Math is Sub­junct­ively Objective

Fol­lowup to: Prob­ab­il­ity is Sub­ject­ively Ob­ject­ive, Can Coun­ter­fac­tu­als Be True?

I am quite con­fid­ent that the state­ment 2 + 3 = 5 is true; I am far less con­fid­ent of what it means for a math­em­at­ical state­ment to be true.

In “The Simple Truth” I defined a pebble-and-bucket sys­tem for track­ing sheep, and defined a con­di­tion for whether a bucket’s pebble level is “true” in terms of the sheep. The bucket is the be­lief, the sheep are the real­ity. I be­lieve 2 + 3 = 5. Not just that two sheep plus three sheep equal five sheep, but that 2 + 3 = 5. That is my be­lief, but where is the real­ity?

So now the one comes to me and says: “Yes, two sheep plus three sheep equals five sheep, and two stars plus three stars equals five stars. I won’t deny that. But this no­tion that 2 + 3 = 5, ex­ists only in your ima­gin­a­tion, and is purely sub­ject­ive.

So I say: Ex­cuse me, what?

And the one says: “Well, I know what it means to ob­serve two sheep and three sheep leave the fold, and five sheep come back. I know what it means to press ‘2’ and ‘+’ and ‘3’ on a cal­cu­lator, and see the screen flash ‘5’. I even know what it means to ask someone ‘What is two plus three?’ and hear them say ‘Five.’ But you in­sist that there is some fact bey­ond this. You in­sist that 2 + 3 = 5.”

Well, it kinda is.

“Per­haps you just mean that when you men­tally visu­al­ize adding two dots and three dots, you end up visu­al­iz­ing five dots. Per­haps this is the con­tent of what you mean by say­ing, 2 + 3 = 5. I have no trouble with that, for brains are as real as sheep.”

No, for it seems to me that 2 + 3 equaled 5 be­fore there were any hu­mans around to do ad­di­tion. When hu­mans showed up on the scene, they did not make 2 + 3 equal 5 by vir­tue of think­ing it. Rather, they thought that ‘2 + 3 = 5’ be­cause 2 + 3 did in fact equal 5.

“Prove it.”

I’d love to, but I’m busy; I’ve got to, um, eat a salad.

“The reason you be­lieve that 2 + 3 = 5, is your men­tal visu­al­iz­a­tion of two dots plus three dots yield­ing five dots. Does this not im­ply that this phys­ical event in your phys­ical brain is the mean­ing of the state­ment ‘2 + 3 = 5’?”

But I hon­estly don’t think that is what I mean. Sup­pose that by an amaz­ing cos­mic co­in­cid­ence, a flurry of neut­ri­nos struck my neur­ons, caus­ing me to ima­gine two dots col­lid­ing with three dots and visu­al­ize six dots. I would then say, ‘2 + 3 = 6’. But this wouldn’t mean that 2 + 3 ac­tu­ally had be­come equal to 6. Now, if what I mean by ‘2 + 3’ con­sists en­tirely of what my mere phys­ical brain merely hap­pens to out­put, then a neut­rino could make 2 + 3 = 6. But you can’t change arith­metic by tam­per­ing with a cal­cu­lator.

“Aha! I have you now!”

Is that so?

“Yes, you’ve given your whole game away!”

Do tell.

“You visu­al­ize a sub­junct­ive world, a coun­ter­fac­tual, where your brain is struck by neut­ri­nos, and says, ‘2 + 3 = 6’. So you know that in this case, your fu­ture self will say that ‘2 + 3 = 6’. But then you add up dots in your own, cur­rent brain, and your cur­rent self gets five dots. So you say: ‘Even if I be­lieved “2 + 3 = 6″, then 2 + 3 would still equal 5.’ You say: ‘2 + 3 = 5 re­gard­less of what any­one thinks of it.’ So your cur­rent brain, com­put­ing the same ques­tion while it ima­gines be­ing dif­fer­ent but is not ac­tu­ally dif­fer­ent, finds that the an­swer seems to be the same. Thus your brain cre­ates the il­lu­sion of an ad­di­tional real­ity that ex­ists out­side it, in­de­pend­ent of any brain.”

Now hold on! You’ve ex­plained my be­lief that 2 + 3 = 5 re­gard­less of what any­one thinks, but that’s not the same as ex­plain­ing away my be­lief. Since 2 + 3 = 5 does not, in fact, de­pend on what any hu­man be­ing thinks of it, there­fore it is right and proper that when I ima­gine coun­ter­fac­tual worlds in which people (in­clud­ing my­self) think ‘2 + 3 = 6’, and I ask what 2 + 3 ac­tu­ally equals in this coun­ter­fac­tual world, it still comes out as 5.

“Don’t you see, that’s just like try­ing to visu­al­ize mo­tion stop­ping every­where in the uni­verse, by ima­gin­ing your­self as an ob­server out­side the uni­verse who ex­per­i­ences time passing while noth­ing moves. But really there is no time without mo­tion.”

I see the ana­logy, but I’m not sure it’s a deep ana­logy. Not everything you can ima­gine see­ing, doesn’t ex­ist. It seems to me that a brain can eas­ily com­pute quant­it­ies that don’t de­pend on the brain.

What? Of course everything that the brain com­putes de­pends on the brain! Everything that the brain com­putes, is com­puted in­side the brain!”

That’s not what I mean! I just mean that the brain can per­form com­pu­ta­tions that refer to quant­it­ies out­side the brain. You can set up a ques­tion, like ‘How many sheep are in the field?’, that isn’t about any par­tic­u­lar per­son’s brain, and whose ac­tual an­swer doesn’t de­pend on any par­tic­u­lar per­son’s brain. And then a brain can faith­fully com­pute that an­swer.

If I count two sheep and three sheep re­turn­ing from the field, and Autrey’s brain gets hit by neut­ri­nos so that Autrey thinks there are six sheep in the fold, then that’s not go­ing to cause there to be six sheep in the fold—right? The whole ques­tion here is just not about what Autrey thinks, it’s about how many sheep are in the fold.

Why should I care what my sub­junct­ive fu­ture self thinks is the sum of 2 + 3, any more than I care what Autrey thinks is the sum of 2 + 3, when it comes to ask­ing what is really the sum of 2 + 3?

“Okay… I’ll take an­other tack. Sup­pose you’re a psy­chi­at­rist, right? And you’re an ex­pert wit­ness in court cases—ba­sic­ally a hired gun, but you try to de­ceive your­self about it. Now wouldn’t it be a bit sus­pi­cious, to find your­self say­ing: ‘Well, the only reason that I in fact be­lieve that the de­fend­ant is in­sane, is be­cause I was paid to be an ex­pert psy­chi­at­ric wit­ness for the de­fense. And if I had been paid to wit­ness for the pro­sec­u­tion, I un­doubtedly would have come to the con­clu­sion that the de­fend­ant is sane. But my be­lief that the de­fend­ant is in­sane, is per­fectly jus­ti­fied; it is jus­ti­fied by my ob­ser­va­tion that the de­fend­ant used his own blood to paint an Elder Sign on the wall of his jail cell.’”

Yes, that does sound sus­pi­cious, but I don’t see the point.

“My point is that the phys­ical cause of your be­lief that 2 + 3 = 5, is the phys­ical event of your brain visu­al­iz­ing two dots and three dots and com­ing up with five dots. If your brain came up six dots, due to a neut­rino storm or whatever, you’d think ‘2 + 3 = 6’. How can you pos­sibly say that your be­lief means any­thing other than the num­ber of dots your brain came up with?”

Now hold on just a second. Let’s say that the psy­chi­at­rist is paid by the judge, and when he’s paid by the judge, he renders an hon­est and neut­ral eval­u­ation, and his eval­u­ation is that the de­fend­ant is sane, just played a bit too much Mythos. So it is true to say that if the psy­chi­at­rist had been paid by the de­fense, then the psy­chi­at­rist would have found the de­fend­ant to be in­sane. But that doesn’t mean that when the psy­chi­at­rist is paid by the judge, you should dis­miss his eval­u­ation as telling you noth­ing more than ‘the psy­chi­at­rist was paid by the judge’. On those oc­ca­sions where the psy­chi­at­rist is paid by the judge, his opin­ion var­ies with the de­fend­ant, and con­veys real evid­ence about the de­fend­ant.

“Okay, so now what’s your point?”

That when my brain is not be­ing hit by a neut­rino storm, it yields hon­est and in­form­at­ive evid­ence that 2 + 3 = 5.

“And if your brain was hit by a neut­rino storm, you’d be say­ing, ‘2 + 3 = 6 re­gard­less of what any­one thinks of it’. Which shows how re­li­able that line of reas­on­ing is.”

I’m not claim­ing that my say­ing ‘2 + 3 = 5 no mat­ter what any­one thinks’ rep­res­ents stronger nu­mer­ical evid­ence than my say­ing ‘2 + 3 = 5’. My say­ing the former just tells you some­thing ex­tra about my epi­stem­o­logy, not num­bers.

“And you don’t think your epi­stem­o­logy is, oh, a little… in­co­her­ent?”

No! I think it is per­fectly co­her­ent to sim­ul­tan­eously hold all of the fol­low­ing:

  • 2 + 3 = 5.

  • If neut­ri­nos make me be­lieve “2 + 3 = 6”, then 2 + 3 = 5.

  • If neut­ri­nos make me be­lieve “2 + 3 = 6”, then I will say “2 + 3 = 6″.

  • If neut­ri­nos make me be­lieve that “2 + 3 = 6”, then I will there­after as­sert that “If neut­ri­nos make me be­lieve ‘2 + 3 = 5’, then 2 + 3 = 6″.

  • The cause of my think­ing that “2 + 3 = 5 in­de­pend­ently of what any­one thinks” is that my cur­rent mind, when it sub­junct­ively re­com­putes the value of 2 + 3 un­der the as­sump­tion that my ima­gined self is hit by neut­ri­nos, does not see the ima­gined self’s be­liefs as chan­ging the dots, and my cur­rent brain just visu­al­izes two dots plus three dots, as be­fore, so that the ima­gin­a­tion of my cur­rent brain shows the same res­ult.

  • If I were ac­tu­ally hit by neut­ri­nos, my brain would com­pute a dif­fer­ent res­ult, and I would as­sert “2 + 3 = 6 in­de­pend­ently of what any­one thinks.”

  • 2 + 3 = 5 in­de­pend­ently of what any­one thinks.

  • Since 2 + 3 will in fact go on equal­ing 5 re­gard­less of what I ima­gine about it or how my brain visu­al­izes cases where my fu­ture self has dif­fer­ent be­liefs, it’s a good thing that my ima­gin­a­tion doesn’t visu­al­ize the res­ult as de­pend­ing on my be­liefs.

“Now that’s just crazy talk!”

No, you’re the crazy one! You’re col­lapsing your levels; you think that just be­cause my brain asks a ques­tion, it should start mix­ing up quer­ies about the state of my brain into the ques­tion. Not every ques­tion my brain asks is about my brain!

Just be­cause some­thing is com­puted in my brain, doesn’t mean that my com­pu­ta­tion has to de­pend on my brain’s rep­res­ent­a­tion of my brain. It cer­tainly doesn’t mean that the ac­tual quant­ity de­pends on my brain! It’s my brain that com­putes my be­liefs about grav­ity, and if neut­ri­nos hit me I will come to a dif­fer­ent con­clu­sion; but that doesn’t mean that I can think dif­fer­ent and fly. And I don’t think I can think dif­fer­ent and fly, either!

I am not a cal­cu­lator who, when someone presses my “2” and “+” and “3” but­tons, com­putes, “What do I out­put when someone presses 2 + 3?” I am a cal­cu­lator who com­putes “What is 2 + 3?” The former is a cir­cu­lar ques­tion that can con­sist­ently re­turn any an­swer—which makes it not very help­ful.

Shouldn’t we ex­pect non-cir­cu­lar ques­tions to be the nor­mal case? The brain evolved to guess at the state of the en­vir­on­ment, not guess at ‘what the brain will think is the state of the en­vir­on­ment’. Even when the brain mod­els it­self, it is try­ing to know it­self, not try­ing to know what it will think about it­self.

Judg­ments that de­pend on our rep­res­ent­a­tions of any­one’s state of mind, like “It’s okay to kiss someone only if they want to be kissed”, are the ex­cep­tion rather than the rule.

Most quant­it­ies we bother to think about at all, will ap­pear to be ‘the same re­gard­less of what any­one thinks of them’. When we ima­gine think­ing dif­fer­ently about the quant­ity, we will ima­gine the quant­ity com­ing out the same; it will feel “sub­junct­ively ob­ject­ive”.

And there’s noth­ing wrong with that! If some­thing ap­pears to be the same re­gard­less of what any­one thinks, then maybe that’s be­cause it ac­tu­ally is the same re­gard­less of what any­one thinks.

Even if you ex­plain that the quant­ity ap­pears to stay the same in my ima­gin­a­tion, merely be­cause my cur­rent brain com­putes it the same way—well, how else would I ima­gine some­thing, ex­cept with my cur­rent brain? Should I ima­gine it us­ing a rock?

“Okay, so it’s pos­sible for some­thing that ap­pears thought-in­de­pend­ent, to ac­tu­ally be thought-in­de­pend­ent. But why do you think that 2 + 3 = 5, in par­tic­u­lar, has some kind of ex­ist­ence in­de­pend­ently of the dots you ima­gine?”

Be­cause two sheep plus three sheep equals five sheep, and this ap­pears to be true in every moun­tain and every is­land, every swamp and every plain and every forest.

And moreover, it is also true of two rocks plus three rocks.

And fur­ther, when I press but­tons upon a cal­cu­lator and ac­tiv­ate a net­work of tran­sist­ors, it suc­cess­fully pre­dicts how many sheep or rocks I will find.

Since all these quant­it­ies, cor­rel­ate with each other and suc­cess­fully pre­dict each other, surely they must have some­thing like a com­mon cause, a sim­il­ar­ity that factors out? So­mething that is true bey­ond and be­fore the con­crete ob­ser­va­tions? So­mething that the con­crete ob­ser­va­tions hold in com­mon? And this com­mon­al­ity is then also the spon­sor of my an­swer, ‘five’, that I find in my own brain.

“But my dear sir, if the fact of 2 + 3 = 5 ex­ists some­where out­side your brain… then where is it?”

Damned if I know.

Part of The Metaethics Sequence

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