"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)

"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?

"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).

"Tom, you appear to have given an argument for never funding anything that has research as a major component."

The utility of funding a specific project goes to zero as the amount of money that project requires per unit of output goes to infinity. Funding one project has an opportunity cost, in the utility equation, of not funding other projects. So at some point, it will make sense (doing the opposite would have a negative expected utility) to contribute to some other project than SIAI. I don't have a clear idea of where that point is, but we've gotten a lot closer in the past two years.

"Why would the amount of research being done stay the same, if the amount of money coming in goes up by a factor of 10?"

Number of publications. And lack of other strong Bayesian evidence. Money does not correlate well with thinking capacity; if you dump $20 million into a startup, its intelligent output will (on average) drop off rapidly.

"I guess they might spend it on advocacy, or buying hardware, or something, but surely what it would take for your comment about utility to be correct would be for them to do *nothing* with it. Why would they do that?"

I have no idea what SIAI's current budget is, or how they spend their money. I'm analyzing it using black-box efficiency, how much goes in versus how much comes out.

How would an artist participate, other than just mailing in a check? Doesn't SIAI have something like $500K worth of checks, from this summer's fundraising alone? If the amount of research being done by SIAI stays the same, and the amount of money coming in goes up by a factor of 10, then the utility of every dollar goes down by a factor of 10; eventually it makes more sense to donate to other groups.

"I challenge the "rules" set out by whomever thinks he's the know-all on what can be done with a compass and straight edge."

I would be interested to see what you can get out of a compass and straightedge if you change the allowable operations. You could wind up with something much more complex than the things the ancient Greeks studied (think of how much more complex a Riemannian manifold is than a Euclidean n-space, once you remove a few of Euclid's axioms).

"So I found this counterexample, and saw that my attempted disproof was false, along with my dreams of fame and glory."

I know how that feels. When I was 14 or so, I took a course on cryptography, and the textbook proclaimed that modular inverses were the basis of public-key algorithms like RSA. I felt that modular inverses were crackable, and I plodded along on the problem for a few weeks, until I finally discovered a polynomial-time algorithms for doing modular inverses. It turned out that I had reinvented Euclid's algorithm, and the textbook authors were idiots.

"On the topic of reversing changes to appreciate their absurdity: movies that were made in say the 40s or 50s, seem much more alien to me than modern movies allegedly set hundreds of years in the future, or in different universes."

Most people do not know enough history (or rather, the specific parts of history) to even realize how absurd the past was. If you read a high-school level history textbook, which is the most information 99% of the public will actually remember (if that), history seems a great deal like the modern day: people had politics and governments and wars and good guys and bad guys and issues and so on, just like they do in the movies. The absurd parts get subtracted out, or added on as "irrelevant" trivia.

"You can actually give a semi-plausible justification of special relativity based on what was known in 1901."

You can give a semi-plausible justification for anything. It was obvious at the time that our knowledge was incomplete, but the specific *way* in which our knowledge was incomplete was still a mystery. It is very easy to invent a plausible-sounding quack theory of physics; that is why we have the Crackpot Index.