In 1919, Sir Arthur Eddington led expeditions to Brazil and to the island of Principe, aiming to observe solar eclipses and thereby test an experimental prediction of Einstein’s novel theory of General Relativity. A journalist asked Einstein what he would do if Eddington’s observations failed to match his theory. Einstein famously replied: “Then I would feel sorry for the good Lord. The theory is correct.”

It seems like a rather foolhardy statement, defying the trope of Traditional Rationality that experiment above all is sovereign. Einstein seems possessed of an arrogance so great that he would refuse to bend his neck and submit to Nature’s answer, as scientists must do. Who can *know* that the theory is correct, in advance of experimental test?

Of course, Einstein did turn out to be right. I try to avoid criticizing people when they are right. If they genuinely deserve criticism, I will not need to wait long for an occasion where they are wrong.

And Einstein may not have been quite so foolhardy as he sounded . . .

To assign more than 50% probability to the correct candidate from a pool of 100,000,000 possible hypotheses, you need at least 27 bits of evidence (or thereabouts). You cannot expect to find the correct candidate without tests that are this strong, because lesser tests will yield more than one candidate that passes all the tests. If you try to apply a test that only has a million-to-one chance of a false positive (~ 20 bits), you’ll end up with a hundred candidates. Just *finding* the right answer, within a large space of possibilities, requires a large amount of evidence.

Traditional Rationality emphasizes justification: “If you want to convince me of X, you’ve got to present me with Y amount of evidence.” I myself often slip into this phrasing, whenever I say something like, “To *justify* believing in this proposition, at more than 99% probability, requires 34 bits of evidence.” Or, “In order to assign more than 50% probability to your hypothesis, you need 27 bits of evidence.” The Traditional phrasing implies that you start out with a hunch, or some private line of reasoning that leads you to a suggested hypothesis, and then you have to gather “evidence” to *confirm* it—to convince the scientific community, or justify saying that you *believe* in your hunch.

But from a Bayesian perspective, you need an amount of evidence roughly equivalent to the complexity of the hypothesis just to locate the hypothesis in theory-space. It’s not a question of justifying anything to anyone. If there’s a hundred million alternatives, you need at least 27 bits of evidence just to focus your attention uniquely on the correct answer.

This is true even if you call your guess a “hunch” or “intuition.” Hunchings and intuitings are real processes in a real brain. If your brain doesn’t have at least 10 bits of genuinely entangled valid Bayesian evidence to chew on, your brain cannot single out a correct 10-bit hypothesis for your attention—consciously, subconsciously, whatever. Subconscious processes can’t find one out of a million targets using only 19 bits of entanglement any more than conscious processes can. Hunches can be mysterious to the huncher, but they can’t violate the laws of physics.

You see where this is going: *At the time of first formulating the hypothesis*—the very first time the equations popped into his head—Einstein must have had, *already in his possession,* sufficient observational evidence to single out the complex equations of General Relativity for his unique attention. Or he couldn’t have gotten them *right.*

Now, how likely is it that Einstein would have *exactly* enough observational evidence to raise General Relativity to the level of his attention, but only justify assigning it a 55% probability? Suppose General Relativity is a 29.3-bit hypothesis. How likely is it that Einstein would stumble across *exactly* 29.5 bits of evidence in the course of his physics reading?

Not likely! If Einstein had enough observational evidence to single out the correct equations of General Relativity in the first place, then he probably had enough evidence to be *damn sure* that General Relativity was true.

In fact, since the human brain is not a perfectly efficient processor of information, Einstein probably had *overwhelmingly more evidence* than would, in principle, be required for a perfect Bayesian to assign massive confidence to General Relativity.

“Then I would feel sorry for the good Lord; the theory is correct.” It doesn’t sound nearly as appalling when you look at it from that perspective. And remember that General Relativity *was* correct, from all that vast space of possibilities.

Most theorists think they have the right theory but are wrong. So just because Einstein was right, that doesn't mean he had good reason to believe he was right. He could have been a lucky draw from the same process.

"And remember that General Relativity was correct, from all the vast space of possibilities."

The Einstein field equation itself is actually extremely simple:

G = 8

piTwhere G is the Einstein tensor and T is the stress-energy tensor. Few serious competitors to GR have emerged for a very good reason; what sane modifications could you make to this equation? G and T have to be directly proportional, because everyone knows that the curvature of spacetime (and hence the effect of gravity) is directly proportional to the quantity of matter/energy. The constant of proportionality is fixed by direct measurement of g. G must vanish when T vanishes, as there must be no gravity in the absence of matter. T itself cannot be modified, because it's the only sane way to measure mass, energy, and momentum in the Lorentzian manifold framework. G cannot be modified, because it must be constructable from the metric tensor (a property of spacetime), it must be directly proportional to the amount of curvature, and it must be invariant with respect to the choice of coordinate system (the full derivation is left as an exercise to the reader in my textbook).

Hanson, that's why I picked Einstein - he'd already been "lucky" once at that point. Also, he would still need quite a lot of evidence just to get to the point of having a

remotechance of being right.McCabe, you're right, it's completely obvious, it makes you wonder why Einstein took ten years to figure it out.

Tom, is that an elaborate joke?

I agree with Tom that there isn't that much room to change the field equations once you have decided on the Riemannian tensor framework: gravity cannot be expressed as first-order differential equations and still fit with observation, while number of objects to build a set of second-order equations is very limited. The equations are the simplest possibility (with the cosmological constant as a slight uglification, but it is just a constant of integration).

But selecting the tensor framework, that is of course where all the bits had to go. It is not an obvious choice at all.

It is interesting to note that Einstein's last paper, "On the relativistic theory of the non-symmetric field" includes a discussion of the "strength" of different theories in terms of how many undetermined degrees of freedom they have. http://books.google.com/books?id=tB9Roi3YnAgC&pg=PA131&lpg=PA131&dq=%22relativistic+theory+of+the+non+symmetric+field%22&source=web&ots=EkMv5tudsI&sig=lkTQE94Ay1h2-qS0mcbGT3xa22M If I recall right, he finds his own theory to be rather flabby.

Um, guys, there are an infinite number of possible hypotheses. Any evidence that corroborates one theory also corroborates (or fails to refute) an infinite number of alternative specifiable accounts of the world.

What evidence does is allow us to say "Whatever the truth is, it must coexist in the same universe with the true nature of this evidence I have accepted. Theory X and its infinite number of variants seems to be ruled out by this evidence (although I may have misinterpreted the theory or the nature of the evidence), whereas Theory Y and its inf... (read more)

Many popular reports of Eddington's test mislead people into thinking it provided significant evidence. See these two Wikipedia pages for reports that the raw evidence was nearly worthless. Einstein may have known how little evidence that test would provide.

"McCabe, you're right, it's completely obvious, it makes you wonder why Einstein took ten years to figure it out."

I never said it was obvious; I said that the equations were a unique solution imposed by various constraints. Proving that the equations are a unique solution is quite difficult; I can't do it, even with a ready-made textbook in front of me. There are many examples of simple, unique-solution equations being very hard to derive- Newton's law of gravity and Maxwell's laws of electromagnetism come to mind.

"But selecting the tensor f... (read more)

The Einstein field equation itself is actually extremely simple:G = 8

piTSure, if we don't mind that G and T take a full page to write out in terms of the derivatives of the metric tensor. By this logic

everyequation is extremely simple -- it simply asserts that A=B for some A,B. :-)http://mathoverflow.net/questions/53122/mathematical-urban-legends

Another urban legend, which I've heard told about various mathematicians, and which Misha Polyak self-effacingly tells about himself (and therefore might even be true), is the following:

As a young postdoc, Misha was giving a talk at a prestigious US university about his new diagrammatic formula for a certain finite type invariant, which had 158 terms. A famous (but unnamed) mathematician was sitting, sleeping, in the front row. "Oh dear, he doesn't like my talk," thought Misha. But then, just as Misha's talk was coming to a close, the famous professor wakes with a start. Like a man possessed, the famous professor leaps up out of his chair, and cries, "By golly! That looks exactly like the Grothendieck-Riemann-Roch Theorem!!!" Misha didn't know what to say. Perhaps, in his sleep, this great professor had simplified Misha's 158 term diagrammatic formula for a topological invariant, and had discovered a deep mathematical connection with algebraic geometry? It was, after all, not impossible. Misha paced in front of the board silently, not knowing quite how to respond. Should he feign understanding, or... (read more)

"Sure, if we don't mind that G and T take a full page to write out in terms of the derivatives of the metric tensor."

The Riemann tensor is a more natural measure of curvature than the metric tensor, and even in that language it's still pretty simple:

8

piT = R (tensor) - .5gR (scalar)where R (tensor) (subscript) ab = Riemann tensor (superscript) c (subscript) acb and R (scalar) = g (superscript) ab * R (tensor) (subscript) ab

You can make any theory seem complicated by writing it out in some nonstandard format. Take Maxwell's equations of electromag... (read more)

I thought that, when you try to apply general relativity to a world described by quantum mechanics, you end up trying to measure curvature of surfaces that do not have a well-defined curvature, much like how the curvature (derivative) of y = |x| is undefined at x=0?

I've heard several different descriptions of the "contradictions" between quantum mechanics and general relativity. One is that the mathematical functions used to define general relativity are undefined on the type of spacetime described by quantum mechanics; naively trying to apply on... (read more)

"If only you had been around to solve the problem instead of Maxwell and Einstein, how much work could have been saved!"

Obvious != simple != easy to learn. You of all people should understand this. You seemed to understand it seven years ago, back during the days of your wild and reckless youth. To quote SitS:

"Let's take a concrete example, the story Flowers for Algernon (later the movie Charly), by Daniel Keyes. (I'm afraid I'll have to tell you how the story comes out, but it's a Character story, not an Idea story, so that shouldn't spoil it.) Flowers for Algernon is about a neurosurgical procedure for intelligence enhancement. This procedure was first tested on a mouse, Algernon, and later on a retarded human, Charlie Gordon. The enhanced Charlie has the standard science-fictional set of superhuman characteristics; he thinks fast, learns a lifetime of knowledge in a few weeks, and discusses arcane mathematics (not shown). Then the mouse, Algernon, gets sick and dies. Charlie analyzes the enhancement procedure (not shown) and concludes that the process is basically flawed. Later, Charlie dies.

That's a science-fictional enhanced human. A real enhanced human ... (read more)

In other words: Einstein also said that God does not play dice with the universe. However, not only does God play dice, but sometimes he ignores the result and just says it worked.

"Fixed by evidence" != "simple". There are few alternatives to Newton's Laws, perhaps, once you (a) invent calculus as the language of description, the interpreter to run the code; (b) observe Kepler's laws; (c) realize that objects in motion remain in motion unless a force acts upon them, as opposed to Aristotle's view, and therefore the law should be written in second derivatives as opposed to third or first derivatives; etc. etc.

Please recall that my original contention was that Einstein must have had enough observational evidence t... (read more)

"Please recall that my original contention was that Einstein must have had enough observational evidence to fix the information inherent in General Relativity as a solution. If you describe ways that the information in General Relativity can be fixed by evidence, you are not contradicting this."

True; why do you have to contradict the main point of a post to comment on it? My point was that the space of possibilities was not vast; it was quite small, given the common-sense rules of gravity and math which were known at the time. Developing GR took ... (read more)

""Fixed by evidence" != "simple"."

This is certainly true in the general case, but all physics theories which I've studied in detail really are simple, in the bits of entropy sense.

That waste of three minutes wasn't your fault. But the decision to sink more time into posting a comment that obviously won't do any good (not least because it's completely unspecific) was.

Guys there's something else worth mentioning here. Einstein had had different conviction about theories. Briefly, in his idealistic ecumenical thoughts, he referred that a real 'Theory'

shouldbe articulated and conceived ipso facto, without any evidence whatsoever, before theobservationscan corroborate the theories predictions.In his own context Einstein you know used to devise the intense 'Thought Experiments', something so insightful, ideation of which can only be possible in an Einstein's brain nerves. The slew of scientific developments taking place... (read more)

Abraham Pais, one of Einstein's many friends, has said that Einstein loved to joke. Are you sure his "sorry for the good Lord" wasn't a bit of humor?

Just as no significant algebra can be both complete and consistent, we can expect that in our future, someone standing on Einstein's shoulders will "correct" his equations the same way that his expanded upon Newton's.

Scientific theories are never proved correct; at best they are merely not disproved by any tests run against them; and have some utility or other attraction (e.g., "beauty.") Odd that this group would say Einstein was proved correct, in an article about how Lord Eddington was merely failing to propose a test with enough power to disprove it.

I would suggest here where Einstein got his evidence. General relativity started from a simple assumption: that inertial mass and gravitational mass are the same. Before Einstein, this was a mere observation, and nobody had really asked themselves why it was so (I'm oversimplifying here of course). But Einstein stated this as a fundamental principle, an axiom if you want. And then he went on to draw what logical conclusion could be drawn out from that basic axiom. Sure it took him ten years, because it wasn't obvious at all, and the mathematical tools to d... (read more)

Something doesn't feel right. Don't people frequently propose complex theories that turn out to be wrong?

Tarleton, people do propose lots of complex wrong theories, but they don't propose literally quintillions of wrong complex theories for every right complex theory. If the ratio is even ten wrong to one right, you can tell the good guessers must have possessed massive evidence - survivorship bias is not remotely enough to account for it. As for the wrong guessers, they are more likely to have suffered from bad evidence or bad thinking, than from having almost exactly enough evidence processed correctly followed by a wrong guess.

I like to think Einstein's confidence came instead from his belief that Relativity suitably justified the KL divergence between experiments in 1905 and physics theory in 1905. He was not necessarily in full possession of whatever evidence was required to narrow the hypothesis space down to relativity (which is a bit of a misformulation, I feel, since this space still contains a number of other theories both equally and more powerful than Physics+Relativity) but instead possessed enough so that in his own mental metropolis jumping he stumbled across Relativ... (read more)

The point of science isn't just to gather evidence. It's to gather evidence without bias. Whatever Einstein was doing looked like really good evidence to him, but he couldn't have been sure it was good evidence. It might have just looked like good evidence. Science works by just ignoring all the evidence that might be biased. You're ignoring a lot of evidence there, but so long as you can gather evidence relatively cheaply, that's not much of a problem.

Suppose Einstein had a 50% chance of being significantly biased. He only gathered 20 bits of evidence ins... (read more)

Einstein didn't come up with General Relativity that way. He didn't even do the hard math himself. He came up with some little truths (e.g. equivalence, speed of light is constant, covariance, must reduce to Newtonian gravity in unexceptional cases), from a handful of results that didn't seem to fit classical theory, and then he found a set of equations that fit.

Newtonian gravity provided heaps of data points and a handful of non-fits. Einstein bootstrapped on prior achievements like Newtonian gravity and special relativity and tweaked them to fit a han... (read more)

It's not an assumption, it's a definition. Whatever is enough to cut your current set of candidates in half is "one bit"- the first bit will eliminate 50,000,000 people, the last bit will eliminate 1. An answer that reduces the set of candidates to .000001 times its original size contains 20 bits of information. (Notice that the

questiondoesn't have bits of information associated with it, since each possible answer reduces the candidate set by a different amount- if they said "no," you acquired only a millionth of a bit of information.)Maybe we should also consider that Einstein fully understood the irony in his statement, and was in a humorous mood. After all, what he would do if the attempt to verify did not succeed was not of any import whatever. It was a typical "sell newspaper" question.

I guess a defense of old Albert would go something like this; the route he took to establish his theory didn't rely upon empirical evidence of the sort Eddington was trying to discover but rather was an elegant way to explain certain unusual properties of light and energy which, once he had formulated his theory, it seemed obvious to him could not be explained any other way. The kind of empirical validation which Eddington was carrying out was a laudable and necessary step in the process of theory confirmation/falstification but nevertheless it is entirely... (read more)

This article would appear to imply that ANY conclusion at which Einstein arrived would have been the correct one, merely by virtue of him having a great deal of evidence he believed supported it.

Once you assume:

1) the equations describing gravity are invariant under all coordinate transformations,

2) energy-momentum is not locally created or destroyed,

3) the equations describing gravity involve only the flow of energy-momentum and the curvature of the spacetime metric (and not powers or products or derivatives of these),

4) the equations reduce to ordinary Newtonian gravity in a suitable limit,

then Einstein's equations for general relativity are the only possible choice... except for one adjustable parameter, the cosmological constant.

(First Einstein said this constant was nonzero, then he said that was the "biggest mistake in his life", and then it turned out he was right in the first place. It's not zero, it's roughly 0.0000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000001. So, a bit of waffling on this issue is understandable.)

It took Einstein about 10 years of hard work to figure this out, with a lot of help from a mathematician Marcel Grossman who taught him the required math. But by the time he talked to that reporter he knew this stuff. That's what gave him his confidence... (read more)