Yesterday I argued that the Powers Beyond Science are actually a standard and necessary part of the social process of science. In particular, scientists must call upon their powers of individual rationality to decide what ideas to test, in advance of the sort of definite experiments that Science demands to bless an idea as confirmed. The ideal of Science does not try to specify this process—we don't suppose that any public authority knows how individual scientists should think—but this doesn't mean the process is unimportant.
A readily understandable, non-disturbing example:
A scientist identifies a strong mathematical regularity in the cumulative data of previous experiments. But the corresponding hypothesis has not yet made and confirmed a novel experimental prediction—which his academic field demands; this is one of those fields where you can perform controlled experiments without too much trouble. Thus the individual scientist has readily understandable, rational reasons to believe (though not with probability 1) something not yet blessed by Science as public knowledge of humankind.
Noticing a regularity in a huge mass of experimental data, doesn't seem all that unscientific. You're still data-driven, right?
But that's because I deliberately chose a non-disturbing example. When Einstein invented General Relativity, he had almost no experimental data to go on, except the precession of Mercury's perihelion. And (AFAIK) Einstein did not use that data, except at the end.
Einstein generated the theory of Special Relativity using Mach's Principle, which is the physicist's version of the Generalized Anti-Zombie Principle. You begin by saying, "It doesn't seem reasonable to me that you could tell, in an enclosed room, how fast you and the room were going. Since this number shouldn't ought to be observable, it shouldn't ought to exist in any meaningful sense." You then observe that Maxwell's Equations invoke a seemingly absolute speed of propagation, c, commonly referred to as "the speed of light" (though the quantum equations show it is the propagation speed of all fundamental waves). So you reformulate your physics in such fashion that the absolute speed of a single object no longer meaningfully exists, and only relative speeds exist. I am skipping over quite a bit here, obviously, but there are many excellent introductions to relativity—it is not like the horrible situation in quantum physics.
Einstein, having successfully done away with the notion of your absolute speed inside an enclosed room, then set out to do away with the notion of your absolute acceleration inside an enclosed room. It seemed to Einstein that there shouldn't ought to be a way to differentiate, in an enclosed room, between the room accelerating northward while the rest of the universe stayed still, versus the rest of the universe accelerating southward while the room stayed still. If the rest of the universe accelerated, it would produce gravitational waves that would accelerate you. Moving matter, then, should produce gravitational waves.
And because inertial mass and gravitational mass were always exactly equivalent—unlike the situation in electromagnetics, where an electron and a muon can have different masses but the same electrical charge—gravity should reveal itself as a kind of inertia. The Earth should go around the Sun in some equivalent of a "straight line". This requires spacetime in the vicinity of the Sun to be curved, so that if you drew a graph of the Earth's orbit around the Sun, the line on the 4D graph paper would be locally flat. Then inertial and gravitational mass would be necessarily equivalent, not just coincidentally equivalent.
(If that did not make any sense to you, there are good introductions to General Relativity available as well.)
And of course the new theory had to obey Special Relativity, and conserve energy, and conserve momentum, etcetera.
Einstein spent several years grasping the necessary mathematics to describe curved metrics of spacetime. Then he wrote down the simplest theory that had the properties Einstein thought it ought to have—including properties no one had ever observed, but that Einstein thought fit in well with the character of other physical laws. Then Einstein cranked a bit, and got the previously unexplained precession of Mercury right back out.
How impressive was this?
Well, let's put it this way. In some small fraction of alternate Earths proceeding from 1800—perhaps even a sizeable fraction—it would seem plausible that relativistic physics could have proceeded in a similar fashion to our own great fiasco with quantum physics.
We can imagine that Lorentz's original "interpretation" of the Lorentz contraction, as a physical distortion caused by movement with respect to the ether, prevailed. We can imagine that various corrective factors, themselves unexplained, were added on to Newtonian gravitational mechanics to explain the precession of Mercury—attributed, perhaps, to strange distortions of the ether, as in the Lorentz contraction. Through the decades, further corrective factors would be added on to account for other astronomical observations. Sufficiently precise atomic clocks, in airplanes, would reveal that time ran a little faster than expected at higher altitudes (time runs slower in more intense gravitational fields, but they wouldn't know that) and more corrective "ethereal factors" would be invented.
Until, finally, the many different empirically determined "corrective factors" were unified into the simple equations of General Relativity.
And the people in that alternate Earth would say, "The final equation was simple, but there was no way you could possibly know to arrive at that answer from just the perihelion precession of Mercury. It takes many, many additional experiments. You must have measured time running slower in a stronger gravitational field; you must have measured light bending around stars. Only then can you imagine our unified theory of ethereal gravitation. No, not even a perfect Bayesian superintelligence could know it!—for there would be many ad-hoc theories consistent with the perihelion precession alone."
In our world, Einstein didn't even use the perihelion precession of Mercury, except for verification of his answer produced by other means. Einstein sat down in his armchair, and thought about how he would have designed the universe, to look the way he thought a universe should look—for example, that you shouldn't ought to be able to distinguish yourself accelerating in one direction, from the rest of the universe accelerating in the other direction.
And Einstein executed the whole long (multi-year!) chain of armchair reasoning, without making any mistakes that would have required further experimental evidence to pull him back on track.
Even Jeffreyssai would be grudgingly impressed. Though he would still ding Einstein a point or two for the cosmological constant. (I don't ding Einstein for the cosmological constant because it later turned out to be real. I try to avoid criticizing people on occasions where they are right.)
What would be the probability-theoretic perspective on Einstein's feat?
Rather than observe the planets, and infer what laws might cover their gravitation, Einstein was observing the other laws of physics, and inferring what new law might follow the same pattern. Einstein wasn't finding an equation that covered the motion of gravitational bodies. Einstein was finding a character-of-physical-law that covered previously observed equations, and that he could crank to predict the next equation that would be observed.
Nobody knows where the laws of physics come from, but Einstein's success with General Relativity shows that their common character is strong enough to predict the correct form of one law from having observed other laws, without necessarily needing to observe the precise effects of the law.
(In a general sense, of course, Einstein did know by observation that things fell down; but he did not get GR by backward inference from Mercury's exact perihelion advance.)
So, from a Bayesian perspective, what Einstein did is still induction, and still covered by the notion of a simple prior (Occam prior) that gets updated by new evidence. It's just the prior was over the possible characters of physical law, and observing other physical laws let Einstein update his model of the character of physical law, which he then used to predict a particular law of gravitation.
If you didn't have the concept of a "character of physical law", what Einstein did would look like magic—plucking the correct model of gravitation out of the space of all possible equations, with vastly insufficient evidence. But Einstein, by looking at other laws, cut down the space of possibilities for the next law. He learned the alphabet in which physics was written, constraints to govern his answer. Not magic, but reasoning on a higher level, across a wider domain, than what a naive reasoner might conceive to be the "model space" of only this one law.
So from a probability-theoretic standpoint, Einstein was still data-driven—he just used the data he already had, more effectively. Compared to any alternate Earths that demanded huge quantities of additional data from astronomical observations and clocks on airplanes to hit them over the head with General Relativity.
There are numerous lessons we can derive from this.
I use Einstein as my example, even though it's cliche, because Einstein was also unusual in that he openly admitted to knowing things that Science hadn't confirmed. Asked what he would have done if Eddington's solar eclipse observation had failed to confirm General Relativity, Einstein replied: "Then I would feel sorry for the good Lord. The theory is correct."
According to prevailing notions of Science, this is arrogance—you must accept the verdict of experiment, and not cling to your personal ideas.
But as I concluded in Einstein's Arrogance, Einstein doesn't come off nearly as badly from a Bayesian perspective. From a Bayesian perspective, in order to suggest General Relativity at all, in order to even think about what turned out to be the correct answer, Einstein must have had enough evidence to identify the true answer in the theory-space. It would take only a little more evidence to justify (in a Bayesian sense) being nearly certain of the theory. And it was unlikely that Einstein only had exactly enough evidence to bring the hypothesis all the way up to his attention.
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 GR 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".
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 post 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.