots\
Which of the two presentations makes the theory `true?'{}'' I have
tried to imagine a presentation in which the list of finite simple
groups is the definition and the definition of simple is the
theorem, but I can't seem to make this work. I just don't know what
he is saying here.


One final gnomic utterance without comment: ``Not only is every
mathematical problem solved, but eventually every mathematical
problem is proved trivial.''


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{\it Schwartz}. Under the pretext of discussing the harm that
mathematics does to science, the author launches an all-out attack
on mathematical physics as an intellectual discipline. He says,
``The mathematician turns the scientist's theoretical assumptions,
i.e., convenient points of analytical emphasis, into axioms, and
then takes these axioms literally. This brings with it the danger
that he may also persuade the scientist to take these axioms
literally.'' Does he really believe that we need to be careful not
to lead gullible physicists astray?


Anyone who has devoted years of effort to mathematical physics is
aware of the abrasive, but at its core mutually respectful,
relationship between theoretical physicists and mathematicians. Our
goals are different, but the goal of the mathematical physicist, to
find what consequences follow rigorously from what explicit
assumptions, is a goal worthy of respect even from those who choose
to follow a different path. The work enriches physics to some extent
and it greatly enriches mathematics.


Schwartz writes, ``The sorry history of the Dirac Delta function
should teach us the pitfalls of rigor.'' (``Pitfalls of rigor''?
Yes, that is what this mathematician says.) He goes on to say,
``This function remained for mathematicians a monstrosity \dots
until it was realized that [it] was not literally a function but a
generalized function.'' {\it Generalized function\/} is Courant
Institute-ese for distribution; implicit in Jack Schwartz's account
is a belittling of Laurent Schwartz's fundamental contribution to
analysis.


Later in the article he belittles the Birkhoff individual ergodic
theorem, writing ``The Birkhoff theorem in fact does us the service
of establishing its own inability to be more than a questionably
relevant superstructure\dots .'' It is correct to say that the
ergodic theorem is not essential to statistical mechanics, as
Khinchin argued before Schwartz. But this theorem is a deep result,
simple to state and pure in its generality. Historically it had its
origin in a central problem of physics and today it plays a central
r\^ole not only in many problems of mathematical physics but in
number theory as well. Schwartz calls it ``intellectual
prestidigitation'' and ``glittering deception.''



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{\it \'Avila del Palacio.} In this essay we find the statement ``The
critical work of Berkeley on Analysis provoked Weierstrass'
mathematical work.'' This is fascinating and I would like to know
more about it. Berkeley's objections to the calculus were absolutely
correct, and think of the time lapse from Berkeley to Weierstrass!
Bishop Berkeley was no unsophisticate in technical mathematics, by
the way. He objected to the procedure of first assuming that
$h\ne0$, drawing certain conclusions, and then setting $h=0$ while
retaining the conclusions. When Newton tried to counter this by
calculating the derivative of $x^2$ by using {\it symmetric\/}
difference quotients, Berkeley said in effect, all right, my friend,
now let's see you do that with $x^3$.


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{\it Pickering.} The account of Hamilton's invention of quaternions
is easy reading. The surrounding sociological argumentation is
harder going but interesting.


\medskip


{\it Glas.} I found this the most interesting essay in the
collection, and the one most likely to change my views on the nature
of mathematics. I'll not try to say how, since I'm still ruminating.
But I will make one technical comment. Citing Popper, Glas poses
three questions: ``Is any even number greater than 2 the sum of two
primes [the Goldbach conjecture]? Is this problem solvable or
unsolvable? And if unsolvable, can its unsolvability be proved?''
The answer to the third question is no. For if the Goldbach
conjecture is false, it is provably false: just exhibit the even
number and check all possibilities. Therefore if the problem is
unsolvable, the conjecture is true. Hence if we could {\it prove\/}
that the problem is unsolvable, we could prove that the conjecture
is true, thereby solving the problem.


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{\it White.} The author's location of mathematical reality in
culture is unsatisfying because he gives no explanation for the {\it
universality\/} of mathematics, which distinguishes it from all
other cultural phenomena, even music.


\medskip


{\it Hersh.} The final essay, by the editor, is a brief and
persuasive contribution to the problem raised by Wigner of the
unreasonable effectiveness of mathematics in the natural sciences. I
would like to add a comment that, taken with a grain of salt, could
supplement Hersh's. It is based on the observation that Wigner's
title is mistaken; it should read {\it physics\/} rather than {\it
natural sciences}.


Mathematics is the invention and investigation of formal patterns,
and good mathematics is the invention and investigation of deep and
beautiful formal patterns. Let us call {\it physics\/} that portion of
science that can be described, to a great extent, by a formal
pattern, and call the rest of science {\it biology}. Then by
definition mathematics is successful in physics.
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[1] Mary Beth Ruskai, The decline of science, {\it Notices Amer. Math.
Soc.} {\bf 45} (1998) 565.


[2] Paul Benacerraf and Hilary Putnam, eds., {\it Philosophy of
Mathematics: Selected Readings}, Cambridge University Press, 1964.


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{\it Princeton University, Princeton, NJ 08544-0001}

 {\it
nelson@math.princeton.edu}


\bye
