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The Ethics of Belief: The Great English Mathematician and Philosopher William Kingdon Clifford on the Discipline of Doubt and How We Can Trust a Truth

“It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence.”

The Ethics of Belief: The Great English Mathematician and Philosopher William Kingdon Clifford on the Discipline of Doubt and How We Can Trust a Truth

“The confidence people have in their beliefs is not a measure of the quality of evidence but of the coherence of the story that the mind has managed to construct,” Nobel-winning psychologist Daniel Kahneman observed in summarizing his pioneering behavioral psychology studies of how and why our minds mislead us. And yet our beliefs are the compass by which we navigate the landscape of reality, steering our actions and thus shaping our impact on that very reality. The great physicist David Bohm captured this inescapable dependency memorably: “Reality is what we take to be true. What we take to be true is what we believe… What we believe determines what we take to be true.”

How, then, do we align our beliefs with truth rather than illusion, so that we may perceive the most accurate representation of reality of which the human mind is capable, in turn guiding our actions toward noble and constructive ends?

That’s what the English mathematician and philosopher William Kingdon Clifford (May 4, 1845–March 3, 1879) explored with uncommon insight and rhetorical elegance nearly a century and a half before the golden age of “alternative facts.”

William Kingdon Clifford by John Collier

By the time tuberculosis claimed his life at the unjust age of thirty-three, Clifford had revolutionized mathematics by developing geometric algebra, had written a book of fairy tales for children, and had become the first person to suggest that gravity might be a function of an underlying cosmic geometry, developing what he called a “space-theory of matter” decades before Einstein transformed our understanding of the universe by bridging space and time into a geometry of spacetime.

But one of Clifford’s most lasting contributions is an essay titled “The Ethics of Belief,” originally published in an 1877 issue of the journal Contemporary Review and later included in Reason and Responsibility: Readings in Some Basic Problems of Philosophy (public library). In it, Clifford probes the nature of right and wrong, the infernal abyss between belief and truth, and our responsibility to the truth despite our habitual human deviations into unreason, delusion, and rationalization.

Clifford, at only thirty-two, begins with a parable containing an ethical thought experiment:

A shipowner was about to send to sea an emigrant-ship. He knew that she was old, and not overwell built at the first; that she had seen many seas and climes, and often had needed repairs. Doubts had been suggested to him that possibly she was not seaworthy. These doubts preyed upon his mind, and made him unhappy; he thought that perhaps he ought to have her thoroughly overhauled and refitted, even though this should put him at great expense. Before the ship sailed, however, he succeeded in overcoming these melancholy reflections. He said to himself that she had gone safely through so many voyages and weathered so many storms that it was idle to suppose she would not come safely home from this trip also. He would put his trust in Providence, which could hardly fail to protect all these unhappy families that were leaving their fatherland to seek for better times elsewhere. He would dismiss from his mind all ungenerous suspicions about the honesty of builders and contractors. In such ways he acquired a sincere and comfortable conviction that his vessel was thoroughly safe and seaworthy; he watched her departure with a light heart, and benevolent wishes for the success of the exiles in their strange new home that was to be; and he got his insurance-money when she went down in mid-ocean and told no tales.

What shall we say of him? Surely this, that he was verily guilty of the death of those men. It is admitted that he did sincerely believe in the soundness of his ship; but the sincerity of his conviction can in no wise help him, because he had no right to believe on such evidence as was before him. He had acquired his belief not by honestly earning it in patient investigation, but by stifling his doubts. And although in the end he may have felt so sure about it that he could not think otherwise, yet inasmuch as he had knowingly and willingly worked himself into that frame of mind, he must be held responsible for it.

Clifford adds a layer of ethical complexity by arguing that even if the ship hadn’t sunk, the shipowner would be guilty of the same error of judgment, for he “would not have been innocent, he would only have been not found out.” He writes:

The question of right or wrong has to do with the origin of his belief, not the matter of it; not what it was, but how he got it; not whether it turned out to be true or false, but whether he had a right to believe on such evidence as was before him.

[…]

For it is not possible so to sever the belief from the action it suggests as to condemn the one without condemning the other. No man holding a strong belief on one side of a question, or even wishing to hold a belief on one side, can investigate it with such fairness and completeness as if he were really in doubt and unbiased; so that the existence of a belief not founded on fair inquiry unfits a man for the performance of this necessary duty.

A century before psychologists came to identify such cognitive flaws as confirmation bias and the backfire effect, Clifford adds:

Nor is it that truly a belief at all which has not some influence upon the actions of him who holds it. He who truly believes that which prompts him to an action has looked upon the action to lust after it, he has committed it already in his heart. If a belief is not realized immediately in open deeds, it is stored up for the guidance of the future. It goes to make a part of that aggregate of beliefs which is the link between sensation and action at every moment of all our lives, and which is so organized and compacted together that no part of it can be isolated from the rest, but every new addition modifies the structure of the whole. No real belief, however trifling and fragmentary it may seem, is ever truly insignificant; it prepares us to receive more of its like, confirms those which resembled it before, and weakens others; and so gradually it lays a stealthy train in our inmost thoughts, which may someday explode into overt action, and leave its stamp upon our character for ever.

In a sentiment evocative of the Indian poet and philosopher Tagore’s reflections on the interdependence of existence, Clifford takes care to highlight the sociological tapestry out of which each strand of our private beliefs is frayed:

No one man’s belief is in any case a private matter which concerns himself alone. Our lives our guided by that general conception of the course of things which has been created by society for social purposes. Our words, our phrases, our forms and processes and modes of thought, are common property, fashioned and perfected from age to age; an heirloom which every succeeding generation inherits as a precious deposit and a sacred trust to be handled on to the next one, not unchanged but enlarged and purified, with some clear marks of its proper handiwork. Into this, for good or ill, is woven every belief of every man who has speech of his fellows. A awful privilege, and an awful responsibility, that we should help to create the world in which posterity will live.

In a passage of astounding pertinence today, as dangerous ideologies divorced from truth offer false comfort in “alternative facts” to the detriment of our common good, Clifford cautions:

Belief, that sacred faculty which prompts the decisions of our will, and knits into harmonious working all the compacted energies of our being, is ours not for ourselves but for humanity. It is rightly used on truths which have been established by long experience and waiting toil, and which have stood in the fierce light of free and fearless questioning. Then it helps to bind men together, and to strengthen and direct their common action. It is desecrated when given to unproved and unquestioned statements, for the solace and private pleasure of the believer; to add a tinsel splendour to the plain straight road of our life and display a bright mirage beyond it; or even to drown the common sorrows of our kind by a self-deception which allows them not only to cast down, but also to degrade us. Whoso would deserve well of his fellows in this matter will guard the purity of his beliefs with a very fanaticism of jealous care, lest at any time it should rest on an unworthy object, and catch a stain which can never be wiped away.

Three centuries after Western philosophy founding father and reason crusader René Descartes asserted that “it is not enough to have a good mind; the main thing is to apply it well,” Clifford adds:

In regard, then, to the sacred tradition of humanity, we learn that it consists, not in propositions or statements which are to be accepted and believed on the authority of the tradition, but in questions rightly asked, in conceptions which enable us to ask further questions, and in methods of answering questions. The value of all these things depends on their being tested day by day. The very sacredness of the precious deposit imposes upon us the duty and the responsibility of testing it, of purifying and enlarging it to the utmost of our power. He who makes use of its results to stifle his own doubts, or to hamper the inquiry of others, is guilty of sacrilege which centuries shall never be able to blot out.

How to purify and enlarge our access to truth is what Carl Sagan outlined a century later in his timeless Baloney Detection Kit, but Clifford himself crystallizes the most effective approach in a wonderfully succinct dictum:

It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence.

Complement with Lewis Thomas, one of the most enchanting writers of the past century, on the transmutation of ignorance into truth, Karl Popper on the crucial difference between truth and certainty, and Laura (Riding) Jackson’s unusual and profound 1967 manifesto for the telling of truth.

BP

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

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