Any accusation of arrogance would have to center around the question, “But Einstein, how did you know you had reasoned correctly?”—to which I can only say: Do not criticize people when they turn out to be right! Wait for an occasion where they are wrong! Otherwise you are missing the chance to see when someone is thinking smarter than you—for you criticize them whenever they depart from a preferred ritual of cognition.
Or consider the famous exchange between Einstein and Niels Bohr on quantum theory—at a time when the then-current, single-world quantum theory seemed to be immensely well-confirmed experimentally; a time when, by the standards of Science, the current (deranged) quantum theory had simply won.
EINSTEIN: “God does not play dice with the universe.”
BOHR: “Einstein, don’t tell God what to do.”
You’ve got to admire someone who can get into an argument with God and win.
If you take off your Bayesian goggles, and look at Einstein in terms of what he actually did all day, then the guy was sitting around studying math and thinking about how he would design the universe, rather than running out and looking at things to gather more data. What Einstein did, successfully, is exactly the sort of high-minded feat of sheer intellect that Aristotle thought he could do, but couldn’t. Not from a probability-theoretic stance, mind you, but from the viewpoint of what they did all day long.
Science does not trust scientists to do this, which is why General Relativity was not blessed as the public knowledge of humanity until after it had made and verified a novel experimental prediction—having to do with the bending of light in a solar eclipse. (It later turned out that particular measurement was not precise enough to verify reliably, and had favored General Relativity essentially by luck.)
However, just because Science does not trust scientists to do something, does not mean it is impossible.
But a word of caution here: The reason why history books sometimes record the names of scientists who thought great high-minded thoughts is not that high-minded thinking is easier, or more reliable. It is a priority bias: Some scientist who successfully reasoned from the smallest amount of experimental evidence got to the truth first. This cannot be a matter of pure random chance: The theory space is too large, and Einstein won several times in a row. But out of all the scientists who tried to unravel a puzzle, or who would have eventually succeeded given enough evidence, history passes down to us the names of the scientists who successfully got there first. Bear that in mind, when you are trying to derive lessons about how to reason prudently.
In everyday life, you want every scrap of evidence you can get. Do not rely on being able to successfully think high-minded thoughts unless experimentation is so costly or dangerous that you have no other choice.
But sometimes experiments are costly, and sometimes we prefer to get there first . . . so you might consider trying to train yourself in reasoning on scanty evidence, preferably in cases where you will later find out if you were right or wrong. Trying to beat low-capitalization prediction markets might make for good training in this?—though that is only speculation.
As of now, at least, reasoning based on scanty evidence is something that modern-day science cannot reliably train modern-day scientists to do at all. Which may perhaps have something to do with, oh, I don’t know, not even trying?
Actually, I take that back. The most sane thinking I have seen in any scientific field comes from the field of evolutionary psychology, possibly because they understand self-deception, but also perhaps because they often (1) have to reason from scanty evidence and (2) do later find out if they were right or wrong. I recommend to all aspiring rationalists that they study evolutionary psychology simply to get a glimpse of what careful reasoning looks like. See particularly Tooby and Cosmides’s “The Psychological Foundations of Culture.”1
As for the possibility that only Einstein could do what Einstein did . . . that it took superpowers beyond the reach of ordinary mortals . . . here we run into some biases that would take a separate essay to analyze. Let me put it this way: It is possible, perhaps, that only a genius could have done Einstein’s actual historical work. But potential geniuses, in terms of raw intelligence, are probably far more common than historical superachievers. To put a random number on it, I doubt that anything more than one-in-a-million g-factor is required to be a potential world-class genius, implying at least six thousand potential Einsteins running around today. And as for everyone else, I see no reason why they should not aspire to use efficiently the evidence that they have.
But my final moral is that the frontier where the individual scientist rationally knows something that Science has not yet confirmed is not always some innocently data-driven matter of spotting a strong regularity in a mountain of experiments. Sometimes the scientist gets there by thinking great high-minded thoughts that Science does not trust you to think.
I will not say, “Don’t try this at home.” I will say, “Don’t think this is easy.” We are not discussing, here, the victory of casual opinions over professional scientists. We are discussing the sometime historical victories of one kind of professional effort over another. Never forget all the famous historical cases where attempted armchair reasoning lost.
*
1. Tooby and Cosmides, “The Psychological Foundations of Culture.”
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That Alien Message
Imagine a world much like this one, in which, thanks to gene-selection technologies, the average IQ is 140 (on our scale). Potential Einsteins are one-in-a-thousand, not one-in-a-million; and they grow up in a school system suited, if not to them personally, then at least to bright kids. Calculus is routinely taught in sixth grade. Albert Einstein, himself, still lived and still made approximately the same discoveries, but his work no longer seems exceptional. Several modern top-flight physicists have made equivalent breakthroughs, and are still around to talk.
(No, this is not the world Brennan lives in.)
One day, the stars in the night sky begin to change.
Some grow brighter. Some grow dimmer. Most remain the same. Astronomical telescopes capture it all, moment by moment. The stars that change change their luminosity one at a time, distinctly so; the luminosity change occurs over the course of a microsecond, but a whole second separates each change.
It is clear, from the first instant anyone realizes that more than one star is changing, that the process seems to center around Earth particularly. The arrival of the light from the events, at many stars scattered around the galaxy, has been precisely timed to Earth in its orbit. Soon, confirmation comes in from high-orbiting telescopes (they have those) that the astronomical miracles do not seem as synchronized from outside Earth. Only Earth’s telescopes see one star changing every second (1,005 milliseconds, actually).
Almost the entire combined brainpower of Earth turns to analysis.
It quickly becomes clear that the stars that jump in luminosity all jump by a factor of exactly 256; those that diminish in luminosity diminish by a factor of exactly 256. There is no apparent pattern in the stellar coordinates. This leaves, simply, a pattern of BRIGHT-dim-BRIGHT-BRIGHT . . .
“A binary message!” is everyone’s first thought.
But in this world there are careful thinkers, of great prestige as well, and they are not so sure. “There are easier ways to send a message,” they post to their blogs, “if you can make stars flicker, and if you want to communicate. Something is happening. It appears, prima facie, to focus on Earth in particular. To call it a ‘message’ presumes a great deal more about the cause behind it. There might be some kind of evolutionary process among, um, things that can make stars flicker, that ends up sensitive to intelligence somehow . . . Yeah, there’s probably something like ‘intelligence’ behind it, but try to appreciate how wide a range of possibilities that really implies. We don’t know this is a message, or that it was sent from the same kind of motivations that might move us. I mean, we would just signal using a big flashlight, we wou
ldn’t mess up a whole galaxy.”
By this time, someone has started to collate the astronomical data and post it to the Internet. Early suggestions that the data might be harmful have been . . . not ignored, but not obeyed, either. If anything this powerful wants to hurt you, you’re pretty much dead (people reason).
Multiple research groups are looking for patterns in the stellar coordinates—or fractional arrival times of the changes, relative to the center of the Earth—or exact durations of the luminosity shift—or any tiny variance in the magnitude shift—or any other fact that might be known about the stars before they changed. But most people are turning their attention to the pattern of BRIGHTs and dims.
It becomes clear almost instantly that the pattern sent is highly redundant. Of the first 16 bits, 12 are BRIGHTs and 4 are dims. The first 32 bits received align with the second 32 bits received, with only 7 out of 32 bits different, and then the next 32 bits received have only 9 out of 32 bits different from the second (and 4 of them are bits that changed before). From the first 96 bits, then, it becomes clear that this pattern is not an optimal, compressed encoding of anything. The obvious thought is that the sequence is meant to convey instructions for decoding a compressed message to follow . . .
“But,” say the careful thinkers, “anyone who cared about efficiency, with enough power to mess with stars, could maybe have just signaled us with a big flashlight, and sent us a DVD?”
There also seems to be structure within the 32-bit groups; some 8-bit subgroups occur with higher frequency than others, and this structure only appears along the natural alignments (32 = 8 + 8 + 8 + 8).
After the first five hours at one bit per second, an additional redundancy becomes clear: The message has started approximately repeating itself at the 16,385th bit.
Breaking up the message into groups of 32, there are 7 bits of difference between the 1st group and the 2nd group, and 6 bits of difference between the 1st group and the 513th group.
“A 2D picture!” everyone thinks. “And the four 8-bit groups are colors; they’re tetrachromats!”
But it soon becomes clear that there is a horizontal/vertical asymmetry: Fewer bits change, on average, between (N,N + 1) versus (N,N + 512). Which you wouldn’t expect if the message was a 2D picture projected onto a symmetrical grid. Then you would expect the average bitwise distance between two 32-bit groups to go as the 2-norm of the grid separation: √(h2 + v2).
There also forms a general consensus that a certain binary encoding from 8-groups onto integers between -64 and 191—not the binary encoding that seems obvious to us, but still highly regular—minimizes the average distance between neighboring cells. This continues to be borne out by incoming bits.
The statisticians and cryptographers and physicists and computer scientists go to work. There is structure here; it needs only to be unraveled. The masters of causality search for conditional independence, screening-off and Markov neighborhoods, among bits and groups of bits. The so-called “color” appears to play a role in neighborhoods and screening, so it’s not just the equivalent of surface reflectivity. People search for simple equations, simple cellular automata, simple decision trees, that can predict or compress the message. Physicists invent entire new theories of physics that might describe universes projected onto the grid—for it seems quite plausible that a message such as this is being sent from beyond the Matrix.
After receiving 32 × 512 × 256 = 4,194,304 bits, around one and a half months, the stars stop flickering.
Theoretical work continues. Physicists and cryptographers roll up their sleeves and seriously go to work. They have cracked problems with far less data than this. Physicists have tested entire theory-edifices with small differences of particle mass; cryptographers have unraveled shorter messages deliberately obscured.
Years pass.
Two dominant models have survived, in academia, in the scrutiny of the public eye, and in the scrutiny of those scientists who once did Einstein-like work. There is a theory that the grid is a projection from objects in a 5-dimensional space, with an asymmetry between 3 and 2 of the spatial dimensions. There is also a theory that the grid is meant to encode a cellular automaton—arguably, the grid has several fortunate properties for such. Codes have been devised that give interesting behaviors; but so far, running the corresponding automata on the largest available computers has failed to produce any decodable result. The run continues.
Every now and then, someone takes a group of especially brilliant young students who’ve never looked at the detailed binary sequence. These students are then shown only the first 32 rows (of 512 columns each), to see if they can form new models, and how well those new models do at predicting the next 224. Both the 3+2 dimensional model, and the cellular automaton model, have been well duplicated by such students; they have yet to do better. There are complex models finely fit to the whole sequence—but those, everyone knows, are probably worthless.
Ten years later, the stars begin flickering again.
Within the reception of the first 128 bits, it becomes clear that the Second Grid can fit to small motions in the inferred 3+2 dimensional space, but does not look anything like the successor state of any of the dominant cellular automaton theories. Much rejoicing follows, and the physicists go to work on inducing what kind of dynamical physics might govern the objects seen in the 3+2 dimensional space. Much work along these lines has already been done, just by speculating on what type of balanced forces might give rise to the objects in the First Grid, if those objects were static—but now it seems not all the objects are static. As most physicists guessed—statically balanced theories seemed contrived.
Many neat equations are formulated to describe the dynamical objects in the 3+2 dimensional space being projected onto the First and Second Grids. Some equations are more elegant than others; some are more precisely predictive (in retrospect, alas) of the Second Grid. One group of brilliant physicists, who carefully isolated themselves and looked only at the first 32 rows of the Second Grid, produces equations that seem elegant to them—and the equations also do well on predicting the next 224 rows. This becomes the dominant guess.
But these equations are underspecified; they don’t seem to be enough to make a universe. A small cottage industry arises in trying to guess what kind of laws might complete the ones thus guessed.
When the Third Grid arrives, ten years after the Second Grid, it provides information about second derivatives, forcing a major modification of the “incomplete but good” theory. But the theory doesn’t do too badly out of it, all things considered.
The Fourth Grid doesn’t add much to the picture. Third derivatives don’t seem important to the 3+2 physics inferred from the Grids.
The Fifth Grid looks almost exactly like it is expected to look.
And the Sixth Grid, and the Seventh Grid.
(Oh, and every time someone in this world tries to build a really powerful AI, the computing hardware spontaneously melts. This isn’t really important to the story, but I need to postulate this in order to have human people sticking around, in the flesh, for seventy years.)
My moral?
That even Einstein did not come within a million light-years of making efficient use of sensory data.
Riemann invented his geometries before Einstein had a use for them; the physics of our universe is not that complicated in an absolute sense. A Bayesian superintelligence, hooked up to a webcam, would invent General Relativity as a hypothesis—perhaps not the dominant hypothesis, compared to Newtonian mechanics, but still a hypothesis under direct consideration—by the time it had seen the third frame of a falling apple. It might guess it from the first frame, if it saw the statics of a bent blade of grass.
We would think of it. Our civilization, that is, given ten years to analyze each frame. Certainly if the average IQ was 140 and Einsteins were common, we would.
Even if we were human-level intelligences in a different sort of physics—minds who had never seen a 3D space pr
ojected onto a 2D grid—we would still think of the 3D → 2D hypothesis. Our mathematicians would still have invented vector spaces, and projections.
Even if we’d never seen an accelerating billiard ball, our mathematicians would have invented calculus (e.g. for optimization problems).
Heck, think of some of the crazy math that’s been invented here on our Earth.
I occasionally run into people who say something like, “There’s a theoretical limit on how much you can deduce about the outside world, given a finite amount of sensory data.”
Yes. There is. The theoretical limit is that every time you see 1 additional bit, it cannot be expected to eliminate more than half of the remaining hypotheses (half the remaining probability mass, rather). And that a redundant message cannot convey more information than the compressed version of itself. Nor can a bit convey any information about a quantity with which it has correlation exactly zero across the probable worlds you imagine.
But nothing I’ve depicted this human civilization doing even begins to approach the theoretical limits set by the formalism of Solomonoff induction. It doesn’t approach the picture you could get if you could search through every single computable hypothesis, weighted by their simplicity, and do Bayesian updates on all of them.
To see the theoretical limit on extractable information, imagine that you have infinite computing power, and you simulate all possible universes with simple physics, looking for universes that contain Earths embedded in them—perhaps inside a simulation—where some process makes the stars flicker in the order observed. Any bit in the message—or any order of selection of stars, for that matter—that contains the tiniest correlation (across all possible computable universes, weighted by simplicity) to any element of the environment gives you information about the environment.
Rationality- From AI to Zombies Page 111
