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Einstein's Arrogance

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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.

Indeed, I think theorists tend to make mistakes of either deductive or inductive bias. They start out tacitly assuming that reality must be some slightly noisy instantiation of a mathematical theorem ... that their favorite equations are *logically* true and for some mucky reason or another we just observe them as being noisily true.

From the post above:

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).

... *or* you just need to be that one guy who made a wild and unjustified guess about where to assign more than 50 % of the probability (despite *not* having bits of evidence to support it) and then be lucky.

This is true even if you call your guess a "hunch" or "intuition".

Only if you make the further assumption that whatever process that generates hunches or intuition must be decision-theoretic. That may not be a bad assumption, but I'm not convinced it's accurate in human beings. From my own readings about Einstein, I think it's more likely that he over-asserted the relevance of differential geometry and justified the pursuit of a theory along those...

Many theories have been defended on grounds of beauty - and been wrong. Heliocentrism was an elegant theory that worked well and explained many things like the absence of naked-eye precession. Just before Einstein, we can find examples:

According to the vortex atomic theory originally proposed by William Thomson in 1867, atoms were nothing but vortical structures in the continuous ether. In this sense the atoms were quasi-material rather than material bodies. As the ultimate and irreducible quality of nature, the ether could exist without matter, but matter could not exist without the ether....By the early 1890s the vortex atomic theory had run out of steam and was abandoned by most researchers as a realistic theory of the constitution of matter. It was never unambiguously proved wrong by experiment, but after twenty years of work it degenerated into mathematics, failing to deliver what it promised of physical results. Physicists simply lost confidence in the theory. On the other hand, although the vortex atom was no longer considered a useful concept in explaining physical phenomena, heuristically and as a mental picture it lived on. Wrong as it was, to many British physicists it remained a methodological guiding principle, the ideal of what a future unified theory of matter and ether should look like. According to Michelson, writing in 1903, it “ought to be true even if it is not” (Kragh 2002: 80).

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I came here with this exact question, and still don't have a good answer. I feel confident that Eliezer is well aware that lucky guesses exist, and that Eliezer is attempting to communicate something in this chapter, but I remain baffled as to what.
Is the idea that, given our current knowledge that the theory was, in fact, correct, the most plausible explanation is that Einstein already had lots of evidence that this theory was true?
I understand that theory-space is massive, but I can locate all kinds of theories just by rolling dice or flipping coins to generate random bits. I can see how this 'random thesis generation method' still requires X number of bits to reach arbitrary theories, but the information required to reach a theory seems orthogonal to the truth. It feels like a stretch to call coin flips "evidence." I'm guessing that's what Robin_Hanson2 means by "lucky draw from the same process"; perhaps there were a few bits selected from observation, and a few others that came from lucky coin flips.
Perhaps a better question would be, given a large array of similar scenarios (someone traveling to look at evidence that will possibly refute a theory), how can I use the insight presented in this chapter to constrain anticipation and attempt to perform better than random in guessing which travelers are likely to see the theory violated, and which travelers are not? Or am i thinking of this the wrong way? I remain genuinely confused here, which i hope is a good sign as far as the search for truth :)

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My original reading was 'there was less arrogance in Einstein's answer than you might think'. After rereading Eliezer's text and the other comments again today, I cannot tell how much arrogance (regarding rationality) we should assume. I think it is worthwhile to compare Einstein not only to a strong Bayesian:
On the one hand, I agree that a impressive-but-still-human Bayesian would probably have accumulated sufficient evidence at the point of having the worked-out theory that a single experimental result against the theory is not enough to outweigh the evidence. In this case there is little arrogance (if I assume the absolute confidence in “Then I would feel sorry for the good Lord. The theory is correct.” to be rhetoric and not meant literally.)
On the other hand, a random person saying 'here is my theory that fundamentally alters the way we have to think of our world' and dismissing a contradicting experiment would be a prime example of arrogance.
Assuming these two cases to be the endpoints of a spectrum, the question becomes where Einstein was located. With special relativity and other significant contributions to physics already at that point in time, I think it is safe to put Einstein into the top tier of physicists. I assume that he did find a strong theory corresponding to his search criteria. But as biases are hard to handle, especially if they concern one's own assumptions about fundamental principles about our world, there remains the possibility that Einstein did not optimize for correspondence-to-reality for finding general relativity but a heuristic that diverged along the way of finding the theory.
As Einstein had already come up with special relativity (which is related and turned out correct), I tend towards assuming that his assumptions about fundamental principles were on an impressive level, too.
With all this i think it is warranted to take his theory of general relativity very seriously even before the experiment. But Einstein's confiden

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True. It is not logically implied from him being right that he had good reason to believe he was right. However, I think it is very strong evidence. Fair warning: I am very new to using Bayes' Theorem, so please make sure to be highly critical of my math, and tell me what, if anything, I'm doing wrong.
First, we must assess the prior probability of Einstein having sufficient evidence, given that he thought he was correct. How often do modern scientists come up with theories that are quickly falsified? Let's be pessimistic and assign 0.001 prior probability for Einstein making his claim with sufficient evidence. That is, only 1 in 1000 credible scientists who publish theories come up with theories that aren’t easily falsified.
What is the probability of him being correct if he had sufficient evidence? Well, if we say that having sufficient evidence means having evidence such that your prediction has P>0.95, then, if someone has sufficient evidence, their prediction must have P>0.95. Let's assign a probability of 0.95.
What is the probability of him being correct if he had insufficient evidence? To be strictly logical about this, we would need to take this probability as 0.95 as well, to avoid a false dichotomy. It is not true that either Einstein had p=0.95 worth of evidence or he had no evidence at all. If we say that he necessarily has p > 0.95 given that he has sufficient evidence, we'd have to say that anything under p=0.95 is insufficient evidence; in which case, to be pessimistic, we'd have to assign the probability of him being correct given insufficient evidence to be infinitesimally less than 0.95. This would result in a likelihood ratio of approximately 1. However, this is only the case if we view "insufficient evidence" and "sufficient evidence" to be distinguished by a sharp point on the real number line. This would contradict common sense; we don't say that p=95 is sufficient but p=94999999 is insufficient. It's a gradient. So we should choose a numbe

"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).

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.

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Not at all obvious, but there are very few hypotheses that could be specified as briefly. What took ten years was figuring out how to get from the very short specification into an algebraic expression that satisfied its constraints.
A bit like, if the theory was 'just' Fermat's Last Theorem, proving it could take a while.

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He was being sarcastic.

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Doesn't that apply to the MWI too?

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...

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There are not an infinite number of possible hypotheses in a great many sensible situations. For example, suppose the question is "who murdered Fred?", because we have already learned that he was murdered. The already known answer: "A human alive at the time he died.", makes the set finite. If we can determine when and where he died, the number of suspects can typically be reduced to dozens or hundreds. Limiting to someone capable of carrying out the means of death may cut 90% of them.
To the extent that "bits" of evidence means things that we don't know yet, the number of bits can be much smaller than suggested. To the extent that "bits" of evidence includes everything we know so far, we all have trillions of bits already in our brains and the minimal number is meaningless.

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What about the aliens who landed on earth, murdered Fred and then went away again? Or the infinite number of other possibilities, each of which has a very small probability?
What confuses me about this is that, if we do accept that there are an infinite number of possibilities, most of the possibilities must have an infinitesimal probability in order for everything to sum to 1. And I don't really understand the concept of an infinitesimal probability -- after all, even my example above must have some finite probability attached?

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Being as, at any one time, the universe only has a finite space about any point that can be reached at sub-speed of light times. As a result there is only a finite amount of matter and, furthermore, possibility that can happen at the point where Fred died. This limits us to finite probabilities of discrete events.
Were your case possible and we were talking about continuous probabilities it would be the case that any one event is impossible; an "area" in probability space between two limiting values (events in probability space) would give you a discrete probability. You're issue is one that I had issues with until I really sat and thought about how integrals work.
FYI: everything I have said is essentially based on my understanding of special relativity, probability and calculus and are more than open to criticism.

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The probability that the universe only has finite space is not exactly 1, is it? Much more might exist than our particular Hubble volume, no? What probability do the, say, world's top 100 physicists assign, on average, to the possibiliy that infinitely much matter exists? And on what grounds?
To my understanding, the universe might be so large that everything that could be described with infinitely many characters actually exists. That kind of "TOE" actually passes the Ockham's razor test excellently; if the universe is that large, then it could (in principle) be exhaustively described by a very simple and short computer program, namely one that produces a string consisting of all the integers in order of size: 110111001011101111000... ad infinitum, translated into any wide-spread language using practially any arbitrarily chosen system for translation. Name anything that could exist in any universe of countably infinite size, and it would be fully described, even at infinitely many places, in the string of characters that such a simple computer program would produce.
Why not assign a pretty large probability to the possibility that the universe is that large, since all other known theories about the size of the universe seem to have a harder time with Ockham's razor?

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"The probability that the universe only has finite space is not exactly 1, is it?"
Nooooo, that's not it. The probability that the reachable space from a particular point within a certain time is finite is effectively one.
So it doesn't matter how large the universe is - the aliens a few trillion ly away cannot have killed Bob.

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Just to point out what may be a nitpick or a clarification. It's perfectly possible for infinity many positive things to sum to a finite number. 1/2+1/4+1/8+...=1.
There can be infinitely many potential murderers. But if the probability of each having done it drops off fast enough you can avoid anything that is literally infinitesimal. Almost all will be less than 1/3^^^^^^3 of course, but that's a perfectly well defined number you know how to do maths with.

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Hate to nitpick myself, but 1/2+1/4+1/8+... diverges (e.g., by the harmonic series test). Sum 1/n^2 = 1/4 + 1/9 + ... = (pi^2)/6 is a more fitting example.
An interesting question, in this context, is what it would mean for infinitely many possibilities to exist in a "finite space about any point that can be reached at sub-speed of light times." Would it be possible under the assumption of a discrete universe (a universe decomposable no further than the smallest, indivisible pieces)? This is an issue we don't have to worry about in dealing with the infinite sums of numbers that converge to a finite number.

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That's not correct at all. sum(1/2^n)[1:infinity] = 1.

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Oops, misread that as sum(1/(2n))[1:infinity] (which it wasn't), my bad.

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How about we put it this way: In the infinite space of possible theories, most of them are far too complex to ever have enough evidence to locate. (If it takes 3^^^3 bits of information to verify the theory... you're never going to verify the theory.)
In realistic circumstances, we have really a quite small list of theories to choose from, because the list of theories that we are capable of comprehending and testing in human lifetimes is itself very small.

-1[anonymous]

Your comments are clogging up the recent comments feed. I normally wouldn't mind, but your comments are often replies to comments made several years ago by users who no longer post. Please be mindful of this when posting. Thanks!

I normally wouldn't mind, but your comments are often replies to comments made several years ago by users who no longer post.

This is fine - if the comments provide useful insight (they don't in this case). We encourage (productive) thread necromancy.

"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...

*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. :-)

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...

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Yeah, but there are only three objects you can write in terms of the metric tensor which transform the “right” way (G, T, and g itself). So the most general equation which satisfies those transformation laws is aG + bT + cg = 0.
Now, a is non-zero (otherwise you get an universe where there's no matter/energy other than “dark energy”), so by redefining b and c we have G + bT + cg = 0; b is negative (because things attract each other rather than repelling each other) and we call it -8piG/c^4 (it's just a matter of choice of units of measurement; we might as well set it to 1); and c is the cosmological constant.

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I doubt that Scott will reply to this, 5 years later and on a different site, so let me try instead.
Hindsight bias? There are plenty of such objects. Google f(R) gravity, for example. There are also many different contractions of powers of products of R, T and G that fit. There is also torsion, and probably other things (supergravity and string theory tend to add a few).
You might want to argue that G=T is "the simplest", but it is anything but, for the reasons Scott explained. Once you find something that works, you call it G and T, write G=T and call it "simple". That's what Einstein did, since his first attempt, R=T, did not work out.

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Interesting...

"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 electromag...

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Your R is actually the Ricci tensor, not the Riemann tensor. The Riemann tensor has four indices, not two. The Ricci tensor is formed by contracting the Riemann tensor on its first and third indices.

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...

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The mathematical inconsistency between quantum mechanics and general relativity illustrates a key point. Most of the time the hypothesis set for new solutions, rather than being infnite, is null. It is often quite easy to illustrate that every available theory is wrong. Even if we know that our theory is clearly inconsistent with reality, we still keep using it until we come up with something better. Even if General Relativity were contradicted by some experimental discovery in 1963, Einstein would still have been lauded as a scientist for finding a theory that fit more data points that the previous one.
In science, and in a lot of other contexts, simply showing that a theory could be right, is much more important the establishing to any degree of statistical significance that it is right.

"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...

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...

"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 ...

""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' *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...

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?

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I've always assumed that it was a joke. If he'd been serious, then he'd have felt sorry for Eddington.

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...

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.

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It's years since this thread came up, but just my two cents on this suggestion.
Correct me if I'm significantly wrong, but I think your premise is that overwhelming evidence is first assembled in a good theoretician's brain, is logically processed into a theory, and then the correct theory is presented and found correct by virtue of this process. The crucial process was that they had to accumulate enough pieces of evidence in accord with the theory to select it, since you believe information theory prohibits any other ways of going about this business.
The thing is, if we follow the line of argument that the number of pieces of information must correspond to the number of possible hypotheses, then surely you would need an infinite number of pieces of information because the number of possible hypotheses (possible statements about the universe) is infinite too.
If you argue that it is a finite number, surely you are suggesting that a gigantic number of hypotheses have been removed in pre-selection based on how relevant they appear. If pre-selection occurs, you must also be open to the possibility that the number of possible hypotheses is far less than an arbitrary 100,000,000 and even single-digits. I think the whole accumulated mountain of science actually exists such that you do not need to generate your entire theory from an infinite number of possibilities, but judge from between a grossly-reduced number, the reducing of the vast number of other possibilities having been done by the work of your predecessors.
So if it satisfies you more, there may have been a huge number of possible theories at the start of humankind, but the combined weight of human experience and endeavour has whittled them down in certain areas to numbers which can be distinguished between by great scientists. In other areas, such as the precise nature of consciousness, we are just as baffled as before!

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See Bayesian probability, Occam's razor, Kolmogorov complexity, Solomonoff induction. You only need to raise a certain hypothesis in probability above the alternatives, not exclude all other hypotheses with certainty.

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...

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...

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...

Similarly, the 27 bit rule for 100,000,000 people assumes that the bits have equal numbers of people who are yes and no on a question. In fact, some bits are more discriminating than others. "Have you ever been elected to an office that requires a statewide vote or been a Vice President?" (perhaps two bits of information), is going to eliminate 99.9999%+ of potential candidates for President, yet work nearly perfectly to dramatically narrow the field from the 100,000,000 eligible candidates. "Do you want to run for President?", cuts another 90%+ of potential candidates.

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 *question* doesn'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...

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.

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Well, Einstein wouldn't arrive at just any conclusion.

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Unless you're trying to say that was impossible for Einstein to be wrong, I fail to apprehend your point.

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The probability distribution over conclusions about physics, representing the probability that a person would arrive at each conclusion, is not the same for Einstein as for an arbitrary person. And this difference is not arbitrary.
Dually, if I told you I had a conclusion about physics in an envelope, and asked what probability you would give that it was a true one, you might give a figure X. If I then told you the conclusion was made by Einstein and asked what probability you now gave that it was a true conclusion, I expect you would give a figure greater than X.

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...

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I can see why Einstein would assume 1), 2) and 4), but what was his motivation for assuming 3)? Just some intuition about simplicity?

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Realize his theory was replacing ether theory. As people learned more, ether theory required increasingly arbitrary "patches" to work. If GR was not simpler then Ether theory, it wasn't a good candidate to replace it, as lorentzian transformations in ether theory still worked mathematically.

Your explanation of "Einstein's Arrogance" feels plainly wrong:

Humans don't surface hypotheses for active consideration via Bayesian means. Perhaps the process by which humans actually generate hypotheses could be coherently formalised in a Bayesian framework, but using that as an explanation for Einstein's Arrogance seems like a non safe procedure.

If Einstein's hypothesis generation process does not actually select from hypothesis space by updating a prior probability distribution on evidence then claiming that Einstein must have had enough information to...

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

Well... actually, one thing I feel pretty confident about is that General relativity is wrong. It is an approximation which works well within a large domain, but at distances and energies where quantum theory is a better description, it does not work. Hence the search for the quantum theory of gravity which has been going on for some time.

In the same way that Newtonian theory is an approximation of Einstein theory for non relativistic speeds, Einstein theory is probably an approximation of this yet to be discovered quantum theory of gravity, which should help us understand how black holes work.

This doesn’t take away from the point your post makes but there’s a small nitpick: there’s no proof that Einstein actually said that. It appears to be one those tongue in cheek stories about Einstein, we don’t have a contemporary source quoting him on that.

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

knowthat 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

findingthe 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

justifybelieving 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" toconfirmit - to convince the scientific community, or justify saying that youbelievein 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 themright.Now, how likely is it that Einstein would have

exactlyenough 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 acrossexactly29.5 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 surethat General Relativity was true.In fact, since the human brain is not a perfectly efficient processor of information, Einstein probably had

overwhelmingly more evidencethan 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

wascorrect, from all the vast space of possibilities.