We’ve already seen how Shakespeare changed everything and how Fibonacci, “the man of numbers,” changed the world. But in this short video, Professor Roger Bowley uses Shakespeare’s iambic pentameter and penchant for the number 14 to show that the bard was quite the man of numbers himself, revealing a relationship between poetry and mathematics much more tightly knit than the standard cultural compartmentalization would have you believe.
Poetry is an extreme form of wordplay, in which numbers dictate form and structure to give more beauty to it.
How a forgotten visionary’s futuristic dream dared generations to reimagine the relationship between nature and human creativity.
By Maria Popova
Nineteen years after the publication of Isaac Newton’s epoch-making Principia — in England, in Latin — the prodigy mathematician Émilie du Châtelet set out to translate his ideas into her native French, making them more comprehensible in the process. Her more-than-translation — which includes several of her mathematical corrections and clarifications of Newton’s imprecisions, and which remains the only comprehensive edition in French to this day — popularized his ideas in France and, from this epicenter of the Enlightenment, spread them centripetally throughout the rest of the Continent, rendering Newton himself an emblem of the Enlightenment the sweep of which he never lived to see.
Not long after Du Châtelet’s untimely death, her legacy reached one of her most gifted compatriots — the visionary architect Étienne-Louis Boullée (February 12, 1728–February 4, 1799), who fell under Newton’s spell. Determined to honor Newton with a worthy cenotaph — a memorial tomb for a person buried elsewhere — he designed a sphere 500 feet in diameter, taller than the Pyramids of Giza, nested into a colossal pedestal and encircled by hundreds of cypress trees, giving it the transfixing illusion of being both half-buried into the Earth and hovering unmoored from gravity. It was also, in essence, the world’s first domed planetarium design.
The cenotaph was a touching gesture in the first place — a Frenchman honoring a genius born of and interred in England, a nation with which Boullée’s own had been in near-ceaseless war for centuries, with those tensions at an all-time high at the time of his design, thanks to the American Revolutionary War. Doubly touching was his choice of a sphere: One of Newton’s most revolutionary contributions — the mathematical inference that because gravity is weaker at the equator, the shape of the Earth must be spherical — had defied France’s greatest son, René Descartes, who maintained that the Earth was egg-shaped. When Boullée was still a boy, a young Frenchman — Émilie du Châtelet’s mathematics tutor — had joined a perilous Arctic expedition to prove Newton correct. Two centuries later, in the wake of the world’s grimmest war yet, a queer Quaker Englishman would do the same, risking his life to defend the epoch-making theory of a German Jew — the theory of relativity that ultimately subverted Newton. Another world war later, Einstein himself would appeal to what he called “the common language of science” — that truth-seeking contact with nature and reality that transcends all borders and all nationalisms, the impulse that animated Boullée’s bold homage to Newton.
While governed by the credo that “our buildings — and our public buildings in particular — should be to some extent poems,” Boullée also believed that science could magnify the poetry of public spaces, which must at bottom reflect the principles of the grand designer: Nature. A century before the teenage Virginia Woolf wrote that “all the Arts… imitate as far as they can the one great truth that all can see,” Boullée insisted:
No idea exists that does not derive from nature… It is impossible to create architectural imagery without a profound knowledge of nature: the Poetry of architecture lies in natural effects. That is what makes architecture an art and that art sublime.
Architecture in the modern sense was then a young art, because the art-science of perspective was so novel. Newton’s optics, derived directly from the laws of nature, had revolutionized it all. Boullée came to define architecture as “the art of creating perspectives by the arrangement of volumes,” but a highly poetic art:
The real talent of an architect lies in incorporating in his work the sublime attraction of Poetry.
The poetry of architecture, he argued, resides in using perspective and light in such a way that “our senses are reminded of nature.” He interpreted the laws of nature, as clarified by Newton’s optics and mathematics, to intimate that no shape embodies this serenade to the senses with greater power and precision than the sphere:
A sphere is, in all respects, the image of perfection. It combines strict symmetry with the most perfect regularity and the greatest possible variety; its form is developed to the fullest extent and is the simplest that exists; its shape is outlined by the most agreeable contour and, finally, the light effects that it produces are so beautifully graduated that they could not possibly be softer, more agreeable or more varied. These unique advantages, which the sphere derives from nature, have an immeasurable hold over our senses.
And so Boullée predicated his cenotaph for Newton on an enormous sphere that would convey his ultimate intent for the temple — to arouse in the visitor’s soul “feelings in keeping with religious ceremonies,” a sense of grandeur leaving them “moved by such an excess of sensibility… that all the faculties of our soul are disturbed to such an extent that we feel it is departing from our body” — an effect always best achieved not by an enormity of sheer size and space but by a considered contrast of scales. No building, he observed, “calls for the Poetry of architecture” more than a memorial to the dead. Believing that architecture, like all art, should ultimately serve to enlarge our sense of aliveness, and that we are never more alive than when we are rooted in our creaturely senses, Boullée insisted that the key to this sense of grandeur lies in applying the principles of nature’s mathematics with poetic subtlety — the principles laid bare in the Principia, the principles that “derive from order, the symbol of wisdom.” He wrote:
Symmetry… is what results from the order that extends in every direction and multiplies them at our glance until we can no longer count them. By extending the sweep of an avenue so that its end is out of sight, the laws of optics and the effects of perspective given an impression of immensity; at each step, the objects appear in a new guise and our pleasure is renewed by a succession of different vistas. Finally, by some miracle which in fact is the result of our own movement but which we attribute to the objects around us, the latter seem to move with us, as if we had imparted Life to them.
But my favorite part of the story is that Boullée found his formative inspiration, not only for the Newton cenotaph and but for his entire creative philosophy, in an unusual encounter with trees — those profoundest of teachers.
One evening, heavy with grief, Boullée went for a walk along the edge of a forest. Under the moonlight, he noticed his shadow. He had seen his shadow a thousand times before, but the peculiar lens of his psychic state rendered it entirely new — a living artwork of “extreme melancholy.” Looking around, he saw the shadows of the trees in this new light, too, etching onto the ground the profound drama of life. The entire scene was suddenly awash in “all that is sombre in nature.” He had seen the state of his soul mirrored back by the natural world, as we so often do in those rawest moments when we are stripped to the base of our being, grounded into our creaturely senses.
This was the moment of Boullée’s artistic awakening — that moment of revelation when, as Virginia Woolf wrote in her exquisite account of her own artistic awakening, something lifts “the cotton wool of daily life” and we see the familiar world afresh. Boullée recounted:
The mass of objects stood out in black against the extreme wanness of the light. Nature offered itself to my gaze in mourning. I was struck by the sensations I was experiencing and immediately began to wonder how to apply this, especially to architecture. I tried to find a composition made up of the effect of shadows. To achieve this, I imagined the light (as I had observed it in nature) giving back to me all that my imagination could think of. That was how I proceeded when I was seeking to discover this new type of architecture.
He called this new architecture “the architecture of shadow.” His vision for Newton’s cenotaph was its grand testament:
I attempted to create the greatest of all effects, that of immensity; for that is what gives us lofty thoughts as we contemplate the Creator and give us celestial sensations.
He attempted, more than that, to honor Newton on his own terms, by the essence of his genius:
O Newton! With the range of your intelligence and the sublime nature of your Genius, you have defined the shape of the earth; I have conceived the idea of enveloping you with your discovery… your own self. How can I find outside you anything worthy of you?
In a further homage to Newton’s legacy, with Boullée regarded as a “divine system” of laws, he chose to suspend a sole spherical lamp over the tomb as the only decoration in the entire monument — anything else, he felt, would be “committing sacrilege.” The contrast of scales — the smaller sphere of the lamp inside the enormous sphere of the building — would dramatize the contrast of light and shadow, just as the moonlight had done that fateful night of artistic revelation by the trees. This would give the visitor the sense that they are “as if by magic floating in the air, borne in the wake of images in the immensity of space.” Boullée considered the play of light the vital element in this enchantment:
It is light that produces impressions which arouse in us various contradictory sensations depending on whether they are brilliant or sombre. If I could manage to diffuse in my temple magnificent light effects I would fill the onlooker with joy; but if, on the contrary, my temple had only sombre effects, I would fill him with sadness. If I could avoid direct light and arrange for its presence without the onlooker being aware of its source, the ensuing effect of mysterious daylight would produce inconceivable impression and, in a sense, a truly enchanting magic quality.
At a time long before readily available electric light and light-projection, he leaned on Newton’s optics to envision something that was part Stonehenge and part Hayden Planetarium. A century and a half before the first modern planetarium dome, Boullée dotted the black interior of his dome with an intricate arrangement of tiny holes reflecting the positions of the constellations and the planets, streaming in daylight to create an enchanting nightscape inside. But unlike the modern counterpart, Boullée’s was a reversible planetarium — at night, the sole spherical light would irradiate the tiny holes from the other direction, making the dome appear as a self-contained universe if viewed from above. This, lest we forget, was the golden age of aeronautics, when hot-air balloons first defied gravity to lift the human animal into the sky.
Too visionary for its era, the cenotaph was never built, but Boullée’s ink-and-wash drawings circulated widely in the final decade of his life, eliciting both gasping admiration and merciless derision — the fate of the true visionary. With the publication of his impassioned and insightful writings nearly two centuries after his death, translated by Helen Rosenau, his vision went on to inspire generations of modern artists and architects with a new way of thinking about the poetry of public spaces and the relationship between nature and human creativity.
No one is more exacting than a man who is not conversant with a given art for he is unable to imagine all the difficulties the artist has to overcome.
His ultimate satisfaction was not the reception or execution of his designs, but the inexhaustible source of their inspiration — the elemental wellspring of the creative impulse behind all art and all science, that richest and readiest reward of our aliveness:
The artist… is always making discoveries and spends his life observing nature.
“From our lonely corner of the cosmos we have used creativity and imagination to shape words and images and structures and sounds to express our longings and frustrations, our confusions and revelations, our failures and triumphs.”
By Maria Popova
“Praised be the fathomless universe, for life and joy, and for objects and knowledge curious,” Walt Whitman wrote as he stood discomposed and delirious before a universe filled with “forms, qualities, lives, humanity, language, thoughts, the ones known, and the ones unknown, the ones on the stars, the stars themselves, some shaped, others unshaped.” And yet the central animating force of our species, the wellspring of our joy and curiosity, the restlessness that gave us Whitman and Wheeler, Keats and Curie, is the very fathoming of this fathomless universe — an impulse itself a marvel in light of our own improbability. Somehow, we went from bacteria to Bach; somehow, we learned to make fire and music and mathematics. And here we are now, walking wildernesses of mossy feelings and brambled thoughts beneath an overstory of one hundred trillion synapses, coruscating with the ultimate question: What is all this?
That is what physicist and mathematician Brian Greene explores with great elegance of thought and poetic sensibility in Until the End of Time: Mind, Matter, and Our Search for Meaning in an Evolving Universe (public library). Nearly two centuries after the word scientistwas coined for the Scottish mathematician Mary Somerville when her unexampled book On the Connexion of the Physical Sciences brought together the separate disciplinary streams of scientific inquiry into a single river of knowledge, Greene draws on his own field, various other sciences, and no small measure of philosophy and literature to examine what we know about the nature of reality, what we suspect about the nature of knowledge, and how these converge to shine a sidewise gleam on our own nature. With resolute scientific rigor and uncommon sensitivity to the poetic syncopations of physical reality, he takes on the questions that bellow through the bone cave atop our shoulders, the cave against whose walls Plato flickered his timeless thought experiment probing the most abiding puzzle: How are we ever sure of reality? — a question that turns the mind into a Rube Goldberg machine of other questions: Why is there something rather than nothing? How did life emerge? What is consciousness?
Although science is Greene’s raw material in this fathoming — its histories, its theories, its triumphs, its blind spots — he emerges, as one inevitably does in contemplating these colossal questions, a testament to Einstein’s conviction that “every true theorist is a kind of tamed metaphysicist.”
Looking back on how he first grew enchanted with what he calls “the romance of mathematics” and its seductive promise to unveil the timeless laws of nature, Greene writes:
Creativity constrained by logic and a set of axioms dictates how ideas can be manipulated and combined to reveal unshakable truths.
The appeal of a law of nature might be its timeless quality. But what drives us to seek the timeless, to search for qualities that may last forever? Perhaps it all comes from our singular awareness that we are anything but timeless, that our lives are anything but forever.
We emerge from laws that, as far as we can tell, are timeless, and yet we exist for the briefest moment of time. We are guided by laws that operate without concern for destination, and yet we constantly ask ourselves where we are headed. We are shaped by laws that seem not to require an underlying rationale, and yet we persistently seek meaning and purpose.
Somewhere along the way of our seeking, at one life-point or another, against one wall or another, we all arrive at what David Foster Wallace, vanquisher of euphemism, called “the recognition that I’m going to die, and die very much alone, and the rest of the world is going to go merrily on without me.” Insisting that from that recognition arises our shimmering capacity for creativity, for beauty, for meaning-making, Greene endeavors to explore “the breathtaking ways in which restless and inventive minds have illuminated and responded to the fundamental transience of everything” — minds ranging from Shakespeare to Wallace, from Sappho to Einstein.
A century after Rachel Carson observed (in a trailblazing essay that pioneered the very genre of poetic science writing in which Greene himself dwells) that “against this cosmic background the lifespan of a particular plant or animal appears, not as drama complete in itself, but only as a brief interlude in a panorama of endless change,” he writes:
In the fullness of time all that lives will die. For more than three billion years, as species simple and complex found their place in earth’s hierarchy, the scythe of death has cast a persistent shadow over the flowering of life. Diversity spread as life crawled from the oceans, strode on land, and took flight in the skies. But wait long enough and the ledger of birth and death, with entries more numerous than stars in the galaxy, will balance with dispassionate precision. The unfolding of any given life is beyond prediction. The final fate of any given life is a foregone conclusion.
Despite how we may distract ourselves from that omnipresent conclusion, we live terrified of our own erasure, but that very terror impels us to more-than-exist — to live, to love, to compose poems and symphonies and equations. With an eye to “the inner life that comes hand in hand with our refined cognitive capacities,” Greene writes:
The mental faculties that allow us to shape and mold and innovate are the very ones that dispel the myopia that would otherwise keep us narrowly focused on the present. The ability to manipulate the environment thoughtfully provides the capacity to shift our vantage point, to hover above the timeline and contemplate what was and imagine what will be. However much we’d prefer it otherwise, to achieve “I think, therefore I am” is to run headlong into the rejoinder “I am, therefore I will die.”
Perhaps our creative forays, from the stags at Lascaux to the equations of general relativity, emerge from the brain’s naturally selected but overly active ability to detect and coherently organize patterns. Perhaps these and related pursuits are exquisite but adaptively superfluous by-products of a sufficiently large brain released from full-time focus on securing shelter and sustenance… What lies beyond question is that we imagine and we create and we experience works, from the Pyramids to the Ninth Symphony to quantum mechanics, that are monuments to human ingenuity whose durability, if not whose content, point toward permanence.
One aspect of Greene’s argument, however, deserves more nuanced consideration: Historically, every time we humans have assumed that a certain feature or faculty is ours alone in the whole of “Creation” — sentience, tools, language, consciousness — we have been wrong. Greene makes the baseline assumption that we alone are aware of our own finitude. “It is only you and I and the rest of our lot,” he asserts, “that can reflect on the distant past, imagine the future, and grasp the darkness that awaits.” But what of elephants and their capacity for grief, deep and documented? What is grief if not a savaging consciousness of the fact that death severs the arrow of time, that what once was — living, beloved — will never again be, while we are left islanded in the present, shipwrecked by an absence?
Still, unblunted by this marginal error of exclusivity is Greene’s astute insight into the elemental equivalence: we are doomed to decay, and so we cope by creating. He highlights two factors that jointly gave rise to the self-awareness seeding our terror and to our wondrous reach for transcendence: entropy and evolution. Across three hundred pages, he fans out the fabric of our present understanding, deftly untangling then interweaving the science of everything from black holes to quanta to DNA, tracing how matter made mind made imagination, probing the pull of eternity and storytelling and the sublime, and arriving at a final chapter lyrically titled “The Nobility of Being,” in which he contemplates how these processes and phenomena, described and discovered by minds honed by millennia of evolution, converge to illuminate our search for meaning:
Most of us deal quietly with the need to lift ourselves beyond the everyday. Most of us allow civilization to shield us from the realization that we are part of a world that, when we’re gone, will hum along, barely missing a beat. We focus our energy on what we can control. We build community. We participate. We care. We laugh. We cherish. We comfort. We grieve. We love. We celebrate. We consecrate. We regret. We thrill to achievement, sometimes our own, sometimes of those we respect or idolize.
Through it all, we grow accustomed to looking out to the world to find something to excite or soothe, to hold our attention or whisk us to someplace new. Yet the scientific journey we’ve taken suggests strongly that the universe does not exist to provide an arena for life and mind to flourish. Life and mind are simply a couple of things that happen to happen. Until they don’t. I used to imagine that by studying the universe, by peeling it apart figuratively and literally, we would answer enough of the how questions to catch a glimpse of the answers to the whys. But the more we learn, the more that stance seems to face in the wrong direction.
Looking for the universe to hug us, its transient conscious squatters, is understandable, but that’s just not what the universe does.
Even so, to see our moment in context is to realize that our existence is astonishing. Rerun the Big Bang but slightly shift this particle’s position or that field’s value, and for virtually any fiddling the new cosmic unfolding will not include you or me or the human species or planet earth or anything else we value deeply.
We exist because our specific particulate arrangements won the battle against an astounding assortment of other arrangements all vying to be realized. By the grace of random chance, funneled through nature’s laws, we are here.
In the final pages, Greene both affirms and refutes Borges’s refutation of time, guiding us, perishable miracles that we are, to the wellspring of meaning in an impartial universe and ending the book with the word — a curious word, improbable for a physicist — on which Whitman perched his entire cosmogony:
Whereas most life, miraculous in its own right, is tethered to the immediate, we can step outside of time. We can think about the past, we can imagine the future. We can take in the universe, we can process it, we can explore it with mind and body, with reason and emotion. From our lonely corner of the cosmos we have used creativity and imagination to shape words and images and structures and sounds to express our longings and frustrations, our confusions and revelations, our failures and triumphs. We have used ingenuity and perseverance to touch the very limits of outer and inner space, determining fundamental laws that govern how stars shine and light travels, how time elapses and space expands — laws that allow us to peer back to the briefest moment after the universe began and then shift our gaze and contemplate its end.
As we hurtle toward a cold and barren cosmos, we must accept that there is no grand design. Particles are not endowed with purpose. There is no final answer hovering in the depths of space awaiting discovery. Instead, certain special collections of particles can think and feel and reflect, and within these subjective worlds they can create purpose. And so, in our quest to fathom the human condition, the only direction to look is inward. That is the noble direction to look. It is a direction that forgoes ready-made answers and turns to the highly personal journey of constructing our own meaning. It is a direction that leads to the very heart of creative expression and the source of our most resonant narratives. Science is a powerful, exquisite tool for grasping an external reality. But within that rubric, within that understanding, everything else is the human species contemplating itself, grasping what it needs to carry on, and telling a story that reverberates into the darkness, a story carved of sound and etched into silence, a story that, at its best, stirs the soul.
Until the End of Time, a splendid and invigorating read in its entirety, left me with the evolutionary miracle of Shelley on my mind — a fragment from the last poetic work he published before he met his own untimely finitude in the entropic spectacle of a sudden storm on the Italian gulf, long before humanity had fathomed entropy and evolution:
Talk no more
Of thee and me, the future and the past…
Earth and ocean,
Space, and the isles of life or light that gem
The sapphire floods of interstellar air,
This firmament pavilioned upon chaos…
Of suns and worlds, and men and beasts, and flowers
With all the violent and tempestuous workings
By which they have been, are, or cease to be,
Is but a vision: all that it inherits
Are motes of a sick eye, bubbles and dreams;
Thought is its cradle and its grave, nor less
The future and the past are idle shadows
Of thought’s eternal flight — they have no being.
Nought is but that it feels itself to be.
An imaginative extension of Euclid’s parallel postulate into life, liberty, and the pursuit of happiness.
By Maria Popova
“The joy of existence must be asserted in each one, at every instant,” Simone de Beauvoir wrote in her paradigm-shifting treatise on how freedom demands that happiness become our moral obligation. A decade and a half later, the mathematician and writer Lillian R. Lieber (July 26, 1886–July 11, 1986) examined the subject from a refreshingly disparate yet kindred angle.
Einstein was an ardent fan of Lieber’s unusual, conceptual books — books discussing serious mathematics in a playful way that bridges science and philosophy, composed in a thoroughly singular style. Like Einstein himself, Lieber thrives at the intersection of science and humanism. Like Edwin Abbott and his classic Flatland, she draws on mathematics to invite a critical shift in perspective in the assumptions that keep our lives small and our world inequitable. Like Dr. Seuss, she wrests from simple verses and excitable punctuation deep, calm, serious wisdom about the most abiding questions of existence. She emphasized that her deliberate line breaks and emphatic styling were not free verse but a practicality aimed at facilitating rapid reading and easier comprehension of complex ideas. But Lieber’s stubborn insistence that her unexampled work is not poetry should be taken with the same grain of salt as Hannah Arendt’s stubborn insistence that her visionary, immensely influential political philosophy is not philosophy.
In her hundred years, Lieber composed seventeen such peculiar and profound books, illustrated with lovely ink drawings by her husband, the artist Hugh Lieber. Among them was the 1961 out-of-print gem Human Values and Science, Art and Mathematics (public library) — an inquiry into the limits and limitless possibilities of the human mind, beginning with the history of the greatest revolution in geometry and ending with the fundamental ideas and ideals of a functioning, fertile democracy.
Lieber paints the conceptual backdrop for the book:
This book is really about
Life, Liberty, and the Pursuit of Happiness,
using ideas from mathematics
to make these concepts less vague.
We shall see first what is meant by
“thinking” in mathematics,
and the light that it sheds on both the
CAPABILITIES and the LIMITATIONS
of the human mind.
And we shall then see what bearing this can have
on “thinking” in general —
even, for example, about such matters as Life, Liberty, and the Pursuit of Happiness!
For we must admit that our “thinking”
about such things,
without this aid,
often leads to much confusion —
mistaking LICENSE for LIBERTY,
often resulting in juvenile delinquency;
mistaking MONEY for HAPPINESS,
often resulting in adult delinquency;
mistaking for LIFE itself
just a sordid struggle for mere existence!
Embedded in the history of mathematics, Lieber argues, is an allegory of what we are capable of as a species and how we can use those capabilities to rise to our highest possible selves. In the first chapter, titled “Freedom and Responsibility,” she chronicles the revolution in our understanding of nature and reality ignited by the advent of non-Euclidean geometry — the momentous event Lieber calls “The Great Discovery of 1826.” She writes:
One of the amazing things
in the history of mathematics
happened at the beginning of the 19th century.
As a result of it,
the floodgates of discovery
were open wide,
and the flow of creative contributions
is still on the increase!
this amazing phenomenon
was due to a mere
CHANGE OF ATTITUDE!
Perhaps I should not say “mere,”
since the effect was so immense —
which only goes to show that
a CHANGE OF ATTITUDE
can be extremely significant
and we might do well to examine our ATTITUDES
toward many things, and people —
this might be the most rewarding,
as it proved to be in mathematics.
In order to fully comprehend a revolutionary change in attitude, Lieber points out, we need to first understand the old attitude — the former worldview — supplanted by the revolution. To appreciate “The Great Discovery of 1826,” we must go back to Euclid:
first put together
the various known facts of geometry
into a SYSTEM,
instead of leaving them as
isolated bits of information —
as in a quiz program!
has served for many centuries
as a MODEL for clear thinking,
and has been and still is
of the greatest value to the human race.
Lieber unpacks what it means to build such a “model for clear thinking” — networked logic that makes it easier to learn and faster to make new discoveries. With elegant simplicity, she examines the essential building blocks of such a system and outlines the basics of mathematical logic:
In constructing a system,
one must begin with
a few simple statements
by means of logic,
one derives the “consequences.”
We can thus
“figure out the consequences”
before they hit us.
And this we certainly need more of!
Thus Euclid stated such
(called “postulates” in mathematics)
“It shall be possible to draw
a straight line joining
any two points,”
and others like it.
he derived many complicated theorems
like the well-known
and many, many others.
And, as we all know,
to “prove” any theorem
one must show how
to “derive” it from the postulates —
every claim made in a “proof”
must be supported by reference to
the postulates or
to theorems which have previously
already been so “proved”
from the postulates.
Of course Theorem #1
must follow from
the postulates ONLY.
Now what about
the postulates themselves?
How can THEY be “proved”?
CANNOT be PROVED at all —
since there is nothing preceding them
from which to derive them!
This may seem disappointing to those who
thought that in
EVERYTHING is proved!
But you can see that
this is IMPOSSIBLE,
even in mathematics,
since EVERY SYSTEM must necessarily
START with POSTULATES,
and these are NOT provable,
since there is nothing preceding them
from which to derive them.
This circularity of certainty permeates all of science. In fact, strangely enough, the more mathematical a science is, the more we consider it a “hard science,” implying unshakable solidity of logic. And yet the more mathematical a mode of thinking, the fuller it is of this circularity reliant upon assumption and abstraction. Euclid, of course, was well aware of this. He reconciled the internal contradiction of the system by considering his unproven postulates to be “self-evident truths.” His system was predicated on using logic to derive from these postulates certain consequences, or theorems. And yet among them was one particular postulate — the famous parallel postulate — which troubled Euclid.
The parallel postulate states that if you were to draw a line between two points, A and B, and then take a third point, C, not on that line, you can only draw one line through C that will be parallel to the line between A and B; and that however much you may extend the two parallel lines in space, they will never cross.
Euclid, however, wasn’t convinced this was a self-evident truth — he thought it ought to be mathematically proven, but he failed to prove it. Generations of mathematicians failed to prove it over the following thirteen centuries. And then, in the early nineteenth century, three mathematicians — Nikolai Lobachevsky in Russia, János Bolyai in Hungary, and Carl Friedrich Gauss in Germany — independently arrived at the same insight: The challenge of the parallel postulate lay not in the proof but, as Lieber puts it, in “the very ATTITUDE toward what postulates are” — rather than considering them to be “self-evident truths” about nature, they should be considered human-made assumptions about how nature works, which may or may not reflect the reality of how nature work.
This may sound like a confounding distinction, but it is a profound one — it allowed mathematicians to see the postulates not as sacred and inevitable but as fungible, pliable, tinkerable with. Leaving the rest of the Euclidean system intact, these imaginative nineteenth-century mathematicians changed the parallel postulate to posit that not one but two lines can be drawn through point C that would be parallel to the line between A and B, and the entire system would still be self-consistent. This resulted in a number of revolutionary theorems, including the notion that the sum of angles in a triangle could be different from 180 degrees — greater if the triangle is drawn on the surface of a sphere, for instance, or lesser if drawn on a concave surface.
It was a radical, thoroughly counterintuitive insight that simply cannot be fathomed, much less diagramed, in flat space. And yet it wasn’t a mere thought experiment, an amusing and suspicious mental diversion. It bust open the floodgates of creativity in mathematics and physics by giving rise to non-Euclidean geometry — an understanding of curved three-dimensional space which we now know is every bit as real as the geometry of flat surfaces, abounding in nature in everything from the blossom of a calla lily to the growth pattern of a coral reef to the fabric of spacetime of which everything that ever was and ever will be is woven. In fact, Einstein himself would not have been able to arrive at his relativity theory, nor bridge space and time into the revolutionary notion of spacetime, without non-Euclidean geometry.
Here, Lieber makes the conceptual leap that marks her books as singular achievements in thought — the leap from mathematics and the understanding of nature to psychology, sociology, and the understanding of human nature. Reflecting on the larger revolution in thought that non-Euclidean geometry embodied in its radical refusal to accept any truth as self-evident, she questions the notion of “eternal verities” — a term popularized by the eighteen-century French philosopher Claude Buffier to signify the aspects of human consciousness that allegedly furnish universal, indubitable moral and humane values. Considering how the relationship between creative limitation, freedom, humility, and responsibility shapes our values, Lieber writes:
Even though mathematics is
only a MAN-MADE enterprise,
man has done very well for himself
in this domain,
where he has
FREEDOM WITH RESPONSIBILITY —
though he has learned the
HUMILITY that goes with
knowing that he does
NOT have access to
“Self-evident truths” and
that he is NOT God —
yet he knows also that
he is not a mouse either,
but a man,
with all the
HUMAN DIGNITY and the
needed to develop
the wonderful domain of
The very dignity and ingenuity driving mathematics, Lieber points out in another lovely conceptual bridging of ideas, is also the motive force behind the central aspiration of human life, the one which Albert Camus saw as our moral obligation — the pursuit of happiness.
In the final chapter, titled “Life, Liberty and the Pursuit of Happiness,” Lieber recounts the principle of metamathematics demanding that a set of postulates within any system not contradict one another in order for the system to be self-consistent, and considers mathematics as a sandbox for the subterranean morality without which human life is unlivable:
[This] means of course that
CANNOT SERVE as an instrument of thought!
Now is not this statement
usually considered to be
a MORAL principle?
without it we cannot have
ANY satisfactory mathematical system,
nor ANY satisfactory system of thought —
indeed we cannot even PLAY a GAME properly
with CONTRADICTORY rules!
In a similar way,
I wish to make the point that
there are other important MORAL ideas
BEHIND THE SCENES,
without which there cannot be
ANY MATHEMATICS or SCIENCE.
And therefore, in this sense,
Science is NOT AT ALL AMORAL —
any more than one could have
a fruitful and non-trivial postulate set
which is not subject to
the METAmathematical demand for
One of these “behind-the-scenes” moral ideas, Lieber argues, is the notion of taking Life itself as a basic postulate:
there can be
no living thing —
no human race —
also of course
no music, no art,
I am not suggesting that we consider here WHETHER life is worth living,
whether it would make more “sense”
to commit suicide,
whether it is all just
“Sound and fury, signifying nothing.”
I am proposing that
LIFE be taken as a POSTULATE,
and therefore not subject to proof,
just like any other postulate.
But I propose to MODIFY this
and take more specifically as
THE PRESERVATION OF
LIFE FOR THE HUMAN RACE
is a goal of human effort.
This does not mean that
we are to go about
wantonly killing animals,
but to do this only when
it is necessary to support
HUMAN life —
for prevention of disease,
Indeed a horse or dog or other animals,
through their friendliness and sincerity,
might actually HELP to sustain
Man’s spirit and faith and even his life.
And I interpret this postulate
also to mean that
or “ganging up” on one little fox —
a hole gang of men and women
(and corrupting even horses and dogs
to help!) —
is really a cowardly act,
so unsportsmanlike that it is amazing
how this activity could ever be called
All this is by way of interpreting
the meaning to be given to Postulate I:
ACCEPTANCE of LIFE for the HUMAN RACE.
Surely everyone will accept the idea that
is definitely present,
behind the scenes,
in science or mathematics.
But this is not all.
For, I take this postulate to mean also
that we are not to limit it to
only a PART of the human race,
as Hitler did,
because this inevitably leads to WAR,
and in this day of
and CBR (chemical, biological, radiological)
this would certainly contradict
would it not?
May I say at the very outset that
the “SYSTEM” suggested here
makes no pretense of finality (!),
remembering how difficult it is,
EVEN in MATHEMATICS,
to have a postulate set which is
Nevertheless, one must go on,
one must TRY,
one must do one’s BEST,
as in mathematics and sciences.
And so, let us continue, in all humility,
to try to make
what can only at best be regarded as
in the hope that the basic idea —
that there is a MORALITY behind the scenes
in Mathematics and Science —
may prove to be helpful
and may be further
improved and strengthened
as time goes on.
Drawing on the consequence of the Second Law of Thermodynamics, which implies for living things an inevitable degradation toward destruction, Lieber offers additional postulates for the moral system that undergirds a thriving democracy:
Each INDIVIDUAL HUMAN BEING
must fight this “degeneration,”
must cling to LIFE as long as possible,
must grow and create —
physically and/or mentally.
And for this we need
We must all have the LIBERTY to
so grow and create,
without of course interfering
with each other’s growth,
This Freedom or LIBERTY
must be accompanied by
if it is not to lead to
individuals or groups
which would of course
CONTRADICT the other postulates.
All this is of course very DIFFICULT to do,
accepting LIFE without whimpering,
growing without interfering with
the growth of others,
it involves what Goethe called
But how can this be done?
It seems clear that we must now add
The PURSUIT of HAPPINESS
is a goal of human effort.
For without some happiness,
or at least the hope of some happiness
(the “pursuit” of happiness)
it would be impossible
to accept “cheerfully”
the program outlined above.
And such acceptance leads to
a calm, sane performance of our work,
in the spirit in which a mathematician
accepts the postulates of a system
and accepts his creative work
based on these —
accepting even the Great Difficulties
which he encounters
and is determined to conquer.
And I finally believe that
the results of such a formulation
will re-discover the conclusions
reached by the
great religious leaders and the
Lieber distills from this conception of the system “some invariants and some differences,” drawing from science and mathematics a working model for democracy:
(1) Invariants: LIFE — which demands
(a) Sufficient and proper food;
(b) Good Health;
(c) Education — both mental and physical;
(d) NO VIOLENCE!
(a real scientist does NOT go
into his laboratory with an AXE
with which to DESTROY his apparatus,
but rather with a well-developed BRAIN,
and lots of PATIENCE
with which to CREATE new things
which will be BENEFICIAL to the
This of course implies PEACE,
and better still
(e) FRIENDSHIP between K and K’!
(f) Humility — remembering that
he will NEVER know THE “truth”
(g) And all this of course
requires a great deal of
(2) Differences which will
NOT PREVENT both K and K’
from studying the universe
WITH EQUAL RIGHT and EQUAL
which is certainly
the clearest concept of
what DEMOCRACY is:
(a) Different coordinate systems
(b) Differences in color of skin!
(c) Different languages — or
other means of communication.
And please do not consider this program
as an unattainable “Utopia,”
for it really WORKS in
Science and Mathematics,
as we have seen,
and can also work in
if we would only
put our BEST EFFORTS into it,
fighting WARS —
(HOT or COLD)
or even PREPARING for wars —
HATING other people,