# Decoherence is Simple

An epis­tle to the physi­cists:

When I was but a lit­tle lad, my father, a PhD physi­cist, warned me sternly against med­dling in the af­fairs of physi­cists; he said that it was hope­less to try to com­pre­hend physics with­out the for­mal math. Pe­riod. No es­cape clauses. But I had read in Feyn­man’s pop­u­lar books that if you re­ally un­der­stood physics, you ought to be able to ex­plain it to a non­physi­cist. I be­lieved Feyn­man in­stead of my father, be­cause Feyn­man had won the No­bel Prize and my father had not.

It was not un­til later—when I was read­ing the Feyn­man Lec­tures, in fact— that I re­al­ized that my father had given me the sim­ple and hon­est truth. No math = no physics.

By vo­ca­tion I am a Bayesian, not a physi­cist. Yet al­though I was raised not to med­dle in the af­fairs of physi­cists, my hand has been forced by the oc­ca­sional gross mi­suse of three terms: sim­ple, falsifi­able, and testable.

The fore­go­ing in­tro­duc­tion is so that you don’t laugh, and say, “Of course I know what those words mean!” There is math here. What fol­lows will be a restate­ment of the points in Belief in the Im­plied In­visi­ble, as they ap­ply to quan­tum physics.

Let’s be­gin with the re­mark that started me down this whole av­enue, of which I have seen sev­eral ver­sions; para­phrased, it runs:

The many-wor­lds in­ter­pre­ta­tion of quan­tum me­chan­ics pos­tu­lates that there are vast num­bers of other wor­lds, ex­ist­ing alongside our own. Oc­cam’s Ra­zor says we should not mul­ti­ply en­tities un­nec­es­sar­ily.

Now it must be said, in all fair­ness, that those who say this will usu­ally also con­fess:

But this is not a uni­ver­sally ac­cepted ap­pli­ca­tion of Oc­cam’s Ra­zor; some say that Oc­cam’s Ra­zor should ap­ply to the laws gov­ern­ing the model, not the num­ber of ob­jects in­side the model.

So it is good that we are all ac­knowl­edg­ing the con­trary ar­gu­ments, and tel­ling both sides of the story—

But sup­pose you had to calcu­late the sim­plic­ity of a the­ory.

The origi­nal for­mu­la­tion of William of Ock­ham stated:

Lex par­si­mo­niae: En­tia non sunt mul­ti­pli­canda praeter ne­ces­si­tatem.

“The law of par­si­mony: En­tities should not be mul­ti­plied be­yond ne­ces­sity.”

But this is qual­i­ta­tive ad­vice. It is not enough to say whether one the­ory seems more sim­ple, or seems more com­plex, than an­other—you have to as­sign a num­ber; and the num­ber has to be mean­ingful, you can’t just make it up. Cross­ing this gap is like the differ­ence be­tween be­ing able to eye­ball which things are mov­ing “fast” or “slow,” and start­ing to mea­sure and calcu­late ve­loc­i­ties.

Sup­pose you tried say­ing: “Count the words—that’s how com­pli­cated a the­ory is.”

Robert Hein­lein once claimed (tongue-in-cheek, I hope) that the “sim­plest ex­pla­na­tion” is always: “The woman down the street is a witch; she did it.” Eleven words—not many physics pa­pers can beat that.

Faced with this challenge, there are two differ­ent roads you can take.

First, you can ask: “The woman down the street is a what?” Just be­cause English has one word to in­di­cate a con­cept doesn’t mean that the con­cept it­self is sim­ple. Sup­pose you were talk­ing to aliens who didn’t know about witches, women, or streets—how long would it take you to ex­plain your the­ory to them? Bet­ter yet, sup­pose you had to write a com­puter pro­gram that em­bod­ied your hy­poth­e­sis, and out­put what you say are your hy­poth­e­sis’s pre­dic­tions—how big would that com­puter pro­gram have to be? Let’s say that your task is to pre­dict a time se­ries of mea­sured po­si­tions for a rock rol­ling down a hill. If you write a sub­rou­tine that simu­lates witches, this doesn’t seem to help nar­row down where the rock rolls—the ex­tra sub­rou­tine just in­flates your code. You might find, how­ever, that your code nec­es­sar­ily in­cludes a sub­rou­tine that squares num­bers.

Se­cond, you can ask: “The woman down the street is a witch; she did what?” Sup­pose you want to de­scribe some event, as pre­cisely as you pos­si­bly can given the ev­i­dence available to you—again, say, the dis­tance/​time se­ries of a rock rol­ling down a hill. You can pref­ace your ex­pla­na­tion by say­ing, “The woman down the street is a witch,” but your friend then says, “What did she do?,” and you re­ply, “She made the rock roll one me­ter af­ter the first sec­ond, nine me­ters af­ter the third sec­ond…” Pre­fac­ing your mes­sage with “The woman down the street is a witch,” doesn’t help to com­press the rest of your de­scrip­tion. On the whole, you just end up send­ing a longer mes­sage than nec­es­sary—it makes more sense to just leave off the “witch” pre­fix. On the other hand, if you take a mo­ment to talk about Gal­ileo, you may be able to greatly com­press the next five thou­sand de­tailed time se­ries for rocks rol­ling down hills.

If you fol­low the first road, you end up with what’s known as Kol­mogorov com­plex­ity and Solomonoff in­duc­tion. If you fol­low the sec­ond road, you end up with what’s known as Min­i­mum Mes­sage Length.

Ah, so I can pick and choose among defi­ni­tions of sim­plic­ity?

No, ac­tu­ally the two for­mal­isms in their most highly de­vel­oped forms were proven equiv­a­lent.

And I sup­pose now you’re go­ing to tell me that both for­mal­isms come down on the side of “Oc­cam means count­ing laws, not count­ing ob­jects.”

More or less. In Min­i­mum Mes­sage Length, so long as you can tell your friend an ex­act recipe they can men­tally fol­low to get the rol­ling rock’s time se­ries, we don’t care how much men­tal work it takes to fol­low the recipe. In Solomonoff in­duc­tion, we count bits in the pro­gram code, not bits of RAM used by the pro­gram as it runs. “En­tities” are lines of code, not simu­lated ob­jects. And as said, these two for­mal­isms are ul­ti­mately equiv­a­lent.

Now be­fore I go into any fur­ther de­tail on for­mal sim­plic­ity, let me digress to con­sider the ob­jec­tion:

So what? Why can’t I just in­vent my own for­mal­ism that does things differ­ently? Why should I pay any at­ten­tion to the way you hap­pened to de­cide to do things, over in your field? Got any ex­per­i­men­tal ev­i­dence that shows I should do things this way?

Yes, ac­tu­ally, be­lieve it or not. But let me start at the be­gin­ning.

The con­junc­tion rule of prob­a­bil­ity the­ory states:

For any propo­si­tions X and Y, the prob­a­bil­ity that “X is true, and Y is true,” is less than or equal to the prob­a­bil­ity that “X is true (whether or not Y is true).” (If this state­ment sounds not ter­ribly profound, then let me as­sure you that it is easy to find cases where hu­man prob­a­bil­ity as­ses­sors vi­o­late this rule.)

You usu­ally can’t ap­ply the con­junc­tion rule di­rectly to a con­flict be­tween mu­tu­ally ex­clu­sive hy­pothe­ses. The con­junc­tion rule only ap­plies di­rectly to cases where the left-hand-side strictly im­plies the right-hand-side. Fur­ther­more, the con­junc­tion is just an in­equal­ity; it doesn’t give us the kind of quan­ti­ta­tive calcu­la­tion we want.

But the con­junc­tion rule does give us a rule of mono­tonic de­crease in prob­a­bil­ity: as you tack more de­tails onto a story, and each ad­di­tional de­tail can po­ten­tially be true or false, the story’s prob­a­bil­ity goes down mono­ton­i­cally. Think of prob­a­bil­ity as a con­served quan­tity: there’s only so much to go around. As the num­ber of de­tails in a story goes up, the num­ber of pos­si­ble sto­ries in­creases ex­po­nen­tially, but the sum over their prob­a­bil­ities can never be greater than 1. For ev­ery story “X and Y,” there is a story “X and ¬Y.” When you just tell the story “X,” you get to sum over the pos­si­bil­ities Y and ¬Y.

If you add ten de­tails to X, each of which could po­ten­tially be true or false, then that story must com­pete with other equally de­tailed sto­ries for pre­cious prob­a­bil­ity. If on the other hand it suffices to just say X, you can sum your prob­a­bil­ity over stories

((X and Y and Z and …) or (X and ¬Y and Z and …) or …) .

The “en­tities” counted by Oc­cam’s Ra­zor should be in­di­vi­d­u­ally costly in prob­a­bil­ity; this is why we pre­fer the­o­ries with fewer of them.

Imag­ine a lot­tery which sells up to a mil­lion tick­ets, where each pos­si­ble ticket is sold only once, and the lot­tery has sold ev­ery ticket at the time of the draw­ing. A friend of yours has bought one ticket for $1—which seems to you like a poor in­vest­ment, be­cause the pay­off is only$500,000. Yet your friend says, “Ah, but con­sider the al­ter­na­tive hy­pothe­ses, ‘To­mor­row, some­one will win the lot­tery’ and ‘To­mor­row, I will win the lot­tery.’ Clearly, the lat­ter hy­poth­e­sis is sim­pler by Oc­cam’s Ra­zor; it only makes men­tion of one per­son and one ticket, while the former hy­poth­e­sis is more com­pli­cated: it men­tions a mil­lion peo­ple and a mil­lion tick­ets!”

To say that Oc­cam’s Ra­zor only counts laws, and not ob­jects, is not quite cor­rect: what counts against a the­ory are the en­tities it must men­tion ex­plic­itly, be­cause these are the en­tities that can­not be summed over. Sup­pose that you and a friend are puz­zling over an amaz­ing billiards shot, in which you are told the start­ing state of a billiards table, and which balls were sunk, but not how the shot was made. You pro­pose a the­ory which in­volves ten spe­cific col­li­sions be­tween ten spe­cific balls; your friend coun­ters with a the­ory that in­volves five spe­cific col­li­sions be­tween five spe­cific balls. What counts against your the­o­ries is not just the laws that you claim to gov­ern billiard balls, but any spe­cific billiard balls that had to be in some par­tic­u­lar state for your model’s pre­dic­tion to be suc­cess­ful.

If you mea­sure the tem­per­a­ture of your liv­ing room as 22 de­grees Cel­sius, it does not make sense to say: “Your ther­mome­ter is prob­a­bly in er­ror; the room is much more likely to be 20 °C. Be­cause, when you con­sider all the par­ti­cles in the room, there are ex­po­nen­tially vastly more states they can oc­cupy if the tem­per­a­ture is re­ally 22 °C—which makes any par­tic­u­lar state all the more im­prob­a­ble.” But no mat­ter which ex­act 22 °C state your room oc­cu­pies, you can make the same pre­dic­tion (for the su­per­vast ma­jor­ity of these states) that your ther­mome­ter will end up show­ing 22 °C, and so you are not sen­si­tive to the ex­act ini­tial con­di­tions. You do not need to spec­ify an ex­act po­si­tion of all the air molecules in the room, so that is not counted against the prob­a­bil­ity of your ex­pla­na­tion.

On the other hand—re­turn­ing to the case of the lot­tery—sup­pose your friend won ten lot­ter­ies in a row. At this point you should sus­pect the fix is in. The hy­poth­e­sis “My friend wins the lot­tery ev­ery time” is more com­pli­cated than the hy­poth­e­sis “Some­one wins the lot­tery ev­ery time.” But the former hy­poth­e­sis is pre­dict­ing the data much more pre­cisely.

In the Min­i­mum Mes­sage Length for­mal­ism, say­ing “There is a sin­gle per­son who wins the lot­tery ev­ery time” at the be­gin­ning of your mes­sage com­presses your de­scrip­tion of who won the next ten lot­ter­ies; you can just say “And that per­son is Fred Smith” to finish your mes­sage. Com­pare to, “The first lot­tery was won by Fred Smith, the sec­ond lot­tery was won by Fred Smith, the third lot­tery was…”

In the Solomonoff in­duc­tion for­mal­ism, the prior prob­a­bil­ity of “My friend wins the lot­tery ev­ery time” is low, be­cause the pro­gram that de­scribes the lot­tery now needs ex­plicit code that sin­gles out your friend; but be­cause that pro­gram can pro­duce a tighter prob­a­bil­ity dis­tri­bu­tion over po­ten­tial lot­tery win­ners than “Some­one wins the lot­tery ev­ery time,” it can, by Bayes’s Rule, over­come its prior im­prob­a­bil­ity and win out as a hy­poth­e­sis.

Any for­mal the­ory of Oc­cam’s Ra­zor should quan­ti­ta­tively define, not only “en­tities” and “sim­plic­ity,” but also the “ne­ces­sity” part.

Min­i­mum Mes­sage Length defines ne­ces­sity as “that which com­presses the mes­sage.”

Solomonoff in­duc­tion as­signs a prior prob­a­bil­ity to each pos­si­ble com­puter pro­gram, with the en­tire dis­tri­bu­tion, over ev­ery pos­si­ble com­puter pro­gram, sum­ming to no more than 1. This can be ac­com­plished us­ing a bi­nary code where no valid com­puter pro­gram is a pre­fix of any other valid com­puter pro­gram (“pre­fix-free code”), e.g. be­cause it con­tains a stop code. Then the prior prob­a­bil­ity of any pro­gram P is sim­ply where is the length of P in bits.

The pro­gram P it­self can be a pro­gram that takes in a (pos­si­bly zero-length) string of bits and out­puts the con­di­tional prob­a­bil­ity that the next bit will be 1; this makes P a prob­a­bil­ity dis­tri­bu­tion over all bi­nary se­quences. This ver­sion of Solomonoff in­duc­tion, for any string, gives us a mix­ture of pos­te­rior prob­a­bil­ities dom­i­nated by the short­est pro­grams that most pre­cisely pre­dict the string. Sum­ming over this mix­ture gives us a pre­dic­tion for the next bit.

The up­shot is that it takes more Bayesian ev­i­dence—more suc­cess­ful pre­dic­tions, or more pre­cise pre­dic­tions—to jus­tify more com­plex hy­pothe­ses. But it can be done; the bur­den of prior im­prob­a­bil­ity is not in­finite. If you flip a coin four times, and it comes up heads ev­ery time, you don’t con­clude right away that the coin pro­duces only heads; but if the coin comes up heads twenty times in a row, you should be con­sid­er­ing it very se­ri­ously. What about the hy­poth­e­sis that a coin is fixed to pro­duce HTTHTT… in a re­peat­ing cy­cle? That’s more bizarre—but af­ter a hun­dred coin­flips you’d be a fool to deny it.

Stan­dard chem­istry says that in a gram of hy­dro­gen gas there are six hun­dred billion trillion hy­dro­gen atoms. This is a startling state­ment, but there was some amount of ev­i­dence that sufficed to con­vince physi­cists in gen­eral, and you par­tic­u­larly, that this state­ment was true.

Now ask your­self how much ev­i­dence it would take to con­vince you of a the­ory with six hun­dred billion trillion sep­a­rately speci­fied phys­i­cal laws.

Why doesn’t the prior prob­a­bil­ity of a pro­gram, in the Solomonoff for­mal­ism, in­clude a mea­sure of how much RAM the pro­gram uses, or the to­tal run­ning time?

The sim­ple an­swer is, “Be­cause space and time re­sources used by a pro­gram aren’t mu­tu­ally ex­clu­sive pos­si­bil­ities.” It’s not like the pro­gram speci­fi­ca­tion, that can only have a 1 or a 0 in any par­tic­u­lar place.

But the even sim­pler an­swer is, “Be­cause, his­tor­i­cally speak­ing, that heuris­tic doesn’t work.”

Oc­cam’s Ra­zor was raised as an ob­jec­tion to the sug­ges­tion that neb­u­lae were ac­tu­ally dis­tant galax­ies—it seemed to vastly mul­ti­ply the num­ber of en­tities in the uni­verse. All those stars!

Over and over, in hu­man his­tory, the uni­verse has got­ten big­ger. A var­i­ant of Oc­cam’s Ra­zor which, on each such oc­ca­sion, would la­bel the vaster uni­verse as more un­likely, would fare less well un­der hu­man­ity’s his­tor­i­cal ex­pe­rience.

This is part of the “ex­per­i­men­tal ev­i­dence” I was al­lud­ing to ear­lier. While you can jus­tify the­o­ries of sim­plic­ity on mathy sorts of grounds, it is also de­sir­able that they ac­tu­ally work in prac­tice. (The other part of the “ex­per­i­men­tal ev­i­dence” comes from statis­ti­ci­ans /​ com­puter sci­en­tists /​ Ar­tifi­cial In­tel­li­gence re­searchers, test­ing which defi­ni­tions of “sim­plic­ity” let them con­struct com­puter pro­grams that do em­piri­cally well at pre­dict­ing fu­ture data from past data. Prob­a­bly the Min­i­mum Mes­sage Length paradigm has proven most pro­duc­tive here, be­cause it is a very adapt­able way to think about real-world prob­lems.)

Imag­ine a space­ship whose launch you wit­ness with great fan­fare; it ac­cel­er­ates away from you, and is soon trav­el­ing at . If the ex­pan­sion of the uni­verse con­tinues, as cur­rent cos­mol­ogy holds it should, there will come some fu­ture point where—ac­cord­ing to your model of re­al­ity—you don’t ex­pect to be able to in­ter­act with the space­ship even in prin­ci­ple; it has gone over the cos­molog­i­cal hori­zon rel­a­tive to you, and pho­tons leav­ing it will not be able to out­race the ex­pan­sion of the uni­verse.

Should you be­lieve that the space­ship liter­ally, phys­i­cally dis­ap­pears from the uni­verse at the point where it goes over the cos­molog­i­cal hori­zon rel­a­tive to you?

If you be­lieve that Oc­cam’s Ra­zor counts the ob­jects in a model, then yes, you should. Once the space­ship goes over your cos­molog­i­cal hori­zon, the model in which the space­ship in­stantly dis­ap­pears, and the model in which the space­ship con­tinues on­ward, give in­dis­t­in­guish­able pre­dic­tions; they have no Bayesian ev­i­den­tial ad­van­tage over one an­other. But one model con­tains many fewer “en­tities”; it need not speak of all the quarks and elec­trons and fields com­pos­ing the space­ship. So it is sim­pler to sup­pose that the space­ship van­ishes.

Alter­na­tively, you could say: “Over nu­mer­ous ex­per­i­ments, I have gen­er­al­ized cer­tain laws that gov­ern ob­served par­ti­cles. The space­ship is made up of such par­ti­cles. Ap­ply­ing these laws, I de­duce that the space­ship should con­tinue on af­ter it crosses the cos­molog­i­cal hori­zon, with the same mo­men­tum and the same en­ergy as be­fore, on pain of vi­o­lat­ing the con­ser­va­tion laws that I have seen hold­ing in ev­ery ex­am­inable in­stance. To sup­pose that the space­ship van­ishes, I would have to add a new law, ‘Things van­ish as soon as they cross my cos­molog­i­cal hori­zon.’ ”

The de­co­her­ence (a.k.a. many-wor­lds) ver­sion of quan­tum me­chan­ics states that mea­sure­ments obey the same quan­tum-me­chan­i­cal rules as all other phys­i­cal pro­cesses. Ap­ply­ing these rules to macro­scopic ob­jects in ex­actly the same way as micro­scopic ones, we end up with ob­servers in states of su­per­po­si­tion. Now there are many ques­tions that can be asked here, such as

“But then why don’t all bi­nary quan­tum mea­sure­ments ap­pear to have 5050 prob­a­bil­ity, since differ­ent ver­sions of us see both out­comes?”

How­ever, the ob­jec­tion that de­co­her­ence vi­o­lates Oc­cam’s Ra­zor on ac­count of mul­ti­ply­ing ob­jects in the model is sim­ply wrong.

De­co­her­ence does not re­quire the wave­func­tion to take on some com­pli­cated ex­act ini­tial state. Many-wor­lds is not spec­i­fy­ing all its wor­lds by hand, but gen­er­at­ing them via the com­pact laws of quan­tum me­chan­ics. A com­puter pro­gram that di­rectly simu­lates quan­tum me­chan­ics to make ex­per­i­men­tal pre­dic­tions, would re­quire a great deal of RAM to run—but simu­lat­ing the wave­func­tion is ex­po­nen­tially ex­pen­sive in any fla­vor of quan­tum me­chan­ics! De­co­her­ence is sim­ply more so. Many phys­i­cal dis­cov­er­ies in hu­man his­tory, from stars to galax­ies, from atoms to quan­tum me­chan­ics, have vastly in­creased the ap­par­ent CPU load of what we be­lieve to be the uni­verse.

Many-wor­lds is not a zillion wor­lds worth of com­pli­cated, any more than the atomic hy­poth­e­sis is a zillion atoms worth of com­pli­cated. For any­one with a quan­ti­ta­tive grasp of Oc­cam’s Ra­zor that is sim­ply not what the term “com­pli­cated” means.

As with the his­tor­i­cal case of galax­ies, it may be that peo­ple have mis­taken their shock at the no­tion of a uni­verse that large, for a prob­a­bil­ity penalty, and in­voked Oc­cam’s Ra­zor in jus­tifi­ca­tion. But if there are prob­a­bil­ity penalties for de­co­her­ence, the lar­ge­ness of the im­plied uni­verse, per se, is definitely not their source!

The no­tion that de­co­her­ent wor­lds are ad­di­tional en­tities pe­nal­ized by Oc­cam’s Ra­zor is just plain mis­taken. It is not sort-of-right. It is not an ar­gu­ment that is weak but still valid. It is not a defen­si­ble po­si­tion that could be shored up with fur­ther ar­gu­ments. It is en­tirely defec­tive as prob­a­bil­ity the­ory. It is not fix­able. It is bad math.

• To­mor­row I will ad­dress my­self to ac­cu­sa­tions I have en­coun­tered that de­co­her­ence is “un­falsifi­able” or “untestable”, as the words “falsifi­able” and “testable” have (even sim­pler) prob­a­bil­ity-the­o­retic mean­ings which would seem to be vi­o­lated by this us­age.

Doesn’t this fol­low triv­ially from the above? No ex­per­i­ment can de­ter­mine whether or not we have souls, but that counts against the idea of souls, not against the idea of their ab­sence. If de­co­her­ence is the sim­pler the­ory, then lack of falsifi­a­bil­ity counts against the other guys, not against it.

• If the ex­pan­sion of the uni­verse con­tinues, as cur­rent cos­mol­ogy holds it should, there will come some fu­ture point where—ac­cord­ing to your model of re­al­ity—you don’t ex­pect to be able to in­ter­act with the space­ship even in prin­ci­ple; it has gone over the cos­molog­i­cal hori­zon rel­a­tive to you, and pho­tons leav­ing it will not be able to out­race the ex­pan­sion of the uni­verse.

IIRC for this to be true the uni­verse’s ex­pan­sion has to ac­cel­er­ate, and the ac­cel­er­a­tion has to stay bounded above zero for­ever. (IIRC this is still con­sid­ered the most prob­a­ble case.)

• No ex­per­i­ment can de­ter­mine whether or not we have souls

Really? Not at­tempted up­load­ing? Micro­phys­i­cal ex­am­i­na­tion of a liv­ing brain? Tests for re­li­able mem­o­ries of past lives, or re­li­able mediums?

• Math­e­mat­ics is just a lan­guage, and any suffi­ciently pow­er­ful lan­guage can say any­thing that can be said in any other lan­guage.

Feyn­man was right—if you can’t ex­plain it in or­di­nary lan­guage, you don’t un­der­stand it at all.

• Or­di­nary lan­guage is not suffi­ciently pow­er­ful.

• Or­di­nary lan­guage in­cludes math­e­mat­ics.

“One, two, three, four” is or­di­nary lan­guage. “The thing turned right” is or­di­nary lan­guage (it’s also mul­ti­pli­ca­tion by -i).

Feyn­man was right, he just ne­glected to spec­ify that the or­di­nary lan­guage needed to ex­plain physics would nec­es­sar­ily in­clude the math sub­set of it.

• If you’re cov­er­ing this later I’ll wait, but I ask now in case my con­fu­sion means I’m mi­s­un­der­stand­ing some­thing.

Why isn’t nearly ev­ery­thing en­tanged with nearly ev­ery­thing else around it by now? Why is there a sig­nifi­cant amount of much quan­tum in­de­pen­dance still around? Or does it just look that way be­cause en­tanged sub­con­figu­ra­tions tend to get split off by decore­hence so branches re­tain a rea­son­able amount of non-en­tan­gled­ness within their branch? Sorry if this is a daft or daftly phrased ques­tion.

• It IS. That was some­thing Eliezer said waay back when, and he was right. En­tan­gle­ment is a very or­di­nary state of af­fairs.

• In­deed, a thor­oughly en­tan­gled world looks clas­si­cal, how­ever para­dox­i­cal it might sound. Re­gard­less of the adopted in­ter­pre­ta­tion.

• To be fair, one could shy away from say­ing all those branches are real due to the difficulty of squar­ing the Born rule with the equal prob­a­bil­ity calcu­la­tions that seem to fol­low from that view. Without some­thing like man­gled wor­lds, one can be tempted by an ob­jec­tive col­lapse view, as that at least gives a co­her­ent ac­count of the Born rule.

• (The other part of the “ex­per­i­men­tal ev­i­dence” comes from statis­ti­ci­ans /​ com­puter sci­en­tists /​ Ar­tifi­cial In­tel­li­gence re­searchers, test­ing which defi­ni­tions of “sim­plic­ity” let them con­struct com­puter pro­grams that do em­piri­cally well at pre­dict­ing fu­ture data from past data. Prob­a­bly the Min­i­mum Mes­sage Length paradigm has proven most pro­duc­tive here, be­cause it is a very adapt­able way to think about real-world prob­lems.)

I once be­lieved that sim­plic­ity is the key to in­duc­tion (it was the topic of my PhD the­sis), but I no longer be­lieve this. I think most re­searchers in ma­chine learn­ing have come to the same con­clu­sion. Here are some prob­lems with the idea that sim­plic­ity is a guide to truth:

(1) Solomonoff/​Gold/​Chaitin com­plex­ity is not com­putable in any rea­son­able amount of time.

(2) The Min­i­mum Mes­sage Length de­pends en­tirely on how a situ­a­tion is rep­re­sented. Differ­ent rep­re­sen­ta­tions lead to rad­i­cally differ­ent MML com­plex­ity mea­sures. This is a gen­eral prob­lem with any at­tempt to mea­sure sim­plic­ity. How do you jus­tify your choice of rep­re­sen­ta­tion? For any two hy­pothe­ses, A and B, it is pos­si­ble to find a rep­re­sen­ta­tion X such that com­plex­ity(A) < com­plex­ity(B) and an­other rep­re­sen­ta­tion Y such that com­plex­ity(A) > com­plex­ity(B).

(3) Sim­plic­ity is merely one type of bias. The No Free Lunch the­o­rems show that there is no a prior rea­son to pre­fer one type of bias over an­other. There­fore there is noth­ing spe­cial about a bias to­wards sim­plic­ity. A bias to­wards com­plex­ity is equally valid a pri­ori.

• Without some­thing like man­gled wor­lds, one can be tempted by an ob­jec­tive col­lapse view, as that at least gives a co­her­ent ac­count of the Born rule.

Does it re­ally ac­count for it in the sense of ex­plain it? I don’t think so. I think it merely says that the col­laps­ing oc­curs in ac­cor­dance with the Born rule. But we can also sim­ply say that many-wor­lds is true and the his­tory of our frag­ment of the mul­ti­verse is con­sis­tent with the Born rule. Ad­mit­tedly, this doesn’t ex­plain why we hap­pen to live in such a frag­ment but merely as­serts that we do, but similarly, the col­lapse view does not (as far as I know) ex­plain why the col­lapse oc­curs in the fre­quen­cies it does but merely as­serts that it does.

• “No math = no physics”

I would say that as a prac­ti­cal mat­ter, this is true, be­cause of­ten, many the­o­ries make the same qual­i­ta­tive pre­dic­tion, but differ­ent quan­ti­ta­tive ones. The effect of grav­ity on light for in­stance. In New­to­nian grav­ity, light af­fected the same as mat­ter, but in Gen­eral Rel­a­tivity, the effect is larger. Another ex­am­ple would be flat-Earth the­ory grav­ity ver­sus New­to­nian. Flat-Earthers would say that the Earth is con­stantly ac­cel­er­at­ing up­wards at 9.8 m/​s^2. To a high level of pre­ci­sion, this matches the idea that ob­jects are at­tracted by G M/​ R^2. Differ­ence be­comes large at high al­ti­tudes (large R), where it is quan­ti­ta­tively differ­ent, but qual­i­ta­tively the same.

One could prob­a­bly get by set­ting up ex­per­i­ments where the only pos­si­ble re­sults are (same, differ­ent), but that’s re­ally the same as defin­ing num­bers of terms of what they lie be­tween; i.e., calcu­lat­ing sqrt(2) by calcu­lat­ing the largest num­ber < sqrt(2) and the small­est num­ber > sqrt(2).

The rate of sci­en­tific progress jumped enor­mously af­ter New­ton, as peo­ple be­gan think­ing more and more quan­ti­ta­tively, and de­vel­oped tools ac­cord­ingly. This is not an ac­ci­dent.

• I just had a thought, prob­a­bly not a good one, about Many Wor­lds. It seems like there’s a par­allel here to the dis­cov­ery of Nat­u­ral Selec­tion and un­der­stand­ing of Evolu­tion.

Dar­win had the key in­sight about how se­lec­tion pres­sure could lead to changes in or­ganisms over time. But it’s taken us over 100 years to get a good han­dle on spe­ci­a­tion and figure out the de­tailed mechanisms of se­lect­ing for ge­netic fit­ness. One could ar­gue that we still have a long way to go.

Similarly, it seems like we’ve had this in­sight that QM leads to Many Wor­lds due to de­co­her­ence. But it could take quite a while for us to get a good han­dle on what hap­pens to wor­lds and figure that de­tailed mechanisms of how they progress.

But it was pretty clear that Dar­win was right long be­fore we had worked the de­tails. So I guess it doesn’t bother me that we haven’t worked out the de­tails of what hap­pens to the Many Wor­lds.

• Eliezer: Could you maybe clar­ify a bit the differ­ence be­tween Kol­mogrov Com­plex­ity and MML? The first is “short­est com­puter pro­gram that out­puts the re­sult” and the sec­ond one is.… “short­est info that can be used to figure out the re­sult”? ie, I’m not quite sure I un­der­stand the differ­ence here.

How­ever, as to the two be­ing equiv­a­lent, I thought I’d seen some­thing about the sec­ond one be­ing used be­cause the first was some­times un­com­putable, in the “to solve it in the gen­eral case, you’d have to have a halt­ing or­a­cle” sense.

Cale­do­nian: Math isn’t so much a lan­guage as var­i­ous no­ta­tions for math are a lan­guage. Ba­si­cally, if you use “non math” to ex­press ex­actly the same thing as math, you ba­si­cally have to turn up the pre­ci­sion and “legaleese” to the point that you prac­tially are de­scribing math, just us­ing a lan­guage not meant for it, right?

• I think you have it back­wards. When we use lan­guage in a very pre­cise and spe­cific way, strip­ping out am­bi­guity and the po­ten­tial for al­ter­nate mean­ings, we call the re­sult ‘math­e­mat­ics’.

• Cale­do­nian: Er… isn’t that what I was say­ing? (That is, that’s ba­si­cally what I meant. What did you think I meant?)

• If you head down this re­duc­tion­ism, oc­cams ra­zor route, doesn’t the con­cpet of a hu­man be­come ex­plana­to­rily re­dun­dant? It will be sim­pler to pre­cisely pre­dict what a hu­man will do with­out in­vok­ing the in­ten­tional stance and just mod­el­ling the un­der­ly­ing physics.

• Nick Tar­leton: sadly, it’s my ex­pe­rience that it’s fu­tile to try and throw flour over the dragon.

• Hang on. @ Cale­do­nian and Psy-Kosh: Surely math­e­mat­i­cal lan­guage is just lan­guage that refers to math­e­mat­i­cal ob­jects—num­bers and such­like. Pre­cise, un­am­bigu­ous lan­guage doesn’t count as math­e­mat­ics un­less it meets this con­di­tion.

• Surely math­e­mat­i­cal lan­guage is just lan­guage that refers to math­e­mat­i­cal ob­jects—num­bers and such­like. Pre­cise, un­am­bigu­ous lan­guage doesn’t count as math­e­mat­ics un­less it meets this con­di­tion.

Is logic math­e­mat­ics? I as­sert that pre­cise, umam­bigu­ous lan­guage nec­es­sar­ily refers to math­e­mat­i­cal ob­jects, be­cause ‘math­e­mat­ics’ is pre­cise, umam­bigu­ous lan­guage. All math is lan­guage. Not all lan­guage is math.

Of course, I also as­sert that math­e­mat­ics is a sub­set of sci­ence, so con­sider that our ba­sic wor­ld­views might be very differ­ent.

• “No math = no physics”

Tell that to Coper­ni­cus, Gilbert, Gal­vani, Volta, Oer­sted and Fara­day, to men­tion a few. And it’s not like even Gal­ileo used much more than the high-school ar­ith­metic of his day for his elu­ci­da­tion of ac­cel­er­a­tion.

• Some physi­cists speak of “el­e­gance” rather than “sim­plic­ity”. This seems to me a bad idea; your judg­ments of el­e­gance are go­ing to be marred by evolved aes­thetic crite­ria that ex­ist only in your head, rather than in the ex­te­rior world, and should only be trusted inas­much as they point to­wards smaller, rather than larger, Kol­mogorov com­plex­ity.

Ex­am­ple:

In the­ory A, the ra­tio of tiny di­men­sion #1 to tiny di­men­sion #2 is finely-tuned to sup­port life.

In the­ory B, the ra­tio of the mass of the elec­tron to the mass of the neu­trino is finely-tuned to sup­port life.

An “el­e­gance” ad­vo­cate might fa­vor A over B, whereas a “sim­plic­ity” ad­vo­cate might be neu­tral be­tween them.

• i came to this dor­mant thread from the fu­ture: http://​​less­wrong.com/​​lw/​​1k4/​​the_con­trar­ian_sta­tus_catch22/​​1ckj.

Seems to me there is a mismap­ping of mul­ti­ple wor­lds wrt quan­tum physics and the mul­ti­ple wor­lds we cre­ate sub­jec­tively. I per­son­ally steer clear of physics and con­cern my­self more with the sub­jec­tive re­al­ities we cre­ate. This seems to me to be more con­gru­ent with the ma­te­rial that eliezer pre­sents here, ie wrt logic and oc­cam’s ra­zor, and what he pre­sents in the ar­ti­cle linked above, ie wrt con­trar­ian dy­nam­ics and feel­ings of satis­fac­tion et al.

• I’m stuck, what does ~ de­note?

For ev­ery story “X∧Y”, there is a story “X∧~Y”. When you just tell the story “X”, you get to sum over the pos­si­bil­ities Y and ~Y.

Y and ~Y, where ~Y does read as … ?

• Not-Y, i.e. Y is false..

• I see, thank you. Does it add too much noise if I ask such ques­tions, should I rather not yet read the se­quences if I some­times have to in­quire about such mat­ters? Or should I ask some­where else?

I was look­ing up this table of math­e­mat­i­cal sym­bols that stated that ~ does read as ‘has dis­tri­bu­tion’ and stopped look­ing any fur­ther since I’m deal­ing with prob­a­bil­ities here. I guess it should have been ob­vi­ous to me to ex­pect a log­i­cal op­er­a­tor. I was only used to the no­ta­tion not and ¬ as the nega­tion of a propo­si­tion. I’ll have to ad­just my per­ceived in­tel­li­gence down­wards.

• There’s no prob­lem with ask­ing a clar­ify­ing ques­tion like that, which might help other lurk­ers and can be an­swered quickly with­out huge amounts of work.

By the way, there’s no need for such self-de­p­re­cat­ing com­ments about your ed­u­ca­tion or in­tel­li­gence. It’s so­cially a bit off-putting to talk about the topic, and it risks com­ing across as dis­inge­nous. Just ask your ques­tions with­out such sup­pli­ca­tion.

• No, these are rea­son­able ques­tions to ask.

• Others that gets used a lot in var­i­ous con­texts are !Y, Y^c (for Y com­ple­ment). There are prob­a­bly more, but I can’t think of them at the mo­ment.

• Y with a bar over it also gets used (though be care­ful as this more com­monly means clo­sure of Y, or, well, quite a few other things...)

• “Not Y”.

The pre­fix tilde, “~”, is com­monly used as an ASCII ap­prox­i­ma­tion for log­i­cal nega­tion, in place of the more math­e­mat­i­cally for­mal no­ta­tion of “¬” (U+00AC, HTML en­tity “¬”, and la­tex “\lnot”) . On that page are a few other com­mon no­ta­tions.

• When you just tell the story “X”, you get to sum over the pos­si­bil­ities Y and ~Y.

Maybe this is a stupid ques­tion, but shouldn’t it mean “When you just tell the story “Y”, you get to sum over the pos­si­bil­ities Y and ~Y.” ?

• I have been work­ing with de­co­her­ence in ex­per­i­men­tal physics. It con­fuses me that you want to use it as a syn­onym for the Many-Wor­lds the­ory.

• MWI is the sup­po­si­tion that there is noth­ing else to the fun­da­men­tal laws of na­ture ex­cept QM. De­co­her­ence is the main tricky point in the bridge be­tween QM and our sub­jec­tive ex­pe­riences.

With de­co­her­ence, a col­lapse pos­tu­late is su­perflu­ous. With de­co­her­ence, you don’t need a Bohmian ‘real thing’ or what­ever he calls it. QM is sim­ply the way things are. You can stick with it, and MWI fol­lows di­rectly.

• The col­lapse pos­tu­late is just a vi­su­al­iza­tion, just like the MWI is.The Born pro­jec­tion rule is the only “real” thing, and it per­sists through MWI or any other “I”. So no, the MWI does not fol­low di­rectly, un­less you strip it of all on­tolog­i­cal mean­ing.

• This isn’t quite what your post was about, but one thing I’ve never un­der­stood is how any­one could pos­si­bly find “the uni­verse is to­tally ran­dom” to be a MORE sim­ple ex­pla­na­tion.

• I did not get a chance to read this en­try un­til four years af­ter it was pub­lished, but it nonethe­less ended up cor­rect­ing a long-held flawed view I had on the Many Wor­lds In­ter­pre­ta­tion. Thank you for open­ing up my eyes to the idea that Oc­cam’s ra­zor ap­plies to rules, and not en­tities in a sys­tem. You have no idea as to how em­bar­rassed I feel for hav­ing so dras­ti­cally mi­s­un­der­stood the con­cept be­fore now.

In­ci­den­tally, I wrote a blog en­try on how this ar­ti­cle changed my mind which seems to have gen­er­ated ad­di­tional dis­cus­sion on this is­sue.

• How do you ex­plain this with many wor­lds, while avoid­ing non-lo­cal­ity? http://​​arxiv.org/​​pdf/​​1209.4191v1.pdf If re­sults such as these are easy to ex­plain/​​pre­dict, can the many wor­lds the­ory gain cred­i­bil­ity by pre­dict­ing such things?

• Glib: start with the ini­tial states. Prop­a­gate time as speci­fied. Ob­serve the re­sult come out. That’s how. MWI is quan­tum me­chan­ics with no mod­ifi­ca­tions, so its pre­dic­tions match what quan­tum me­chan­ics pre­dicts, and quan­tum me­chan­ics hap­pens to be lo­cal.

Ful­ller: The first mo­ment of de­co­her­ence is when pho­ton 1 is mea­sured. At this point, we’ve split the world ac­cord­ing to its po­lariza­tion.

Then we have pho­tons 2 and 3 in­terfere. They are then shortly mea­sured, so we’ve split the world again in sev­eral parts. This al­lows post-se­lec­tion for when the two pho­tons com­ing out go to differ­ent de­tec­tors. When that crite­rion is met, then we’re in the sub­space with pho­ton 2 be­ing a match for pho­ton 3, which is the same sub­space as 4 be­ing a match for pho­ton 1.

When they mea­sure pho­ton 4, it pro­ceeds to match pho­ton 1.

Un­der MWI, this is so un­sur­pris­ing that I’m hav­ing a hard time jus­tify­ing perform­ing the ex­per­i­ment ex­cept as a way of show­ing off cool things we can do with co­her­ence.

Now, as for whether this was lo­cal, note that the pro­ce­dure in­volved ig­nor­ing the events we don’t want. That sort of thing is al­lowed to be non-lo­cal since it’s an ar­ti­fact of data anal­y­sis. It’s not like pho­ton 1 made pho­ton 4 do any­thing. THAT would be non-lo­cal. But the force was ap­plied by post-se­lec­tion.

Of COURSE… you don’t NEED to look at it through the lens of MWI to get that. Even Copen­hagen would come up with the right an­swer to that, I think. Ac­tu­ally, I’m not sure why it would be a sur­pris­ing re­sult even un­der Copen­hagen. Post-se­lec­tion cre­ates cor­re­la­tions! News at 11.

• Un­der MWI, this is so un­sur­pris­ing that I’m hav­ing a hard time jus­tify­ing perform­ing the ex­per­i­ment ex­cept as a way of show­ing off cool things we can do with co­her­ence.

Then the jus­tifi­ca­tion is sim­ple; it ei­ther pro­vides ev­i­dence in favour of MWI (and Copen­hagen and any other the­ory that pre­dicts the ex­pected re­sult) or it shat­ters all of them.

Scien­tists have to do ex­per­i­ments to which the an­swer is ob­vi­ous—failing to do so leads to the situ­a­tion where ev­ery­body knows that heav­ier ob­jects fall faster than lighter ob­jects be­cause no­body ac­tu­ally checked that.

• ev­ery­body knows that heav­ier ob­jects fall faster than lighter objects

Be­cause this hap­pens to be mostly true. Air re­sis­tance is a thing.

ac­tu­ally checked that

Ac­tu­ally, if I re­mem­ber the high school physics anec­dote cor­rectly, the trou­ble for the idea that heavy ob­jects fall faster than light ones be­gan when a cer­tain sci­en­tist asked a hy­po­thet­i­cal ques­tion: what would hap­pen if you drop a light and a heavy ob­ject at the same time, but con­nect them with a string?

• Be­cause this hap­pens to be mostly true. Air re­sis­tance is a thing.

Got noth­ing to do with weight, though. An acre of tis­sue pa­per, spread out flat, will still fall more slowly than a ten cent coin dropped edge-first.

Ac­tu­ally, if I re­mem­ber the high school physics anec­dote cor­rectly, the trou­ble for the idea that heavy ob­jects fall faster than light ones be­gan when a cer­tain sci­en­tist asked a hy­po­thet­i­cal ques­tion: what would hap­pen if you drop a light and a heavy ob­ject at the same time, but con­nect them with a string?

Well, yes. (Gal­ileo, wasn’t it?) Doesn’t af­fect my point, though—the ba­sics do need to be checked oc­ca­sion­ally.

• Got noth­ing to do with weight, though.

Noth­ing? Are you quite sure about that? :-)

• Hmmm… let me con­sider it.

In an air­less void, the an­swer is no—the mass terms of the force-due-to-grav­ity and the ac­cel­er­a­tion-due-to-force equa­tions can­cel out, and weight has noth­ing to do with the speed of the fal­ling ob­ject.

In the pres­ence of air re­sis­tance, how­ever… the force from air re­sis­tance de­pends on how much air the ob­ject hits (which in turn de­pends on the shape of the ob­ject), and how fast rel­a­tive to the ob­ject the air is mov­ing. The force ap­plied by air re­sis­tance is in­de­pen­dent of the mass (but de­pen­dent on the shape and speed of the ob­ject) - but the ac­cel­er­a­tion caused by that force is de­pen­dant on the mass (f=ma). There­fore, the ac­cel­er­a­tion due to air re­sis­tance does de­pend par­tially on the mass of the ob­ject.

Okay, so not quite “noth­ing”, but mass is not the most im­por­tant fac­tor to con­sider in these equa­tions...

• I don’t know how you de­cide what’s more and what’s less im­por­tant in physics equa­tions :-/​

If I tell you I dropped a sphere two inches in di­ame­ter from 200 feet up, can you calcu­late its speed at the mo­ment it hits the ground? Without know­ing its weight, I don’t think you can.

• I don’t know how you de­cide what’s more and what’s less im­por­tant in physics equa­tions :-/​

Pre­dic­tive power. The more ac­cu­rate a pre­dic­tion I can make with­out know­ing the value of a given vari­able, the less im­por­tant that vari­able is.

If I tell you I dropped a sphere two inches in di­ame­ter from 200 feet up, can you calcu­late its speed at the mo­ment it hits the ground? Without know­ing its weight, I don’t think you can.

Ugh, im­pe­rial mea­sures. Do you mind if I work with a five-cen­time­tre sphere dropped from 60 me­tres?

A sphere is quite an aero­dy­namic shape; so I ex­pect, for most masses, that air fric­tion will have a small to neg­ligible im­pact on the sphere’s fi­nal ve­loc­ity. I know that the ac­cel­er­a­tion due to grav­ity is 9.8m/​s^2, and so I turn to the equa­tions of mo­tion; v^2 = v_0^2+2*a*s (where v_0 is the start­ing ve­loc­ity). Start­ing ve­loc­ity v_0 is 0, a is 9.8, s is 60m; thus v^2 = (0*0)+(2*9.8*60) = 1176, there­fore v = about 34.3m/​s. Lit­tle slower than that be­cause of air re­sis­tance, but prob­a­bly not too much slower. (You’ll also no­tice that I’m not us­ing the ra­dius of the sphere any­where in this calcu­la­tion). It’s an ap­prox­i­ma­tion, yes, but it’s prob­a­bly fairly ac­cu­rate… good enough for many, though not all pur­poses.

Now, if I know the mass but not the shape, it’s a lot harder to jus­tify the “ig­nore air re­sis­tance” step...

• A sphere is quite an aero­dy­namic shape

(Eng­ineer with a back­ground in fluid dy­nam­ics here.)

A sphere is quite un­aero­dy­namic. Its drag co­effi­cient is about 10 times higher than that of a stream­lined body (at a rele­vant Reynolds num­ber). You have bound­ary layer sep­a­ra­tion off the back of the sphere, which re­sults in a large wake and con­se­quently high drag.

The speed as a func­tion of time for an ob­ject with a con­stant drag co­effi­cient drop­ping ver­ti­cally is known and it is a di­rect func­tion of mass. If I learned any­thing from mak­ing potato guns, it’s that in gen­eral, dra­gless calcu­la­tions are pretty in­ac­cu­rate. You’ll get the trend right in many cases with a dra­gless calcu­la­tion, but in gen­eral it’s best to not as­sume drag is neg­ligible un­less you’ve done the math or ex­per­i­ment to show that it is in a par­tic­u­lar case.

• A sphere is quite un­aero­dy­namic.

Huh. I thought the fact that it got con­tinu­ally and mono­ton­i­cally big­ger un­til a given point and then mono­ton­i­cally smaller meant at least some aero­dy­nam­ics in the shape. I did not even con­sider the wake...

The speed as a func­tion of time for an ob­ject with a con­stant drag co­effi­cient drop­ping ver­ti­cally is known and it is a di­rect func­tion of mass.

Well. I stand cor­rected, then. Ev­i­dently drag has a far big­ger effect than I gave it credit for.

...pro­por­tional to the square root of the mass, given all oher fac­tors are un­changed, I see.

• It’s bet­ter than a flat plate per­pen­dicu­lar to the flow. Most peo­ple seem to not ex­pect that the back of the ob­ject af­fects the drag, but there’s a large low pres­sure zone due to the wake. With high pres­sure in the front and low pres­sure in the back (along with a some­what neg­ligible skin fric­tion con­tri­bu­tion), the drag is con­sid­er­able. So you need to tar­get both the front and back to have a low drag shape. Most aero­dy­namic shapes trade pres­sure drag for skin fric­tion drag, as the lat­ter is small (if the Reynolds num­ber is high).

• For “an aero­dy­namic shape” my in­tu­ition first gives me a stylized drop: hemi­spheric in the front and a long tail thin­ning to a point in the back. But af­ter a cou­ple of sec­onds it de­cides that a spin­dle shape would prob­a­bly be bet­ter :-)

• The “teardrop” shape is pretty good, though the name is a fair bit mis­lead­ing as droplets al­most never look like that. Their shape varies in time de­pend­ing on the flow con­di­tions.

Not quite sure what you mean by spin­dle shape, but I’m sure a va­ri­ety of shapes like that could be pretty good. For the front, it’s im­por­tant to not have a flat tip. For the back, you’d want a grad­ual de­cay of the ra­dius to pre­vent the fluid from sep­a­rat­ing off the back, cre­at­ing a large wake. Th­ese are the heuris­tics.

Which shape ob­jects have min­i­mum drag is a fairly in­ter­est­ing sub­ject. The shape with min­i­mum wave drag (i.e., su­per­sonic flow) is known, but I’m not sure there are any gen­eral proofs for other flow regimes. Per­haps it doesn’t mat­ter much, as we already know a bunch of shapes with low drag. The real prob­lem seems to be get­ting these shapes adopted, as (for ex­am­ple) cars don’t seem to be bought on ra­tio­nal bases like en­g­ineer­ing. This should not be sur­pris­ing.

• cars don’t seem to be bought on ra­tio­nal bases like en­g­ineer­ing.

Of course, but I don’t see it as a bad thing. Typ­i­cally when peo­ple buy cars they have a col­lec­tion of must-haves and then from the short list of cars match­ing the must-haves, they pick what they like. I think it’s a perfectly fine method of pick­ing cars. Com­pare to pick­ing clothes, for ex­am­ple...

• You’re do­ing the mid­dle-school physics “an ob­ject dropped in vac­uum” calcu­la­tion :-) If you want to get a num­ber that takes air re­sis­tance into ac­count you need col­lege-level physics.

So, since you’ve men­tioned ac­cu­racy, how ac­cu­rate your 34.3 m/​s value is? Can you give me some kind of con­fi­dence in­ter­vals?

• You’re do­ing the mid­dle-school physics “an ob­ject dropped in vac­uum” calcu­la­tion :-)

Yes, ex­actly. Be­cause for many ev­ery­day situ­a­tions, it’s close enough.

So, since you’ve men­tioned ac­cu­racy, how ac­cu­rate your 34.3 m/​s value is? Can you give me some kind of con­fi­dence in­ter­vals?

No, I can’t. In or­der to do that, I would need, first of all, to know how to do the air re­sis­tance calcu­la­tion—I can prob­a­bly look that up, but it’s go­ing to be com­pli­cated—and, im­por­tantly, some sort of prob­a­bil­ity dis­tri­bu­tion for the pos­si­ble masses of the ball (know­ing the ra­dius might help in es­ti­mat­ing this).

Of course, the greater the mass of the ball, the more ac­cu­rate my value is, be­cause the air ri­sis­tance will have less effect; in the limit, if the ball is a hy­dro­gen bal­loon, I ex­pect it to float away and never ac­tu­ally hit the ground at all, while in the other limit, if the ball is a tiny black hole, I ex­pect it to smash into the ground at ex­actly the calcu­lated value (and then keep go­ing).

• No, I can’t.

And thus we get back to the ques­tion of what’s im­por­tant in physics equa­tions.

But let’s do a nu­mer­i­cal ex­am­ple for fun.

Our ball is 5 cm in di­ame­ter, so its vol­ume is about 65.5 cm3. Let’s make it out of wood, say, bam­boo. Its den­sity is about 0.35 g/​cm3 so the ball will weigh about 23 g.

Let’s calcu­late its ter­mi­nal ve­loc­ity, that is, the speed at which drag ex­actly bal­ances grav­ity. The for­mula is v = sqrt(2mg/​(pAC)) where m is mass (0.023 kg) , g is the same old 9.8, p is air den­sity which is about 1.2 kg/​m3, A is pro­jected area and since we have a sphere it’s 19.6 cm2 or 0.00196 m2, and C is the drag co­effi­cient which for a sphere is 0.47.

v = sqrt( 2 0.023 9.8 /​ (1.2 0.00196 0.47)) = 20.2 m/​s

So the ter­mi­nal ve­loc­ity of a 5 cm di­ame­ter bam­boo ball is about 20 m/​s. That is quite a way off your es­ti­mate of 34.3 and we got there with­out us­ing things like hol­low balls or aero­gel :-)

• To be fair, a light ball is ex­actly where my es­ti­mate is known to be least ac­cu­rate. Let’s con­sider, rather, a ball with a den­sity of 1 - one that nei­ther floats nor sinks in wa­ter. (Since, in my ex­pe­rience, many things sink in wa­ter and many, but not quite as many, things float in it, I think it makes a rea­son­able guess for the av­er­age den­sity of all pos­si­ble balls). Then you have m=0.0655kg, and thus:

v = sqrt( 2 0.0655 9.8 /​ (1.2 0.00196 0.47)) = 34.0785 m/​s

...okay, if it was fal­ling in a vac­uum it would have reached that speed, but it’s had air re­sis­tance all the way down, so it’s prob­a­bly not even close to that. (And it it had been dropped from, say, 240m, then I would have calcu­lated a value of close on 70 m/​s, which would have been even more wildly out).

So, I will ad­mit, it turns out that mass is a good deal more im­por­tant than I had ex­pected—also, air re­sis­tance has a larger effect than I had an­ti­ci­pated.

• while in the other limit, if the ball is a tiny black hole, I ex­pect it to smash into the ground at ex­actly the calcu­lated value

Nope, be­cause in that case, your value of g would be sig­nifi­cantly higher than 9.8 m/​s^2.

• Wikipe­dia has a longer ver­sion of the thought ex­per­i­ment.

• The prob­lem is, we’ve done much, MUCH more stringent tests than this. It’s like, af­ter check­ing the be­hav­ior of pen­du­lums and fal­ling ob­jects of vary­ing weights and lengths and ar­eas, over vast spans of time and all re­gions of the globe, and in cen­trifuges, and on pul­leys… we went on to then check if two iden­ti­cal ob­jects would fall at the same speed if we dropped one when the other landed.

Any­way, I didn’t say it shouldn’t be done. I sup­port ba­sic ex­per­i­ments on QM, but I’d like them to push the en­velope in in­ter­est­ing ways rather than, well, this.

• Karl Pop­per alread dealy with the prob­lem of Oc­cam’s Ra­zor not be­ing us­able based on com­plex­ity. He re­cast it as pre­dic­tive util­ity. When one does that, the pre­dic­tion of Many Wor­lds is untestable in pin­ci­ple—IE not ever wrong.