Perpetual Motion Beliefs

Yes­ter­day’s post con­cluded:

To form ac­cu­rate be­liefs about some­thing, you re­ally do have to ob­serve it. It’s a very phys­i­cal, very real pro­cess: any ra­tio­nal mind does “work” in the ther­mo­dy­namic sense, not just the sense of men­tal effort… So un­less you can tell me which spe­cific step in your ar­gu­ment vi­o­lates the laws of physics by giv­ing you true knowl­edge of the un­seen, don’t ex­pect me to be­lieve that a big, elab­o­rate clever ar­gu­ment can do it ei­ther.

One of the chief morals of the math­e­mat­i­cal anal­ogy be­tween ther­mo­dy­nam­ics and cog­ni­tion is that the con­straints of prob­a­bil­ity are in­escapable; prob­a­bil­ity may be a “sub­jec­tive state of be­lief”, but the laws of prob­a­bil­ity are harder than steel.

Peo­ple learn un­der the tra­di­tional school reg­i­men that the teacher tells you cer­tain things, and you must be­lieve them and re­cite them back; but if a mere stu­dent sug­gests a be­lief, you do not have to obey it. They map the do­main of be­lief onto the do­main of au­thor­ity, and think that a cer­tain be­lief is like an or­der that must be obeyed, but a prob­a­bil­is­tic be­lief is like a mere sug­ges­tion.

They look at a lot­tery ticket, and say, “But you can’t prove I won’t win, right?” Mean­ing: “You may have calcu­lated a low prob­a­bil­ity of win­ning, but since it is a prob­a­bil­ity, it’s just a sug­ges­tion, and I am al­lowed to be­lieve what I want.”

Here’s a lit­tle ex­per­i­ment: Smash an egg on the floor. The rule that says that the egg won’t spon­ta­neously re­form and leap back into your hand is merely prob­a­bil­is­tic. A sug­ges­tion, if you will. The laws of ther­mo­dy­nam­ics are prob­a­bil­is­tic, so they can’t re­ally be laws, the way that “Thou shalt not mur­der” is a law… right?

So why not just ig­nore the sug­ges­tion? Then the egg will un­scram­ble it­self… right?

It may help to think of it this way—if you still have some lin­ger­ing in­tu­ition that un­cer­tain be­liefs are not au­thor­i­ta­tive:

In re­al­ity, there may be a very small chance that the egg spon­ta­neously re­forms. But you can­not ex­pect it to re­form. You must ex­pect it to smash. Your manda­tory be­lief is that the egg’s prob­a­bil­ity of spon­ta­neous re­for­ma­tion is ~0. Prob­a­bil­ities are not cer­tain­ties, but the laws of prob­a­bil­ity are the­o­rems.

If you doubt this, try drop­ping an egg on the floor a few decillion times, ig­nor­ing the ther­mo­dy­namic sug­ges­tion and ex­pect­ing it to spon­ta­neously re­assem­ble, and see what hap­pens. Prob­a­bil­ities may be sub­jec­tive states of be­lief, but the laws gov­ern­ing them are stronger by far than steel.

I once knew a fel­low who was con­vinced that his sys­tem of wheels and gears would pro­duce re­ac­tion­less thrust, and he had an Ex­cel spread­sheet that would prove this—which of course he couldn’t show us be­cause he was still de­vel­op­ing the sys­tem. In clas­si­cal me­chan­ics, vi­o­lat­ing Con­ser­va­tion of Mo­men­tum is prov­ably im­pos­si­ble. So any Ex­cel spread­sheet calcu­lated ac­cord­ing to the rules of clas­si­cal me­chan­ics must nec­es­sar­ily show that no re­ac­tion­less thrust ex­ists—un­less your ma­chine is com­pli­cated enough that you have made a mis­take in the calcu­la­tions.

And similarly, when half-trained or tenth-trained ra­tio­nal­ists aban­don their art and try to be­lieve with­out ev­i­dence just this once, they of­ten build vast ed­ifices of jus­tifi­ca­tion, con­fus­ing them­selves just enough to con­ceal the mag­i­cal steps.

It can be quite a pain to nail down where the magic oc­curs—their struc­ture of ar­gu­ment tends to morph and squirm away as you in­ter­ro­gate them. But there’s always some step where a tiny prob­a­bil­ity turns into a large one—where they try to be­lieve with­out ev­i­dence—where they step into the un­known, think­ing, “No one can prove me wrong”.

Their foot nat­u­rally lands on thin air, for there is far more thin air than ground in the realms of Pos­si­bil­ity. Ah, but there is an (ex­po­nen­tially tiny) amount of ground in Pos­si­bil­ity, and you do have an (ex­po­nen­tially tiny) prob­a­bil­ity of hit­ting it by luck, so maybe this time, your foot will land in the right place! It is merely a prob­a­bil­ity, so it must be merely a sug­ges­tion.

The ex­act state of a glass of boiling-hot wa­ter may be un­known to you—in­deed, your ig­no­rance of its ex­act state is what makes the molecules’ ki­netic en­ergy “heat”, rather than work wait­ing to be ex­tracted like the mo­men­tum of a spin­ning fly­wheel. So the wa­ter might cool down your hand in­stead of heat­ing it up, with prob­a­bil­ity ~0.

De­cide to ig­nore the laws of ther­mo­dy­nam­ics and stick your hand in any­way, and you’ll get burned.

“But you don’t know that!”

I don’t know it with cer­tainty, but it is manda­tory that I ex­pect it to hap­pen. Prob­a­bil­ities are not log­i­cal truths, but the laws of prob­a­bil­ity are.

“But what if I guess the state of the boiling wa­ter, and I hap­pen to guess cor­rectly?”

Your chance of guess­ing cor­rectly by luck, is even less than the chance of the boiling wa­ter cool­ing your hand by luck.

“But you can’t prove I won’t guess cor­rectly.”

I can (in­deed, must) as­sign ex­tremely low prob­a­bil­ity to it.

“That’s not the same as cer­tainty, though.”

Hey, maybe if you add enough wheels and gears to your ar­gu­ment, it’ll turn warm wa­ter into elec­tric­ity and ice cubes! Or, rather, you will no longer see why this couldn’t be the case.

“Right! I can’t see why couldn’t be the case! So maybe it is!”

Another gear? That just makes your ma­chine even less effi­cient. It wasn’t a per­pet­ual mo­tion ma­chine be­fore, and each ex­tra gear you add makes it even less effi­cient than that.

Each ex­tra de­tail in your ar­gu­ment nec­es­sar­ily de­creases the joint prob­a­bil­ity. The prob­a­bil­ity that you’ve vi­o­lated the Se­cond Law of Ther­mo­dy­nam­ics with­out know­ing ex­actly how, by guess­ing the ex­act state of boiling wa­ter with­out ev­i­dence, so that you can stick your finger in with­out get­ting burned, is, nec­es­sar­ily, even less than the prob­a­bil­ity of stick­ing in your finger into boiling wa­ter with­out get­ting burned.

I say all this, be­cause peo­ple re­ally do con­struct these huge ed­ifices of ar­gu­ment in the course of be­liev­ing with­out ev­i­dence. One must learn to see this as analo­gous to all the wheels and gears that fel­low added onto his re­ac­tion­less drive, un­til he fi­nally col­lected enough com­pli­ca­tions to make a mis­take in his Ex­cel spread­sheet.