Brain Pickings

Posts Tagged ‘books’

25 SEPTEMBER, 2012

Measurement: Exploring the Whimsy of Math through Playful Patterns, Shape and Motion

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“What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty.”

Math, as Vi Hart’s stop-motion doodles, Robin Moore’s string portraits, and Anatolii Fomenko’s stunning black-and-white illustrations have previously shown us, can be the conduit of great fun and great beauty. In Measurement (public library), mathematician Paul Lockhart invites us to “make patterns of shape and motion, and then [try] to understand our patterns and measure them.” (Because, lest we forget, we have a natural penchant for patterns.) What results as we step away from physical reality and immerse ourselves in the imaginary — and imaginative — world of mathematical reality is a thing of infinite beauty and infinite fascination.

Lockhart writes in the introduction:

Physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in the universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line.

Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I can never hold a circle in my hand, but I can hold one in my mind.

[…]

The point is I get to have them both — physical reality and mathematical reality. Both are beautiful and interesting… The former is important to me because I am in i, the latter because it is in me.

Lockhart underpins this excitement with a fair warning:

Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively.

And yet, he’s quick to reassure that the gold standard of math, not unlike that of science, isn’t the answer but the driver of curiosity:

What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty.

But what makes Lockhart particularly compelling is his ability to relate mathematics to parallel concepts from disciplines and aspects of life that are more familiar, more comfortable, more ingrained in our everyday understanding of the world — for instance, in comparing math to storytelling:

A mathematical argument [is] otherwise known as a proof. A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. A proof should be lovely as well as logical.

In a nod to the value of “useless” knowledge and figuring things out, Lockhart argues:

People don’t do mathematics because it’s useful. They do it because it’s interesting … The point of a measurement problem is not what the measurement is; it’s how to figure out what it is.

He ultimately offers several pieces of advice on engaging with math:

  1. The best problems are your own. Mathematical reality is yours — it’s in your head for you to explore any time you feel like it… Don’t be afraid that you can’t answer your own questions — that’s the natural state of the mathematician.
  2. Collaborate. Work together and share the joys and frustrations. It’s a lot like playing music together.
  3. Improve your proofs. Just because you have an explanation doesn’t mean it’s the best explanation. Can you eliminate any unnecessary clutter or complexity? Can you find an entirely different approach that gives you deeper insight? Prove, prove, and prove again. Painters, sculptors, and poets do the same thing.
  4. Let a problem take you where it takes you. If you come across a river in the jungle, follow it!
  5. Critique your work. Subject your arguments to scathing criticism by yourself and others. That’s what all artists do, especially mathematicians… For a piece of mathematics to fully qualify as such, it has to stand up to two very different kids of criticism: it must be logically sound and convincing as a rational argument, and it must also be elegant, revelatory, and emotionally satisfying. [But don't] worry about trying to hold yourself to some impossibly high standard of aesthetic excellence.

Lockhart elaborates on the latter point with a poignant reflection that applies to math just as much as it does to life itself:

Part of the problem is that we are so concerned with our ideas being simple and beautiful that when we do have a pretty idea, we want so much to believe it. We want it to be true so badly that we don’t always give it the careful scrutiny that we should read. It’s the mathematical version of ‘rapture of the deep.’ Divers see beautiful sights that they forget to come up for air. Well, logic is our air, and careful reasoning is how we breathe.

Indeed, much of math sounds an awful lot like the art of living itself: Take for instance, that same old fear of failure that often stands in the way of creativity, which also holds us back from immersing ourselves in the art of figuring things out:

The important thing is not to be afraid. So you try some crazy idea, and it doesn’t work. That puts you in some pretty good company! Archimedes, Gauss, you and I — we’re all groping our way through mathematical reality, trying to understand what is going on, making guesses, trying out ideas, mostly failing. And then every once in a while, you succeed… And that feeling of unlocking an eternal mystery is what keeps you going back to the jungle to get scratched up all over again.

Here’s a little teaser for the whimsical jungle of mathematical reality and the logical aesthetic of math:

Measurement, from Harvard University Press, comes seven years after Lockhart’s exquisite critique of math’s tragic fate in contemporary education, A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.

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24 SEPTEMBER, 2012

Anaïs Nin on the Meaning of Life & the Dangers of the Internet (1946)

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“We believe we are in touch with a greater amount of people… This is the illusion which might cheat us of being in touch deeply with the one breathing next to us.”

Last week’s widely reverberating meditations on the meaning of life by cultural icons like Charles Bukowski, Annie Dillard, Arthur C. Clarke, and John Cage reminded me of a passage from the altogether sublime The Diary of Anais Nin, Vol. 4: 1944-1947 (public library) — the same tome that gave us this poignant reflection on why emotional excess is essential to creativity.

In an entry from May 1946, Anaïs Nin once again challenges our presentism bias by thinking deeply and timelessly about issues we tend to believe we’re brushing up against for the very first time, from the pitfalls of always-on communication technology to the pace of modern life to the venom of procrastination.

Even more interesting than the striking similarity between what Nin admonishes against and the present dynamics of the internet is the fact that she essentially describes Marshall McLuhan’s seminal concept of the global village… a decade and a half before he coined it.

The secret of a full life is to live and relate to others as if they might not be there tomorrow, as if you might not be there tomorrow. It eliminates the vice of procrastination, the sin of postponement, failed communications, failed communions. This thought has made me more and more attentive to all encounters, meetings, introductions, which might contain the seed of depth that might be carelessly overlooked. This feeling has become a rarity, and rarer every day now that we have reached a hastier and more superficial rhythm, now that we believe we are in touch with a greater amount of people, more people, more countries. This is the illusion which might cheat us of being in touch deeply with the one breathing next to us. The dangerous time when mechanical voices, radios, telephones, take the place of human intimacies, and the concept of being in touch with millions brings a greater and greater poverty in intimacy and human vision.

For more on Nin’s timeless insights on life, see Lisa Congdon’s stunning hand-lettered diary quotes.

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21 SEPTEMBER, 2012

Hermann Hesse on What Trees Teach Us About Belonging and Life

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“When we have learned how to listen to trees, then the brevity and the quickness and the childlike hastiness of our thoughts achieve an incomparable joy.”

I woke up this morning to discover a tiny birch tree rising amidst my city quasi-garden, having overcome unthinkable odds to float its seed over heaps of concrete and glass, and begin a life in a meager oasis of soil. And I thought, my god*, what a miracle. What magic. What a reminder that life does not await permission to be lived.

This little wonder reminded me of a beautiful passage, perhaps one of the most beautiful passages I’ve ever read, from Hermann Hesse’s Bäume: Betrachtungen und Gedichte [Trees: Reflections and Poems] (public library), originally published in 1984, that touches on some of life’s most essential livingness — home and belonging, truth and beauty, happiness.

For me, trees have always been the most penetrating preachers. I revere them when they live in tribes and families, in forests and groves. And even more I revere them when they stand alone. They are like lonely persons. Not like hermits who have stolen away out of some weakness, but like great, solitary men, like Beethoven and Nietzsche. In their highest boughs the world rustles, their roots rest in infinity; but they do not lose themselves there, they struggle with all the force of their lives for one thing only: to fulfill themselves according to their own laws, to build up their own form, to represent themselves. Nothing is holier, nothing is more exemplary than a beautiful, strong tree. When a tree is cut down and reveals its naked death-wound to the sun, one can read its whole history in the luminous, inscribed disk of its trunk: in the rings of its years, its scars, all the struggle, all the suffering, all the sickness, all the happiness and prosperity stand truly written, the narrow years and the luxurious years, the attacks withstood, the storms endured. And every young farmboy knows that the hardest and noblest wood has the narrowest rings, that high on the mountains and in continuing danger the most indestructible, the strongest, the ideal trees grow.

Trees are sanctuaries. Whoever knows how to speak to them, whoever knows how to listen to them, can learn the truth. They do not preach learning and precepts, they preach, undeterred by particulars, the ancient law of life.

A tree says: A kernel is hidden in me, a spark, a thought, I am life from eternal life. The attempt and the risk that the eternal mother took with me is unique, unique the form and veins of my skin, unique the smallest play of leaves in my branches and the smallest scar on my bark. I was made to form and reveal the eternal in my smallest special detail.

A tree says: My strength is trust. I know nothing about my fathers, I know nothing about the thousand children that every year spring out of me. I live out the secret of my seed to the very end, and I care for nothing else. I trust that God is in me. I trust that my labor is holy. Out of this trust I live.

When we are stricken and cannot bear our lives any longer, then a tree has something to say to us: Be still! Be still! Look at me! Life is not easy, life is not difficult. Those are childish thoughts. . . . Home is neither here nor there. Home is within you, or home is nowhere at all.

A longing to wander tears my heart when I hear trees rustling in the wind at evening. If one listens to them silently for a long time, this longing reveals its kernel, its meaning. It is not so much a matter of escaping from one’s suffering, though it may seem to be so. It is a longing for home, for a memory of the mother, for new metaphors for life. It leads home. Every path leads homeward, every step is birth, every step is death, every grave is mother.

So the tree rustles in the evening, when we stand uneasy before our own childish thoughts: Trees have long thoughts, long-breathing and restful, just as they have longer lives than ours. They are wiser than we are, as long as we do not listen to them. But when we have learned how to listen to trees, then the brevity and the quickness and the childlike hastiness of our thoughts achieve an incomparable joy. Whoever has learned how to listen to trees no longer wants to be a tree. He wants to be nothing except what he is. That is home. That is happiness.

Artwork by Bryan Nash Gill from his book 'Woodcut.' Click image for more.

* As Anaïs Nin wrote in her correspondence with Henry Miller, “I spell ‘god’ with a small ‘g’ because I do not believe in him, but I love to swear by him.”

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