Appalachians

I've been reading about geology, the most epic of the sciences.

Long ago, the continent of Africa collided with what’s now the east cost of North America. The collision lasted for, oh, eighty million years and left behind... the Appalachian mountains.

The Appalachians you see now are not the originals, though. Those were some serious, Himalayan-scale mountains, but time and water wore them away to level ground. The current Appalachians are made of more of the same rock that was left underground, untouched by erosion, and uplifted quite recently, no more than sixty million years ago.

The Earth is at a rolling boil.

Neurology on the edge

Benjamin Libet conducted this experiment in the 1970s. Apart from one or two electrodes on the scalp, there's really nothing creepy about the experiment. Until you read about the results.

Libet asked his experimental subjects to move one hand at an arbitrary moment decided by them, and to report when they made the decision (they timed the decision by noticing the position of a dot circling a clock face). At the same time the electrical activity of their brain was monitored. Now it had already been established by much earlier research that consciously-chosen actions are preceded by a pattern of activity known as a Readiness Potential (or RP). The surprising result was that the reported time of each decision was consistently a short period (some tenths of a second) after the RP appeared.

—from “Pre-empted decisions”, a page on Conscious Entities

The RP starts to ramp up as much as 0.3 seconds before the reported decision time. It continues to increase after that, leading up to actual hand movement about 0.2 seconds later.

What does this mean? All the test subjects, of course, felt they had consciously chosen to move. But if unconscious brain activity precedes the conscious experience of decision-making, then surely we must conclude that the decision is not consciously made. Effects don't precede causes.

Now there are countless philosophical objections to this conclusion. Some philosophers claim that to interpret this result at all competently, you have to be well-versed in the philosophy of the mind. Which seems reasonable enough, but it's a deep field with centuries of literature in many languages. So this prerequisite rules out anyone who has spent his life studying anything as patently irrelevant as mere neurology. To say nothing of random bloggers.

I'll wade in anyway, of course. Just don't get the impression I know anything about this subject. I don't.

There's a really nice, compelling interpretation that permits free will. It goes like this. The way the mind interprets time is anything but objective. In the Libet experiment, what's happening is that the mind shifts the experience of deciding to move forward in time and shifts the experience of motion backward in time, effectively bringing them closer together. So the conscious decision to act actually does cause the RP ramp-up. But the subject incorrectly reports the decision as having happened later, because his brain has deceived him about the timing.

Why would the brain do this? It seems likely to me that there's an evolutionary benefit to perceiving decision, action, and effect as a single event. I don't think we're equipped to deal with that kind of time lag consciously. Just think—there's a half-second lag between when you decide to move a muscle and when it moves. Have you ever played the piano? If you were aware of this lag all the time, could you do that? Could you run? Could you fight?

An article, “Free Will and Free Won't” (in American Scientist; $12 to download the PDF from their site), puts the Libet experiment alongside four or five other rather clever experiments into will and consciousness. Then it starts talking about alien hand syndrome. The brain is strange.

Fiction, meet science

At some point, Dartmouth University offered a semester course on Renaissance Math in Fiction and Drama. From the site:

This course explores scientific developments in Renaissance astronomy and their portrayal in literature past and present. By reading some of the writings by Copernicus, Galileo and the prolific Kepler, we will attempt to draw a portrait of scientific upheaval during that period. The science fiction of the Renaissance offers a window into the popular response to these developments, as do various commentaries of the time. Dramatic pieces both recent and of that period show the artistic reconstruction of scientific events, sometimes through a very modern lens.

“Science fiction of the Renaissance”? There's not a huge amount of this, as it turns out, but one amazing, atypical example is Johannes Kepler's Somnium, which was at once a fanciful journey to the moon and a serious thought experiment in support of Copernican heliocentrism. Wow.

A complex story, part 4

(See parts 1, 2, and 3.)

Everybody literally sees the world from a different point of view. Each person is standing in a different location and looking out in a different direction from everyone else. But all viewpoints share certain similarities. If you and I are near one another, we'll see the same events happen in the same order, and although we may differ in our use of the words “right” and “left”, if we're watching something from opposite sides, we'll at least agree on the distances between things. If I see two people holding hands, you'll never see them on separate sides of the street at the same time, no matter where you're standing. All the different viewpoints preserve certain observed properties: distances, angles, durations, causality, and so on.

Mathematically, we can write this in two equations. For each of us, every event has a measurable position in space (x, y, z) and time (t). If we put my observations on the left-hand side and yours on the right, they will match.

We agree on distances: x² + y² + z² = x'² + y'² + z'²

We agree on durations: t = t'

Even if I'm in a car doing eighty and you're sitting on the sidewalk enjoying an ice cream cone, we'll agree on the distances between and durations of any events we both happen to witness as I zoom by.

...Or so everyone thought. Don't get me wrong, this is a lovely picture. Mathematically, it's your basic three-dimensional Euclidean geometry, plus a separate dimension for time. All our viewpoints are identical except for a bit of spacial displacement and rotation. There's only one problem. This isn't how the universe really behaves.

1887 was the year of the famous Michelson-Morley experiment, which blew this nice, simple Newtonian view all to hell. For twenty years, confusion reigned. By 1905, a mere eyeblink in academic terms, physics had righted itself, now with a totally new model of space and time.

The new theory was called special relativity. It was built on brilliant new insights from Hendrik Lorentz, Henri Poincaré, and Albert Einstein. And it went something like this: Two observers traveling at incredible velocities (relative to one another) actually do not agree on distances, angles, durations, or even the relative time-order of events. But they will agree on something even more fundamental: the basic laws of nature, including laws of motion, causality, and—in particular—the speed of light.

This had the advantage of being, you know, consistent with experiment. But geometrically, it was awfully weird. It wrecked the two equations above. Individual viewpoints were not simple spacial rotations and translations of one another. They were, uh, Lorentz transformations. Yeah. It was two more years before geometry caught up with physics.

In 1907, Hermann Minkowski discovered a kind of geometry (a four-dimensional manifold) that exactly describes the spacetime of special relativity. That is, Minkowski space is the actual geometry of the universe around us, according to relativity. Minkowski's geometry succeeded by treating space and time as interrelated. For example, in Minkowski space:

We may not agree on the spatial distance between two events: x² + y² + z² ≠ x'² + y'² + z'²

We may not agree on the passage of time: t ≠ t'

But we will agree on a particular mathematical combination of the two: x² + y² + z² - ct² = x'² + y'² + z'² - ct'²

(Here c is the speed of light.)

Now comes the controversial, beautiful part. Define a variable w as ict. We're going to use w as our time coordinate, instead of t. Then the last equation above becomes:

x² + y² + z² + w² = x'² + y'² + z'² + w'²

This looks a lot like our original equation for distance. And in fact this equation describes basic Euclidean geometry in four dimensions. Time becomes just another spacial dimension. All viewpoints are again simple rotations and translations of one another—not in three-dimensional space, but in four-dimensional spacetime.

Here the role of the complex numbers is to provide a new way of looking at the geometry of the universe.

But... what does it all mean? Is time really an imaginary dimension? What does it mean for three dimensions to be real numbers and one to be an imaginary number? These questions are, in a way, the same questions RT asked me months ago, the questions that got me interested in telling this story. What are the imaginary numbers? Do they exist? Do they appear in nature? I don't think anyone really knows. Einstein found the ict trick interesting at least (he mentions it twice in his short book Relativity, which by the way I enthusiastically recommend), but some physicists think it's a red herring. Maybe we're just dressing the universe up to look more comfortable and familiar.