The New York Review of Books (2022-01-13)

January 13, 2022 41

middle”) is available. Here Ording

quotes the Hungarian mathematician

Georg Pólya, who noted that this kind

of indirect proof “has some resem-

blance to irony which is the favorite

procedure of the satirist.”

Similarly, chapter 14, “Contrapos-

itive”—which also relies on indirect

argument—makes use of the fact that

proving an “If this, then that” statement

is equivalent to proving “If not that,

then not this.” These introductions to

the mechanics of logic are for me like

little mathematical madeleines. I recall

my own early difficulties with the lan-

guage of mathematical proofs. It took

me several weeks in my first pure math

class to understand that the technical

meaning of “A if and only if B” is both

that A implies B and that B implies A,

and not just that the author was being

emphatic about B following from A.

There are also several proofs here—

chapter 0, “Omitted”; chapter 44,

“Omitted with Condescension”; and

chapter 94, “Authority”—that aren’t

really proofs at all: each in its own way

says “this is obvious” and doesn’t actu-

ally do the work of demonstrating any-

thing. Ording uses “Omitted” to talk

about the idea that choosing to prove

something is part of the “aesthetic” of

writing mathematics or about mathe-

matics. He uses the other two chapters

to call out some of the ways in which

pedagogy can be lazy or even cruel.

“Authority” produces a proof that

says only that “it follows from Euler,”

one of the great mathematicians of all

time, whose collected works fill ninety

volumes. Good luck finding that ref-

erence. I can’t count the hours of self-

flagellation I’ve experienced laboring

(and swearing) over the word “obvious”

in a textbook or research paper. These

madeleines are migraines. As Ording

says in the commentary accompanying

“Authority,” “When the authority of

reference is oneself, a proof by author-

ity becomes a proof by intimidation.”

As the saying goes, sometimes words

are not enough, even in mathematics.

Mathematical inspiration and argumen-

tation often make use of the visual—

which for some mathematicians might

not rise to the level of a proof, but could

very well be more convincing for those

who blanch at symbols. These make for

some beautiful variations. A favorite is

chapter 10, “Wordless,” a visual proof—

or “Proof Without Words”—based on

Ording’s Euclidean chapter 52, “Antiq-

uity.”^7 It uses drawings of cubes, squares,

rectangles, columns, and lines, and

makes tangible the manipulations of al-

gebra that to many can seem inscrutable.

This is also called a “Look- see! Proof.”

A paper- folding diagram in chapter

39, “Origami,” shows that abstract ar-

guments can be made physical, and the

photograph that is chapter 29, “Model,”

shows a delicate and beautiful paper-

model embodiment of a solution. As

one who has always had trouble “see-

ing” mathematics, I found these picto-

rial arguments a delight.

Some mathematical styles have to do

with genre. An algebraic style of ar-

gument—such as chapters 23 and 24,

“Symmetry” and “Another Symme-

try”—will almost surely include a lot of

formula rewriting and make use of sym-

metries, sometimes symbolic (“cancel

out the x’s”), but other times reflected

in a pictorial or geometric representa-

tion. “If a problem will divide along

a line or axis of symmetry,” Ording

notes, “it means there’s a good chance

that a solution to one part of the prob-

lem will extend... to the problem as a

whole.” A geometric proof will frame

the assertion using the familiar lines,

curves, and notions of distance in a

two- dimensional or three- dimensional

space, or if necessary might even re-

quire analogous “objects” that reside in

spaces of higher dimensions or having

exotic curved or crenellated structures.

Topological arguments—as in chap-

ter 51—can have a geometric feel but

allow the freedom to slide, squash, and

squeeze the shapes. An analytic proof,

such as chapter 33, “Calculus,” will usu-

ally involve techniques that derive from

or are related to calculus. On the whole,

it is not uncommon in mathematics to

use newly developed ideas and subject

areas to reprove old and seemingly

settled problems; as with many kinds

of writing, old themes revisited in new

styles have the power to reveal previ-

ously unseen subtleties or even produce

simpler arguments than the original.

Some of Ording’s variations give

readers a glimpse of the culture of math-

ematics as a practice. Chapter 48, “Com-

puter Assisted,” shows how computers

are now used not just to solve equations

but to assist with or even fully produce

proofs. Doron Zeil berger, a mathema-

tician at Rutgers, has even listed his

computer program “Shalosh B. Echad”

(a Hebrew- English mashup based on

the model name of Zeilberger’s first

computer) as a coauthor on some of his

published papers. More recently, the

same technologies that are producing

machine- written news articles are also

being used to solve math problems.

In a different direction, computers

can now provide the modern mathema-

tician with a platform for experimenta-

tion, giving mathematics more of the

laboratory feel of the physical or life

sciences. For many this has changed

the way mathematics is approached—

see, for example, chapter 76, “Ex-

perimental,” and chapter 77, “Monte

Carlo”—as well as the way in which

mathematics is written. The latter is

the source of another variation, chap-

ter 35, “Typeset,” which gives a win-

dow into the TeX typesetting software

package (invented by Donald Knuth, a

computer scientist at Stanford) that has

revolutionized the production of math-

ematics papers and the pace of modern

mathematical research.

This acceleration in the production

of new mathematical ideas is transfor-

mative for the discipline. Historically,

mathematics has been among the slow-

est of the sciences. Articles submitted

for peer review can take years before

they are returned with comments that

may then need to be addressed and the

articles returned for second reviews

before acceptance for publication—if

accepted at all. But nowadays math-

ematicians often deposit preliminary

versions of their papers, known as

“preprints,” on arXiv.org, a digital de-

pository and preprint server, which ac-

celerates the availability of the work at

the expense of detailed review; Ording

refers to this in chapter 37. Some math-

ematicians—most likely those with

tenure—may not even go to the trouble

of pursuing official publication.

Despite all the new ways of doing

mathematical research, most mathema-

ticians are still attached to the chalk-

board—a picture of which is shown

in chapter 21, displaying Cardano’s

method for finding the solutions to

the book’s ur- equation. “More than a

teaching aid, a blackboard is a medium

for mathematics,” Ording writes, al-

though it can also be “a source of anx-

iety” for a student “when called to the

board.” But that’s often where the hard

work is done: scribbling, erasing, work-

ing out examples in front of colleagues,

even sketching cartoons as inspiration.

Around 3,500 years ago in Mesopota-

mia traders scratched marks on clay to

work out the proto- problems of mathe-

matics. In that sense, little has changed.

For further evidence, I recommend

Jessica Wynne’s Do Not Erase: Math-

ematicians and Their Chalkboards,^8 a

collection of photographs of the chalk-

boards of mathematicians from around

the world, each accompanied by a re-

flection or explanation from its creator.

The chalkboard is just one meeting

place for ideas. Ording points out that

“many university mathematics depart-

ments hold tea in the afternoon,” creat-

ing “a venue for informal discussion.”

Ours at Dartmouth is every weekday,

usually at 3:30—come visit. Chapter

65, “Tea,” is an imagined exchange

during such a get- together. The partic-

ipants include a person referred to as

Lambda, who has just proved Ording’s

theorem by complicated means—see

chapter 64, “Research Seminar”—

and a few attendees, buttonholing (à

la Queneau?) Lambda for details or

maybe just to strut their stuff. Ording

cleverly has tea follow a seminar, as is

often the case in real life.

Group proof efforts of course no lon-

ger require contributors to be in the

same room. Chapter 25, “Open Col-

laborative,” is an imagined exchange

on a blog thread, and those interested

in this lively online research culture

might make an excursion to the Math-

ematics Stack Exchange website; it’s

not just students hunting for homework

solutions. An updated version of Ord-

ing’s book might have to add an entry:

chapter 100, “Zoom.”

This is of course but a sampling of

Ording’s mathematical buffet. The

more linguistically inspired variations

in the book feel like direct tributes to

Exercises in Style. Chapter 9, “Mono-

syllabic,” is a proof using words of only

one syllable. Chapter 81, “Doggerel,” is

written in verse and resembles, Ording

notes, a “stanza from Lewis Carroll’s

double acrostic, The First Riddle.”

Chapter 98, “Mondegreen,” is—as per

the definition of the word—a homo-

phonic rewriting of the proposition:

Their omelette: eggs, beer, eel.

If eggs cued my nose, six eggs

queered.

Plus nine eggs.

My nose for equals—zero.

Then, egg sequels one or four....

“Whenever I see the ubiquitous ru-

bric ‘Theorem. Let... , ’” Ording ex-

plains, “I cannot help but hear ‘their

omelette.’” These proofs’ playfulness

at the boundary of sense and nonsense

surely expands the limits of mathemat-

ical exposition as well as readerly re-

sponse to mathematical ideas. I think

Queneau would have been proud. Q

“Szilárd Borbély was one of

the best and most original

poets and novelists of his

generation—and Ottilie Mulzet

is a wonderful translator of

his work.” —George Szirtes

Available from booksellers and http://www.nyrb.com

Szilárd Borbély spent his childhood in a

tiny impoverished village in northeastern

Hungary, where the archaic peasant world

of Eastern Europe coexisted with the col-

lectivist ideology of a new Communist

state. Close to the Soviet border and far

from any metropolitan center, the village

was a world apart: life was harsh, monot-

onous, and often brutal, and the Borbélys,

outsiders and “class enemies,” were shunned.

In a Bucolic Land, Borbély’s final, posthu-

mously published book of poems, combines

autobiography, ethnography, classical my-

thology, and pastoral idyll in a remarkable

central poetic sequence about the starkly

precarious and yet strangely numinous

liminal zone of his youth. This is framed

by elegies for a teacher in which the poet

meditates on the nature of language and

speech and on the adequacy of words to

speak of and for the dead. Ottilie Mulzet’s

English translation conveys the full power

of a writer of whom László Krasznahorkai

has said, “He was a poet—a great poet—

who shatters us.”

“Borbély’s poetry, prose, and essays try

to bring the readers closer to the lives of

those who cannot speak of their trauma

or suffering. They can be uneducated and

poor villagers, survivors of the Holocaust,

women grieving after a miscarriage, or

victims of terrible aggression. Through

Borbély’s texts we readers become

increasingly less cruel-hearted."

—László Bedecs, Asymptote

IN A BUCOLIC LAND

SZILÁRD BORBÉLY

Translated from the Hungarian by

Ottilie Mulzet

Paperback • $16.00

Also available as an e-book

On sale January 11, 2022

ALSO BY

SZILÁRD BORBÉLY

BERLIN-HAMLET

(^7) For more on visual proofs, I heartily rec-
ommend Roger Nelsen’s Proofs Without
Words, volumes 1, 2, and 3, all published
by the American Mathematical Society.^8 Princeton University Press, 2021.
Rockmore 39 41 .indd 41 12 / 15 / 21 6 : 01 PM