BOOK REVIEW
Serving a Silicon Master
Doron Zeilberger
Mathematics by Experiment: Plausible Reasoning in the 21st
Century. Jonathan Borwein and David Bailey. x + 288 pp. A K
Peters, 2004. $45.
Experimentation in Mathematics: Computational Paths to
Discovery. Jonathan Borwein, David Bailey and Roland
Girgensohn. x + 357 pp. A K Peters, 2004. $49
Once upon a time, in ancient Greece, science was platonic and a
priori. The Sun revolved around the Earth in a perfect circle,
because the circle is such a perfect figure; there were four
elements, because four is such a nice number, and so forth. Then
along came Bacon, Boyle, Galileo, Kepler, Lavoisier, Newton and
their buddies, and revolutionized science, making it experimental
and empirical.
But math remains a priori and platonic to this day. Kant
even went to excruciating lengths to "show" that geometry,
although synthetic, is nevertheless a priori. Sure, all
mathematicians, great and small, conducted experiments (until
recently, using paper and pencil), but they kept their diaries and
notebooks well hidden in the closet.
But stand by for a paradigm shift: Thanks to Its Omnipotence the
Computer, math—that last stronghold of dear Plato—is
becoming (overtly!) experimental, a posteriori and even contingent.
But what are poor pure mathematicians to do? Their professional
weltanschauung—in other words, their philosophy—and more
important, their working habits—in other words, their
methodology—never prepared them for serving this new silicon
master. Some of them, like the conceptual genius Alexander
Grothendieck, even consider the computer (seriously!) the devil. But
although many pure mathematicians strongly dislike and mistrust the
computer, some others have already started to see the light. For
example, the great noncommutative geometer Alain Connes stated in a
recent talk that his computer had confirmed a certain conjecture of
his for 30 special cases, and consequently he is absolutely certain
that the general conjecture is correct.
Mathematicians who want to jump on this bandwagon (which, unlike
most bandwagons, is here to stay) had better read both
Mathematics by Experiment and Experimentation in
Mathematics. Traditionalists may get annoyed, since the authors
(Jonathan Borwein, David Bailey and Roland Girgensohn) don't make
any bones about "math by experiment" being truly a
paradigm shift. They even dedicate a whole section to the Kuhnian
notion of paradigm shift, quoting Max Planck ("the transfer of
allegiance from paradigm to paradigm is a conversion experience that
cannot be forced") to make the point that we can't hasten
acceptance of the new perspective, we can only be patient and wait
for the old guard to die.
These are such fun books to read! Actually, calling them
books does not do them justice. They have the liveliness
and feel of great Web sites, with their bite-size
fascinating factoids and their many human- and math-interest stories
and other gems. But do not be fooled by the lighthearted, immensely
entertaining style. You are going to learn more math (experimental
or otherwise) than you ever did from any two single volumes. Not
only that, you will learn by osmosis how to become an experimental mathematician.
One of the many highlights is a detailed behind-the-scenes account
of the discovery of the amazing Borwein--Bailey-Plouffe (BBP)
formula for π:
(By the way, the Bailey is David, but the Borwein is Jonathan's
brother Peter. Simon Plouffe, a latter-day Ramanujan, is the
webmaster of the celebrated Inverse Symbolic Calculator site.)
The BBP formula allows one to compute the billion-and-first digit of
π (in base 2) without computing the first billion digits. It was
discovered with the aid of the so-called PSLQ algorithm of Helaman
Ferguson (who is also an "experimental mathematician" in
another sense, being a noted mathematical sculptor). Once the
formula is known, the proof is an elementary calculus
exercise, but the haystack of such formulas is infinitely large, and
to find the one that was "just right" required ingenious
experimental mathematics, which the authors generously share with
the readers.
There is also a very interesting chapter on normality, which
attempts to tackle the famous, notoriously difficult problem of
proving that the decimal (or any base) expansion of famous constants
such as e and π behaves "randomly." Aside from
some constructive-but-contrived numbers (for example, the
Champernowne constant 0.12...891011...9899100101102...) and
natural-but-nonconstructive numbers (such as Chaitin's Ω),
there are no known examples. Who knows? Perhaps the experimental
approach outlined here will lead to the ultimate solution.
When you do experiments, serendipitous mistakes may lead to
breakthroughs. In Experimentation in Mathematics, the
authors describe an "electronic Petri dish" that was
obtained by typing infty (the TeX symbol for infinity)
rather than the correct infinity, during a Maple
session. The software interpreted the word as a mere symbol and gave
an actual (unexpectedly symbolic!) answer. This led to a beautiful
conjecture that was later proved by Gert Almkvist and Andrew
Granville. Although this particular discovery is not quite
penicillin, we should expect in the future many more such
serendipitous discoveries generated by "errors."
Like all successful accounts of new, rapidly growing areas, these
books are going to be victims of their own success, because the
further development that they are going to inspire will render them
obsolete very fast. I am sure that in five years they will seem very
naive; all their "controversial" pronouncements are
destined to become true-but-trite. But then again, these two lovely
books have the making of classics, and they will always be fun to
read, even if they come to seem quaint rather than avant-garde.
But I've said enough! Buy these books, and read them, if possible, from
beginning to end. Or simply open either of them at any random page and
read just that page; you will most likely learn something new,
fascinating and potentially useful. If you get hooked enough to read the
books cover-to-cover, then you are ready to become a full-fledged
experimental mathematician.—Doron Zeilberger, Mathematics,
Rutgers University, New Brunswick/Piscataway Campus, New Jersey