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Love and Math: Equations as an Equalizer for Humanity

“Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries.”

French polymath Henri Poincaré saw in mathematics a metaphor for how creativity works, while autistic savant Daniel Tammet believes that math expands our circle of empathy. So how can a field so diverse in its benefits and so rich in human value remain alienating to so many people who subscribe to the toxic cultural mythology that in order to appreciate its beauty, one needs a special kind of “mathematical mind”? That’s precisely what renowned mathematician Edward Frenkel sets out to debunk in Love and Math: The Heart of Hidden Reality (public library) — a quest to unravel the secrets of the “hidden parallel universe of beauty and elegance, intricately intertwined with ours,” premised on the idea that math is just as valuable a part of our cultural heritage as art, music, literature, and the rest of the humanities we so treasure.

Frenkel makes the same case for math that philosopher Judith Butler made for reading and the humanities, arguing for it as a powerful equalizer of humanity:

Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere — no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given away to the “initiated few.” It belongs to all of us.

Math also helps lift our blinders and break the shackles of our own prejudices:

Mathematics is a way to break the barriers of the conventional, an expression of unbounded imagination in the search for truth. Georg Cantor, creator of the theory of infinity, wrote: “The essence of mathematics lies in its freedom.” Mathematics teaches us to rigorously analyze reality, study the facts, follow them wherever they lead. It liberates us from dogmas and prejudice, nurtures the capacity for innovation.

BEAUTY OF MATHEMATICS by Yann Pineill & Nicolas Lefaucheux

To illustrate why our aversion to math is a product of our culture’s bias rather than of math’s intrinsic whimsy, Frenkel offers an analogy:

What if at school you had to take an “art class” in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. You would probably say something like this: “Learning art at school was a waste of my time. If I ever need to have my fence painted, I’ll just hire people to do this for me.” Of course, this sounds ridiculous, but this is how math is taught, and so in the eyes of most of us it becomes the equivalent of watching paint dry. While the paintings of the great masters are readily available, the math of the great masters is locked away.

Countering these conventional attitudes toward math, Frenkel argues that it isn’t necessary to immerse yourself in the field for years of rigorous study in order to appreciate its far-reaching power and beauty:

Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.

[…]

There is a common fallacy that one has to study mathematics for years to appreciate it. Some even think that most people have an innate learning disability when it comes to math. I disagree: most of us have heard of and have at least a rudimentary understanding of such concepts as the solar system, atoms and elementary particles, the double helix of DNA, and much more, without taking courses in physics and biology. And nobody is surprised that these sophisticated ideas are part of our culture, our collective consciousness. Likewise, everybody can grasp key mathematical concepts and ideas, if they are explained in the right way. . . .

The problem is: while the world at large is always talking about planets, atoms, and DNA, chances are no one has ever talked to you about the fascinating ideas of modern math, such as symmetry groups, novel numerical systems in which 2 and 2 isn’t always 4, and beautiful geometric shapes like Riemann surfaces. It’s like they keep showing you a little cat and telling you that this is what a tiger looks like. But actually the tiger is an entirely different animal. I’ll show it to you in all of its splendor, and you’ll be able to appreciate its “fearful symmetry,” as William Blake eloquently said.

Drawing from Soviet artist and mathematician Anatolii Fomenko’s ‘Mathematical Impressions.’ Click image for more.

And as if a mathematician quoting Blake weren’t already an embodiment that boldly counters our cultural stereotypes, Frenkel adds even more compelling evidence from his own journey: Born in Soviet Russia where mathematics had become “an outpost of freedom in the face of an oppressive regime,” discriminatory policies denied him entrance into Moscow State University. But already enamored with math, he secretly snuck into lectures and seminars, read books well into the night, and gave himself the education the system had attempted to bar him from. A young self-taught mathematician, he began publishing provocative papers, one of which was smuggled abroad and gained international acclaim. Soon, he was invited as a visiting professor at Harvard. He was only twenty-one.

The point of this biographical anecdote, of course, isn’t that Frenkel is brilliant, though he certainly is — it’s that the love math ignites in those willing to surrender to its siren call can stir hearts, move minds, and change lives. Frenkel puts it beautifully, returning to math’s equalizing quality:

Mathematics is the source of timeless profound knowledge, which goes to the heart of all matter and unites us across cultures, continents, and centuries. My dream is that all of us will be able to see, appreciate, and marvel at the magic beauty and exquisite harmony of these ideas, formulas, and equations, for this will give so much more meaning to our love for this world and for each other.

Love and Math goes on to explore the alchemy of that magic through its various facets, including one of the biggest ideas that ever came from mathematics — the Langlands Program, launched in the 1960s by Robert Langlands, the mathematician who currently occupies Einstein’s office at Princeton, and considered by many the Grand Unified Theory of mathematics. Complement it with Paul Lockhart’s exploration of the whimsy of math and Daniel Tammet on the poetry of numbers.

Thanks, Kirstin

BP

The Three-Body Problem: French Polymath Paul Valéry on the Trifecta of Creaturely Realities We Inhabit and Strive to Integrate

“Everything that is masks for us something that might be.”

“It is in the thousands of days of trying, failing, sitting, thinking, resisting, dreaming, raveling, unraveling that we are at our most engaged, alert, and alive… The body becomes irrelevant,” Dani Shapiro wrote in her beautiful meditation on the pleasures and perils of the creative life. And yet the body is the single most relevant, persistent, and unrelenting reality of our lives, a constant companion, on whom “we” — as much as we’re able to separate the “we” from the “it” — depend as much as “it” depends on “us,” an often ambivalent and conflicted codependence that endures, whether we like it or not, for as long as we are alive. Even consciousness itself can’t transcend the nesting-doll physical reality of the body that includes the brain that includes the mind that contemplates itself. But what, exactly, is the body as a conscious experience beyond a biological mass?

That’s precisely what legendary French polymath Paul Valéry (October 30, 1871–July 20, 1945) explores in his 1943 essay “Some Simple Reflections on the Body,” found in the altogether fantastic 1989 anthology Zone 4: Fragments for a History of the Human Body, Part 2 (public library), in which he poses “the three-body problem” — the trifecta of bodily realities that we each inhabit and struggle to integrate.

Illustration from The Human Body, 1959

He begins with the First Body, which possesses us more than we possess it and serves as a reference point to the world:

The [First Body] is the privileged object of which, at each instant, we find ourselves in possession, although our knowledge of it — like everything that is inseparable from the instant — may be extremely variable and subject to illusions. Each of us calls this object My Body, but we give it no name in ourselves, that is to say, in it. We speak of it to others as of a thing that belongs to us; but for us it is not entirely a thing; and it belongs to us a little less than we belong to it. . . .

It is for each of us, in essence, the most important object in the world, standing in opposition to the world, on which, however, it knows itself to be closely dependent. We can say that the world is based on it and exists in reference to it; or just as accurately, with a simple change in the adjustment of our intellectual vision, that the selfsame body is only an infinitely negligible, unstable event in the world.

There’s a particular amorphousness to this First Body:

The thing itself is formless: all we know of it by sight is the few mobile parts that are capable of coming within the conspicuous zone of the space which makes up this My Body, a strange, asymmetrical space in which distances are exceptional relations. I have no idea of the spatial relations between “My Forehead” and “My Foot,” between “My Knee” and “My Back.” … This gives rise to strange discoveries. My right hand is generally unaware of my left. To take one hand in the other is to take hold of an object that is not-I. These oddities must play a part in sleep and, if such things as dreams exist, must provide them with infinite combinations.

This First Body, Valéry argues, is “our most redoubtable antagonist,” for “it carries within it all constancy and all variation.” Then we get to the Second Body — the package of physical concreteness we present to others, as well as to ourselves:

Our Second Body is the one which others see, and an approximation of which confronts us in the mirror or in portraits. It is the body which has a form and is apprehended by the arts, the body on which materials, ornaments, armor sit, which love sees or wants to see, and yearns to touch.

Nude female anatomical figure, artist unknown, c. 1550, from The Art of Medicine

This Second Body, with its cruel concreteness, is also the one that causes us distress — the part of our mortality paradox that makes it so burdensome and so distressing:

This is the body that was so dear to Narcissus, but that drives many to despair, and is a source of gloom to almost all of us once the time comes when we cannot help admitting that the aged creature in the glass, whom we do not accept, stands in some terrible close though incomprehensible relation to ourselves.

In some ways, this Second Body serves as surface protection for what goes on inside — that which we long so desperately to understand and make palpable, yet which remains largely mysterious and intangible:

One can live without ever having seen oneself, without knowing the color of one’s skin.

Man as Industrial Palace of Industry by Fritz Kahn, 1926

This brings us to the Third Body, that of medicine’s fascination and the one best captured in the industrial-age vision for the body as a machine:

[The Third Body] has unity only in our thought, since we know it only for having dissected and dismembered it. To know it is to have reduced it to parts and pieces.

Complete Notes on the Dissection of Cadavers by Kaishi Hen, 1772, from Hidden Treasure

And yet, Valéry suggests there is more to the human body than the abstract, the superficial, and the mechanical. He thus proposes a Fourth Body, which is distinct from the other three and is at once a Real Body and an Imaginary Body — a body of possibility:

My Fourth Body is neither more nor less distinct than is a whirlpool from the liquid in which it is formed. . . . The mind’s knowledge is a product of what this Fourth Body is not. Necessarily and irrevocably everything that is masks for us something that might be.

As Valéry brushes up against the inherent contradictions of this proposition, he hears “the Voice of the Absurd” within himself admonishing:

Think carefully: Where do you expect to find answers to these philosophical questions? Your images, your abstractions, derive only from the properties and experiences of your Three Bodies. But the first offers you nothing by moments; the second a few visions; and the third, at the cost of ruthless dissections and complicated preparations, a mass of figures more indecipherable than Etruscan texts. Your mind, with its language, pulverizes, mixes and rearranges all this and from it, by the abuse, if you will, of its habitual questionnaire, evolves its notorious problems; but it can give them a shadow of meaning only by tacitly presupposing a certain Nonexistence — of which my Fourth Body is a kind of incarnation.

Fragments for a History of the Human Body is excellent in its entirety. Complement it with Nancy Etcoff’s exploration of the science of beauty, which revisits Valéry’s theories with the lens of modern cognitive science and neurobiology.

BP

Inclining the Mind Toward “Sudden Illumination”: French Polymath Henri Poincaré on How Creativity Works

“The subliminal self is in no way inferior to the conscious self; it is not purely automatic; it is capable of discernment; it has tact, delicacy; it knows how to choose, to divine.”

Inclining the Mind Toward “Sudden Illumination”: French Polymath Henri Poincaré on How Creativity Works

In his fantastic 1939 Technique for Producing Ideas, James Webb Young extolled “unconscious processing” — a period marked by “no effort of a direct nature” toward the objective of your creative pursuit — as the essential fourth step of his five-step outline of the creative process. The idea dates back to William James, who coined the concept of fringe consciousness. T. S. Eliot called this mystical yet vital part of creativity “idea incubation,” which Malcolm Cowley echoed in the second stage of his anatomy of the writing process. John Cleese similarly stressed the importance of time in creative work.

From French polymath and pioneering mathematician Henri Poincaré (April 29, 1854–July 17, 1912) — whose famous words on the nature of invention inspired the survey that gave us a glimpse of how Einstein’s genius works — comes a fascinating testament to the powerful role of this unconscious incubation in the creative process.

Henri Poincaré, 1879.

In a chapter titled “Mathematical Creation” from his 1904 tome The Foundations of Science: Science and Hypothesis, the Value of Science, Science and Method (public library; free download), Poincaré observes a process profoundly applicable not only to mathematics, but to just about any creative discipline:

I wanted to represent these functions by the quotient of two series; this idea was perfectly conscious and deliberate; the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and succeeded without difficulty in forming the series I have called thetafuchsian.

Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the result at my leisure.

With a true scientist’s insistence on empirical evidence, Poincaré ensures this was a pattern rather than mere one-off coincidence by citing another example of the process at work:

I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty, that the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidian geometry.

He puts forth a notion contemporary researchers on creative productivity have since affirmed and speaks to the value of what we too tragically term “procrastination,” which is in fact a valuable part of ideation:

Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable, and traces of it would be found in other cases where it is less evident. Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind. It might be said that the conscious work has been more fruitful because it has been interrupted and the rest has given back to the mind its force and freshness.

What’s more, Poincaré observes, not only circumstances but also substances help prime the mind for such moments of “sudden illumination”:

One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. It seems, in such cases, that one is present at his own unconscious work, made partially perceptible to the over-excited consciousness, yet without having changed its nature. Then we vaguely comprehend what distinguishes the two mechanisms or, if you wish, the working methods of the two egos.

Still, Poincaré is careful to point out that without a foundation of prior conscious work — something Alexander Flexner touched on in his timeless meditation on the usefulness of useless knowledge — these “sudden illuminations” wouldn’t take place:

There is another remark to be made about the conditions of this unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations (and the examples already cited sufficiently prove this) never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come, where the way taken seems totally astray. These efforts then have not been as sterile as one thinks; they have set agoing the unconscious machine and without them it would not have moved and would have produced nothing.

He concludes:

The subliminal self is in no way inferior to the conscious self; it is not purely automatic; it is capable of discernment; it has tact, delicacy; it knows how to choose, to divine. What do I say? It knows better how to divine than the conscious self, since it succeeds where that has failed.

It appears, then, that in order for us to lubricate the machinery of unconscious ideation, we have to first prime the mind with directed conscious work, then relieve it of its standard inhibitions, whether by a distracting situation like a trip or a vacation, or by a stimulating substance like caffeine. (Curoiusly, more than a century after Poincaré’s remarks, scientists are beginning to suspect caffeine has the opposite effect and cramps creativity by making us too focused.)

Complement Poincaré’s insights from The Foundations of Science with Young’s indispensable 5-step “technique” for ideation and this excellent contemporary field guide to creativity.

BP

Synesthesia and the Poetry of Numbers: Autistic Savant Daniel Tammet on Literature, Math, and Empathy, by Way of Borges

“Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view.”

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:

Imagine.

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.

Sketches from synesthetic artist and musician Michal Levy’s animated visualization of John Coltrane’s ‘Giant Steps.’ Click for details.

Tammet, above all, is enchanted by the mesmerism of the unknown, which lies at the heart of science and the heart of poetry:

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.

Daniel Tammet. Portrait by Jerome Tabet.

Sampling from Jorge Luis Borges’s humorous fictional taxonomy of animals, inspired by the work of nineteenth-century German mathematician Georg Cantor, Tammet points to the deeper insight beneath our efforts to itemize and organize the universe — something Umberto Eco knew when he proclaimed that “the list is the origin of culture” and Susan Sontag intuited when she reflected on why lists appeal to us. Tammet writes:

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.

The rest of the essays in Thinking In Numbers, ranging from fascinating biographical anecdotes to speculative fiction imagining young Shakespeare’s first arithmetic lessons in zero, are equal parts mind-bending and soul-stirring, and altogether delightful in innumerable ways. Complement it with Paul Lockhart’s multisensory exploration of the whimsy of math, then revisit the extraordinary feats of other autistic savants, from Gregory Blackstock’s astonishing visual taxonomies to Gilles Trehin’s remarkable imaginary city.

BP

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