This is the knot of reducibility. Can the behavior of a system be understood in terms of—that is, be reduced to—the behavior of its components? This question, in one form or another, pervades science. Systems biologists vs. biochemists, cognitive scientists vs. neuroscientists, Durkheim vs. Bentham, Gould vs. Dawkins, Aristotle vs. Democritus—the gulf (epistemological, ontological, and methodological) between the holist vs. reductionist stances lies at the root of many of science’s greatest disputes. It is also the source of advances, as one stance is abandoned in favor of another. Indeed, holist and reductionist research programs often exist side by side in uneasy truce (think of any biology department). But when, as so often, the truce breaks down and open warfare resumes, it’s clear that what’s needed is a way of rationally partitioning the creative forces operating at different levels.
That is what Price gave. His equation applies only to variation-selection systems, but if you think about it, most order-creating systems are variation-selection systems. Returning to our musical world: Who really shapes it? Beethoven’s epigones tweaking their MIDI files? Adolescents downloading in the solitude of their bedrooms? The massed impulses of the public? I think that Price’s equation can explain. It certainly has some explaining to do.
UNCONSCIOUS INFERENCES
GERD GIGERENZER
Psychologist; director of the Center for Adaptive Behavior and Cognition at the Max Planck Institute for Human Development, Berlin; author, Gut Feelings: The Intelligence of the Unconscious
Optical illusions are a pleasure to look at, puzzling, and robust. Even if you know better, you still are caught in the illusion. Why do they exist? Are they merely mental quirks? The physicist and physiologist Hermann von Helmholtz (1821–1894) provided us with a beautiful explanation of the nature of perception and how it generates perceptual illusions of depth, space, and other properties. Perception requires smart bets called unconscious inferences.
In Volume III of his Physiological Optics, Helmholtz recounts a childhood experience:
I can recall when I was a boy going past the garrison chapel in Potsdam, where some people were standing in the belfry. I mistook them for dolls and asked my mother to reach up and get them for me, which I thought she could do. The circumstances were impressed on my memory, because it was by this mistake that I learned to understand the law of foreshortening in perspective.
This childhood experience taught Helmholtz that information available from the retina and other sensory organs is not sufficient to reconstruct the world. Size, distance, and other properties need to be inferred from uncertain cues, which in turn have to be learned by experience. Based on this experience, the brain draws unconscious inferences about what a sensation means. In other words, perception is a kind of bet about what’s really out there.
But how exactly does this inference work? Helmholtz drew an analogy with probabilistic syllogisms. The major premise is a collection of experiences that are long out of consciousness; the minor premise is the present sensory impression. Consider the “dots illusion” of V. S. Ramachandran and colleagues at the Center for Brain and Cognition, University of California–San Diego:
The dots in the left picture appear concave, receding into the surface away from the observer, while those on the right side appear convex, curved toward the observer. If you turn the page around, the inward dots will pop out and vice versa. In fact, the two pictures are identical, except for being rotated 180 degrees. The illusion of concave and convex dots occurs because our brain makes unconscious inferences.
Major premise:
A shade on the upper part of a dot is nearly always associated with a concave shape.
Minor premise:
The shade is in the upper part.
Unconscious inference:
The shape of the dot is concave.
Our brains assume a three-dimensional world, and the major premise guesses the third dimension from two ecological structures:
Light comes from above, and
There is only one source of light.
These two structures dominated most of human and mammalian history, in which the sun and the moon were the only sources of light, and the first also holds approximately for artificial light today. Helmholtz would have favored the view that the major premise is learned from individual experience; others have favored evolutionary learning. In both cases, visual illusions are seen as the product of unconscious inferences based on evidence that is usually reliable but can be misleading in special circumstances.
The concept of unconscious inference can also explain phenomena from other sensory modalities. A remarkable instance where a major premise suddenly becomes incorrect is the case of a person whose leg has been amputated. Although the major premise (“A stimulation of certain nerves is associated with that toe”) no longer holds, patients nevertheless feel pain in toes that are no longer there. The “phantom limb” also illustrates our inability to correct unconscious inferences despite our knowledge. Helmholtz’s concept has given us a new perspective on perception in particular and cognition in general:
Cognition is inductive inference. Today, the probabilistic syllogism has been replaced by statistical and heuristic models of inference, inspired by Thomas Bayes and Herbert Simon, respectively.
Rational inferences need not be conscious. Gut feelings and intuition work with the same inductive inferences as conscious intelligence.
Illusions are a necessary consequence of intelligence. Cognition requires going beyond the information given, to make bets and therefore to risk errors. Would we be better off without visual illusions? We would in fact be worse off—like a person who never says anything to avoid making any mistakes. A system that makes no errors is not intelligent.
SNOWFLAKES AND THE MULTIVERSE
MARTIN J. REES
Former president of the Royal Society; emeritus professor of cosmology and astrophysics, University of Cambridge; master, Trinity College; author, From Here to Infinity: A Vision for the Future of Science
An astonishing concept has entered the mainstream of cosmological thought: Physical reality could be hugely more extensive than the patch of space and time traditionally called “the universe.” A further Copernican demotion may loom ahead. We’ve learned that we live in just one planetary system among billions, in one galaxy among billions. But now that’s not all. The entire panorama that astronomers can observe could be a tiny part of the aftermath of “our” Big Bang, which is itself just one bang among a perhaps infinite ensemble.
Our cosmic environment could be richly textured but on scales so vast that our purview is restricted to a tiny fragment. We’re not aware of the “big picture,” any more than a plankton whose universe was a liter of water would be aware of the world’s topography and biosphere. It is obviously sensible for cosmologists to start off by exploring the simplest models. But there is no more reason to expect simplicity on the grandest scale than in the terrestrial environment, where intricate complexity prevails.
Moreover, string theorists suspect—for reasons quite independent of cosmology—that there may be an immense variety of “vacuum states.” Were this correct, different universes could be governed by different physics. Some of what we call laws of nature may, in this grander perspective, be local bylaws, consistent with some overarching theory governing the ensemble, but not uniquely fixed by that theory. More specifically, some aspects may be arbitrary and others not. As an analogy (which I owe to the astrobiologist and cosmologist Paul Davies), consider the form of snowflakes. Their ubiquitous sixfold symmetry is a direct consequence of the properties and shape of water molecules. But snowflakes display an immense variety of patterns, because each is molded by its distinctive history and microenvironment; how each flake grows is sensitive to the fortuitous temperature and humidity changes during its growth.
If physicists achieved a fundamental theory, it would tell us which aspects of nature are direct consequences of the bedrock theory (just as the symmetrical template of s
nowflakes is due to the basic structure of a water molecule) and which cosmic numbers are (like the distinctive pattern of a particular snowflake) the outcome of environmental contingencies.
Our domain wouldn’t then be just a random one. It would belong to the unusual subset where there was a “lucky draw” of cosmic numbers conducive to the emergence of complexity and consciousness. Its seemingly designed or fine-tuned features wouldn’t be surprising. We may, by the end of this century, be able to say with confidence whether we live in a multiverse and how much variety its constituent universes display. The answer to this question will, I think, determine crucially how we should interpret the “biofriendly” universe in which we live (and which we share with any aliens whom we might one day contact).
It may disappoint some physicists if some of the key numbers they’re trying to explain turn out to be mere environmental contingencies, no more “fundamental” than the parameters of Earth’s orbit around the sun. But that disappointment would surely be outweighed by the revelation that physical reality was grander and richer than hitherto envisioned.
EINSTEIN’S PHOTONS
ANTON ZEILINGER
Physicist, University of Vienna; scientific director, Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences; author, Dance of the Photons: From Einstein to Quantum Teleportation
My favorite deep, elegant, and beautiful explanation is Albert Einstein’s 1905 proposal that light consists of energy quanta, today called photons. It is little known, even among physicists, but extremely interesting how Einstein came to this conclusion. It’s often thought that he invented the concept to explain the photoelectric effect. Certainly that is part of Einstein’s 1905 publication, but only toward its end. The idea itself is much deeper, more elegant—and, yes, more beautiful.
Imagine a closed container whose walls are at some temperature. The walls are glowing, and as they emit radiation, they also absorb radiation. After some time, there will be an equilibrium distribution of radiation inside the container. This was already well known before Einstein. Max Planck had introduced the idea of quantization that explained the energy distribution of the radiation inside such a volume. Einstein went a step further. He studied how orderly the distribution of the radiation is inside such a container.
For physicists, entropy is a measure of disorder. And the Austrian physicist Ludwig Boltzmann showed that the entropy of a system is a measure of how probable its state is. To take a simple example, it is much more probable that books, notes, pencils, photos, pens, etc. are scattered all over my desk than that they form orderly stacks. Or if we consider a million atoms inside a container, it is much more probable that they are more or less equally distributed throughout the container’s volume than all collected in one corner. In both cases, the first state is the less orderly, and when the atoms fill a larger volume they have an even higher entropy.
Einstein realized that the entropy of radiation (including light) changes with the volume of its container in the same mathematical way as for atoms; in both cases, the entropy increases with the logarithm of the volume. For Einstein, this could not be just a coincidence. Since we can understand the entropy of the gas because it consists of atoms, radiation, too, consists of particles—which he called energy quanta (today called photons).
Einstein immediately applied his idea to the photoelectric effect. But he also realized a fundamental conflict of the idea of energy quanta with the well-studied and observed phenomenon of interference.
The problem was how to understand the two-slit interference pattern. This is the phenomenon that, according to Richard Feynman, contains “the only mystery” of quantum physics. The challenge is very simple. When we shine a beam of photons at a plate in which there are two slits, and both slits are open, we obtain bright and dark stripes on an observation screen behind the plate; these are the interference fringes. When we have only one slit open, we get no stripes, no interference fringes, but instead a broad distribution of photons. This result can easily be understood, given the wave picture of light: Waves pass through each of the two slits, alternately extinguishing and reinforcing each other on the observation screen. That way, we obtain dark and bright fringes.
But what to expect if the light beam’s intensity is so low that only one photon at a time passes through the apparatus? Following Einstein’s realist position, it would be natural to assume that a photon has to pass through either one open slit or the other, but not both. We can do the experiment by sending photons in, one at a time. According to Einstein, no interference fringes should appear, because a single photon, as a particle, would have to “choose” one open slit or the other, and thus there would be no reinforcement or extinguishing, as there was in the wave picture. This was indeed Einstein’s opinion, and he suggested that the fringes appear only if many photons are passing through at the same time and somehow interact with each other such that they make up the interference pattern.
Today, we know from many experiments that the interference pattern arises even at such low intensities that only one photon per second passes through the apparatus. If we wait long enough and look at the distribution of all of them on the observation screen, we get the interference pattern. The modern explanation is that the interference pattern arises only if there is no information, anywhere in the universe, about which slit the particle passes through (the colloquial statement that a photon passes through both slits at once has to be taken with a grain of salt). But even as Einstein was wrong here, his idea of the energy quanta of light, i.e., photons, pointed far into the future.
In a letter to his friend Conrad Habicht in the same year of 1905, the miraculous year wherein he also published his special theory of relativity, he called the paper on photons “revolutionary.” As far as is known, this was the only work of his that he ever called revolutionary, and therefore it is quite fitting that in 1921 it brought him a Nobel Prize. That the situation was not as clear a few years earlier is witnessed by a famous letter signed by Planck, Walther Nernst, Heinrich Rubens, and Emil Warburg, suggesting Einstein for membership in the Prussian Academy of Sciences in 1913. They wrote: “That he might have in his speculations, occasionally, overshot the target, as for example in his light-quantum hypothesis, should not be counted against him too much, because without occasionally taking a risk, even in the most exact science no real innovation can be introduced.” Einstein’s deep, elegant, and beautiful explanation in 1905 of the entropy of radiation by proposing light quanta makes a strong case for the usefulness of occasional speculation.
GO SMALL
JEREMY BERNSTEIN
Professor of physics, emeritus, Stevens Institute of Technology; former staff writer, The New Yorker; author, Quantum Leaps
When confronted with a question like this, the temptation is to “go big” and respond with something, say, from Einstein’s theory of relativity. Instead I will go small. When Planck introduced his quantum of action at the turn of the 20th century, he realized that this allowed for a new set of natural units. For example, the Planck time is the square root of Planck’s constant times the gravitational constant divided by the fifth power of the speed of light. It is the smallest unit of time anyone talks about, but is it a “time”? The problem is that these constants are just that. They are the same to a resting observer as to a moving one. But the time is not. I posed this as a “divinette” to my “coven,” and Freeman Dyson came up with a beautiful answer. He tried to construct a clock that would measure it. Using the quantum uncertainties, he showed that it would be consumed by a black hole of its own making. No measurement is possible. The Planck time ain’t a time—or it may be beyond time.
WHY IS OUR WORLD COMPREHENSIBLE?
ANDREI LINDE
Father of eternal chaotic inflation; professor of physics, Stanford University
“The most incomprehensible thing about the world is that it is comprehensible.” So said Albert Einstein. A similar problem was noted by Eugene Wigner, who said
that the unreasonable efficiency of mathematics is “a wonderful gift which we neither understand nor deserve.”
Why do we live in a comprehensible universe, with certain rules that can be efficiently used for predicting our future?
Of course, one could always respond that this is “just so”—that God created the universe and made it simple enough so that we could comprehend it. But shall we give up so easily? Let us consider several other questions of a similar type. Why is our universe so large? Why don’t parallel lines intersect? Why do different parts of the universe look so similar? For a long time, such questions seemed too metaphysical to be considered seriously. Now we know that inflationary cosmology provides a possible answer to all of them.
To understand the issue, let’s consider some examples of an incomprehensible universe, where mathematics is inefficient. Suppose the universe is in a state of the so-called Planck density: r ~ 1094 g/cm3, which is 94 orders of magnitude greater than the density of water. According to the theory of quantum gravity, quantum fluctuations of spacetime in this regime are so large that all measuring sticks are rapidly bending, shrinking, and extending in a chaotic and unpredictable way—faster than you could measure distance with them. All clocks are destroyed faster than you could measure time with them. All records of previous events become erased, so that you cannot remember anything, record it, and predict the future. The universe is incomprehensible to anybody living there (if life is possible there at all), and the laws of mathematics cannot be efficiently used.
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