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Mathematician Lillian Lieber on Infinity, Art, Science, the Meaning of Freedom, and What It Takes to Be a Finite But Complete Human Being

Mathematics and poetry converge in an ode to the “sweet reasonableness” at the heart of a psychologically balanced character.

Mathematician Lillian Lieber on Infinity, Art, Science, the Meaning of Freedom, and What It Takes to Be a Finite But Complete Human Being

“We’re all intrinsically of the same substance,” astrophysicist Janna Levin wrote in her exquisite inquiry into whether the universe is infinite or finite. “The fabric of the universe is just a coherent weave from the same threads that make our bodies. How much more absurd it becomes to believe that the universe, space and time could possibly be infinite when all of us are finite.” How, then, do we set aside this instinctual absurdity in order to grapple with the concept of infinity, which pushes our creaturely powers of comprehension past their limit so violently?

That’s what the mathematician and writer Lillian R. Lieber (July 26, 1886–July 11, 1986) set out to explore more than half a century earlier in the unusual and wonderful 1953 gem Infinity: Beyond the Beyond the Beyond (public library) — one of seventeen marvelous books she published in her hundred years, inviting the common reader into science with uncommon ingenuity and irresistible warmth. Emanating from Lieber’s discussion of infinity is a larger message about what it means, and what it takes, to be a finite but complete and balanced human being.

Lillian R. Lieber

Lieber belongs to the “enchanter” category of great writers and was among the first generation of women mathematicians to hold academic positions in her role chairing the Department of Mathematics at Long Island University. She had a peculiar style resembling poetry, though she insisted it was not free verse but, rather, a deliberate way of breaking lines in order to speed up reading and intensify comprehension. (Curiously, I find her style to have precisely the opposite effect, which is why I’ve enjoyed it so tremendously — it does what poetry does, which is slow down the spinning world and dilate the pupil of attention so that the infinite becomes comprehensible.)

Populating her books is the character of T.C. Mits, “the Celebrated Man-in-the-Street,” and his mate, Wits, “the Woman-in-the-Street.” Accompanying Lieber’s writing are original line drawings by her own mate, the illustrator Hugh Gray Lieber.

Lieber’s work was so influential in elevating the popular science genre that even Albert Einstein himself heartily praised her book on relativity, yet many of her books have fallen out of print — no doubt because the depth, complexity, and visionary insurgency of her style don’t conform to the morass of formulaic mediocrity passing for popular science writing today.

Lieber frames the premise of Infinity in the charming opening verse — or, as she insisted, decidedly not-verse — of the second chapter:

Of course you know that
the Infinite
is a subject which
has always been of the deepest interest
to all people —
to the religious,
to poets,
to philosophers,
to mathematicians,
as well as to
T.C. Mits
(The Celebrated Man-in-the-Street)
and to his mate,
Wits
(the Woman-in-the-Street).
And it probably interests you,
or you would not be reading this book.

But it is in the first chapter, titled “Our Good Friend, Sam,” that Lieber’s genius for science, metaphor, and wordplay shines most brilliantly as she takes on everything from the symbiotic relationship between art and science to free will to the vital difference between common sense and truth to the evils of antisemitism and all exclusionary ideologies. (It is self-evident to point out that Lieber, a Jewish woman writing shortly after WWII in a climate of acute antisemitism and sexism, was, like any artist, bringing all of herself to her art.)

Lieber writes:

For those who have not met SAM before,
I wish to summarize
VERY BRIEFLY
what his old acquaintances
may already know,
and then to tell to all of you
MORE about him.
In the first place,
the name “SAM”
was first derived from
Science, Art, Mathematics;
but I now find
the following interpretation
much more helpful:
the “S” stands for
OUR CONTACT WITH THE OUTSIDE WORLD;
please note that
I do NOT say
that “S” represents “facts” or “reality”,
for
the only knowledge we can have of
the outside world
is through our own senses or
“extended” senses —
like microscopes and telescopes et al
which help us to see better,
or radios, etc., which
help us to hear sounds
which we would otherwise
not be aware of at all,
and so on and so on.

But of course
there may be
many, many more things
in the world
which we do not yet perceive
either directly through our senses
or with the aid of
our wonderful inventions.
And so it would be
Quite arrogant
to speak as if we knew
what the outside world “really” is.
That is why I wish to give to “S”
the more modest interpretation
and emphasize that
it represents merely
that PART of the OUTSIDE world
which we are able to contact, —
and therefore even “S” has
a “human” element in it.

Next:
the “A” in SAM represents
our INTUITION,
our emotions, —
loves, hates, fears, etc. —
and of course is also
a “human” element.

And the “M” represents
our ability to draw inferences,
and hence includes
mathematics, logic, “common sense”,
and other ways in which
we mentally derive the “consequences”
before they hit us.
So the “M” too is
a “human” element.

Thus SAM is entirely human
though not an individual human being.

Furthermore,
a Scientist utilizes the SAM within him,
for he must make
“observations” (“S”),
he must use his “intuition” (“A”)
to help him formulate
a good set of basic postUlates,
from which his “reasoning powers” (“M”)
will then help him to
derive conclusions
which in turn must again be
“tested” (“S” again!) to see
if they are “correct”.

Perhaps you are thinking that
SAM and the Scientist
are really one and the same,
and that all I am doing is
to recommend that we all become
Scientists!
But you will soon see that
this is not the case at all.
For,
in the first place,
it too often happens, —
alas and alack! —
that when a Scientist is
not actually engaged in doing
his scientific work,
he may “slip” and not use
his “S”, his “A”, and his “M”,
so carefully,
will bear watching,
like the rest of us.

In a sentiment which physicist and poet Alan Lightman would come to echo decades later in his beautiful meditation on the creative sympathies of art and science, Lieber adds:

So, you see,
being a SAMite and being a Scientist
are NOT one and the same.

Besides,
a SAMite may not be a Scientist at all,
but an Artist!
For an Artist, too, must use
his “S” in order to “observe” the world,
his “A” (“intuition”) to sense
some basic ways to translate his
“observations”,
and his “M”
to derive his “results” in the form of
drawings, music, and so on.
Thus an Artist, too,
WHEN AT HIS BEST,
is a SAMite.

Perhaps Lieber’s most interesting, layered, and timelessly relevant discussion is of the concept of freedom, its misconceptions and mutations, and its implication for our private, public, and political lives:

Now consider a person
who is SOMETIMES or OFTEN like this:
SaM.
He is evidently relying very heavily on
his “intuition”, his “hunches”, his “emotions”,
hardly checking to see whether
the “observations” of the outside world (“S”)
and his own reasoning powers (“M”)
show his “hunch” to be correct or not!
And so,
precious as our “intuition” may be,
it can go terribly “haywire”
if not checked and double-checked
by “S” and “M”.
Thus, a person who
habitually behaves like this
is allowing his “S” and “M” to
become practically atrophied,
and is the wild, “over-emotional” type,
who is not only a nuisance to have around,
but is hurting himself and
not allowing himself to become
adjusted to the world he lives in.
Such a person,
with an exaggerated “A”,
and atrophied “S” and “M”,
has a feeling of “freedom”,
of not being held down by “S” and “M”
(“facts” and “reason”) ;
but, as you can easily see
this makes for Anarchy,
for a lack of “self-control” —
and can lead
to fatty degeneration from
feeling “free” to eat all he wants;
to the D.T.’s from
feeling “free” to drink all he wants;
to accidents because
he feels “free” to drive as fast as he wants
and to “hog” the road;
to a sadistic lack of
consideration for others
by feeling “free” to
kick them in the teeth for “nuttin'”;
to antisocial “black market” practices
from a similar feeling of “freedom”,
giving “free” reign to the “A”
without the necessary consideration of “facts” (“S”) and “reason” (“M”).
Needless to say this is a
PATHOLOGICAL FREEDOM
as against
a NORMAL, HEALTHY FREEDOM of
the well-balanced SAM
which is so necessary in society
in which EACH individual
must be guided by the SAM within himself
in order to avoid conflict with
the SAM in someone else.
This is something that
a bully does not understand —
that if he acts like a pathological sAm,
he induces sAmite-ism in others,
as in mob violence;
this is indeed a horrible “ism”
that can destroy a society as well as
individuals in it.

Lieber proceeds to build on this taxonomy of psychological imbalances, reminiscent of neuroscience founding father Santiago Ramón y Cajal ‘s taxonomy of the “diseases of the will.” She turns to the next imbalance — the person blinded by isolated facts, unable to integrate them into an understanding of the big picture:

Similarly,
there is the Sam type:
he may be called the “tourist” type —
running around seeing this and that
but without the “imagination” (“A”)
or the reasoning power (“M”)
to put his observations together
with either heart (“A”) or mind (“M”),
but is concerned only with
ISOLATED BITS OF INFORMATION:
he is like the man who,
seeing a crowd had gathered,
wanted to know what happened.
and, when someone told him
“Ein Mann hat sich dem Kopf zerbrochen”
(It happened to be in Germany),
corrected the speaker’s grammar
and said “DEN Kopf!”
He knew his bit of grammar,
but what an inadequate reaction
under the circumstances.
don’t you think?

Next comes the flawed rationalizer, who misuses the tools of logic against reason:

And there is also the saM type —
one who can reason (“M”)
but starts with perhaps
some postulate (“A”) favoring murder.
Such a man would make
a wonderfully “rational”
homicidal maniac or crook
who could plan you a murder
calmly and rationally enough
to surprise any who are not familiar with
this sAM type of pathological case.

Lieber returns to the core purpose of her SAM metaphor and its relationship to the central question of the book:

Thus SAM gives us a way of
examining our own behavior
and that of others,
taking into account the “facts” (“S”),
and using imagination and sympathy (“A”)
in a rational way (“M”).

Are you perhaps thinking,
“Well, this may be interesting,
but
why all this talk about SAM,
when you are writing a book about
Infinity?”
To which the answer is:
The yearning for Infinity,
for Immortality,
is an “intuitive” yearning (“A”):
we look for support for it
in the physical world (“S”),
we try to reason about it (“M”), —
but only when we turn
the full light of SAM upon it
are we able to make
genuine progress in considering
Infinity.

In a brilliant and necessary caveat reminiscent of mathematician Kurt Gödel’s world-changing incompleteness theorems, which unsettled some of our most elemental assumptions by demonstrating the limits of logic turned unto itself, Lieber adds:

There is only one more point
I must make here:
Namely, that
even being a well-balanced
SAMite —
and not a pathological anti-SAMite
like SAM, etc. etc. —
is NECESSARY but NOT SUFFICIENT.
You will probably agree that
it is further necessary
to have our SAM up-to-date.
For he is a GROWING boy,
and what was good enough for him in 1800
is utterly inadequate in 1953;
and unless the “S” is up-to-date
and the postulates (“A”)
and reasoning (“M”)
are appropriately MODERN,
we cannot make proper
ADJUSTMENT in the world TODAY.
And ADJUSTMENT is what we must have.
For adjustment means
SURVIVAL,
and that is a MINIMUM demand —
for, without survival
we need not bother to study anything
we just won’t be here to tell the tale.

In a passage of piercing pertinence today, as we watch various oppressive ideologies and tyrannical regimes engulf the globe, Lieber concludes by returning to the subject of freedom, its malformations, and its redemptions:

And so let me summarize
by saying that the
ANTI-SAMITES
hurt not only themselves,
by getting “ulcers”, nervous breakdowns,
drinking excessively, etc. etc.,
but hurt others also,
for from their ranks are recruited
those who advocate war and destruction,
the homicidal maniacs, the greedy crooks,
the gamblers, the drunken drivers,
the liars, et al.

[…]

Just a word more about
FREEDOM —
you have seen above
the pathological idea of freedom,
but when you consider this important concept
from SAM’s WEll-BALANCED viewpoint,
you will see that,
from this point of view,
the “feeling” of freedom (“A”),
being supported on one side by “S”
(the “facts” of the outside world),
and on the other by “M”
(“sweet reasonableness”) —
is definitely NOT the
ANARCHICAL freedom of SAM,
but is a sort of
CONTROLLED FREEDOM —
controlled by facts and reason
and is therefore SELF-controlled
(by the SAM within us)
and hence implies
VOLUNTARY COOPERATION rather than FORCE.
Thus anyone who demands
“freedom unlimited” as his right,
is a pathological SAM,
a destructive creature;
whereas,
in mathematics
you will find the
CONTROLLED FREEDOM of SAM
and you will feel refreshed to see
how genuine progress can be made
with this kind of freedom.

Infinity: Beyond the Beyond the Beyond is a thoroughly magnificent read in its totality. Pair it with the lovely children’s book Infinity and Me, then complement this particular fragment with Simone de Beauvoir, writing shortly before Lieber, on art, science, and freedom, and James Baldwin, writing shortly thereafter, on freedom and how we imprison ourselves.

HT Natalie Wolchover

BP

The Trailblazing 18th-Century French Mathematician Émilie du Châtelet on Jealousy and the Metaphysics of Love

“It is the privilege of affection to see a friend in all the situations of his soul.”

The Trailblazing 18th-Century French Mathematician Émilie du Châtelet on Jealousy and the Metaphysics of Love

“Anxiety is love’s greatest killer,” Anaïs Nin admonished.“It makes others feel as you might when a drowning man holds on to you. You want to save him, but you know he will strangle you with his panic.” No form of anxiety sinks the buoyancy of love more readily than jealousy. The Swiss philosopher Henri-Frédéric Amiel put it best in his reflections on love and its demons: “Jealousy… is precisely love’s contrary… the most passionate form of egotism, the glorification of a despotic, exacting, and vain ego, which can neither forget nor subordinate itself.”

Indeed, this corrosive yet common human experience is one which responds better to being befriended rather than forcefully subordinated, for the more one denies and resists it, the more it persists. How to accept it as natural and, in that acceptance, let it dissolve is what the trailblazing French mathematician Émilie du Châtelet (December 17, 1706–September 10, 1749) explores in a letter to one of her lovers, found in her Selected Philosophical and Scientific Writings (public library).

Émilie du Châtelet (Portrait by Nicolas de Largillière)

What made Du Châtelet particularly extraordinary is that her rigorous scientific mind came coupled with immensely sensitive insight into the workings of the human heart. In the late spring of 1735, the 29-year-old mathematician — who had enchanted Voltaire two years earlier and would soon popularize Newton and lead the way for women in science — writes to Louis François Armand de Vignerot du Plessis, Duke of Richelieu, a notorious playboy:

There is much difference between jealousy and the fear of not being loved enough: one can brave the one when one feels that one does not merit it, but one cannot help being touched and distressed by the other. Jealousy is an annoying feeling, and the fear of it a delicate anxiety, against which there are fewer weapons and fewer remedies, other than to go to be happy… There, in truth, is the metaphysics of love, and this is where the excess of this passion leads. All this appears to me as the clearest and most natural thing in the world.

In the same letter, Du Châtelet models the counterpoint to jealousy’s contracted clutch — the largeness of heart and generosity of spirit that loves another unconditionally in their imperfect entirety, excludes nothing from the scope of that love, and longs to partake in the other’s completeness. Using the French word amitie, which connotes affectionate friendship and which Du Châtelet imbues with distinct romantic hues in her correspondence, she addresses her lover:

It is the privilege of affection to see a friend in all the situations of his soul. I love you sad, gay, lively, blocked; I want my friendly feelings to add to your pleasures and diminish your troubles, and I want to share them.

Complement Du Châtelet’s altogether electrifying Selected Philosophical and Scientific Writings with her prescient 18th-century reflection on gender in science and the nature of genius, then revisit philosopher Martha Nussbaum on jealousy as illuminated by anger and forgiveness.

BP

How the French Mathematician Sophie Germain Paved the Way for Women in Science and Endeavored to Save Gauss’s Life

“The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare… since the charms of this sublime science in all their beauty reveal themselves only to those who have the courage to fathom them.”

How the French Mathematician Sophie Germain Paved the Way for Women in Science and Endeavored to Save Gauss’s Life

A century after the trailblazing French mathematician Émilie du Châtelet popularized Newton and paved the path for women in science, and a few decades before the word “scientist” was coined for the Scottish mathematician Mary Somerville, Sophie Germain (April 1, 1776–June 27, 1831) gave herself an education using her father’s books and became a brilliant mathematician, physicist, and astronomer, who pioneered elasticity theory and made significant contributions to number theory.

In lieu of a formal education, unavailable to women until more than a century later, Germain supplemented her reading and her natural gift for science by exchanging letters with some of the era’s most prominent mathematicians. Among her famous correspondents was Carl Friedrich Gauss, considered by many scholars the greatest mathematician who ever lived. Writing under the male pseudonym M. LeBlanc — “fearing the ridicule attached to a female scientist,” as she herself later explained — Germain began sharing with Gauss some of her theorem proofs in response to his magnum opus Disquisitiones Arithmeticae.

Sophie Germain

Their correspondence began in 1804, at the peak of the French occupation of Prussia. In 1806, Germain received news that Napoleon’s troops were about to enter Gauss’s Prussian hometown of Brunswick. Terrified that her faraway mentor might suffer the fate of Archimedes, who was killed when Roman forces conquered Syracuse after a two-year siege, she called on a family friend — the French military chief M. Pernety — to find Gauss in Brunswick and ensure his safety. Pernety tasked one of his battalion commanders with traveling two hundred miles to the occupied Brunswick in order to carry out the rescue mission.

But Gauss, it turned out, was unscathed by the war. In a letter from November 27 of 1806, included in the altogether fascinating Sophie Germain: An Essay in the History of the Theory of Elasticity (public library), the somewhat irate battalion commander reports to his chief:

Just arrived in this town and have bruised myself with your errand. I have asked several persons for the address of Gauss, at whose residence I was to gather some news on your and Sophie Germain’s behalf. M. Gauss replied that he did not have the honor of knowing you or Mlle. Germain… After I had spoken of the different points contained in your order, he seemed a little confused and asked me to convey his thanks for your consideration on his behalf.

Carl Friedrich Gauss (Portrait by Jensen)

Upon receiving the comforting if somewhat comical news, Germain felt obliged to write to Gauss and clear his confusion about his would-be savior’s identity. After coming out as the woman behind the M. LeBlanc persona in a letter from February 20 of 1807, she tells Gauss:

The appreciation I owe you for the encouragement you have given me, in showing me that you count me among the lovers of sublime arithmetic whose mysteries you have developed, was my particular motivation for finding out news of you at a time when the troubles of the war caused me to fear for your safety; and I have learned with complete satisfaction that you have remained in your house as undisturbed as circumstances would permit. I hope, however, that these events will not keep you too long from your astronomical and especially your arithmetical researches, because this part of science has a particular attraction for me, and I always admire with new pleasure the linkages between truths exposed in your book.

Gauss responds a few weeks later:

Mademoiselle,

Your letter … was for me the source of as much pleasure as surprise. How pleasant and heartwarming to acquire a friend so flattering and precious. The lively interest that you have taken in me during this war deserves the most sincere appreciation. Your letter to General Pernety would have been most useful to me, if I had needed special protection on the part of the French government.

Happily, the events and consequences of war have not affected me so much up until now, although I am convinced that they will have a large influence on the future course of my life. But how I can describe my astonishment and admiration on seeing my esteemed correspondent M. LeBlanc metamorphosed into this celebrated person, yielding a copy so brilliant it is hard to believe? The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare: this is not surprising, since the charms of this sublime science in all their beauty reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius. Nothing could prove me in a more flattering and less equivocal way that the attractions of that science, which have added so much joy to my life, are not chimerical, than the favor with which you have honored it.

The scientific notes which your letters are so richly filled have given me a thousand pleasures. I have studied them with attention, and I admire the ease with which you penetrate all branches of arithmetic, and the wisdom with which you generalize and perfect. I ask you to take it as proof of my attention if I dare to add a remark to your last letter.

With this, Gauss extends the gift of constructive criticism on some mathematical solutions Germain had shared with him — the same gift which trailblazing feminist Margaret Fuller bestowed upon Thoreau, which shaped his career. Although Gauss eventually disengaged from the exchange, choosing to focus on his scientific work rather than on correspondence, he remained an admirer of Germain’s genius. He advocated for the University of Gottingen to award her a posthumous honorary degree, for she had accomplished, despite being a woman and therefore ineligible for actually attending the University, “something worthwhile in the most rigorous and abstract of sciences.”

She was never awarded the degree.

Red fish pond in front of the girls’ school named after Germain

After the end of their correspondence, Germain heard that the Paris Academy of Sciences had announced a prix extraordinaire — a gold medal valued at 3,000 francs, roughly $600 then or about $11,000 now — awarded to whoever could explain an exciting new physical phenomenon scientists had found in the vibration of thin elastic surfaces. The winning contestant would have to “give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence.”

The problem appeared so difficult that it discouraged all other mathematicians except Germain and the esteemed Denis Poisson from tackling it. But Poisson was elected to the Academy shortly after the award was announced and therefore had to withdraw from competing. Only Germain remained willing to brave the problem. She began work on it in 1809 and submitted her paper in the autumn of 1811. Despite being the only entrant, she lost — the jurors ruled that her proofs were unconvincing.

Germain persisted — because no solution had been accepted, the Academy extended the competition by two years, and she submitted a new paper, anonymously, in 1813. It was again rejected. She decided to try a third time and shared her thinking with Poisson, hoping he would contribute some useful insight. Instead, he borrowed heavily from her ideas and published his own work on elasticity, giving Germain no credit. Since he was the editor of the Academy’s journal, his paper was accepted and printed in 1814.

Still, Germain persisted. On January 8, 1816, she submitted a third paper under her own name. Her solution was still imperfect, but the jurors decided that it was as good as it gets given the complexity of the problem and awarded her the prize, which made her the first woman to win an accolade from the Paris Academy of Sciences.

But even with the prize in tow, Germain was not allowed to attend lectures at the Academy — the only women permitted to audit were the wives of members. She decided to self-publish her winning essay, in large part in order to expose Poisson’s theft and point out errors in his proof. She went on to do foundational mathematical work on elasticity, as well as work in philosophy and psychology a century before the latter was a formal discipline. Like Rachel Carson, Germain continued to work as she was dying of breast cancer. A paper she published shortly before her terminal diagnosis precipitated the discovery the laws of movement and equilibrium of elastic solids.

Her unusual life and enduring scientific legacy are discussed in great detail in the biography Sophie Germain. Complement it with the stories of how Ada Lovelace became the world’s first computer programmer, how physicist Lise Meitner discovered nuclear fission, was denied the Nobel Prize, but led the way for women in science anyway, and how Harvard’s unsung 19th-century female astronomers revolutionized our understanding of the universe decades before women could vote.

BP

The Invention of Zero: How Ancient Mesopotamia Created the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form

“If you look at zero you see nothing; but look through it and you will see the world.”

The Invention of Zero: How Ancient Mesopotamia Created the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form

If the ancient Arab world had closed its gates to foreign travelers, we would have no medicine, no astronomy, and no mathematics — at least not as we know them today.

Central to humanity’s quest to grasp the nature of the universe and make sense of our own existence is zero, which began in Mesopotamia and spurred one of the most significant paradigm shifts in human consciousness — a concept first invented (or perhaps discovered) in pre-Arab Sumer, modern-day Iraq, and later given symbolic form in ancient India. This twining of meaning and symbol not only shaped mathematics, which underlies our best models of reality, but became woven into the very fabric of human life, from the works of Shakespeare, who famously winked at zero in King Lear by calling it “an O without a figure,” to the invention of the bit that gave us the 1s and 0s underpinning my ability to type these words and your ability to read them on this screen.

Mathematician Robert Kaplan chronicles nought’s revolutionary journey in The Nothing That Is: A Natural History of Zero (public library). It is, in a sense, an archetypal story of scientific discovery, wherein an abstract concept derived from the observed laws of nature is named and given symbolic form. But it is also a kind of cross-cultural fairy tale that romances reason across time and space

Art by Paul Rand from Little 1 by Ann Rand, a vintage concept book about the numbers

Kaplan writes:

If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating the odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else – and all of their parts swing on the smallest of pivots, zero

With these mental devices we make visible the hidden laws controlling the objects around us in their cycles and swerves. Even the mind itself is mirrored in mathematics, its endless reflections now confusing, now clarifying insight.

[…]

As we follow the meanderings of zero’s symbols and meanings we’ll see along with it the making and doing of mathematics — by humans, for humans. No god gave it to us. Its muse speaks only to those who ardently pursue her.

With an eye to the eternal question of whether mathematics is discovered or invented — a question famously debated by Kurt Gödel and the Vienna Circle — Kaplan observes:

The disquieting question of whether zero is out there or a fiction will call up the perennial puzzle of whether we invent or discover the way of things, hence the yet deeper issue of where we are in the hierarchy. Are we creatures or creators, less than – or only a little less than — the angels in our power to appraise?

Art by Shel Silverstein from The Missing Piece Meets the Big O

Like all transformative inventions, zero began with necessity — the necessity for counting without getting bemired in the inelegance of increasingly large numbers. Kaplan writes:

Zero began its career as two wedges pressed into a wet lump of clay, in the days when a superb piece of mental engineering gave us the art of counting.

[…]

The story begins some 5,000 years ago with the Sumerians, those lively people who settled in Mesopotamia (part of what is now Iraq). When you read, on one of their clay tablets, this exchange between father and son: “Where did you go?” “Nowhere.” “Then why are you late?”, you realize that 5,000 years are like an evening gone.

The Sumerians counted by 1s and 10s but also by 60s. This may seem bizarre until you recall that we do too, using 60 for minutes in an hour (and 6 × 60 = 360 for degrees in a circle). Worse, we also count by 12 when it comes to months in a year, 7 for days in a week, 24 for hours in a day and 16 for ounces in a pound or a pint. Up until 1971 the British counted their pennies in heaps of 12 to a shilling but heaps of 20 shillings to a pound.

Tug on each of these different systems and you’ll unravel a history of customs and compromises, showing what you thought was quirky to be the most natural thing in the world. In the case of the Sumerians, a 60-base (sexagesimal) system most likely sprang from their dealings with another culture whose system of weights — and hence of monetary value — differed from their own.

Having to reconcile the decimal and sexagesimal counting systems was a source of growing confusion for the Sumerians, who wrote by pressing the tip of a hollow reed to create circles and semi-circles onto wet clay tablets solidified by baking. The reed eventually became a three-sided stylus, which made triangular cuneiform marks at varying angles to designate different numbers, amounts, and concepts. Kaplan demonstrates what the Sumerian numerical system looked like by 2000 BCE:

This cumbersome system lasted for thousands of years, until someone at some point between the sixth and third centuries BCE came up with a way to wedge accounting columns apart, effectively symbolizing “nothing in this column” — and so the concept of, if not the symbol for, zero was born. Kaplan writes:

In a tablet unearthed at Kish (dating from perhaps as far back as 700 BC), the scribe wrote his zeroes with three hooks, rather than two slanted wedges, as if they were thirties; and another scribe at about the same time made his with only one, so that they are indistinguishable from his tens. Carelessness? Or does this variety tell us that we are very near the earliest uses of the separation sign as zero, its meaning and form having yet to settle in?

But zero almost perished with the civilization that first imagined it. The story follows history’s arrow from Mesopotamia to ancient Greece, where the necessity of zero awakens anew. Kaplan turns to Archimedes and his system for naming large numbers, “myriad” being the largest of the Greek names for numbers, connoting 10,000. With his notion of orders of large numbers, the great Greek polymath came within inches of inventing the concept of powers, but he gave us something even more important — as Kaplan puts it, he showed us “how to think as concretely as we can about the very large, giving us a way of building up to it in stages rather than letting our thoughts diffuse in the face of immensity, so that we will be able to distinguish even such magnitudes as these from the infinite.”

“Archimedes Thoughtful” by Domenico Fetti, 1620

This concept of the infinite in a sense contoured the need for naming its mirror-image counterpart: nothingness. (Negative numbers were still a long way away.) And yet the Greeks had no word for zero, though they clearly recognized its spectral presence. Kaplan writes:

Haven’t we all an ancient sense that for something to exist it must have a name? Many a child refuses to accept the argument that the numbers go on forever (just add one to any candidate for the last) because names run out. For them a googol — 1 with 100 zeroes after it — is a large and living friend, as is a googolplex (10 to the googol power, in an Archimedean spirit).

[…]

By not using zero, but naming instead his myriad myriads, orders and periods, Archimedes has given a constructive vitality to this vastness — putting it just that much nearer our reach, if not our grasp.

Ordinarily, we know that naming is what gives meaning to existence. But names are given to things, and zero is not a thing — it is, in fact, a no-thing. Kaplan contemplates the paradox:

Names belong to things, but zero belongs to nothing. It counts the totality of what isn’t there. By this reasoning it must be everywhere with regard to this and that: with regard, for instance, to the number of humming-birds in that bowl with seven — or now six — apples. Then what does zero name? It looks like a smaller version of Gertrude Stein’s Oakland, having no there there.

Zero, still an unnamed figment of the mathematical imagination, continued its odyssey around the ancient world before it was given a name. After Babylon and Greece, it landed in India. The first surviving written appearance of zero as a symbol appeared there on a stone tablet dated 876 AD, inscribed with the measurements of a garden: 270 by 50, written as “27°” and “5°.” Kaplan notes that the same tiny zero appears on copper plates dating back to three centuries earlier, but because forgeries ran rampant in the eleventh century, their authenticity can’t be ascertained. He writes:

We can try pushing back the beginnings of zero in India before 876, if you are willing to strain your eyes to make out dim figures in a bright haze. Why trouble to do this? Because every story, like every dream, has a deep point, where all that is said sounds oracular, all that is seen, an omen. Interpretations seethe around these images like froth in a cauldron. This deep point for us is the cleft between the ancient world around the Mediterranean and the ancient world of India.

But if zero were to have a high priest in ancient India, it would undoubtedly be the mathematician and astronomer Āryabhata, whose identity is shrouded in as much mystery as Shakespeare’s. Nonetheless, his legacy — whether he was indeed one person or many — is an indelible part of zero’s story.

Āryabhata (art by K. Ganesh Acharya)

Kaplan writes:

Āryabhata wanted a concise way to store (not calculate with) large numbers, and hit on a strange scheme. If we hadn’t yet our positional notation, where the 8 in 9,871 means 800 because it stands in the hundreds place, we might have come up with writing it this way: 9T8H7Te1, where T stands for ‘thousand’, H for “hundred” and Te for “ten” (in fact, this is how we usually pronounce our numbers, and how monetary amounts have been expressed: £3.4s.2d). Āryabhata did something of this sort, only one degree more abstract.

He made up nonsense words whose syllables stood for digits in places, the digits being given by consonants, the places by the nine vowels in Sanskrit. Since the first three vowels are a, i and u, if you wanted to write 386 in his system (he wrote this as 6, then 8, then 3) you would want the sixth consonant, c, followed by a (showing that c was in the units place), the eighth consonant, j, followed by i, then the third consonant, g, followed by u: CAJIGU. The problem is that this system gives only 9 possible places, and being an astronomer, he had need of many more. His baroque solution was to double his system to 18 places by using the same nine vowels twice each: a, a, i, i, u, u and so on; and breaking the consonants up into two groups, using those from the first for the odd numbered places, those from the second for the even. So he would actually have written 386 this way: CASAGI (c being the sixth consonant of the first group, s in effect the eighth of the second group, g the third of the first group)…

There is clearly no zero in this system — but interestingly enough, in explaining it Āryabhata says: “The nine vowels are to be used in two nines of places” — and his word for “place” is “kha”. This kha later becomes one of the commonest Indian words for zero. It is as if we had here a slow-motion picture of an idea evolving: the shift from a “named” to a purely positional notation, from an empty place where a digit can lodge to “the empty number”: a number in its own right, that nudged other numbers along into their places.

Kaplan reflects on the multicultural intellectual heritage encircling the concept of zero:

While having a symbol for zero matters, having the notion matters more, and whether this came from the Babylonians directly or through the Greeks, what is hanging in the balance here in India is the character this notion will take: will it be the idea of the absence of any number — or the idea of a number for such absence? Is it to be the mark of the empty, or the empty mark? The first keeps it estranged from numbers, merely part of the landscape through which they move; the second puts it on a par with them.

In the remainder of the fascinating and lyrical The Nothing That Is, Kaplan goes on to explore how various other cultures, from the Mayans to the Romans, contributed to the trans-civilizational mosaic that is zero as it made its way to modern mathematics, and examines its profound impact on everything from philosophy to literature to his own domain of mathematics. Complement it with this Victorian love letter to mathematics and the illustrated story of how the Persian polymath Ibn Sina revolutionized modern science.

BP

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