Einstein’s Speed

Yes­ter­day I ar­gued that the Pow­ers Beyond Science are ac­tu­ally a stan­dard and nec­es­sary part of the so­cial pro­cess of sci­ence. In par­tic­u­lar, sci­en­tists must call upon their pow­ers of in­di­vi­d­ual ra­tio­nal­ity to de­cide what ideas to test, in ad­vance of the sort of definite ex­per­i­ments that Science de­mands to bless an idea as con­firmed. The ideal of Science does not try to spec­ify this pro­cess—we don’t sup­pose that any pub­lic au­thor­ity knows how in­di­vi­d­ual sci­en­tists should think—but this doesn’t mean the pro­cess is unim­por­tant.

A read­ily un­der­stand­able, non-dis­turb­ing ex­am­ple:

A sci­en­tist iden­ti­fies a strong math­e­mat­i­cal reg­u­lar­ity in the cu­mu­la­tive data of pre­vi­ous ex­per­i­ments. But the cor­re­spond­ing hy­poth­e­sis has not yet made and con­firmed a novel ex­per­i­men­tal pre­dic­tion—which his aca­demic field de­mands; this is one of those fields where you can perform con­trol­led ex­per­i­ments with­out too much trou­ble. Thus the in­di­vi­d­ual sci­en­tist has read­ily un­der­stand­able, ra­tio­nal rea­sons to be­lieve (though not with prob­a­bil­ity 1) some­thing not yet blessed by Science as pub­lic knowl­edge of hu­mankind.

Notic­ing a reg­u­lar­ity in a huge mass of ex­per­i­men­tal data, doesn’t seem all that un­scien­tific. You’re still data-driven, right?

But that’s be­cause I de­liber­ately chose a non-dis­turb­ing ex­am­ple. When Ein­stein in­vented Gen­eral Rel­a­tivity, he had al­most no ex­per­i­men­tal data to go on, ex­cept the pre­ces­sion of Mer­cury’s per­ihe­lion. And (AFAIK) Ein­stein did not use that data, ex­cept at the end.

Ein­stein gen­er­ated the the­ory of Spe­cial Rel­a­tivity us­ing Mach’s Prin­ci­ple, which is the physi­cist’s ver­sion of the Gen­er­al­ized Anti-Zom­bie Prin­ci­ple. You be­gin by say­ing, “It doesn’t seem rea­son­able to me that you could tell, in an en­closed room, how fast you and the room were go­ing. Since this num­ber shouldn’t ought to be ob­serv­able, it shouldn’t ought to ex­ist in any mean­ingful sense.” You then ob­serve that Maxwell’s Equa­tions in­voke a seem­ingly ab­solute speed of prop­a­ga­tion, c, com­monly referred to as “the speed of light” (though the quan­tum equa­tions show it is the prop­a­ga­tion speed of all fun­da­men­tal waves). So you re­for­mu­late your physics in such fash­ion that the ab­solute speed of a sin­gle ob­ject no longer mean­ingfully ex­ists, and only rel­a­tive speeds ex­ist. I am skip­ping over quite a bit here, ob­vi­ously, but there are many ex­cel­lent in­tro­duc­tions to rel­a­tivity—it is not like the hor­rible situ­a­tion in quan­tum physics.

Ein­stein, hav­ing suc­cess­fully done away with the no­tion of your ab­solute speed in­side an en­closed room, then set out to do away with the no­tion of your ab­solute ac­cel­er­a­tion in­side an en­closed room. It seemed to Ein­stein that there shouldn’t ought to be a way to differ­en­ti­ate, in an en­closed room, be­tween the room ac­cel­er­at­ing north­ward while the rest of the uni­verse stayed still, ver­sus the rest of the uni­verse ac­cel­er­at­ing south­ward while the room stayed still. If the rest of the uni­verse ac­cel­er­ated, it would pro­duce grav­i­ta­tional waves that would ac­cel­er­ate you. Mov­ing mat­ter, then, should pro­duce grav­i­ta­tional waves.

And be­cause in­er­tial mass and grav­i­ta­tional mass were always ex­actly equiv­a­lent—un­like the situ­a­tion in elec­tro­mag­net­ics, where an elec­tron and a muon can have differ­ent masses but the same elec­tri­cal charge—grav­ity should re­veal it­self as a kind of in­er­tia. The Earth should go around the Sun in some equiv­a­lent of a “straight line”. This re­quires space­time in the vicinity of the Sun to be curved, so that if you drew a graph of the Earth’s or­bit around the Sun, the line on the 4D graph pa­per would be lo­cally flat. Then in­er­tial and grav­i­ta­tional mass would be nec­es­sar­ily equiv­a­lent, not just co­in­ci­den­tally equiv­a­lent.

(If that did not make any sense to you, there are good in­tro­duc­tions to Gen­eral Rel­a­tivity available as well.)

And of course the new the­ory had to obey Spe­cial Rel­a­tivity, and con­serve en­ergy, and con­serve mo­men­tum, etcetera.

Ein­stein spent sev­eral years grasp­ing the nec­es­sary math­e­mat­ics to de­scribe curved met­rics of space­time. Then he wrote down the sim­plest the­ory that had the prop­er­ties Ein­stein thought it ought to have—in­clud­ing prop­er­ties no one had ever ob­served, but that Ein­stein thought fit in well with the char­ac­ter of other phys­i­cal laws. Then Ein­stein cranked a bit, and got the pre­vi­ously un­ex­plained pre­ces­sion of Mer­cury right back out.

How im­pres­sive was this?

Well, let’s put it this way. In some small frac­tion of al­ter­nate Earths pro­ceed­ing from 1800—per­haps even a size­able frac­tion—it would seem plau­si­ble that rel­a­tivis­tic physics could have pro­ceeded in a similar fash­ion to our own great fi­asco with quan­tum physics.

We can imag­ine that Lorentz’s origi­nal “in­ter­pre­ta­tion” of the Lorentz con­trac­tion, as a phys­i­cal dis­tor­tion caused by move­ment with re­spect to the ether, pre­vailed. We can imag­ine that var­i­ous cor­rec­tive fac­tors, them­selves un­ex­plained, were added on to New­to­nian grav­i­ta­tional me­chan­ics to ex­plain the pre­ces­sion of Mer­cury—at­tributed, per­haps, to strange dis­tor­tions of the ether, as in the Lorentz con­trac­tion. Through the decades, fur­ther cor­rec­tive fac­tors would be added on to ac­count for other as­tro­nom­i­cal ob­ser­va­tions. Suffi­ciently pre­cise atomic clocks, in air­planes, would re­veal that time ran a lit­tle faster than ex­pected at higher al­ti­tudes (time runs slower in more in­tense grav­i­ta­tional fields, but they wouldn’t know that) and more cor­rec­tive “ethe­real fac­tors” would be in­vented.

Un­til, fi­nally, the many differ­ent em­piri­cally de­ter­mined “cor­rec­tive fac­tors” were unified into the sim­ple equa­tions of Gen­eral Rel­a­tivity.

And the peo­ple in that al­ter­nate Earth would say, “The fi­nal equa­tion was sim­ple, but there was no way you could pos­si­bly know to ar­rive at that an­swer from just the per­ihe­lion pre­ces­sion of Mer­cury. It takes many, many ad­di­tional ex­per­i­ments. You must have mea­sured time run­ning slower in a stronger grav­i­ta­tional field; you must have mea­sured light bend­ing around stars. Only then can you imag­ine our unified the­ory of ethe­real grav­i­ta­tion. No, not even a perfect Bayesian su­per­in­tel­li­gence could know it!—for there would be many ad-hoc the­o­ries con­sis­tent with the per­ihe­lion pre­ces­sion alone.”

In our world, Ein­stein didn’t even use the per­ihe­lion pre­ces­sion of Mer­cury, ex­cept for ver­ifi­ca­tion of his an­swer pro­duced by other means. Ein­stein sat down in his arm­chair, and thought about how he would have de­signed the uni­verse, to look the way he thought a uni­verse should look—for ex­am­ple, that you shouldn’t ought to be able to dis­t­in­guish your­self ac­cel­er­at­ing in one di­rec­tion, from the rest of the uni­verse ac­cel­er­at­ing in the other di­rec­tion.

And Ein­stein ex­e­cuted the whole long (multi-year!) chain of arm­chair rea­son­ing, with­out mak­ing any mis­takes that would have re­quired fur­ther ex­per­i­men­tal ev­i­dence to pull him back on track.

Even Jeffreys­sai would be grudg­ingly im­pressed. Though he would still ding Ein­stein a point or two for the cos­molog­i­cal con­stant. (I don’t ding Ein­stein for the cos­molog­i­cal con­stant be­cause it later turned out to be real. I try to avoid crit­i­ciz­ing peo­ple on oc­ca­sions where they are right.)

What would be the prob­a­bil­ity-the­o­retic per­spec­tive on Ein­stein’s feat?

Rather than ob­serve the planets, and in­fer what laws might cover their grav­i­ta­tion, Ein­stein was ob­serv­ing the other laws of physics, and in­fer­ring what new law might fol­low the same pat­tern. Ein­stein wasn’t find­ing an equa­tion that cov­ered the mo­tion of grav­i­ta­tional bod­ies. Ein­stein was find­ing a char­ac­ter-of-phys­i­cal-law that cov­ered pre­vi­ously ob­served equa­tions, and that he could crank to pre­dict the next equa­tion that would be ob­served.

No­body knows where the laws of physics come from, but Ein­stein’s suc­cess with Gen­eral Rel­a­tivity shows that their com­mon char­ac­ter is strong enough to pre­dict the cor­rect form of one law from hav­ing ob­served other laws, with­out nec­es­sar­ily need­ing to ob­serve the pre­cise effects of the law.

(In a gen­eral sense, of course, Ein­stein did know by ob­ser­va­tion that things fell down; but he did not get GR by back­ward in­fer­ence from Mer­cury’s ex­act per­ihe­lion ad­vance.)

So, from a Bayesian per­spec­tive, what Ein­stein did is still in­duc­tion, and still cov­ered by the no­tion of a sim­ple prior (Oc­cam prior) that gets up­dated by new ev­i­dence. It’s just the prior was over the pos­si­ble char­ac­ters of phys­i­cal law, and ob­serv­ing other phys­i­cal laws let Ein­stein up­date his model of the char­ac­ter of phys­i­cal law, which he then used to pre­dict a par­tic­u­lar law of grav­i­ta­tion.

If you didn’t have the con­cept of a “char­ac­ter of phys­i­cal law”, what Ein­stein did would look like magic—pluck­ing the cor­rect model of grav­i­ta­tion out of the space of all pos­si­ble equa­tions, with vastly in­suffi­cient ev­i­dence. But Ein­stein, by look­ing at other laws, cut down the space of pos­si­bil­ities for the next law. He learned the alpha­bet in which physics was writ­ten, con­straints to gov­ern his an­swer. Not magic, but rea­son­ing on a higher level, across a wider do­main, than what a naive rea­soner might con­ceive to be the “model space” of only this one law.

So from a prob­a­bil­ity-the­o­retic stand­point, Ein­stein was still data-driven—he just used the data he already had, more effec­tively. Com­pared to any al­ter­nate Earths that de­manded huge quan­tities of ad­di­tional data from as­tro­nom­i­cal ob­ser­va­tions and clocks on air­planes to hit them over the head with Gen­eral Rel­a­tivity.

There are nu­mer­ous les­sons we can de­rive from this.

I use Ein­stein as my ex­am­ple, even though it’s cliche, be­cause Ein­stein was also un­usual in that he openly ad­mit­ted to know­ing things that Science hadn’t con­firmed. Asked what he would have done if Ed­ding­ton’s so­lar eclipse ob­ser­va­tion had failed to con­firm Gen­eral Rel­a­tivity, Ein­stein replied: “Then I would feel sorry for the good Lord. The the­ory is cor­rect.”

Ac­cord­ing to pre­vailing no­tions of Science, this is ar­ro­gance—you must ac­cept the ver­dict of ex­per­i­ment, and not cling to your per­sonal ideas.

But as I con­cluded in Ein­stein’s Ar­ro­gance, Ein­stein doesn’t come off nearly as badly from a Bayesian per­spec­tive. From a Bayesian per­spec­tive, in or­der to sug­gest Gen­eral Rel­a­tivity at all, in or­der to even think about what turned out to be the cor­rect an­swer, Ein­stein must have had enough ev­i­dence to iden­tify the true an­swer in the the­ory-space. It would take only a lit­tle more ev­i­dence to jus­tify (in a Bayesian sense) be­ing nearly cer­tain of the the­ory. And it was un­likely that Ein­stein only had ex­actly enough ev­i­dence to bring the hy­poth­e­sis all the way up to his at­ten­tion.

Any ac­cu­sa­tion of ar­ro­gance would have to cen­ter around the ques­tion, “But Ein­stein, how did you know you had rea­soned cor­rectly?”—to which I can only say: Do not crit­i­cize peo­ple when they turn out to be right! Wait for an oc­ca­sion where they are wrong! Other­wise you are miss­ing the chance to see when some­one is think­ing smarter than you—for you crit­i­cize them when­ever they de­part from a preferred rit­ual of cog­ni­tion.

Or con­sider the fa­mous ex­change be­tween Ein­stein and Niels Bohr on quan­tum the­ory—at a time when the then-cur­rent, sin­gle-world quan­tum the­ory seemed to be im­mensely well-con­firmed ex­per­i­men­tally; a time when, by the stan­dards of Science, the cur­rent (de­ranged) quan­tum the­ory had sim­ply won.

Ein­stein: “God does not play dice with the uni­verse.”
Bohr: “Ein­stein, don’t tell God what to do.”

You’ve got to ad­mire some­one who can get into an ar­gu­ment with God and win.

If you take off your Bayesian gog­gles, and look at Ein­stein in terms of what he ac­tu­ally did all day, then the guy was sit­ting around study­ing math and think­ing about how he would de­sign the uni­verse, rather than run­ning out and look­ing at things to gather more data. What Ein­stein did, suc­cess­fully, is ex­actly the sort of high-minded feat of sheer in­tel­lect that Aris­to­tle thought he could do, but couldn’t. Not from a prob­a­bil­ity-the­o­retic stance, mind you, but from the view­point of what they did all day long.

Science does not trust sci­en­tists to do this, which is why Gen­eral Rel­a­tivity was not blessed as the pub­lic knowl­edge of hu­man­ity un­til af­ter it had made and ver­ified a novel ex­per­i­men­tal pre­dic­tion—hav­ing to do with the bend­ing of light in a so­lar eclipse. (It later turned out that par­tic­u­lar mea­sure­ment was not pre­cise enough to ver­ify re­li­ably, and had fa­vored GR es­sen­tially by luck.)

How­ever, just be­cause Science does not trust sci­en­tists to do some­thing, does not mean it is im­pos­si­ble.

But a word of cau­tion here: The rea­son why his­tory books some­times record the names of sci­en­tists who thought great high-minded thoughts, is not that high-minded think­ing is eas­ier, or more re­li­able. It is a pri­or­ity bias: Some sci­en­tist who suc­cess­fully rea­soned from the small­est amount of ex­per­i­men­tal ev­i­dence got to the truth first. This can­not be a mat­ter of pure ran­dom chance: The the­ory space is too large, and Ein­stein won sev­eral times in a row. But out of all the sci­en­tists who tried to un­ravel a puz­zle, or who would have even­tu­ally suc­ceeded given enough ev­i­dence, his­tory passes down to us the names of the sci­en­tists who suc­cess­fully got there first. Bear that in mind, when you are try­ing to de­rive les­sons about how to rea­son pru­dently.

In ev­ery­day life, you want ev­ery scrap of ev­i­dence you can get. Do not rely on be­ing able to suc­cess­fully think high-minded thoughts un­less ex­per­i­men­ta­tion is so costly or dan­ger­ous that you have no other choice.

But some­times ex­per­i­ments are costly, and some­times we pre­fer to get there first… so you might con­sider try­ing to train your­self in rea­son­ing on scanty ev­i­dence, prefer­ably in cases where you will later find out if you were right or wrong. Try­ing to beat low-cap­i­tal­iza­tion pre­dic­tion mar­kets might make for good train­ing in this?—though that is only spec­u­la­tion.

As of now, at least, rea­son­ing based on scanty ev­i­dence is some­thing that mod­ern-day sci­ence can­not re­li­ably train mod­ern-day sci­en­tists to do at all. Which may per­haps have some­thing to do with, oh, I don’t know, not even try­ing?

Ac­tu­ally, I take that back. The most sane think­ing I have seen in any sci­en­tific field comes from the field of evolu­tion­ary psy­chol­ogy, pos­si­bly be­cause they un­der­stand self-de­cep­tion, but also per­haps be­cause they of­ten (1) have to rea­son from scanty ev­i­dence and (2) do later find out if they were right or wrong. I recom­mend to all as­piring ra­tio­nal­ists that they study evolu­tion­ary psy­chol­ogy sim­ply to get a glimpse of what care­ful rea­son­ing looks like. See par­tic­u­larly Tooby and Cos­mides’s “The Psy­cholog­i­cal Foun­da­tions of Cul­ture”.

As for the pos­si­bil­ity that only Ein­stein could do what Ein­stein did… that it took su­per­pow­ers be­yond the reach of or­di­nary mor­tals… here we run into some bi­ases that would take a sep­a­rate post to an­a­lyze. Let me put it this way: It is pos­si­ble, per­haps, that only a ge­nius could have done Ein­stein’s ac­tual his­tor­i­cal work. But po­ten­tial ge­niuses, in terms of raw in­tel­li­gence, are prob­a­bly far more com­mon than his­tor­i­cal su­per­achiev­ers. To put a ran­dom num­ber on it, I doubt that any­thing more than one-in-a-mil­lion g-fac­tor is re­quired to be a po­ten­tial world-class ge­nius, im­ply­ing at least six thou­sand po­ten­tial Ein­steins run­ning around to­day. And as for ev­ery­one else, I see no rea­son why they should not as­pire to use effi­ciently the ev­i­dence that they have.

But my fi­nal moral is that the fron­tier where the in­di­vi­d­ual sci­en­tist ra­tio­nally knows some­thing that Science has not yet con­firmed, is not always some in­no­cently data-driven mat­ter of spot­ting a strong reg­u­lar­ity in a moun­tain of ex­per­i­ments. Some­times the sci­en­tist gets there by think­ing great high-minded thoughts that Science does not trust you to think.

I will not say, “Don’t try this at home.” I will say, “Don’t think this is easy.” We are not dis­cussing, here, the vic­tory of ca­sual opinions over pro­fes­sional sci­en­tists. We are dis­cussing the some­time his­tor­i­cal vic­to­ries of one kind of pro­fes­sional effort over an­other. Never for­get all the fa­mous his­tor­i­cal cases where at­tempted arm­chair rea­son­ing lost.