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.
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,
A 19th-century love letter to the most limitless medium of thought.
By Maria Popova
Mathematics is at once the most precise and the most abstract instrument of thought — a convergence of symbol and sentience utterly poetic in its ability to convey the most complex underlying laws of the universe in stunning simplicity of expression. It mirrors the world back to itself both condensed and expanded, granting us an enlarged understanding through the art of distillation. Ada Lovelace considered mathematics the “poetical science” and, in contemplating the nature of the imagination, called it “the language of the unseen relations between things.” Perhaps E = mc2 is the greatest line of poetry ever written, then. At the very least, it inhabits the same world as “To be, or not to be”; it is the mathematical counterpart to “the still point of the turning world.” So is it any wonder that mathematics renders some of humanity’s most potent minds nothing short of besotted?
Hardly anyone has captured the mesmerism of mathematics more beautifully than the pioneering 19th-century English mathematician James Joseph Sylvester (September 3, 1814–March 15, 1897) in a magnificent speech he delivered on February 22, 1877 in Baltimore.
Sylvester considers this difficult art of conceptual condensation:
It is the constant aim of the mathematician to reduce all his expressions to the lowest terms, to retrench every superfluous word and phrase, and to condense the Maximum of meaning into the Minimum of language.
And yet the joy of mathematics, he argues, isn’t an esoteric pursuit reserved for academically trained mathematicians — rather, it is a supreme and universal delight of the human mind at play with itself:
I have reason to think that the taste for mathematical science, even in its most abstract form, is much more widely diffused than is generally supposed…
This wide appeal of the mathematical spirit, Sylvester observes, stems from its immensity of scope and its infinite range of intimacies with the nature of the world:
Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze: it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence.
Every science becomes more perfect, approaches more closely to its own ideal, in proportion as it imitates or imbibes the mathematical form and spirit.
“Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view.”
By Maria Popova
Daniel Tammet was born with an unusual mind — he was diagnosed with high-functioning autistic savant syndrome, which meant his brain’s uniquely wired circuits made possible such extraordinary feats of computation and memory as learning Icelandic in a single week and reciting the number pi up to the 22,514th digit. He is also among the tiny fraction of people diagnosed with synesthesia — that curious crossing of the senses that causes one to “hear” colors, “smell” sounds, or perceive words and numbers in different hues, shapes, and textures. Synesthesia is incredibly rare — Vladimir Nabokov was among its few famous sufferers — which makes it overwhelmingly hard for the majority of us to imagine precisely what it’s like to experience the world through this sensory lens. Luckily, Tammet offers a fascinating first-hand account in Thinking In Numbers: On Life, Love, Meaning, and Math (public library) — a magnificent collection of 25 essays on “the math of life,” celebrating the magic of possibility in all its dimensions. In the process, he also invites us to appreciate the poetics of numbers, particularly of ordered sets — in other words, the very lists that dominate everything from our productivity tools to our creative inventories to the cheapened headlines flooding the internet.
Reflecting on his second book, Embracing the Wide Sky: A Tour Across the Horizons of the Mind, and the overwhelming response from fascinated readers seeking to know what it’s really like to experience words and numbers as colors and textures — to experience the beauty that a poem and a prime number exert on a synesthete in equal measure — Tammet offers an absorbing simulation of the synesthetic mind:
Close your eyes and imagine a space without limits, or the infinitesimal events that can stir up a country’s revolution. Imagine how the perfect game of chess might start and end: a win for white, or black, or a draw? Imagine numbers so vast that they exceed every atom in the universe, counting with eleven or twelve fingers instead of ten, reading a single book in an infinite number of ways.
Such imagination belongs to everyone. It even possesses its own science: mathematics. Ricardo Nemirovsky and Francesca Ferrara, who specialize in the study of mathematical cognition, write that “like literary fiction, mathematical imagination entertains pure possibilities.” This is the distillation of what I take to be interesting and important about the way in which mathematics informs our imaginative life. Often we are barely aware of it, but the play between numerical concepts saturates the way we experience the world.
The fact that we have never read an endless book, or counted to infinity (and beyond!) or made contact with an extraterrestrial civilization (all subjects of essays in the book) should not prevent us from wondering: what if? … Literature adds a further dimension to the exploration of those pure possibilities. As Nemirovsky and Ferrara suggest, there are numerous similarities in the patterns of thinking and creating shared by writers and mathematicians (two vocations often considered incomparable.)
In fact, this very link between mathematics and fiction, between numbers and storytelling, underpins much of Tammet’s exploration. Growing up as one of nine siblings, he recounts how the oppressive nature of existing as a small number in a large set spurred a profound appreciation of numbers as sensemaking mechanisms for life:
Effaced as individuals, my brothers, sisters, and I existed only in number. The quality of our quantity became something we could not escape. It preceded us everywhere: even in French, whose adjectives almost always follow the noun (but not when it comes to une grande famille). … From my family I learned that numbers belong to life. The majority of my math acumen came not from books but from regular observations and day-to-day interactions. Numerical patterns, I realized, were the matter of our world.
This awareness was the beginning of Tammet’s synesthetic sensibility:
Like colors, the commonest numbers give character, form, and dimension to our world. Of the most frequent — zero and one — we might say that they are like black and white, with the other primary colors — red, blue, and yellow — akin to two, three, and four. Nine, then, might be a sort of cobalt or indigo: in a painting it would contribute shading, rather than shape. We expect to come across samples of nine as we might samples of a color like indigo—only occasionally, and in small and subtle ways. Thus a family of nine children surprises as much as a man or woman with cobalt-colored hair.
Borges here also makes several thought-provoking points. First, though a set as familiar to our understanding as that of “animals” implies containment and comprehension, the sheer number of its possible subsets actually swells toward infinity. With their handful of generic labels (“mammal,” “reptile,” “amphibious,” etc.), standard taxonomies conceal this fact. To say, for example, that a flea is tiny, parasitic, and a champion jumper is only to begin to scratch the surface of all its various aspects.
Second, defining a set owes more to art than it does to science. Faced with the problem of a near endless number of potential categories, we are inclined to choose from a few — those most tried and tested within our particular culture. Western descriptions of the set of all elephants privilege subsets like “those that are very large,” and “those possessing tusks,” and even “those possessing an excellent memory,” while excluding other equally legitimate possibilities such as Borges’s “those that at a distance resemble flies,” or the Hindu “those that are considered lucky.”
Reading Borges invites me to consider the wealth of possible subsets into which my family “set” could be classified, far beyond those that simply point to multiplicity.
Tammet circles back to the shared gifts of literature and mathematics, which both help cultivate our capacity for compassion:
Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people.