math

Two Sides of the Same Coin

Excerpt from "The Joy of Math: Keith Devlin on Learning and What It Means To Be Human," On Being, September 19, 2013:

People get turned off mathematics in various ways. If you teach it as sort of just stuff you need to know to balance your checkbook — which is nonsense because none of us balance our checkbooks; computers do that for us. On the other hand, because language is so important to us as living creatures, everyone is interested in language one way or another, be we language mavens or just interested in listening to the radio or reading or novels. You know, language is a fundamental part of what we ask.

In fact, in a book I wrote in 2000, called The Math Gene, I actually made a case based on sort of rational reconstruction of human evolutionary development, that mathematics and language are actually two sides of the same coin in terms of evolutionary development. Human beings, when we developed the capacity for language — and nobody knows when that was; it might be as recent as 50,000 years ago — but when our ancestors developed language capacity, at that moment they developed the capacity for mathematics. It's the same capacity. It just plays out in different ways.

...

A lot of the problem in mathematics is that an awful lot of what goes on in the school system is basically trying to train the mind to do what a $10 calculator can do: follow rules and algorithms and procedures. And one thing that we do know is, that the human brain does not find that natural. The human brain is analogical, not logical. And so, when we try to force it to be procedural and exact, the brain simply doesn't like it.

It was important for many thousands of years to be able to do computation and calculation because that was the basis of commerce and trade and buying and selling. And you had to do it in your head or with an abacus board or something. So for hundreds of years, it was actually important to train the mind to follow rules to do computations and get the right answer. Well, now we've automated that. And we carry around devices in our pockets that can do that. Which means that we can spend more time letting the brain do things that the brain is really well suited for that computers can't do very well: making value judgments, making analogical leaps.

Flat, Round, or Wavy

"Take your finger and draw this line: summer, fall, winter, spring...noon, dusk, dark, dawn...

Have you ever seen those stratus clouds that go on parallel stripes across the sky? Did you know that's a continuous sheet of cloud that's dipping in and out of the condensation layer?

What if every seemingly isolated object was actually just where the continuous wave of that object poked through into our world? The Earth is neither flat nor round; it's wavy."

~ Reuben Margolin

More videos of Reuben Margolin's waves.

Balance in Nature

Shinzen Young, in reponse to a Brain Pickings post on seventeen historically significant mathematical equations:

Just for the record, here's my all-time favorite equation:

First, let me admit that the way I just wrote it involves some abuse of notation. Properly, it should be written this way:

 

But I think the former form is justified for the visual effect.

To the eye, it seems to equate two closed curves that have symmetry: A regular triangle, with 3-fold rotational symmetry (the minimum possible) and the circle with infinite rotational symmetry (the maximum possible).

But as a mathematical formula, it represents the "generalized Laplacian equation."

This equation is one of the broadest statements of balance in nature. Phenomena as different as three-dimensional thermodynamic equilibrium and four-dimensional relativistic motion can be described by this equation.

To me, it's a reminder that "mutually-canceling polarities" play a fundamental role both in the physical world as described abstractly by scientists and in the spiritual world as described concretely by mystics. 

See also:

 

Using Technology to Humanize the Classroom

“By removing the one size fits all lecture from the classroom and letting students have a self-paced lecture at home, and then when you go to the classroom, letting them do work, having the teacher walk around, having the peers actually be able to interact with each other, these teachers have used technology to humanize the classroom.”

~ Sal Khan, of Khan Academy

The Best Our Species Can Do

lambda=hmv

Duality
by Georges Whitesides, from No Small Matter: Science on the Nanoscale

We’re burdened by a curious conditioning that blinds us to one of the greatest — perhaps the greatest — of  art forms. We live for poetry; we live in terror of equations.

We see a poem, and we try it on for size: we read a line or two; we roll it around in our mind; we see how it fits and tastes and sounds. We may not like it, and let it drop, but we enjoy the encounter and look forward to the next. We see an equation, and it is as if we’d glimpsed a tarantula in the baby’s crib. We panic.

An equation can be a thing of such beauty and subtlety that only a poem can equal it. As an evocation of reality — as the shortest of descriptions, but describing worlds — it is hard to beat the most artful of poems and, equally, of equations. They are the best our species can do.

Equations are the poetry that we use to describe the behavior of electrons and atoms, just as we use poems to describe ourselves. Equations may be all we have: sometimes word fail, since words best describe what we have experienced, and behaviors at the smallest scale are forever beyond our direct experience.

Consider Margaret Atwood:

You fit into me
Like a hook in an eye

A fish hook
An open eye

Consider Louis de Broglie (a twentieth-century physicist, and an architect of quantum mechanics):

λ = h/mv

Read the equation as if it were poetry — a condensed description of a reality we can only see from the corner of our eye. The “equals” sign is the equivalent of “is,” and makes the equation a sentence: “A moving object is a wave.” Huh? What did you just say? How can that be?

It’s an idea worth trying on for size. Poetry describes humanity with a human voice; equations describe a reality beyond the reach of words. Playing a fugue, and tasting fresh summer tomatoes, and writing poetry, and falling in love all ultimately devolve into molecules and electrons, but we cannot yet (and perhaps, ever) trace that path from one end (from molecules) to other (us). Not with poetry, nor with equations. But each guides us part way.

Of course, not all equations are things of beauty: some are porcupines, some are plumber’s helpers, and some are tarantulas.

*     *     *     *

I’m a chemist. My universe is nuclei and electrons, and the almost endless ways they can assemble. Atoms are just at the border between ordinary, macroscopic matter and matter dominated by the Alice-in-Wonderland rules of quantum mechanics. Electrons, in particular, have the unnerving property of having mass and charge but no extent — no size. There’s no tiny BB down in their core, as there is a nucleus sitting at the center of an atom. “Ah,” you say, “that’s strange. If there’s nothing there, what is it that has a mass? And what’s charged?” Good question…

…As a chemist, I’ve come to uneasy terms with the weirdness of electrons and photons, and with their ability to meld into the ordinariness of macroscopic things. But sometimes, lying awake in a strange hotel room at 4 a.m., considering what I might say that I really understand about anything, I fret that the answer is: almost nothing.

Selling a Product to a Market that Doesn’t Want It

“Can I ask you to please recall a time when you really loved something, a movie, an album, a song or a book, and you recommended it wholeheartedly to someone you also really liked, and you anticipated that reaction, you waited for it, and it came back, and the person hated it. So, by way of introduction, that is the exact same state in which I spent every working day of the last six years. I teach high school math. I sell a product to a market that doesn't want it, but is forced by law to buy it. I mean, that's kind of—it's just a losing proposition.”

~ Dan Meyer, from “Math Class Needs a Makeover,” TED Talks (May 2010)

A Strange and Hidden Relationship

 

“In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane to a generating straight line of that surface. Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola. The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles.”

Keep Your Temper

Excerpt from “Algebra in Wonderland,” by Melanie Bayley, New York Times (March 7, 2010):

Alice has slid down from a world governed by the logic of universal arithmetic to one where her size can vary from nine feet to three inches. She thinks this is the root of her problem: “Being so many different sizes in a day is very confusing.” No, it isn’t, replies the Caterpillar, who comes from the mad world of symbolic algebra. He advises Alice to “Keep your temper.”

In Dodgson’s day, intellectuals still understood “temper” to mean the proportions in which qualities were mixed — as in “tempered steel” — so the Caterpillar is telling Alice not to avoid getting angry but to stay in proportion, even if she can’t “keep the same size for 10 minutes together!” Proportion, rather than absolute length, was what mattered in Alice’s above-ground world of Euclidean geometry.

Scene from Jan Švankmajer's Alice.

Mapping Algorithms

“I'm a composer, orchestrally-trained, and the inventor of the AlloSphere. With my visual artist colleagues, we map complex mathematical algorithms that unfold in time and space, visually and sonically. Our scientist colleagues are finding new patterns in the information. And our engineering colleagues are making one of the largest dynamically varying computers in the world for this kind of data exploration. I'm going to fly you into five research projects in the AlloSphere that are going to take you from biological macroscopic data all the way down to electron spin.”

~ Composer JoAnn Kuchera-Morin, founder of the Center for Research in Electronic Art Technology (CREATE)

Waiting to Be Discovered

Dr. George Ellis, Professor of Applied Mathematics at the University of Cape Town and the author of On the Moral Nature of the Universe: Cosmology, Theology, and Ethics in conversation with Krista Tippett on Speaking of Faith (5/10/07):

Archimedes Thoughtful by Fetti (1620)

Mathematicians discover the nature of mathematics despite what they want. What I mean by that is something like the following. It was a great shock to mathematicians when they discovered that the square root of two is irrational. That's not something that they wanted. The number pi is irrational. That's also not something mathematicians wanted. What I'm pointing out here is that mathematics exists and is discovered. It's not invented by humans. It's something which is discovered. Therefore, in some sense, it exists in order to be discovered.

The view on ethics I take as an ethical realist is it's the same, the nature is sitting there in some sense waiting to be discovered. And the deep nature of ethics...is what we call kenotic ethics.

[Kenosis is] a Greek word meaning letting go or giving up, and it's used in the Bible in Philippians. It's central to my understanding of Christianity, and there's a spectrum which goes through in practical terms from forgiveness, which is a crucial part in which you are giving up the need for revenge. And it goes through to self-sacrifice on behalf of others, which is what Gandhi was about, Martin Luther King was about. And to me, that's the really, really deep transformative principle, which was also in the life of Christ, of course, when he sacrificed himself on behalf of others.

I think it's important to say that to me, kenosis is a generic principle which is much wider than just ethics. For instance, it's actually central — this emptying oneself — it's actually central to education and learning, because if you go into learning any subject with a preconceived notion, you can't learn. You have to empty your mind of your preconceived notion that you can see something new.

In ethics, though the key point about kenosis is the willingness to give up, which makes way for contact with the human part of the other person. And it's a kind of moral jujitsu in that they're expecting you to react in the way that they want you to react. They are your enemy and they want you to be their enemy. And if you refuse to be their enemy, then they don't know how to handle it.