January 13, 2022 39Prove It!
Dan Rockmore
99 Variations on a Proof
by Philip Ording.
Princeton University Press,
260 pp., $24.95; $19.95 (paper)Mathematics is writing. For all the
quantification it makes possible and all
the technological and scientific discov-
eries it has helped to produce, it is ul-
timately words upon words. There is a
bedrock of definitions (“A point is that
which has no part,” says Euclid) cross-
cut by axioms (“A straight line segment
can be drawn joining any two points”),
whose only restrictions are that they
not contradict one another. From this
starting material we derive the terse
assertions of consequences that are
known as theorems and lemmas and
corollaries. Were a theorem the result
of a prosecutor’s successful argument,
then the lemmas would be the settled
issues of fact, the corollary the sentenc-
ing. The arguments are the key. These
are the “proofs”—sometimes called
“demonstrations”—without which the
assertions are just so much blather. The
proofs actually conjure mathematics
into existence.
Stephen King has described writing
as an act of telepathy across time and
space.^1 This is surely true of mathe-
matical writing: I’ve figured out that
the square root of two is irrational—
and this is my proof. What follows is as
close to the written transmission of one
person’s pure thought as any writing
can be. But even in mathematical writ-
ing, it’s not just what you say but how
you say it—better known as style.
This is the subject of Philip Ording’s
99 Variations on a Proof, which sur-
veys different styles that can be used to
prove a single theorem—a mathemat-
ical version of musical variations on a
theme. In fact, there are one hundred
numbered entries, beginning—accord-
ing to the mathematician’s style—with
chapter 0, a little math joke in itself.
The entries range from the technical to
the whimsical, from real proofs to more
meditative reflections on the process
and activity of mathematical research.
But Ording’s book is more than just a
survey. Each proof is accompanied by a
brief commentary, outlining the inspi-
ration of the variation and reflecting on
the culture of mathematics through the
ages. The less mathematically inclined
reader might benefit from reading
these commentaries first.
If one has any familiarity with math-
ematical proofs, it probably comes from
a year- long sojourn into Euclidean ge-
ometry in middle or high school. For
many, this is a jarring scholastic expe-
rience sandwiched between two years
of algebra. Until that moment mathe-
matics has been about finding the an-
swer to a question: for example, “Sally
rows downstream for 20 miles with
a current of 2 miles per hour and the
trip upstream takes 5 hours. How fast
does Sally row?” In geometry, however,
you’re told the answer—“If in a trian-
gle two angles equal each other, then
the sides opposite the equal angles alsoequal each other”—and then asked to
find a reason, and explain it. QED.
Students learn in geometry that a
mathematical proof has three simple
ingredients: definitions, axioms, and
deductions. The ways in which they log-
ically combine—the recipe—depends
on what you are cooking up. Definitions
and axioms are statements whose truths
are both obvious and incontrovertible
(as in the example above involving
the connection of any two points with
a straight line), while deductions are
statements that follow from the defi-
nitions and axioms, or even previously
proven facts (which are themselves de-
ductions), arrived at through the step-
by- step application of deductive logic.
Proofs are usually expressed in two
columns, a sort of logical accounting
ledger with assertions on the left and
justifications on the right.
Such classical proofs read a little like
a distillation of the Socratic method
(“Well surely if you grant me that,
then you must grant me this!”), and
they can sound a bit like a prosecutor
leading a witness to the nub of revela-
tion. Many may be surprised to know
that mathematical proofs often are
text- heavy, even if symbol- laden, which
is itself a form of stylistic evolution.
In Carl Boyer’s timeless A History of
Mathematics^2 we learn that in early
Egyptian mathematics the unknown
was referred to as “aha” for a “heap”
of to- be- determined size—the sound
of discovery and what was waiting to
be discovered were one and the same.
The letter x has been around as a sym-
bol for millennia, including to mark
the unknown on treasure maps, but it
wasn’t until the seventeenth centurythat it began to represent the unknown
in mathematical compositions.
A mathematical assertion often can
be proved by many different kinds
of arguments. Each proof should be
correct and clear, and perhaps even
captivating or charming. The way it is
written may appeal to one kind of intel-
lect or another (or sometimes none at
all). But the style of the argument—the
writing—is not just about convincing
the reader; it is also about the manner
in which the reader is convinced. One
truth, but in the range of tellings, differ-
ent aspects, implications, inspirations,
and connections are revealed—just
like great literature, but with “How do
I love thee? Let me count the ways”
traded for “How do I count these? Let
me show you the ways I love best.”Philip Ording is a professor of math-
ematics at Sarah Lawrence College.
His interests include geometry and to-
pology (especially the mathematics of
knots), as well as the intersections of
mathematics and the humanities. The
organization of the book pays homage
to its inspiration: the French polymath
Raymond Queneau’s Exercises in Style
(1947), which was itself inspired by
Bach’s The Art of Fugue. Queneau’s
book, Ording explains,takes the same simple story—that
of a peculiar individual who is first
seen in a dispute on a bus, and then
later in conversation with a friend
about the position of a coat but-
ton—and transforms it in ninety-
nine different ways.Queneau was one of the founders of Ou-
lipo, an avant- garde literary collective
whose members included writers and
artists ranging from Italo Calvino to
Marcel Duchamp, as well as the math-ematician Claude Berge, a pioneer in
graph theory—a mathematical theory
of connectivity that has found direct ap-
plication in the broad and burgeoning
study of networks, and even in formal
literary constructions. “Oulipo” is short
for “Ouvroir de littérature potentielle,”
which translates to “workshop of poten-
tial literature.” It’s an appropriate name
for a group that together investigated
what literature could be rather than
what it was, fueled by experimentation
in literary production, often involving
formal or rule- based systems.
Some connect Oulipo’s birth to
Bourbaki, a collective of French math-
ematicians also united by a shared be-
lief in the importance of formalism who
strove to rebuild all of mathematics
from the ground up, axioms first. Que-
neau is known to have attended at least
one of their meetings, and the manifes-
tos of Oulipo co- founder François Le
Lionnais directly refer to the highly ab-
stract elements on which Bourbaki was
founded. Ironically, what was an avant-
garde approach to literature was seen
by many as reactionary when it came
to mathematics.^3 Bourbaki’s stripped-
down axiomatic framing waxed briefly
and has since waned.
Queneau didn’t just admire mathe-
matics. He published two peer- reviewed
papers, both on “s- sequences”: lists of
numbers characterized by conditions
such that any number in the list depends
in certain ways on numbers that precede
it.^4 You don’t need to squint to see this
as an abstraction of writing. In one of
Queneau’s last works, Les fondements
de la littérature d’après David Hilbert, he
pushes the analogy, turning the axioms
of geometry (as written by the math-
ematician David Hilbert, a towering
figure and great fan of formalism) into
axioms for literature. Points become
words, lines become sentences, and
planes (infinite flat sheets comprising
lines and points) become paragraphs.
Each of Queneau’s ninety- nine varia-
tions in Exercises in Style demonstrates
that even the simplest set of facts can
be brought differently to life by a range
of storytelling styles. In this way it is
something like a literary version of
Euclid’s Elements, whose many more
mathematical “stories” derive from five
(or ten, depending on how you count)
simple axioms, each even plainer than
the few quotidian activities that define
Queneau’s peevish hero.
The variations in Queneau’s Exer-
cises in Style include a sonnet, a book
blurb, and a one- act opera. There is a
musical score. An entry titled “Zoolog-
ical” is animated with animals (“In the
dog days while I was in a bird cage... ”).
And of course there is mathematics.
“Negativities” is the story of what itEmma Kunz: Work No. 004, undatedEmma Kunz Stiftung(^1) Stephen King, On Writing: A Memoir
of the Craft (Scribner, 2000); reissued
in 2020 with contributions from King’s
sons, Joe Hill and Owen King.
(^2) Originally published in 1968 and most
recently issued in an updated version
coauthored with Uta C. Merzbach
(Wiley, 2010).
(^3) For a fascinating recounting of the
interactions between and coevolution
of Oulipo and Bourbaki, see Amir D.
Aczel, The Artist and the Mathemati-
cian (Thunder’s Mouth, 2006).
(^4) The well- known Fibonacci sequence is
an example of a sequence—although not
an s- sequence —wherein each number of
the list depends on previous numbers. In
this case, each number (after starting
with a pair of ones) is given by the sum
of the two numbers that precede it.
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