Einstein's Arrogance

Prerequisite:  How Much Evidence Does It Take?

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 Reality 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," doesn't sound nearly as appalling when you look at it from that perspective.  And remember that General Relativity was correct, from all the vast space of possibilities.

Comments

[Robin Hanson]

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.

[Tom McCabe]

"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*pi*T

where 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).

[Eliezer Yudkowsky]

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 remote chance of being right.

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

[Constant]

Tom, is that an elaborate joke?

[Anders Sandberg]

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.

[James Bach]

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 infinite number of variants seems not yet to be ruled out."

Yeah, I realize this is a complicated way to phrase it. The reason I like to phrase it this way is to point out that Einstein did not have merely 29 "bits" of evidence, he had VAST evidence, based on an entire lifetime of neuron-level programming, that automatically focused his mind on a productive way of thinking about the universe. He was imagining and eliminating vast swaths of potential theories of the universe, as are we all, from his earliest days in the womb. This is hardly surprising, considering that humans are the result of an evolutionary process that systematically killed the creatures who couldn't map the universe sufficiently well.

We can never know if we are getting to the right hypothesis. What we can say is that we have arrived at a hypothesis that is isomorphic with the truth, as we understand that hypothesis, over the span of evidence we think we have and think we understand. Always the next bit of evidence we discover may turn what we think we knew upside down. All knowledge is defeasible.

[Peter McCluskey]

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.

[Tom McCabe]

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

I agree that it is not at all obvious, but the search space doesn't seem to be all that large- how many mathematical toys are there which could form a viable framework for gravity? The difficulty seems to be in understanding the math well enough to determine whether it can represent real-world phenomena. Differential geometry is not a simple Bayesian hypothesis like "the cat is blue"; to figure out whether piece of evidence Q supports a geometric theory of gravity, you have to understand what a geometric theory of gravity would look like (in Bayesian terms, which outcomes it would predict), which is quite difficult.

"Tom, is that an elaborate joke?"

No. What makes you think that?

[Scott Aaronson]

The Einstein field equation itself is actually extremely simple:

G = 8*pi*T

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. By this logic every equation is extremely simple -- it simply asserts that A=B for some A,B. :-)

[Tom McCabe]

"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*pi*T = R (tensor) - .5*g*R (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 electromagnetism in tensor form:

dF = 0
d*F = 4*pi*J

Now differential form:

(divergence) E = p
(divergence) B = 0
(curl) E = -dB/dt
(curl) B = J + dE/dt

Now integral form:

(flux E over closed surface A) = q
(flux B over closed surface A) = 0
(line integral of E over closed loop l) = - d (flux of B over surface enclosed by l)/dt
(line integral of B over closed loop l) = (current I passing through surface enclosed by l) + d (flux of E over surface enclosed by l)/dt

Now in action-at-a-distance form:

E = (sum q) -q/4/pi * ((r' unit vector from q)/r'/r' + r' * d/dt ((r' unit vector from q)/r'/r') + d^2/dt^2 (r' unit vector from q))
B = (sum q) E x -(r' unit vector from q)

[Eliezer Yudkowsky]

Tom, I expect to hear from you soon on the many new amazing physics discoveries you will generate using your insight that most previous physics problems have had obvious unique solutions. If only you had been around to solve the problem instead of Maxwell and Einstein, how much work could have been saved!

[Doug S.]

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 one to the other requires you to find limits that do not exist (or something like that). Another explanation said that yes, you can create a quantum theory of gravity using a "naive" approach, but such a theory requires an infinite number of arbitrary physical constants and is therefore completely useless because 1) you can't actually measure an infinite number of physical constants and 2) if you don't measure them, the proper "choice" of constants can give you any result whatsoever, so it can't make any predictions about the actual universe.

By the way, has anyone else here had the thought that the reason quantum mechanics and general relativity are contradictory yet seem to predict reality perfectly is that "there's a bug in the code"?

[Tom McCabe]

"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 would not have been taken by surprise. A real enhanced human would realize that any simple intelligence enhancement will be a net evolutionary disadvantage - if enhancing intelligence were a matter of a simple surgical procedure, it would have long ago occurred as a natural mutation. This goes double for a procedure that works on rats! (As far as I know, this never occurred to Keyes. I selected Flowers, out of all the famous stories of intelligence enhancement, because, for reasons of dramatic unity, this story shows what happens to be the correct outcome.)

Note that I didn't dazzle you with an abstruse technobabble explanation for Charlie's death; my explanation is two sentences long and can be understood by someone who isn't an expert in the field. It's the simplicity of smartness that's so impossible to convey in fiction, and so shocking when we encounter it in person. All that science fiction can do to show intelligence is jargon and gadgetry. A truly ultrasmart Charlie Gordon wouldn't have been taken by surprise; he would have deduced his probable fate using the above, very simple, line of reasoning. He would have accepted that probability, rearranged his priorities, and acted accordingly until his time ran out - or, more probably, figured out an equally simple and obvious-in-retrospect way to avoid his fate. If Charlie Gordon had really been ultrasmart, there would have been no story. "

We know that Newton's theory of gravity was hard to invent; it *must* not have been obvious, because nobody had solved it until Newton, and he was lauded as a hero for his great theory. And yet, it is so simple that we teach it to high school students, and some of them actually understand it. Newton's equation is also a unique solution; the constant of proportionality is fixed by experiment, the m/r^2 term is fixed by the need to include Kepler's laws (which were well known at the time), and extra terms are excluded, because F must vanish when M2 vanishes, or else you violate the laws of motion which Newton had just discovered.

[Doug S.]

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.

[Eliezer Yudkowsky]

"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 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.

You are also falling prey to hindsight by not making an equal effort to consider how you could have justified alternatives as unique obvious solutions using subsets of other knowledge known at the time, rather than the particular aspects that now obviously seem so prominent.

[Tom McCabe]

"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 years, not because Einstein has to sort through ten million different versions of the theory, but because developing a single version of the theory is difficult.

"You are also falling prey to hindsight by not making an equal effort to consider how you could have justified alternatives as unique obvious solutions using subsets of other knowledge known at the time, rather than the particular aspects that now obviously seem so prominent."

This is mathematically impossible unless you assume false knowledge. If equations (A, B, C, D, E) are known at the time of Newton, and Newton's theory of gravity is unique if you assume A, C and D, then any alternative theory of gravity must contradict A, C, or D. Suppose that you can construct an alternative theory of gravity, which is unique assuming equations B and E. If you assume that both B and E are true, then the alternative theory of gravity must be true, hence Newton's theory must be false, hence either A, C, or D must be false. We know now that A, C, and D are all true, therefore, either B or E must be false.

[Tom McCabe]

""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.

[Dale]

Reading that worthless tripe took up three minutes of my life that I'll never get back. Thanks reddit!

[g]

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.

[Ashish]

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' should be articulated and conceived ipso facto, without any evidence whatsoever, before the observations can 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 majorly before/after Relativity era had a different flavor: from Photoelectric effect tests to sprouting of Quantum Physics theories...you name it, all involved observation-research-theory evolution.

However, in only the case of General Relativity, there were NO experimental foundation put forth with the theory. The REAL evidences (1959+), [ not the Eddington one...that was essentially farce and exaggerated! ] came must after theory postulation (1919), and are still coming...
This is where Einstein is marvelous than it is thought to be. The beauty of a Theory is determined by its life it lives...Newton's ones lived three centuries. Einstein one has survived one...and counting.

In fact it has been referred that Einstein work on General Relativity during 1914-19 is a period of 'the greatest intellectual human endeavor by a single brain'
Refer Clark's Biography for more.

So there we are Eliezer, its not just about bits and observation for something to be conceived and articulated.

+Thanks.

[Shakespeare's Fool]

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?

[WaltFrench]

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.

[Francois Ouellette]

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 do the work were relatively new and obscure. But Einstein never faltered from his initial hypothesis.

And there WAS overwhelming evidence that inertial mass and gravitational mass were the same. Nobody knew for sure if they were EXACTLY the same, but they were sufficiently similar to support Einstein's hypothesis that they were, indeed, exactly the same.

So in Einstein's mind, the fact that posing gravitational mass equal to inertial mass led, logically, to the final conclusion in terms of general relativity, plus the fact that a vast amount of evidence pointed to the two being indeed equal, all that was enough for him to have confidence in the theory. Eddington's measurement was a very difficult one, and the results far from conclusive as has been shown elsewhere. Einstein had every reason to believe that a failure by Eddington to confirm his theory would in no way falsify it (mind you, this was way before Popper and Kuhn!...).

That's my two bit of explanation here. I used to be much more familiar with the history of general relativity but that was some time ago. Maybe re-reading Pais would help confirm or refute this idea.

[Nick Tarleton]

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

[Eliezer Yudkowsky]

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.