# Logical Pinpointing

Fol­lowup to: Causal Refer­ence, Proofs, Im­pli­ca­tions and Models

The fact that one ap­ple added to one ap­ple in­vari­ably gives two ap­ples helps in the teach­ing of ar­ith­metic, but has no bear­ing on the truth of the propo­si­tion that 1 + 1 = 2.

-- James R. New­man, The World of Mathematics

Pre­vi­ous med­i­ta­tion 1: If we can only mean­ingfully talk about parts of the uni­verse that can be pinned down by chains of cause and effect, where do we find the fact that 2 + 2 = 4? Or did I just make a mean­ingless noise, there? Or if you claim that “2 + 2 = 4”isn’t mean­ingful or true, then what al­ter­nate prop­erty does the sen­tence “2 + 2 = 4″ have which makes it so much more use­ful than the sen­tence “2 + 2 = 3”?

Pre­vi­ous med­i­ta­tion 2: It has been claimed that logic and math­e­mat­ics is the study of which con­clu­sions fol­low from which premises. But when we say that 2 + 2 = 4, are we re­ally just as­sum­ing that? It seems like 2 + 2 = 4 was true well be­fore any­one was around to as­sume it, that two ap­ples equalled two ap­ples be­fore there was any­one to count them, and that we couldn’t make it 5 just by as­sum­ing differ­ently.

Speak­ing con­ven­tional English, we’d say the sen­tence 2 + 2 = 4 is “true”, and any­one who put down “false” in­stead on a math-test would be marked wrong by the schoolteacher (and not with­out jus­tice).

But what can make such a be­lief true, what is the be­lief about, what is the truth-con­di­tion of the be­lief which can make it true or al­ter­na­tively false? The sen­tence ‘2 + 2 = 4’ is true if and only if… what?

In the pre­vi­ous post I as­serted that the study of logic is the study of which con­clu­sions fol­low from which premises; and that al­though this sort of in­evitable im­pli­ca­tion is some­times called “true”, it could more speci­fi­cally be called “valid”, since check­ing for in­evita­bil­ity seems quite differ­ent from com­par­ing a be­lief to our own uni­verse. And you could claim, ac­cord­ingly, that “2 + 2 = 4” is ‘valid’ be­cause it is an in­evitable im­pli­ca­tion of the ax­ioms of Peano Arith­metic.

And yet think­ing about 2 + 2 = 4 doesn’t re­ally feel that way. Figur­ing out facts about the nat­u­ral num­bers doesn’t feel like the op­er­a­tion of mak­ing up as­sump­tions and then de­duc­ing con­clu­sions from them. It feels like the num­bers are just out there, and the only point of mak­ing up the ax­ioms of Peano Arith­metic was to al­low math­e­mat­i­ci­ans to talk about them. The Peano ax­ioms might have been con­ve­nient for de­duc­ing a set of the­o­rems like 2 + 2 = 4, but re­ally all of those the­o­rems were true about num­bers to be­gin with. Just like “The sky is blue” is true about the sky, re­gard­less of whether it fol­lows from any par­tic­u­lar as­sump­tions.

So com­par­i­son-to-a-stan­dard does seem to be at work, just as with phys­i­cal truth… and yet this no­tion of 2 + 2 = 4 seems differ­ent from “stuff that makes stuff hap­pen”. Num­bers don’t oc­cupy space or time, they don’t ar­rive in any or­der of cause and effect, there are no events in num­ber­land.

Med­i­ta­tion: What are we talk­ing about when we talk about num­bers? We can’t nav­i­gate to them by fol­low­ing causal con­nec­tions—so how do we get there from here?

...
...
...

“Well,” says the math­e­mat­i­cal lo­gi­cian, “that’s in­deed a very im­por­tant and in­ter­est­ing ques­tion—where are the num­bers—but first, I have a ques­tion for you. What are these ‘num­bers’ that you’re talk­ing about? I don’t be­lieve I’ve heard that word be­fore.”

Yes you have.

“No, I haven’t. I’m not a typ­i­cal math­e­mat­i­cal lo­gi­cian; I was just cre­ated five min­utes ago for the pur­poses of this con­ver­sa­tion. So I gen­uinely don’t know what num­bers are.”

But… you know, 0, 1, 2, 3...

“I don’t rec­og­nize that 0 thingy—what is it? I’m not ask­ing you to give an ex­act defi­ni­tion, I’m just try­ing to figure out what the heck you’re talk­ing about in the first place.”

Um… okay… look, can I start by ask­ing you to just take on faith that there are these thin­gies called ‘num­bers’ and 0 is one of them?

“Of course! 0 is a num­ber. I’m happy to be­lieve that. Just to check that I un­der­stand cor­rectly, that does mean there ex­ists a num­ber, right?”

Um, yes. And then I’ll ask you to be­lieve that we can take the suc­ces­sor of any num­ber. So we can talk about the suc­ces­sor of 0, the suc­ces­sor of the suc­ces­sor of 0, and so on. Now 1 is the suc­ces­sor of 0, 2 is the suc­ces­sor of 1, 3 is the suc­ces­sor of 2, and so on in­definitely, be­cause we can take the suc­ces­sor of any num­ber -

“In other words, the suc­ces­sor of any num­ber is also a num­ber.”

Ex­actly.

“And in a sim­ple case—I’m just try­ing to vi­su­al­ize how things might work—we would have 2 equal to 0.”

What? No, why would that be -

“I was vi­su­al­iz­ing a case where there were two num­bers that were the suc­ces­sors of each other, so SS0 = 0. I mean, I could’ve vi­su­al­ized one num­ber that was the suc­ces­sor of it­self, but I didn’t want to make things too triv­ial—”

No! That model you just drew—that’s not a model of the num­bers.

“Why not? I mean, what prop­erty do the num­bers have that this model doesn’t?”

Be­cause, um… zero is not the suc­ces­sor of any num­ber. Your model has a suc­ces­sor link from 1 to 0, and that’s not al­lowed.

“I see! So we can’t have SS0=0. But we could still have SSS0=S0.”

What? How -

No! Be­cause -

(con­sults text­book)

- if two num­bers have the same suc­ces­sor, they are the same num­ber, that’s why! You can’t have 2 and 0 both hav­ing 1 as a suc­ces­sor un­less they’re the same num­ber, and if 2 was the same num­ber as 0, then 1′s suc­ces­sor would be 0, and that’s not al­lowed! Be­cause 0 is not the suc­ces­sor of any num­ber!

“I see. Oh, wow, there’s an awful lot of num­bers, then. The first chain goes on for­ever.”

It sounds like you’re start­ing to get what I—wait. Hold on. What do you mean, the first chain -

“I mean, you said that there was at least one start of an in­finite chain, called 0, but—”

I mis­spoke. Zero is the only num­ber which is not the suc­ces­sor of any num­ber.

“I see, so any other chains would ei­ther have to loop or go on for­ever in both di­rec­tions.”

Wha?

“You said that zero is the only num­ber which is not the suc­ces­sor of any num­ber, that the suc­ces­sor of ev­ery num­ber is a num­ber, and that if two num­bers have the same suc­ces­sor they are the same num­ber. So, fol­low­ing those rules, any suc­ces­sor-chains be­sides the one that start at 0 have to loop or go on for­ever in both di­rec­tions—”

There aren’t sup­posed to be any chains be­sides the one that starts at 0! Argh! And now you’re go­ing to ask me how to say that there shouldn’t be any other chains, and I’m not a math­e­mat­i­cian so I can’t figure out ex­actly how to -

“Hold on! Calm down. I’m a math­e­mat­i­cian, af­ter all, so I can help you out. Like I said, I’m not try­ing to tor­ment you here, just un­der­stand what you mean. You’re right that it’s not triv­ial to for­mal­ize your state­ment that there’s only one suc­ces­sor-chain in the model. In fact, you can’t say that at all in­side what’s called first-or­der logic. You have to jump to some­thing called sec­ond-or­der logic that has some re­mark­ably differ­ent prop­er­ties (ha ha!) and make the state­ment there.”

What the heck is sec­ond-or­der logic?

“It’s the logic of prop­er­ties! First-or­der logic lets you quan­tify over all ob­jects—you can say that all ob­jects are red, or all ob­jects are blue, or ‘x: red(x)→¬blue(x)‘, and so on. Now, that ‘red’ and ‘blue’ we were just talk­ing about—those are prop­er­ties, func­tions which, ap­plied to any ob­ject, yield ei­ther ‘true’ or ‘false’. A prop­erty di­vides all ob­jects into two classes, a class in­side the prop­erty and a com­ple­men­tary class out­side the prop­erty. So ev­ery­thing in the uni­verse is ei­ther blue or not-blue, red or not-red, and so on. And then sec­ond-or­der logic lets you quan­tify over prop­er­ties—in­stead of look­ing at par­tic­u­lar ob­jects and ask­ing whether they’re blue or red, we can talk about prop­er­ties in gen­eral—quan­tify over all pos­si­ble ways of sort­ing the ob­jects in the uni­verse into classes. We can say, ‘For all prop­er­ties P’, not just, ‘For all ob­jects X’.”

Okay, but what does that have to do with say­ing that there’s only one chain of suc­ces­sors?

“To say that there’s only one chain, you have to make the jump to sec­ond-or­der logic, and say that for all prop­er­ties P, if P be­ing true of a num­ber im­plies P be­ing true of the suc­ces­sor of that num­ber, and P is true of 0, then P is true of all num­bers.”

Um… huh. That does sound rem­i­nis­cent of some­thing I re­mem­ber hear­ing about Peano Arith­metic. But how does that solve the prob­lem with chains of suc­ces­sors?

“Be­cause if you had an­other sep­a­rated chain, you could have a prop­erty P that was true all along the 0-chain, but false along the sep­a­rated chain. And then P would be true of 0, true of the suc­ces­sor of any num­ber of which it was true, and not true of all num­bers.”

I… huh. That’s pretty neat, ac­tu­ally. You thought of that pretty fast, for some­body who’s never heard of num­bers.

“Thank you! I’m an imag­i­nary fic­tion­al­ized rep­re­sen­ta­tion of a very fast math­e­mat­i­cal rea­soner.”

Any­way, the next thing I want to talk about is ad­di­tion. First, sup­pose that for ev­ery x, x + 0 = x. Next sup­pose that if x + y = z, then x + Sy = Sz -

“There’s no need for that. We’re done.”

What do you mean, we’re done?

“Every num­ber has a suc­ces­sor. If two num­bers have the same suc­ces­sor, they are the same num­ber. There’s a num­ber 0, which is the only num­ber that is not the suc­ces­sor of any other num­ber. And ev­ery prop­erty true at 0, and for which P(Sx) is true when­ever P(x) is true, is true of all num­bers. In com­bi­na­tion, those premises nar­row down a sin­gle model in math­e­mat­i­cal space, up to iso­mor­phism. If you show me two mod­els match­ing these re­quire­ments, I can perfectly map the ob­jects and suc­ces­sor re­la­tions in them. You can’t add any new ob­ject to the model, or sub­tract an ob­ject, with­out vi­o­lat­ing the ax­ioms you’ve already given me. It’s a uniquely iden­ti­fied math­e­mat­i­cal col­lec­tion, the ob­jects and their struc­ture com­pletely pinned down. Ergo, there’s no point in adding any more re­quire­ments. Any mean­ingful state­ment you can make about these ‘num­bers’, as you’ve defined them, is already true or already false within that pin­pointed model—its truth-value is already se­man­ti­cally im­plied by the ax­ioms you used to talk about ‘num­bers’ as op­posed to some­thing else. If the new ax­iom is already true, adding it won’t change what the pre­vi­ous ax­ioms se­man­ti­cally im­ply.”

Whoa. But don’t I have to define the + op­er­a­tion be­fore I can talk about it?

“Not in sec­ond-or­der logic, which can quan­tify over re­la­tions as well as prop­er­ties. You just say: ‘For ev­ery re­la­tion R that works ex­actly like ad­di­tion, the fol­low­ing state­ment Q is true about that re­la­tion.’ It would look like, ‘ re­la­tions R: (∀x∀y∀z: (R(x, 0, z)↔(x=z)) ∧ (R(x, Sy, z)↔R(Sx, y, z))) → Q)’, where Q says what­ever you meant to say about +, us­ing the to­ken R. Oh, sure, it’s more con­ve­nient to add + to the lan­guage, but that’s a mere con­ve­nience—it doesn’t change which facts you can prove. Or to say it out­side the sys­tem: So long as I know what num­bers are, you can just ex­plain to me how to add them; that doesn’t change which math­e­mat­i­cal struc­ture we’re already talk­ing about.”

...Gosh. I think I see the idea now. It’s not that ‘ax­ioms’ are math­e­mat­i­ci­ans ask­ing for you to just as­sume some things about num­bers that seem ob­vi­ous but can’t be proven. Rather, ax­ioms pin down that we’re talk­ing about num­bers as op­posed to some­thing else.

“Ex­actly. That’s why the math­e­mat­i­cal study of num­bers is equiv­a­lent to the log­i­cal study of which con­clu­sions fol­low in­evitably from the num­ber-ax­ioms. When you for­mal­ize logic into syn­tax, and prove the­o­rems like ‘2 + 2 = 4’ by syn­tac­ti­cally de­riv­ing new sen­tences from the ax­ioms, you can safely in­fer that 2 + 2 = 4 is se­man­ti­cally im­plied within the math­e­mat­i­cal uni­verse that the ax­ioms pin down. And there’s no way to try to ‘just study the num­bers with­out as­sum­ing any ax­ioms’, be­cause those ax­ioms are how you can talk about num­bers as op­posed to some­thing else. You can’t take for granted that just be­cause your mouth makes a sound ‘NUM-burz’, it’s a mean­ingful sound. The ax­ioms aren’t things you’re ar­bi­trar­ily mak­ing up, or as­sum­ing for con­ve­nience-of-proof, about some pre-ex­is­tent thing called num­bers. You need ax­ioms to pin down a math­e­mat­i­cal uni­verse be­fore you can talk about it in the first place. The ax­ioms are pin­ning down what the heck this ‘NUM-burz’ sound means in the first place—that your mouth is talk­ing about 0, 1, 2, 3, and so on.”

Could you also talk about uni­corns that way?

“I sup­pose. Uni­corns don’t ex­ist in re­al­ity—there’s noth­ing in the world that be­haves like that—but they could nonethe­less be de­scribed us­ing a con­sis­tent set of ax­ioms, so that it would be valid if not quite true to say that if a uni­corn would be at­tracted to Bob, then Bob must be a vir­gin. Some peo­ple might dis­pute whether uni­corns must be at­tracted to vir­gins, but since uni­corns aren’t real—since we aren’t lo­cat­ing them within our uni­verse us­ing a causal refer­ence—they’d just be talk­ing about differ­ent mod­els, rather than ar­gu­ing about the prop­er­ties of a known, fixed math­e­mat­i­cal model. The ‘ax­ioms’ aren’t mak­ing ques­tion­able guesses about some real phys­i­cal uni­corn, or even a math­e­mat­i­cal uni­corn-model that’s already been pin­pointed; they’re just fic­tional premises that make the word ‘uni­corn’ talk about some­thing in­side a story.”

But when I put two ap­ples into a bowl, and then put in an­other two ap­ples, I get four ap­ples back out, re­gard­less of any­thing I as­sume or don’t as­sume. I don’t need any ax­ioms at all to get four ap­ples back out.

“Well, you do need ax­ioms to talk about four, SSSS0, when you say that you got ‘four’ ap­ples back out. That said, in­deed your ex­pe­rienced out­come—what your eyes see—doesn’t de­pend on what ax­ioms you as­sume. But that’s be­cause the ap­ples are be­hav­ing like num­bers whether you be­lieve in num­bers or not!”

The ap­ples are be­hav­ing like num­bers? What do you mean? I thought num­bers were this ethe­real math­e­mat­i­cal model that got pin­pointed by ax­ioms, not by look­ing at the real world.

“When­ever a part of re­al­ity be­haves in a way that con­forms to the num­ber-ax­ioms—for ex­am­ple, if putting ap­ples into a bowl obeys rules, like no ap­ple spon­ta­neously ap­pear­ing or van­ish­ing, which yields the high-level be­hav­ior of num­bers—then all the math­e­mat­i­cal the­o­rems we proved valid in the uni­verse of num­bers can be im­ported back into re­al­ity. The con­clu­sion isn’t ab­solutely cer­tain, be­cause it’s not ab­solutely cer­tain that no­body will sneak in and steal an ap­ple and change the phys­i­cal bowl’s be­hav­ior so that it doesn’t match the ax­ioms any more. But so long as the premises are true, the con­clu­sions are true; the con­clu­sion can’t fail un­less a premise also failed. You get four ap­ples in re­al­ity, be­cause those ap­ples be­hav­ing nu­mer­i­cally isn’t some­thing you as­sume, it’s some­thing that’s phys­i­cally true. When two clouds col­lide and form a big­ger cloud, on the other hand, they aren’t be­hav­ing like in­te­gers, whether you as­sume they are or not.”

But if the awe­some hid­den power of math­e­mat­i­cal rea­son­ing is to be im­ported into parts of re­al­ity that be­have like math, why not rea­son about ap­ples in the first place in­stead of these ethe­real ‘num­bers’?

“Be­cause you can prove once and for all that in any pro­cess which be­haves like in­te­gers, 2 thin­gies + 2 thin­gies = 4 thin­gies. You can store this gen­eral fact, and re­call the re­sult­ing pre­dic­tion, for many differ­ent places in­side re­al­ity where phys­i­cal things be­have in ac­cor­dance with the num­ber-ax­ioms. More­over, so long as we be­lieve that a calcu­la­tor be­haves like num­bers, press­ing ‘2 + 2’ on a calcu­la­tor and get­ting ‘4’ tells us that 2 + 2 = 4 is true of num­bers and then to ex­pect four ap­ples in the bowl. It’s not like any­thing fun­da­men­tally differ­ent from that is go­ing on when we try to add 2 + 2 in­side our own brains—all the in­for­ma­tion we get about these ‘log­i­cal mod­els’ is com­ing from the ob­ser­va­tion of phys­i­cal things that allegedly be­have like their ax­ioms, whether it’s our neu­rally-pat­terned thought pro­cesses, or a calcu­la­tor, or ap­ples in a bowl.”

I… think I need to con­sider this for a while.

“Be my guest! Oh, and if you run out of things to think about from what I’ve said already—”

Hold on.

“—try pon­der­ing this one. Why does 2 + 2 come out the same way each time? Never mind the ques­tion of why the laws of physics are sta­ble—why is logic sta­ble? Of course I can’t imag­ine it be­ing any other way, but that’s not an ex­pla­na­tion.”

Are you sure you didn’t just de­gen­er­ate into talk­ing bloody non­sense?

“Of course it’s bloody non­sense. If I knew a way to think about the ques­tion that wasn’t bloody non­sense, I would already know the an­swer.”

Hu­mans need fan­tasy to be hu­man.

“Tooth fairies? Hog­fathers? Lit­tle—”

Yes. As prac­tice. You have to start out learn­ing to be­lieve the lit­tle lies.

“So we can be­lieve the big ones?”

Yes. Jus­tice. Mercy. Duty. That sort of thing.

“They’re not the same at all!”

You think so? Then take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of jus­tice, one molecule of mercy.

- Su­san and Death, in Hog­father by Terry Pratchett

So far we’ve talked about two kinds of mean­ingful­ness and two ways that sen­tences can re­fer; a way of com­par­ing to phys­i­cal things found by fol­low­ing pinned-down causal links, and log­i­cal refer­ence by com­par­i­son to mod­els pinned-down by ax­ioms. Is there any­thing else that can be mean­ingfully talked about? Where would you find jus­tice, or mercy?

Main­stream sta­tus.

Part of the se­quence Highly Ad­vanced Episte­mol­ogy 101 for Beginners

Next post: “Causal Uni­verses

Pre­vi­ous post: “Proofs, Im­pli­ca­tions, and Models

• “—try pon­der­ing this one. Why does 2 + 2 come out the same way each time? Never mind the ques­tion of why the laws of physics are sta­ble—why is logic sta­ble? Of course I can’t imag­ine it be­ing any other way, but that’s not an ex­pla­na­tion.”

Noth­ing in the pro­cess de­scribed, of pin­point­ing the nat­u­ral num­bers, makes any refer­ence to time. That is why it is tem­po­rally sta­ble: not be­cause it has an on­go­ing ex­is­tence which is mys­te­ri­ously un­af­fected by the pas­sage of time, but be­cause time has no con­nec­tion with it. When­ever you look at it, it’s the same, iden­ti­cal thing, not a later, mirac­u­lously pre­served ver­sion of the thing.

• What if 2 + 2 varies over some­thing other than time that nonethe­less cor­re­lates with time in our uni­verse? Sup­pose 2 + 2 comes out to 4 the first 1 trillion times the op­er­a­tion is performed by hu­mans, and to 5 on the 1 trillion and first time.

I sup­pose you could raise the same ex­pla­na­tion: the defi­ni­tion of 2 + 2 makes no refer­ence to how many times it has been ap­plied. I be­lieve the same can be said for any other rea­son you may give for why 2 + 2 might cease to equal 4.

• Where that is the case, your method of map­ping from the re­al­ity to ar­ith­metic is not a good model of that pro­cess—no more, no less.

• I couldn’t agree more. The time­less­ness of maths should be read nega­tively, as in­de­pence of any­thing else, not as de­pen­dence on a time­less realm.

• I love the el­e­gance of this an­swer, up­vot­ing.

• “—try pon­der­ing this one. Why does 2 + 2 come out the same way each time? Never mind the ques­tion of why the laws of physics are sta­ble—why is logic sta­ble? Of course I can’t imag­ine it be­ing any other way, but that’s not an ex­pla­na­tion.”

I have re­cently had a thought rele­vant to the topic; an op­er­a­tion that is not sta­ble.

In cer­tain con­texts, the op­er­a­tion d is used, where XdY means “take a set of X fair dice, each die hav­ing Y sides (num­bered 1 to Y), and throw them; add to­gether the num­bers on the up­per­most faces”. Us­ing this defi­ni­tion, 2d2 has value ‘2’ 25% of the time, value ‘3’ 50% of the time, and value ‘4’ 25% of the time. The pro­ce­dure is always iden­ti­cal, and so there’s noth­ing in the pro­cess which makes any refer­ence to time, but the re­sult can differ (though note that ‘time’ is still not a pa­ram­e­ter in that re­sult). If the op­er­a­tion ‘+’ is re­placed by the op­er­a­tion ‘d’ - well, then that is one other way that can be imag­ined.

Edited to add: It has been pointed out that XdY is a con­stant prob­a­bil­ity dis­tri­bu­tion. The un­sta­ble op­er­a­tion to which I re­fer is the op­er­a­tion of tak­ing a sin­gle ran­dom in­te­ger sam­ple, in a fair man­ner, from that dis­tri­bu­tion.

• The ran­dom is not in the dice, it is in the throw, and that pro­ce­dure is never iden­ti­cal. Also, XdY is a dis­tri­bu­tion, always the same, and the dice are just a rel­a­tively fair way of pick­ing a sam­ple.

• Aren’t you just con­fus­ing dis­tri­bu­tions (2d2) and sam­ples (‘3’) here?

• But the ques­tion isn’t, “Why don’t they change over time,” but rather, “why are they the same on each oc­ca­sion”. It makes no refer­ence to oc­ca­sion? Sure, but even so, why doesn’t 2 + 2 = a ran­dom num­ber each time? Why is the same iden­ti­cal thing the same?

• I’m not sure what the eti­quette is of re­spond­ing to re­tracted com­ments, but I’ll have a go at this one.

Why is the same iden­ti­cal thing the same?

That’s what I mean when I say they are iden­ti­cal. It’s not an­other, sep­a­rate thing, ex­ist­ing on a sep­a­rate oc­ca­sion, dis­tinct from the first but stand­ing in the re­la­tion of iden­tity to it. In math­e­mat­ics, you can step into the same river twice. Even aliens in dis­tant galax­ies step into the same river.

How­ever, there is some­thing else in­volved with the sta­bil­ity, which ex­ists in time, and which is ca­pa­ble of be­ing im­perfectly sta­ble: one­self. 2+2=4 is im­mutable, but my judge­ment that 2+2 equals 4 is muta­ble, be­cause I change over time. If it seems im­pos­si­ble to be­come con­fused about 2+2=4, just think of de­gen­er­a­tive brain dis­eases. Or be­ing asleep and dream­ing that 2+2 made 5.

• So the ques­tion be­comes, “If “2+2” is just an­other way of say­ing “4″, what is the point of hav­ing two ex­pres­sions for it?”

My an­swer: As hu­mans, we of­ten de­sire to split a group of large, dis­tinct ob­jects into smaller groups of large, dis­tinct ob­jects, or to put two smaller groups of large, dis­tinct, ob­jects, to­gether. So, when we say “2 + 2 = 4”, what we are re­ally ex­press­ing is that a group of 4 ob­jects can be trans­formed into a group of 2 ob­jects and an­other group of 2 ob­jects, by mov­ing the ob­jects apart (and vice versa). Shar­ing re­sources with fel­low hu­mans is fun­da­men­tal to hu­man in­ter­ac­tion. The rea­son I say, “large, dis­tinct ob­jects” is that the rules of ad­di­tion do not hold for ev­ery­thing. For ex­am­ple, when you add “1” par­ti­cle of mat­ter to “1″ par­ti­cle of an­ti­mat­ter, you get “0” par­ti­cles of both mat­ter and an­ti­mat­ter.

Num­bers, and, yes, even logic, only ex­ist fun­da­men­tally in the mind. They are good de­scrip­tions that cor­re­spond to re­al­ity. The sound­ness the­o­rem for logic (which is not prov­able in the same logic it is de­scribing) is what re­ally be­gins to hint at logic’s cor­re­spon­dence to the real world. The sound­ness the­o­rem re­lies on the fact that all of the ax­ioms are true and that in­fer­ence rules are truth-pre­serv­ing. The Peano ax­ioms and logic are use­ful be­cause, given the com­monly known mean­ing we as­sign to the sym­bols of those sys­tems, the ax­ioms do prop­erly de­scribe our ob­ser­va­tions of re­al­ity and the in­fer­ence rules do lead to con­clu­sions that con­tinue to cor­re­spond to our ob­ser­va­tions of re­al­ity (in (one of) the cor­rect do­main(s), groups of large, dis­tinct, ob­jects). We ob­serve that quan­tity is pre­served re­gard­less of group­ing; this is the as­so­ci­a­tive prop­erty (here’s an­other way of look­ing at it).

The math­e­mat­i­cal proof of the sound­ness the­o­rem is use­less for con­vinc­ing the hard skep­tic, be­cause it uses math­e­mat­i­cal in­duc­tion it­self! The prin­ci­ple of math­e­mat­i­cal in­duc­tion is called such be­cause it was for­mu­lated in­duc­tively. When it comes to the large num­bers, no one has ob­served these quan­tities. But, for all quan­tities we have ob­served so far, math­e­mat­i­cal in­duc­tion has held. We use de­duc­tion to ap­ply in­duc­tion, but that doesn’t make the in­duc­tion any less in­duc­tive to be­gin with. We use the real num­ber sys­tem to make pre­dic­tions in physics. If we have the lux­ury of mak­ing an ob­ser­va­tion, we should go ahead and up­date. For com­pa­nies with limited re­sources that are try­ing to de­velop a use­ful product to sell to make money, and even more so for Friendly AI (a mis­take could end hu­man civ­i­liza­tion), it’s nice to have a good idea of what an out­come will be be­fore it hap­pens. Bayes’ rule pro­vides a sys­tem­atic way of work­ing with this un­cer­tainty. Maybe, one day, when I put two ap­ples next to two ap­ples on my kitchen table, there will be five (the or­der in which I move the ap­ples around will af­fect their quan­tity), but, if I had to bet one way or the other, I as­sure you that my money is on this not hap­pen­ing.

• The pre­sen­ta­tion of the nat­u­ral num­bers is meant to be stan­dard, in­clud­ing the (well-known and proven) idea that it re­quires sec­ond-or­der logic to pin them down. There’s some fur­ther con­tro­versy about sec­ond-or­der logic which will be dis­cussed in a later post.

I’ve seen some (old) ar­gu­ments about the mean­ing of ax­io­m­a­tiz­ing which did not re­solve in the an­swer, “Be­cause oth­er­wise you can’t talk about num­bers as op­posed to some­thing else,” so AFAIK it’s the­o­ret­i­cally pos­si­ble that I’m the first to spell out that idea in ex­actly that way, but it’s an ob­vi­ous-enough idea and there’s been enough de­bate by philo­soph­i­cally in­clined math­e­mat­i­ci­ans that I would be gen­uinely sur­prised to find this was the case.

On the other hand, I’ve surely never seen a gen­eral ac­count of mean­ingful­ness which puts log­i­cal pin­point­ing alongside causal link-trac­ing to delineate two differ­ent kinds of cor­re­spon­dence within cor­re­spon­dence the­o­ries of truth. To what­ever ex­tent any of this is a stan­dard po­si­tion, it’s not nearly widely-known enough or ex­plic­itly taught in those terms to gen­eral math­e­mat­i­ci­ans out­side model the­ory and math­e­mat­i­cal logic, just like the stan­dard po­si­tion on “proof”. Nor does any of it ap­pear in the S. E. P. en­try on mean­ing.

• Very nice post!

Bug: Higher-or­der logic (a stan­dard term) means “in­finite-or­der logic” (not a stan­dard term), not “logic of or­der greater 1″ (also not a stan­dard term). (For what­ever rea­son, nei­ther the Wikipe­dia nor the SEP en­try seem to come out and say this, but ev­ery refer­ence I can re­mem­ber used the terms like that, and the us­age in SEP seems to im­ply it too, e.g. “This sec­ond-or­der ex­press­ibil­ity of the power-set op­er­a­tion per­mits the simu­la­tion of higher-or­der logic within sec­ond or­der.”)

• A few points:

i) you don’t ac­tu­ally need to jump di­rectly to sec­ond or­der logic in to get a cat­e­gor­i­cal ax­iom­a­ti­za­tion of the nat­u­ral num­bers. There are sev­eral weaker ways to do the job: L_omega_omega (which al­lows in­fini­tary con­junc­tions), adding a prim­i­tive finite­ness op­er­a­tor, adding a prim­i­tive an­ces­tral op­er­a­tor, al­low­ing the omega rule (i.e. from the in­finitely many premises P(0), P(1), … P(n), … in­fer AnP(n)). Se­cond or­der logic is more pow­er­ful than these in that it gives a quasi cat­e­gor­i­cal ax­iom­a­ti­za­tion of the uni­verse of sets (i.e. of any two mod­els of ZFC_2, they are ei­ther iso­mor­phic or one is iso­mor­phic to an ini­tial seg­ment of the other).

ii) al­though there is a minor­ity view to the con­trary, it’s typ­i­cally thought that go­ing sec­ond or­der doesn’t help with de­ter­mi­nate­ness wor­ries (i.e. roughly what you are talk­ing about with re­gard to “pin­ning down” the nat­u­ral num­bers). The point here is that go­ing sec­ond or­der only works if you in­ter­pret the sec­ond or­der quan­tifiers “fully”, i.e. as rang­ing over the whole power set of the do­main rather than some proper sub­set of it. But the prob­lem is: how can we rule out non-full in­ter­pre­ta­tions of the quan­tifiers? This seems like just the same sort of prob­lem as rul­ing out non-stan­dard mod­els of ar­ith­metic (“the same sort”, not the same, be­cause for the rea­sons men­tioned in (i) it is ac­tu­ally more stringent of a con­di­tion.) The point is if you for some rea­son doubt that we have a cat­e­gor­i­cal grasp of the nat­u­ral num­bers, you are cer­tainly not go­ing to grant that we can en­force a full in­ter­pre­ta­tion of the sec­ond or­der quan­tifiers. And al­though it seems in­tu­itively ob­vi­ous that we have a cat­e­gor­i­cal grasp of the nat­u­ral num­bers, care­ful con­sid­er­a­tion of the first in­com­plete­ness the­o­rem shows that this is by no means clear.

iii) Given that cat­e­goric­ity re­sults are only up to iso­mor­phism, I don’t see how they help you pin down talk of the nat­u­ral num­bers them­selves (as op­posed to any old omega_se­quence). At best, they help you pin down the struc­ture of the nat­u­ral num­bers, but tak­ing this in­sight into ac­count is eas­ier said than done.

• iii) Given that cat­e­goric­ity re­sults are only up to iso­mor­phism, I don’t see how they help you pin down talk of the nat­u­ral num­bers them­selves (as op­posed to any old omega_se­quence). At best, they help you pin down the struc­ture of the nat­u­ral num­bers, but tak­ing this in­sight into ac­count is eas­ier said than done.

Gen­er­ally, things be­ing iden­ti­cal up to iso­mor­phism is con­sid­ered to make them the same thing in all senses that mat­ter. If some­thing has all the same prop­er­ties as the nat­u­ral num­bers, in ev­ery re­spect and ev­ery par­tic­u­lar, then that’s no differ­ent from merely chang­ing the names. This is a pretty ba­sic math­e­mat­i­cal con­cept, and that you aren’t fa­mil­iar with it makes me ques­tion the rest of this com­ment as well.

• I think philoso­phers who think that the cat­e­goric­ity of sec­ond-or­der Peano ar­ith­metic al­lows us to re­fer to the nat­u­ral num­bers uniquely tend to also re­ject the causal the­ory of refer­ence, pre­cisely be­cause the causal the­ory of refer­ence is usu­ally put as re­quiring all refer­ence to be causally guided. Among those, lots of peo­ple more-or-less think that refer­ences can be fixed by some kinds of de­scrip­tion, and I think log­i­cal de­scrip­tions of this kind would be pretty un­con­tro­ver­sial.

OTOH, for some rea­son ev­ery­one in philos­o­phy of maths is aller­gic to sec­ond-or­der logic (blame Quine), so the cat­e­goric­ity ar­gu­ment doesn’t always hold wa­ter. For some dis­cus­sion, there’s a sec­tion in the SEP en­try on Philos­o­phy of Math­e­mat­ics.

(To give one of the rea­sons why peo­ple don’t like SOL: to in­ter­pret it fully you seem to need set the­ory. Prop­er­ties ba­si­cally be­have like sets, and so you can make SOL state­ments that are valid iff the Con­tinuum Hy­poth­e­sis is true, for ex­am­ple. It seems wrong that logic should de­pend on set the­ory in this way.)

• This is a fa­cepalm “Duh” mo­ment, I hear this crit­i­cism all the time but it does not mean that “logic” de­pends on “set the­ory”. There is a con­fu­sion here be­tween what can be STATED and what can be KNOWN. The crit­i­cism only has any force if you think that all “log­i­cal truths” ought to be rec­og­niz­able so that they can be effec­tively enu­mer­ated. But the crit­ics don’t mind that for any effec­tive enu­mer­a­tion of the­o­rems of ar­ith­metic, there are true state­ments about in­te­gers that won’t be in­cluded—we can’t KNOW all the true facts about in­te­gers, so the crit­i­cism of sec­ond-or­der logic boils down to say­ing that you don’t like us­ing the word “logic” to be ap­plied to any sys­tem pow­er­ful enough to EXPRESS quan­tified state­ments about the in­te­gers, but only to sys­tems weak enough that all their con­se­quences can be enu­mer­ated.

This de­mand is un­rea­son­able. Even if logic is only about “cor­rect rea­son­ing”, the usual frame­work given by SOL does not pre­sume any du­bi­ous prin­ci­ples of rea­son­ing and ZF proves its con­sis­tency. The ex­is­tence of propo­si­tions which are not de­duc­tively set­tled by that frame­work but which can be given math­e­mat­i­cal in­ter­pre­ta­tions means noth­ing more than that our reper­toire of “tech­niques of cor­rect rea­son­ing”, which has grown over the cen­turies, isn’t nec­es­sar­ily fi­nal­ized.

• What about Steven Lands­burg’s fre­quent crow­ing on the Pla­ton­ic­ity of math and how num­bers are real be­cause we can “di­rectly per­ceive them”? How does this re­late to it?

EDIT: Well, he replies here.

While I greatly sym­pa­thize with the “Pla­ton­ic­ity of math”, I can’t shake the idea that my rea­son­ing about num­bers isn’t any kind of di­rect per­cep­tion, but just rea­son­ing about an in-mem­ory rep­re­sen­ta­tion of a model that is ul­ti­mately based on all the other sys­tems that be­have like num­bers.

I find the ar­gu­ments about how not all true state­ments re­gard­ing the nat­u­ral num­bers can be in­ferred via first-or­der logic te­dious. It doesn’t seem like our un­der­stand­ing of the nat­u­ral num­bers is par­tic­u­larly im­pov­er­ished be­cause of it.

• “Be­cause oth­er­wise you can’t talk about num­bers as op­posed to some­thing else,”

The Ab­stract Alge­bra course I took pre­sented it in this fash­ion. I have a hard time see­ing how you could even have ab­stract alge­bra with­out this no­tion.

• so AFAIK it’s the­o­ret­i­cally pos­si­ble that I’m the first to spell out that idea in ex­actly that way

I re­mem­ber ex­plain­ing the Ax­iom of Choice in this way to a fel­low un­der­grad­u­ate on my in­te­gra­tion the­ory course in late 2000. But of course it never oc­curred to me to write it down, so you only have my word for this :-)

• 2 Nov 2012 2:37 UTC
0 points
Parent

This post definitely de­serves a lot of credit.

• I’ve seen some (old) ar­gu­ments about the mean­ing of ax­io­m­a­tiz­ing which did not re­solve in the an­swer, “Be­cause oth­er­wise you can’t talk about num­bers as op­posed to some­thing else,” so AFAIK it’s the­o­ret­i­cally pos­si­ble that I’m the first to spell out that idea in ex­actly that way, but it’s an ob­vi­ous-enough idea and there’s been enough de­bate by philo­soph­i­cally in­clined math­e­mat­i­ci­ans that I would be gen­uinely sur­prised to find this was the case.

If mem­ory serves, Hofs­tadter uses roughly this ex­pla­na­tion in GEB.

• This is pretty close to how I re­mem­ber the dis­cus­sion in GEB. He has a good dis­cus­sion of non-Eu­clidean ge­om­e­try. He em­pha­sizes that origi­nally the nega­tion of Par­allel Pos­tu­late was viewed as ab­surd, but that now we can un­der­stand that the non-Eu­clidean ax­ioms are perfectly rea­son­able state­ments which de­scribe some­thing other than plane ge­om­e­try we are used to. Later he has a bit of a dis­cus­sion of what a model of PA + NOT(CON(PA)) would look like. I re­mem­ber find­ing it pretty con­fus­ing, and I didn’t re­ally know what he was get­ting at un­til I red some ac­tual logic the­ory text­books. But he did get across the idea that the ax­ioms would still de­scribe some­thing, but that some­thing would be larger and stranger than the in­te­gers we think we know.

• ???

IRC, Hofs­tadter is a firm for­mal­ist, and I don’t see how that square with EYs ap­par­ent Cor­re­spon­dence The­ory. At least i don’t see the point in cor­re­spon­dence if hat is be­ing cor­re­sponded to is it­self gen­er­ated by ax­ioms.

• Thanks for post­ing this. My in­tended com­ments got pretty long, so I con­verted them to a blog post here. The gist is that I don’t think you’ve solved the prob­lem, partly be­cause sec­ond or­der logic is not logic (as ex­plained in my post) and partly be­cause you are rely­ing on a the­o­rem (that sec­ond or­der Peano ar­ith­metic has a unique model) which re­lies on set the­ory, so you have “solved” the prob­lem of what it means for num­bers to be “out there” only by re­duc­ing it to the ques­tion of what it means for sets to be “out there”, which is, if any­thing, a greater mys­tery.

• You just say: ‘For ev­ery re­la­tion R that works ex­actly like ad­di­tion, the fol­low­ing state­ment S is true about that re­la­tion.’ It would look like, ‘∀ re­la­tions R: (∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)→R(x, Sy, Sz))) → S)’, where S says what­ever you meant to say about +, us­ing the to­ken R.

The ex­pres­sion ‘(∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)→R(x, Sy, Sz)))’ is true for ad­di­tion, but also for many other re­la­tions, such as a ‘∀x∀y∀z: R(x, y, z)’ re­la­tion.

• I’m not sure that adding the con­junc­tion (R(x,y,z)&R(x,y,w)->z=w) would have made things clearer...I thought it was ob­vi­ous the hy­po­thet­i­cal math­e­mat­i­cian was just ex­plain­ing what kind of steps you need to “taboo ad­di­tion”

• Yes, the ed­u­ca­tional goal of that para­graph is to “taboo ad­di­tion”. Nonethe­less, the taboo­ing should be done cor­rectly. If it is too difficult to do, then it is Eliezer’s prob­lem for choos­ing a difficult ex­am­ple to illus­trate a con­cept.

This may sound like nit­pick­ing, but this web­site has a goal is to teach peo­ple ra­tio­nal­ity skills, as op­posed to “guess­ing the teacher’s pass­word”. The ar­ti­cle spends five screens ex­plain­ing why de­tails are so im­por­tant when defin­ing the con­cept of a “num­ber”, and the reader is sup­posed to un­der­stand it. So it’s un­for­tu­nate if that ex­pla­na­tion is fol­lowed by an­other ex­am­ple, which ac­ci­den­tally gets the similar de­tails wrong. My ob­jec­tions against the wrong for­mula are very similar to the in-story math­e­mat­i­cian’s ob­jec­tions to the defi­ni­tions of “num­ber”; the defi­ni­tion is too wide.

Your sug­ges­tion: ‘∀x∀y∀z∀w: R(x, 0, x) ∧ (R(x, y, z)↔R(x, Sy, Sz)) ∧ ((R(x, y, z)∧R(x, y, w))→z=w)’

My al­ter­na­tive: ‘∀x∀y∀z: (R(x, 0, z)↔(x=z)) ∧ (R(x, y, z)↔R(x, Sy, Sz)) ∧ (R(x, y, z)↔R(Sx, y, Sz))’.

Both seem cor­rect, and any­one knows a shorter (or a more leg­ible) way to ex­press it, please con­tribute.

• Shorter (but not nec­es­sar­ily more leg­ible): ∀x∀y∀z: (R(x, 0, z)↔(x=z)) ∧ (R(x, Sy, z)↔R(Sx, y, z)).

• Done!

• Perfect!

• Both seem cor­rect, and any­one knows a shorter (or a more leg­ible) way to ex­press it, please con­tribute.

The ver­sion in the ar­ti­cle now, ∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)↔R(x, Sy, Sz)), is bet­ter than be­fore, but it leaves open the pos­si­bil­ity that R(0,0,7) as well as R(0,0,0). One more pos­si­bil­ity is:

“Not in sec­ond-or­der logic, which can quan­tify over func­tions as well as prop­er­ties. (...) It would look like, ‘∀ func­tions f: ((∀x∀y: f(x, 0) = x ∧ f(x, Sy) = Sf(x, y)) → Q)’ (...)”

(I guess I’m not en­tirely in fa­vor of this ver­sion—ETA: com­pared to Kindly’s fix—be­cause quan­tify­ing over re­la­tions surely seems like a smaller step from quan­tify­ing over prop­er­ties than does quan­tify­ing over func­tions, if you’re new to this, but still thought it might be worth point­ing out in a com­ment.)

• Your idea of pin­ning down the nat­u­ral num­bers us­ing sec­ond or­der logic is in­ter­est­ing, but I don’t think that it re­ally solves the prob­lem. In par­tic­u­lar, it shouldn’t be enough to con­vince a for­mal­ist that the two of you are talk­ing about the same nat­u­ral num­bers.

Even in sec­ond or­der PA, there will still be state­ments that are in­de­pen­dent of the ax­ioms, like “there doesn’t ex­ist a num­ber cor­re­spond­ing to a Godel en­cod­ing of a proof that 0=S0 un­der the ax­ioms of sec­ond or­der PA”. Thus un­less you are as­sum­ing full se­man­tics (i.e. that for any col­lec­tion of num­bers there is a cor­re­spond­ing prop­erty), there should be dis­tinct mod­els of sec­ond or­der PA for which the ve­rac­ity of the above state­ment differs.

Thus it seems to me that all you have done with your ap­peal to sec­ond or­der logic is to change my ques­tions about “what is a num­ber?” into ques­tions about “what is a prop­erty?” In any case, I’m still not to­tally con­vinced that it is pos­si­ble to pin down The Nat­u­ral Num­bers ex­actly.

• I’m as­sum­ing full se­man­tics for sec­ond-or­der logic (for any col­lec­tion of num­bers there is a cor­re­spond­ing prop­erty be­ing quan­tified over) so the ax­ioms have a se­man­tic model prov­ably unique up to iso­mor­phism, there are no non­stan­dard mod­els, the Com­plete­ness The­o­rem does not hold and some truths (like Godel’s G) are se­man­ti­cally en­tailed with­out be­ing syn­tac­ti­cally en­tailed, etc.

• OK then. As soon as you can ex­plain to me ex­actly what you mean when you say “for any col­lec­tion of num­bers there is a cor­re­spond­ing prop­erty be­ing quan­tified over”, I will be satis­fied. In par­tic­u­lar, what do you mean when you say “any col­lec­tion”?

• If you’re already fine with the al­ter­nat­ing quan­tifiers of first-or­der logic, I don’t see why al­low­ing branch­ing quan­tifiers would cause a prob­lem. I could de­scribe sec­ond or­der logic in terms of branch­ing quan­tifiers.

• Huh. That’s in­ter­est­ing. Are you say­ing that you can ac­tu­ally pin down The Nat­u­ral Num­bers ex­actly us­ing some “first or­der logic with branch­ing quan­tifiers”? If so, I would be in­ter­ested in see­ing it.

• Sure:

It is not the case that: there ex­ists a z such that for ev­ery x and x’, there ex­ists a y de­pend­ing only on x and a y’ de­pend­ing only on x’ such that Q(x,x’,y,y’,z) is true

where Q(x,x’,y,y’,z) is ((x=x’ ) → (y=y’ )) ∧ ((Sx=x’ ) → (y=y’ )) ∧ ((x=0) → (y=0)) ∧ ((x=z) → (y=1))

• Cool. I agree that this is po­ten­tially less prob­le­matic than the sec­ond or­der logic ap­proach. But it does still man­age to en­code the idea of a func­tion in it im­plic­itly when it talks about “y de­pend­ing only on x”, it es­sen­tially re­quires that y is a func­tion of x, and if it’s un­clear ex­actly which func­tions are al­lowed, you will have prob­lems. I guess first or­der logic has this prob­lem to some de­gree, but with al­ter­nat­ing quan­tifiers, the func­tions that you might need to define seem closer to the type that should nec­es­sar­ily ex­ist.

• Are you claiming that this term is am­bigu­ous? In what spe­cially fa­vored set the­ory, in what spe­cially fa­vored col­lec­tion of al­lowed mod­els, is it am­bigu­ous? Maybe the model of set the­ory I use has only one set of al­low­able ‘col­lec­tions of num­bers’ in which case the term isn’t am­bigu­ous. Now you could claim that other pos­si­ble mod­els ex­ist, I’d just like to know in what math­e­mat­i­cal lan­guage you’re claiming these other mod­els ex­ist. How do you as­sert the am­bi­guity of sec­ond-or­der logic with­out us­ing sec­ond-or­der logic to frame the sur­round­ing set the­ory in which it is am­bigu­ous?

• I’m not en­tirely sure what you’re get­ting at here. If we start re­strict­ing prop­er­ties to only cut out sets of num­bers rather than ar­bi­trary col­lec­tions, then we’ve already given up on full se­man­tics.

If we take this leap, then it is a the­o­rem of set the­ory that all set-the­o­retic mod­els of the of the nat­u­ral num­bers are iso­mor­phic. On the other hand, since not all state­ments about the in­te­gers can be ei­ther proven or dis­proven with the ax­ioms of set the­ory, there must be differ­ent mod­els of set the­ory which have differ­ent mod­els of the in­te­gers within them (in fact, I can build these two mod­els within a larger set the­ory).

On the other hand, if we con­tinue to use full se­man­tics, I’m not sure how you clar­ify to be what you mean when you say “a prop­erty ex­ists for ev­ery col­lec­tion of num­bers”. Tel­ling me that I should already know what a col­lec­tion is doesn’t seem much more rea­son­able than tel­ling me that I should already know what a nat­u­ral num­ber is.

• On the other hand, since not all state­ments about the in­te­gers can be ei­ther proven or dis­proven with the ax­ioms of set the­ory, there must be differ­ent mod­els of set the­ory which have differ­ent mod­els of the in­te­gers within them

Doesn’t the proof of the Com­plete­ness The­o­rem /​ Com­pact­ness The­o­rem in­ci­den­tally in­voke sec­ond-or­der logic it­self? (In the very quiet way that e.g. any as­sump­tion that the stan­dard in­te­gers even ex­ist in­vokes sec­ond-or­der logic.) I’m not sure but I would ex­pect it to, since oth­er­wise the no­tion of a “con­sis­tent” the­ory is en­tirely de­pen­dent on which mod­els your set the­ory says ex­ist and which proofs your in­te­ger the­ory says ex­ist. Per­haps my fa­vorite model of set the­ory has only one model of set the­ory, so I think that only one model ex­ists. Can you prove to me that there are other mod­els with­out in­vok­ing sec­ond-or­der logic im­plic­itly or ex­plic­itly in any called-on lemma? Keep in mind that all math­e­mat­i­ci­ans speak sec­ond-or­der logic as English, so check­ing that all proofs are first-or­der doesn’t seem easy.

• I am ad­mit­tedly in a lit­tle out of my depth here, so the fol­low­ing could rea­son­ably be wrong, but I be­lieve that the Com­pact­ness The­o­rem can be proved within first or­der set the­ory. Given a con­sis­tent the­ory, I can use the ax­iom of choice to ex­tend it to a max­i­mal con­sis­tent set of state­ments (i.e. so that for ev­ery P ei­ther P or (not P) is in my set). Then for ev­ery state­ment that I have of the form “there ex­ists x such that P(x)”, I in­tro­duce an el­e­ment x to my model and add P(x) to my list of true state­ments. I then re-ex­tend to a max­i­mal set of state­ments, and add new vari­ables as nec­es­sary, un­til I can­not do this any longer. What I am left with is a model for my the­ory. I don’t think I in­voked sec­ond or­der logic any­where here. In par­tic­u­lar, what I did amounts to a con­struc­tion within set the­ory. I sup­pose it is the case that some set the­o­ries will have no mod­els of set the­ory (be­cause they prove that set the­ory is in­con­sis­tent), while oth­ers will con­tain in­finitely many.

My in­tu­ition on the mat­ter is that if you can state what you are try­ing to say with­out sec­ond or­der logic, you should be able to prove it with­out sec­ond or­der logic. You need sec­ond or­der logic to even make sense of the idea of the stan­dard nat­u­ral num­bers. The Com­pact­ness The­o­rem can be stated in first or­der set the­ory, so I ex­pect the proof to be for­mal­iz­able within first or­der set the­ory.

• I’m not en­tirely sure what you’re get­ting at here. If we start re­strict­ing prop­er­ties to only cut out sets of num­bers rather than ar­bi­trary col­lec­tions, then we’ve already given up on full se­man­tics.

If we take this leap, then it is a the­o­rem of set the­ory that all set-the­o­retic mod­els of the of the nat­u­ral num­bers are iso­mor­phic. On the other hand, since not all state­ments about the in­te­gers can be ei­ther proven or dis­proven with the ax­ioms of set the­ory, there must be differ­ent mod­els of set the­ory which have differ­ent mod­els of the in­te­gers within them (in fact if you give me an in­ac­cessible car­di­nal, I build these two mod­els within a larger set the­ory).

On the other hand, if we con­tinue to use full se­man­tics, I’m not sure how you clar­ify to be what you mean when you say “a prop­erty ex­ists for ev­ery col­lec­tion of num­bers”. Tel­ling me that I should already know what a col­lec­tion is doesn’t seem much more rea­son­able than tel­ling me that I should already know what a nat­u­ral num­ber is.

• I think this is his way of con­nect­ing num­bers to the pre­vi­ous posts. If “a prop­erty” is defined as a causal re­la­tion, which all prop­er­ties are, then I think this makes sense. It doesn’t provide some sort of ul­ti­mate meta­phys­i­cal jus­tifi­ca­tion for num­bers or prop­er­ties or any­thing, but it clar­ifies con­nec­tions be­tween the two and such a jus­tifi­ca­tion isn’t re­ally pos­si­ble any­ways.

• I don’t think that I un­der­stand what you mean here.

How can these prop­er­ties rep­re­sent causal re­la­tions? They are things that are satis­fied by some num­bers and not by oth­ers. Since num­bers are aphys­i­cal, how do we re­late this to causal re­la­tions.

On the other hand, even with a satis­fac­tory an­swer to the above ques­tion, how do we know that “be­ing in the first chain” is ac­tu­ally a prop­erty, since oth­er­wise we still can’t show that there is only one chain.

• Since num­bers are aphys­i­cal, how do we re­late this to causal re­la­tions?

You just begged the ques­tion. Eliezer an­swered you in the OP:

Be­cause you can prove once and for all that in any pro­cess which be­haves like in­te­gers, 2 thin­gies + 2 thin­gies = 4 thin­gies. You can store this gen­eral fact, and re­call the re­sult­ing pre­dic­tion, for many differ­ent places in­side re­al­ity where phys­i­cal things be­have in ac­cor­dance with the num­ber-ax­ioms. More­over, so long as we be­lieve that a calcu­la­tor be­haves like num­bers, press­ing ‘2 + 2’ on a calcu­la­tor and get­ting ‘4’ tells us that 2 + 2 = 4 is true of num­bers and then to ex­pect four ap­ples in the bowl. It’s not like any­thing fun­da­men­tally differ­ent from that is go­ing on when we try to add 2 + 2 in­side our own brains—all the in­for­ma­tion we get about these ‘log­i­cal mod­els’ is com­ing from the ob­ser­va­tion of phys­i­cal things that allegedly be­have like their ax­ioms, whether it’s our neu­rally-pat­terned thought pro­cesses, or a calcu­la­tor, or ap­ples in a bowl.

• I can’t think of an ex­am­ple, but I’m think­ing that if a prop­erty ex­isted then it would be a causal re­la­tion. A prop­erty wouldn’t rep­re­sent a causal re­la­tion, it would be one. I wasn’t think­ing math­e­mat­i­cally but in­stead in terms of a more com­mon­place un­der­stand­ing of prop­er­ties as things like red and yel­low and blue.

The ar­gu­ment made by the sim­ple idea of truth might be a way to get us from phys­i­cal states (which are causal re­la­tions) to num­bers. If you be­lieve that count­ing sheep is a valid op­er­a­tion, then quan­tify­ing color also seems fine. The rea­son I spoke in terms of causal re­la­tions is be­cause I be­lieve un­der­stand­ing qual­ities as causal re­la­tions be­tween things al­lows us to de­duce prop­er­ties about things through a com­bi­na­tion of Sal­monoff In­duc­tion and the method de­scribed in this post.

Are you ques­tion­ing the idea that num­bers or prop­er­ties are a qual­ity about ob­jects? If so, what are they?

I’m feel­ing con­fused though. If the defi­ni­tion of prop­erty used here doesn’t con­nect to or means some­thing com­pletely differ­ent than facts about ob­jects, then I’m way off base. I might also be off base for other rea­sons. Not sure.

• I am ques­tion­ing the idea that num­bers (at least the things that this post refers to as num­bers) are a qual­ity about ob­jects. Num­bers, as they are de­scribed here, are an ab­stract log­i­cal con­struc­tion.

• How come we never see any­thing phys­i­cal that be­haves like any of of the non-stan­dard mod­els of first or­der PA? Given that’s the case, it seems like we can com­mu­ni­cate the idea of num­bers to other hu­mans or even aliens by say­ing “the only model of first or­der PA that ever shows up in re­al­ity”, so we don’t need sec­ond or­der logic (or the other log­i­cal ideas men­tioned in the com­ments) just to talk about the nat­u­ral num­bers?

• The nat­u­ral num­bers are sup­posed to be what you get if you start count­ing from 0. If you start count­ing from 0 in a non­stan­dard model of PA you can’t get to any of the non­stan­dard bits, but first-or­der logic just isn’t ex­pres­sive enough to al­low you to talk about “the set of all things that I get if I start count­ing from 0.” This is what al­lows non­stan­dard mod­els to ex­ist, but they ex­ist only in a some­what del­i­cate math­e­mat­i­cal sense and there’s no rea­son that you should ex­pect any phys­i­cal phe­nomenon cor­re­spond­ing to them.

If I wanted to com­mu­ni­cate the idea of num­bers to aliens, I don’t think I would even talk about logic. I would just start count­ing with what­ever was available, e.g. if I had two rocks to smash to­gether I’d smash the rocks to­gether once, then twice, etc. If the aliens don’t get it by the time I’ve smashed the rocks to­gether, say, ten times, then they’re ei­ther so bad at in­duc­tion or so un­fa­mil­iar with count­ing that we prob­a­bly can’t mean­ingfully com­mu­ni­cate with them any­way.

• The Pirahã are un­fa­mil­iar with count­ing and we still can kind-of mean­ingfully com­mu­ni­cate with them. I agree with the rest of the com­ment, though.

• I was ready to re­ply “bul­lshit”, but I guess if their lan­guage doesn’t have any car­di­nal or or­di­nal num­ber terms …

Still, they could count with beads or rocks, à la the magic sheep-count­ing bucket.

It’s un­der­stand­able why they wouldn’t re­ally need count­ing given their lifestyle. But I won­der what they do (or did) when a neigh­bor­ing tribe at­tacks or en­croaches on their ter­ri­tory? Their lan­guage ap­par­ently does have words for ‘small amount’ and ‘large amount’, but how would they de­cide how many war­riors to send to meet an op­pos­ing band?

• Still, they could count with beads or rocks, à la the magic sheep-count­ing bucket.

Here’s a de­cent ar­gu­ment that they prob­a­bly don’t have words for num­bers be­cause they don’t count, rather than the other way round, con­tra pop-Whorfi­anism. (Other­wise I guess they’d just bor­row the words for num­bers from Por­tuguese or some­thing, as they prob­a­bly did with per­sonal pro­nouns from Tupi.)

• This is what al­lows non­stan­dard mod­els to ex­ist, but they ex­ist only in a some­what del­i­cate math­e­mat­i­cal sense and there’s no rea­son that you should ex­pect any phys­i­cal phe­nomenon cor­re­spond­ing to them.

Is it just co­in­ci­dence that these non­stan­dard mod­els don’t show up any­where in the em­piri­cal sci­ences, but real num­bers and com­plex num­bers do? I’m won­der­ing if there is some sort of deeper rea­son… Maybe you were hint­ing at some­thing by “del­i­cate”?

If I wanted to com­mu­ni­cate the idea of num­bers to aliens, I don’t think I would even talk about logic.

Good point. I guess I was try­ing to make the point that Eliezer seems a bit ob­sessed with log­i­cal pin­point­ing (aka cat­e­goric­ity) in this post. (“You need ax­ioms to pin down a math­e­mat­i­cal uni­verse be­fore you can talk about it in the first place.”) Be­fore we achieved cat­e­goric­ity, we already knew what math­e­mat­i­cal struc­ture we wanted to talk about, and af­ter­wards, it’s still use­ful to add more ax­ioms if we want to prove more the­o­rems.

• Is it just co­in­ci­dence that these non­stan­dard mod­els don’t show up any­where in the em­piri­cal sci­ences, but real num­bers and com­plex num­bers do?

The pro­cess by which the con­cepts “nat­u­ral /​ real /​ com­plex num­bers” vs. “non­stan­dard mod­els of PA” were gen­er­ated is very differ­ent. In the first case, math­e­mat­i­ci­ans were try­ing to model var­i­ous as­pects of the world around them (e.g. count­ing and physics). In the sec­ond case, math­e­mat­i­ci­ans were try­ing to pin­point some­thing else they already un­der­stood and ended up not quite get­ting it be­cause of log­i­cal sub­tleties.

I’m not sure how to ex­plain what I mean by “del­i­cate.” It roughly means “un­likely to have been in­de­pen­dently in­vented by alien math­e­mat­i­ci­ans.” In or­der for alien math­e­mat­i­ci­ans to in­de­pen­dently in­vent the no­tion of a non­stan­dard model of PA, they would have to have in­de­pen­dently de­cided that writ­ing down the first-or­der Peano ax­ioms is a good idea, and I just don’t find this all that likely. On the other hand, there are var­i­ous routes alien math­e­mat­i­ci­ans might take to­wards in­de­pen­dently in­vent­ing the com­plex num­bers, such as figur­ing out quan­tum me­chan­ics.

Be­fore we achieved cat­e­goric­ity, we already knew what math­e­mat­i­cal struc­ture we wanted to talk about, and af­ter­wards, it’s still use­ful to add more ax­ioms if we want to prove more the­o­rems.

I guess Eliezer’s in­tended re­sponse here is some­thing like “but when you want to ex­plain to an AI what you mean by the nat­u­ral num­bers, you can’t just say The Things You Use To Count With, You Know, Those.”

• How come we never see any­thing phys­i­cal that be­haves like any of of the non-stan­dard mod­els of first or­der PA?

Umm… wouldn’t they be con­sid­ered “stan­dard” in this case? I.e. match­ing some real-world ex­pe­rience?

Let’s imag­ine a coun­ter­fac­tual world in which some of our “stan­dard” mod­els ap­pear non-stan­dard. For ex­am­ple, in a purely dis­crete world (like the one con­sist­ing solely of causal chains, as EY once sug­gested), con­ti­nu­ity would be a non-stan­dard ob­ject in­vented by math­e­mat­i­ci­ans. What makes con­ti­nu­ity “stan­dard” in our world is, dis­ap­point­ingly, our limited vi­sual acu­ity.

Another ex­am­ple: in a world simu­lated on a 32-bit in­te­ger ma­chine, nat­u­ral num­bers would be con­sid­ered non-stan­dard, given how all ac­tual num­bers wrap around af­ter 2^32-1.

Ex­er­cise for the reader: imag­ine a world where a cer­tain non-stan­dard model of first or­der PA would be viewed as stan­dard.

• This is ba­si­cally the theme of the next post in the se­quence. :)

• How come we never see any­thing phys­i­cal that be­haves like any of of the non-stan­dard mod­els of first or­der PA?

Qiaochu’s an­swer: be­cause PA isn’t unique. There are other (stronger/​weaker) ax­iom­a­ti­za­tions of nat­u­ral num­bers that would lead to other non­stan­dard mod­els. I don’t think that an­swer works, be­cause we don’t see non­stan­dard mod­els of these other the­o­ries ei­ther.

wedrifid’s an­swer: be­cause PA was de­signed to talk about nat­u­ral num­bers, not other things in re­al­ity that hu­mans can tell apart from nat­u­ral num­bers.

My an­swer: be­cause PA was de­signed to talk about nat­u­ral num­bers, and we prov­ably did a good job. PA has many mod­els, but only one com­putable model. Since re­al­ity seems to be com­putable, we don’t ex­pect to see non­stan­dard mod­els of PA in re­al­ity. (Though that leaves the mys­tery of whether/​why re­al­ity is com­putable.)

• So this is where (one of the in­spira­tions for) Eliezer’s meta-ethics comes from! :)

A quick re­fresher from a former com­ment:

Cog­ni­tivism: Yes, moral propo­si­tions have truth-value, but not all peo­ple are talk­ing about the same facts when they use words like “should”, thus cre­at­ing the illu­sion of dis­agree­ment.

… and now from this post:

Some peo­ple might dis­pute whether uni­corns must be at­tracted to vir­gins, but since uni­corns aren’t real—since we aren’t lo­cat­ing them within our uni­verse us­ing a causal refer­ence—they’d just be talk­ing about differ­ent mod­els, rather than ar­gu­ing about the prop­er­ties of a known, fixed math­e­mat­i­cal model.

(This lit­tle re­al­iza­tion also holds a key to re­solv­ing the last med­i­ta­tion, I sup­pose.)

I’ve heard peo­ple say the meta-ethics se­quence was more or less a failure since not that many peo­ple re­ally un­der­stood it, but if these last posts were taken as a pereq­ui­site read­ing, it would be at least a bit eas­ier to un­der­stand where Eliezer’s com­ing from.

• I’ve heard peo­ple say the meta-ethics se­quence was more or less a failure since not that many peo­ple re­ally un­der­stood it, but if these last posts were taken as a pereq­ui­site read­ing, it would be at least a bit eas­ier to un­der­stand where Eliezer’s com­ing from.

Agreed, and dis­ap­pointed that this com­ment was down­voted.

• Re­quest­ing feed­back:

“When­ever a part of re­al­ity be­haves in a way that con­forms to the num­ber-ax­ioms—for ex­am­ple, if putting ap­ples into a bowl obeys rules, like no ap­ple spon­ta­neously ap­pear­ing or van­ish­ing, which yields the high-level be­hav­ior of num­bers—then all the math­e­mat­i­cal the­o­rems we proved valid in the uni­verse of num­bers can be im­ported back into re­al­ity. The con­clu­sion isn’t ab­solutely cer­tain, be­cause it’s not ab­solutely cer­tain that no­body will sneak in and steal an ap­ple and change the phys­i­cal bowl’s be­hav­ior so that it doesn’t match the ax­ioms any more. But so long as the premises are true, the con­clu­sions are true; the con­clu­sion can’t fail un­less a premise also failed. You get four ap­ples in re­al­ity, be­cause those ap­ples be­hav­ing nu­mer­i­cally isn’t some­thing you as­sume, it’s some­thing that’s phys­i­cally true. When two clouds col­lide and form a big­ger cloud, on the other hand, they aren’t be­hav­ing like in­te­gers, whether you as­sume they are or not.”

This is ex­actly what I ar­gued and grounded back in this ar­ti­cle.

Speci­fi­cally, that the two premises:

1) rocks be­have iso­mor­phi­cally to num­bers, and
2) un­der the ax­ioms of num­bers, 2+2 = 4

jointly im­ply that adding two rocks to two rocks gets four rocks. (See the cute di­a­gram.)

And yet the re­sponse on that ar­ti­cle (which had an ar­ray of other im­pli­ca­tions and rec­on­cili­a­tions) was pretty nega­tive. What gives?

Fur­ther­more, in dis­cus­sions about this in per­son, Eliezer_Yud­kowsky has (IIRC and I’m pretty sure I do) in­voked the “hey, adding two ap­ples to two ap­ples gets four ap­ples” ar­gu­ment to jus­tify the truth of 2+2=4, in di­rect con­tra­dic­tion of the above point. What gives on that?

• This is a re­ally good post.

If I can bother your math­e­mat­i­cal lo­gi­cian for just a mo­ment...

Hey, are you con­scious in the sense of be­ing aware of your own aware­ness?

Also, now that Eliezer can’t eth­i­cally de­in­stan­ti­ate you, I’ve got a few more ques­tions =)

You’ve given a not-iso­mor­phic-to-num­bers model for all the pre­fixes of the ax­ioms. That said, I’m still not clear on why we need the sec­ond-to-last ax­iom (“Zero is the only num­ber which is not the suc­ces­sor of any num­ber.”) -- once you’ve got the fi­nal ax­iom (re­cur­sion), I can’t seem to vi­su­al­ize any not-iso­mor­phic-to-num­bers mod­els.

Also, how does one go about prov­ing that a par­tic­u­lar set of ax­ioms has all its mod­els iso­mor­phic? The fact that I can’t think of any al­ter­na­tives is (ob­vi­ously, given the above) not quite suffi­cient.

Oh, and I re­mem­ber this story some­body on LW told, there were these num­bers peo­ple talked about called...um, I’m just gonna call them mimsy num­bers, and one day this math­e­mat­i­cian comes to a sem­i­nar on mimsy num­bers and pre­sents a proof that all mimsy num­bers have the Jaber­wock prop­erty, and all the math­e­mat­i­ci­ans nod and de­clare it a very fine find­ing, and then the next week, he comes back, and pre­sents a proof that no mimsy num­bers have the Jaber­wock prop­erty, and then ev­ery­one sud­denly loses in­ter­est in mimsy num­bers...

Point be­ing, noth­ing here definitely jus­tifies think­ing that there are num­bers, be­cause some­one could come along to­mor­row and prove ~(2+2=4) and we’d be done talk­ing about “num­bers”. But I feel re­ally re­ally con­fi­dent that that won’t ever hap­pen and I’m not quite sure how to say whence this con­fi­dence. I think this might be similar to your last ques­tion, but it seems to dodge RichardKen­n­away’s ob­jec­tion.

• I’m still not clear on why we need the sec­ond-to-last ax­iom (“Zero is the only num­ber which is not the suc­ces­sor of any num­ber.”)

I guess it is not nec­es­sary. It was just an illus­tra­tion of a “quick fix”, which was later shown to be in­suffi­cient.

• You just say: ‘For ev­ery re­la­tion R that works ex­actly like ad­di­tion, the fol­low­ing state­ment S is true about that re­la­tion.’ It would look like, ‘∀ re­la­tions R: (∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)→R(x, Sy, Sz))) → S)’, where S says what­ever you meant to say about +, us­ing the to­ken R.

I would change the state­ment to be some­thing other than ‘S’, say ‘Q’, as S is already used for ‘suc­ces­sor’.

• First post in this se­quence that lives up to the stan­dard of the old clas­sics. Love it.

• Yeah, but I’ve found the pre­vi­ous posts much more use­ful for com­ing up with clear ex­pla­na­tions aimed at non-LWers, and I pre­sume they’d make a bet­ter in­tro­duc­tion to some of the core LW epistemic ra­tio­nal­ity than just throw­ing “The Sim­ple Truth” at them.

• It’s a pretty hard bal­ance to strike that’s prob­a­bly differ­ent for ev­ery­one, be­tween in­com­pre­hen­si­bil­ity and bor­ing­ness.

• I already more-or-less knew most of the stuff in the pre­vi­ous posts in this se­quences and still didn’t find them bor­ing.

• Agree. When I first read The Sim­ple Truth, I thought Eliezer was en­dors­ing prag­ma­tism over cor­re­spon­dence.

• I’m still won­der­ing what The Sim­ple Truth is about. My best guess is that it is a cri­tique of in­straw­man­tal­ism.

• 2 Nov 2012 3:54 UTC
2 points
Parent

In my opinion, Causal Di­a­grams and Causal Models is far su­pe­rior to Time­less Causal­ity.

I am not say­ing that there is any­thing wrong with “Time­less Causal­ity”, or any of Eliezer’s old posts, but this se­quence goes into enough depth of ex­pla­na­tion that even some­one who has not read the older se­quences on Less Wrong would have a good chance of un­der­stand­ing it.

• Terry Tao’s 2007 post on non­firstorder­iz­abil­ity and branch­ing quan­tifiers gives an in­ter­est­ing view of the bound­ary be­tween first- and sec­ond-or­der logic. Key quote:

Mov­ing on to a more com­pli­cated ex­am­ple, if Q(x,x’,y,y’) is a qua­ter­nary re­la­tion on four ob­jects x,x’,y,y’, then we can ex­press the statement

For ev­ery x and x’, there ex­ists a y de­pend­ing only on x and a y’ de­pend­ing on x and x’ such that Q(x,x’,y,y’) is true

...but it seems that one can­not express

For ev­ery x and x’, there ex­ists a y de­pend­ing only on x and a y’ de­pend­ing only on x’ such that Q(x,x’,y,y’) is true

in first or­der logic!

The post and com­ments give some well-known the­o­rems that turn out to rely on such “branch­ing quan­tifiers”, and an en­cod­ing of the pred­i­cate “there are in­finitely many X” which can­not be done in first-or­der logic.

• I think it’s worth men­tion­ing ex­plic­itly that the sec­ond-or­der ax­iom in­tro­duced is in­duc­tion.

• I’m a lit­tle con­fused as to which of two po­si­tions this is ad­vo­cat­ing:

1. Num­bers are real, se­ri­ous things, but the way that we pick them out is by hav­ing a cat­e­gor­i­cal set of ax­ioms. They’re in­ter­est­ing to talk about be­cause lots of things in the world be­have like them (to some de­gree).

2. Math­e­mat­i­cal talk is ac­tu­ally talk about what fol­lows from cer­tain ax­ioms. This is in­ter­est­ing to talk about be­cause lots of things obey the ax­ioms and so ex­hibit the the­o­rems (to some de­gree).

Both of these have some prob­lems. The first one re­quires you to have weird, non-phys­i­cal num­bery-things. Not only this, but they’re a spe­cial ex­cep­tion to the the­ory of refer­ence that’s been de­vel­oped so far, in that you can re­fer to them with­out hav­ing a causal con­nec­tion.

The sec­ond one (which is similar to what I my­self would es­pouse) doesn’t have this prob­lem, be­cause it’s just talk­ing about what fol­lows log­i­cally from other stuff, but you do then have to ex­plain why we seem to be talk­ing about num­bers. And also what peo­ple were do­ing talk­ing about ar­ith­metic be­fore they knew about the Peano ax­ioms. But the real bug­bear here is that you then can’t re­ally ex­plain logic as part of math­e­mat­ics. The usual ana­ly­i­sis of logic that we do in maths with the do­main, in­ter­pre­ta­tion, etc. can’t be the whole deal if we’re cash­ing out the math­e­mat­ics in terms of log­i­cal im­pli­ca­tion! You’ve got to say some­thing else about logic.

(I think the an­swer is, loosely, that

1. the “num­bers” we talk about are mainly fic­tional aides to us­ing the sys­tem, and

2. the situ­a­tion of pre-ax­iom speak­ers is much like that of English speak­ers who nonethe­less can’t ex­plain English gram­mar.

3. I have no idea what to say about logic! )

I’m cu­ri­ous which of these (or nei­ther) is the cor­rect in­ter­pre­ta­tion of the post, and if it’s one of them, what Eliezer’s an­swers are… but per­haps they’re com­ing in an­other post.

• I’m not sure ex­actly what Eliezer in­tends, but I’ll put in my two cents:

A proof is sim­ply a game of sym­bol ma­nipu­la­tion. You start with some sym­bols, say ‘(’, ‘)’, ‘¬’, ‘→’, ‘↔’, ‘∀’, ‘∃’, ‘P’, ‘Q’, ‘R’, ‘x’, ‘y’, and ‘z’. Call these sym­bols the alpha­bet. Some se­quences of sym­bols are called well-formed for­mu­las, or wffs for short. There are rules to tell what se­quences of sym­bols are wffs, these are called a gram­mar. Some wffs are called ax­ioms. There is an­other im­por­tant sym­bol that is not one of the sym­bols you chose—this is the ‘⊢’ sym­bol. A dec­la­ra­tion is the ‘⊢’ sym­bol fol­lowed by a wff. A le­gal dec­la­ra­tion is ei­ther the ‘⊢’ sym­bol fol­lowed by an ax­iom or the re­sult of an in­fer­ence rule. An in­fer­ence rule is a rule that de­clares that a dec­la­ra­tion of a cer­tain form is le­gal, given that cer­tain dec­la­ra­tions of other forms are le­gal. A fa­mous in­fer­ence rule called modus po­nens is part of a for­mal sys­tem called first-or­der logic. This rule says: “If ‘⊢ P’ and ‘⊢ (P → Q)’ (where P and Q are re­placed with some wffs) are valid dec­la­ra­tions, then ‘⊢ Q’ is also a valid dec­la­ra­tion.” By the way, a for­mal sys­tem is just a spe­cific alpha­bet, gram­mar, set of ax­ioms, and set of in­fer­ence rules. You also might like to note that if ‘⊢ P’ (where P is re­placed with some wff) is a valid dec­la­ra­tion, then we also call P a the­o­rem. So now we know some­thing: In a for­mal sys­tem, all ax­ioms are the­o­rems.

The sec­ond thing to note is that a for­mal sys­tem does not nec­es­sar­ily have any­thing to do with even propo­si­tional logic (let alone first- or sec­ond-or­der logic!). Con­sider the MIU sys­tem (open link in WordPad, on Win­dows), for ex­am­ple. It has four in­fer­ence rules for just mess­ing around with the or­der of the let­ters, ‘M’, ‘I’, and ‘U’! That doesn’t have to do with the real world or even math, does it?

The third thing to note is that, though a for­mal sys­tem can tell us what wffs are the­o­rems, it can­not (di­rectly) tell us what wffs are not the­o­rems. And hence we have the MU puz­zle. This asks whether “MU” is a the­o­rem in the MIU sys­tem. If it is, then you only need the MIU sys­tem to demon­strate this, but if it is not, you need to use rea­son­ing from out­side of that sys­tem.

As other com­menters have already noted, math­e­mat­i­ci­ans are not think­ing about ZFC set the­ory when they prove things (that’s not a bad thing; they’d never man­age to prove any new re­sults if they had to start from foun­da­tions for ev­ery proof!). How­ever, math­e­mat­i­ci­ans should be fairly con­fi­dent that the proofs they cre­ate could be re­duced down to proofs from the low-level ax­ioms. So Eliezer is definitely right to be wor­ried when a math­e­mat­i­cian says “A proof is a so­cial con­struct – it is what we need it to be in or­der to be con­vinced some­thing is true. If you write some­thing down and you want it to count as a proof, the only real is­sue is whether you’re com­pletely con­vinc­ing.”. A proof is a so­cial con­struct, but it is one, very, very spe­cific kind of so­cial con­struct. The ax­ioms and in­fer­ence rules of first-or­der Peano ar­ith­metic are sym­bolic rep­re­sen­ta­tions of our most fun­da­men­tal no­tion of what the nat­u­ral num­bers are. The rea­son for propo­si­tional logic, first-or­der logic, sec­ond-or­der logic, Peano ar­ith­metic, and the sci­en­tific method is that hu­mans have lit­tle things called “cog­ni­tive bi­ases”. We are con­vinced by way too many things that should be ut­terly un­con­vinc­ing. To say that a proof is a con­vinc­ing so­cial con­struct is...tech­ni­cally...cor­rect (oh how it pains me to say that!)...but that very vague part of what it means for some­thing to be a proof seems to im­ply that a proof is the ut­ter an­tithe­sis of what it was meant for! A math­e­mat­i­cal proof should be the most con­vinc­ing so­cial con­struct we have, be­cause of how it is con­structed.

First-or­der Peano ar­ith­metic has just a few sim­ple ax­ioms, and a cou­ple sim­ple in­fer­ence rules, and its sym­bols have a clear in­tended in­ter­pre­ta­tion (in terms of the nat­u­ral num­bers (which char­ac­ter­ize parts of the web of causal­ity as already ex­plained in the OP)). The truth of a few sim­ple ax­ioms and val­idity of a cou­ple sim­ple in­fer­ence rules can be eval­u­ated with­out our cog­ni­tive bi­ases get­ting in the way. On the other hand, it’s prob­a­bly not a good idea to make “There is a prime num­ber larger than any given nat­u­ral num­ber.” an ax­iom of a for­mal sys­tem about the nat­u­ral num­bers, be­cause it is not an im­me­di­ate part of our in­tu­itive un­der­stand­ing of how causal sys­tems that be­have ac­cord­ing to the rules of the nat­u­ral num­bers be­have. We as hu­mans would have to be very, very, con­fused if a the­o­rem of first-or­der Peano ar­ith­metic (be­cause we are so sure that its ax­ioms are true and its in­fer­ence rules are valid) turned out to be the nega­tion of an­other the­o­rem of Peano ar­ith­metic, but not so con­fused if the same hap­pened for ZFC set the­ory, be­cause we do not so read­ily ob­serve in­finite sets in our day-to-day ex­pe­rience. The ax­ioms and in­fer­ence rules of first-or­der Peano ar­ith­metic more di­rectly cor­re­spond to our phys­i­cal re­al­ity than those of ZFC set the­ory do (and the ax­ioms and in­fer­ence rules of the MIU sys­tem have noth­ing to do with our phys­i­cal re­al­ity at all!). If a con­tra­dic­tion in first-or­der Peano ar­ith­metic were found, though, life would go on. First-or­der Peano ar­ith­metic does have a lot to do with our phys­i­cal re­al­ity, but not all of it does. It in­ducts to num­bers like 3^^^3 that we will prob­a­bly never in­ter­act with. The ul­tra­fini­tists would be shout­ing “Told you so!”

Now I have said enough to give my di­rect re­sponse to the com­ment I am re­ply­ing to. First of all, the di­chotomy be­tween “logic” and “math­e­mat­ics” can be dis­solved by refer­ring to “for­mal sys­tems” in­stead. A for­mal sys­tem is ex­actly as en­twined with re­al­ity as its ax­ioms and in­fer­ence rules are. In terms of in­stru­men­tal ra­tio­nal­ity, the more ex­otic the­o­rems of ZFC set the­ory (and MIU) re­ally don’t help us, un­less we in­trin­si­cally en­joy con­sid­er­ing the ques­tion “What if there were (even though we have no ev­i­dence that this is the case) a pla­tonic realm of sets? How would it be­have?”

When used as means to an end, the point of a for­mal sys­tem is to cor­rect for our cog­ni­tive bi­ases. In other words, the defi­ni­tion of a proof should state that a proof is a “con­vinc­ing demon­stra­tion that should be con­vinc­ing”, to be­gin with. I sus­pect Eliezer is so con­cerned with the Peano ax­ioms be­cause com­puter pro­grams hap­pen to ev­i­dently be­have in a very, very math­e­mat­i­cal way, and he be­lieves that even­tu­ally a com­puter pro­gram will de­cide the fate of hu­man­ity. I share his con­cerns; I want a math­e­mat­i­cal ar­gu­ment that the Gen­eral Ar­tifi­cial In­tel­li­gence that will be cre­ated will be Friendly, not any­thing that might “con­vince” a few un­in­formed gov­ern­ment offi­cials.

• A few things:

1. I don’t think we dis­agree about the so­cial con­struct thing: see my other com­ment where I’m talk­ing about that.

2. It sounds like you pretty much come down in favour of the sec­ond po­si­tion that I ar­tic­u­lated above, just with a for­mal­ist twist. Math­e­mat­i­cal talk is about what fol­lows from the ax­ioms; ob­vi­ously only cer­tain sets of ax­ioms are worth in­ves­ti­gat­ing, as they’re the ones that ac­tu­ally line up with sys­tems in the world. I agree so far, but you think that there is no no­tion of logic be­yond the syn­tac­tic?

First of all, the di­chotomy be­tween “logic” and “math­e­mat­ics” can be dis­solved by refer­ring to “for­mal sys­tems” in­stead.

Aren’t you just drop­ping the dis­trinc­tion be­tween syn­tax and se­man­tics here? One of the big points of the last few posts has been that we’re in­ter­ested in the se­man­tic im­pli­ca­tions, and the for­mal sys­tems are a (sound) syn­tac­tic means of reach­ing true con­clu­sions. From your post it sounds like you’re a pretty se­ri­ous for­mal­ist, though, so that may not be a big deal to you.

• 4 Nov 2012 5:18 UTC
0 points
Parent

Definitely po­si­tion two.

I would de­scribe first-or­der logic as “a for­mal en­cap­su­la­tion of hu­man­ity’s most fun­da­men­tal no­tions of how the world works”. If it were shown to be in­con­sis­tent, then I could still fall back to some­thing like in­tu­ition­is­tic logic, but from that point on I’d be pretty skep­ti­cal about how much I could re­ally know about the world, be­yond that which is com­pletely ob­vi­ous (grav­ity, etc.).

What did I say that im­plied that I “think that there is no no­tion of logic be­yond the syn­tac­tic”? I think of “logic” and “proof” as com­pletely syn­tac­tic pro­cesses, but the premises and con­clu­sions of a proof have to have se­man­tic mean­ing; oth­er­wise, why would we care so much about prov­ing any­thing? I may have im­plied some­thing that I didn’t be­lieve, or I may have in­con­sis­tent be­liefs re­gard­ing math and logic, so I’d ac­tu­ally ap­pre­ci­ate it if you pointed out where I con­tra­dicted what I just said in this com­ment (if I did).

• Look­ing back, it’s hard to say what gave me that im­pres­sion. I think I was mostly just con­fused as to why you were spend­ing quite so much time go­ing over the syn­tax stuff ;) And

First of all, the di­chotomy be­tween “logic” and “math­e­mat­ics” can be dis­solved by refer­ring to “for­mal sys­tems” in­stead.

made me think that you though that all log­i­cal/​math­e­mat­i­cal talk was just talk of for­mal sys­tems. That can’t be true if you’ve got some se­man­tic story go­ing on: then the syn­tax is im­por­tant, but mainly as a way to reach se­man­tic truths. And the se­man­tics don’t have to men­tion for­mal sys­tems at all. If you think that the se­man­tics of logic/​math­e­mat­ics is re­ally about syn­tax, then that’s what I’d think of as a “for­mal­ist” po­si­tion.

• Oh, I think I may un­der­stand your con­fu­sion, now. I don’t think of math­e­mat­ics and logic as equals! I am more con­fi­dent in first-or­der logic than I am in, say, ZFC set the­ory (though I am ex­tremely con­fi­dent in both). How­ever, for­mal sys­tem-space is much larger than the few for­mal sys­tems we use to­day; I wanted to em­pha­size that. Logic and set the­ory were se­lected for be­cause they were use­ful, not be­cause they are the only pos­si­ble for­mal ways of think­ing out there. In other words, I was try­ing to right the wrong ques­tion, why do math­e­mat­ics and logic tran­scend the rest of re­al­ity?

• In con­trast with my es­teemed col­league RichardKen­n­away, I think it’s mostly #2. Be­fore the Peano ax­ioms, peo­ple talk­ing about num­bers might have been talk­ing about any of a large class of things which dis­crete ob­jects in the real world mostly model. It was hard to make progress in math past a cer­tain level un­til some­one pointed out ax­io­mat­i­cally ex­actly which things-that-dis­crete-ob­jects-in-the-real-world-mostly-model it would be most pro­duc­tive to talk about.

Con­cor­dantly, the situ­a­tion of pre-ax­iom speak­ers is much like that of peo­ple from Scot­land try­ing to talk to peo­ple from the Amer­i­can South and peo­ple from Bos­ton, when none of them knows the rules of their gram­mar. Edit: Or, to be more pre­cise, it’s like two scots speak­ers as fluent as Ka­woomba talk­ing about whether a soli­tary, fallen tree made a “sound,” with­out defin­ing what they mean by sound.

• Aye, right. Yer bum’s oot the win­dae, lad­die. Ye dinna need tae been lairnin a wee Scots tae un­ner­stan, it’s gaein be awricht! Ane leid is enough.

• What about “both ways si­mul­ta­neously, the dis­tinc­tion left am­bigu­ous most of the time be­cause it isn’t use­ful”?

• EY seems to be taken with the re­sem­blance be­tween a causal di­a­gram and the ab­stract struc­ture of ax­ioms, in­fer­ences and the­o­rems in math­e­mat­cal logic. But there are differ­ences: with causal­ity, our ev­i­dence is the lat­est causal out­put, the leaf nodes. We have to trace back to the Big Bang from them.How­ever, in maths we start from ax­ioms, and can­not get di­rectly to the the­o­rems or leaf nodes. We could see this pro­cess as ex­plor­ing a pre-ex­ist­ing ter­ri­tory, but it is hard to see what this adds, since the ax­ioms and rules of in­fer­ence are suffi­cient for truth, and it is hard to see, in EY’s pre­sen­ta­tion how liter­ally he takes the idea.

• Er, no, causal mod­els and log­i­cal im­pli­ca­tions seem to me very differ­ent in how they prop­a­gate mod­u­larly. Unify­ing the two is go­ing to be trou­ble­some.

• We could see this pro­cess as ex­plor­ing a pre-ex­ist­ing ter­ri­tory, but it is hard to see what this adds, since the ax­ioms and rules of in­fer­ence are suffi­cient for truth, and it is hard to see, in EY’s pre­sen­ta­tion how liter­ally he takes the idea.

It’s use­ful for rea­son­ing heuris­ti­cally about con­jec­tures.

• Could I have an ex­am­ple?

ax­ioms pin down that we’re talk­ing about num­bers as op­posed to some­thing else.

as:

ax­ioms pin down that we’re talk­ing about some sys­tem that be­haves like num­bers as op­posed to some­thing else.

Lots of things in both real and imag­ined wor­lds be­have like num­bers. It’s most con­ve­nient to pick one of them and call them “The Num­bers” but this is re­ally just for the sake of con­ve­nience and doesn’t nec­es­sar­ily give them ele­vated philo­soph­i­cal sta­tus. That would be my po­si­tion any­way.

• The Peano Arith­metic talks about the Suc­ces­sor func­tion, and jazz. Did you know that the set of finite strings of a sin­gle sym­bol alpha­bet also satis­fies the Peano Ax­ioms? Did you know that in ZFC, defin­ing the set all sets con­tain­ing only other mem­bers of the par­ent set with lower car­di­nal­ity, and then say­ing {} is a mem­ber obeys the Peano Ax­ioms? Did you know that say­ing you have a Com­mu­ta­tive Monoid with right di­vi­sion, that mul­ti­pli­ca­tion with some­thing other than iden­tity always yields a new el­e­ment and that the set {1} is pro­duc­tive, obey the Peano Ax­ioms? Did you know the even nat­u­rals obey the Peano Ax­ioms? Did you know any fully or­dered set with in­fi­mum, but no supre­mum obey the Ax­ioms?

There is no such thing as “Num­bers,” only things satis­fy­ing the Peano Ax­ioms.

• Did you know that the set of finite strings of a sin­gle sym­bol alpha­bet also satis­fies the Peano Ax­ioms?

Surely the set of finite strings in an alpha­bet of no-mat­ter-how-many-sym­bols satis­fies the Peano ax­ioms? e.g. us­ing the English alpha­bet (with A=0, B=S(A), C=S(B)....AA=S(Z), AB=S(AA), etc would make a base-26 sys­tem).

• Sin­gle sym­bol alpha­bet is more in­ter­est­ing, (empty string = 0, suces­sor func­tion = ap­pend an­other sym­bol) the sys­tem you de­scribe is more suc­cinctly de­scribed us­ing a con­cate­na­tion op­er­a­tor:

• 0 = 0, 1 = S0, 2 = S1 … 9 = S8.

• For All b in {0,1,2,3,4,5,6,7,8,9}, a in N: ab = a x S9 + b

From these defi­ni­tions we get, ex­am­ple-wise:

• 10 = 1 x S9 + 0 = SSSSSSSSSS0

• I’m not quite sure what you’re say­ing here—that “Num­bers” don’t ex­ist as such but “the even nat­u­rals” do ex­ist?

• We don’t know whether the uni­verse is finite or not. If it is finite, then there is noth­ing in it that fully mod­els the nat­u­ral num­bers. Would we then have to say that the num­bers did not ex­ist? If the sys­tem that we’re refer­ring to isn’t some phys­i­cal thing, what is it?

• Finite sub­sets of the nat­u­rals still be­have like nat­u­rals.

• Not pre­cisely. In many ways, yes, but for ex­am­ple they don’t model the ax­iom of PA that says that ev­ery num­ber has a suc­ces­sor.

• I’ve re­al­ised that I’m slightly more con­fused on this topic than I thought.

As non-log­i­cally om­ni­scient be­ings, we need to keep track of hy­po­thet­i­cal uni­verses which are not just phys­i­cally differ­ent from our own, but which don’t make sense—i.e. they con­tain log­i­cal con­tra­dic­tions that we haven’t no­ticed yet.

For ex­am­ple, let T be a Tur­ing ma­chine where we haven’t yet es­tab­lished whether or not T halts. Then one of the fol­low­ing is true but we don’t know which one:

• (a) The uni­verse is in­finite and T halts

• (b) The uni­verse is in­finite and T does not halt

• (c) The uni­verse is finite and T halts

• (d) The uni­verse is finite and T does not halt

If we then dis­cover that T halts, we not only as­sign zero prob­a­bil­ity to (b) and (d), we strike them off the list en­tirely. (At least that’s how I imag­ine it, I haven’t yet heard any­one de­scribe ap­proaches to log­i­cal un­cer­tainty).

But it feels like there should also be (e) - “the uni­verse is finite and the ques­tion of whether or not T halts is mean­ingless”. If we were to dis­cover that we lived in (e) then all in­finite uni­verses would have to be struck off our list of mean­ingful hy­po­thet­i­cal uni­verses, since we are view­ing hy­po­thet­i­cal uni­verses as math­e­mat­i­cal ob­jects.

But it’s hard to imag­ine what would con­sti­tute ev­i­dence for (or against) (e). So af­ter 5 min­utes of pon­der­ing, that more or less maps out my cur­rent state of con­fu­sion.

• I think you’re con­fused if you think the fini­tude of the uni­verse mat­ters in an­swer­ing the math­e­mat­i­cal ques­tion of whether T halts. An­swer­ing that ques­tion may be of in­ter­est for then figur­ing out whether cer­tain things in our uni­verse that be­have like Turn­ing ma­chines be­have in cer­tain ways, but the math­e­mat­i­cal ques­tion is in­de­pen­dent.

Your con­fu­sion is that you think there need to be ob­jects of some kind that cor­re­spond to math­e­mat­i­cal struc­tures that we talk about. Then you’ve got to figure out what they are, and that seems to be tricky how­ever you cut it.

• I agree that the fini­tude of the uni­verse doesn’t mat­ter in an­swer­ing the math­e­mat­i­cal ques­tion of whether T halts. I was pon­der­ing whether the fini­tude of the uni­verse had some bear­ing on whether the ques­tion of T halt­ing is nec­es­sar­ily mean­ingful (in an in­finite uni­verse it surely is mean­ingful, in a finite uni­verse it very likely is but not so ob­vi­ously so).

• Surely if the in­fini­tude of the uni­verse doesn’t af­fect that state­ment’s truth, it can’t af­fect that state­ment’s mean­ingful­ness? Seems pretty ob­vi­ous to me that the mean­ing is the same in a finite and an in­finite uni­verse: you’re talk­ing about the math­e­mat­i­cal con­cept of a Tur­ing ma­chine in both cases.

• Con­di­tional on the state­ment be­ing mean­ingful, in­fini­tude of the uni­verse doesn’t af­fect the state­ment’s truth. If the mean­ingful­ness is in ques­tion then I’m con­fused so wouldn’t as­sign very high or low prob­a­bil­ities to any­thing.

Essen­tially:

• I have a very strong in­tu­ition that there is a unique (up to iso­mor­phism) math­e­mat­i­cal struc­ture called the “non-nega­tive in­te­gers”

• I have a weaker in­tu­ition that state­ments in sec­ond-or­der logic have a unique mean­ingful interpretation

• I have a strong in­tu­ition that model se­man­tics of first-or­der logic is meaningful

• I have a very strong in­tu­ition that the uni­verse is real in some sense

It’s pos­si­ble that my in­tu­ition might be wrong though. I can pic­ture the in­te­gers in my mind but my pic­ture isn’t com­pletely ac­cu­rate—they ba­si­cally come out as a line of dots with a “go­ing on for­ever” con­cept at the end. I can carry on pul­ling dots out of the “go­ing on for­ever”, but I can’t ever pull all of them out be­cause there isn’t room in my mind.

Any at­tempt to cap­ture the in­te­gers in first-or­der logic will per­mit non­stan­dard mod­els. From the van­tage point of ZF set the­ory there is a sin­gle “stan­dard” model, but I’m not sure this helps—there are just non­stan­dard mod­els of set the­ory in­stead. Similarly I’m not sure sec­ond-or­der logic helps as you pretty much need set the­ory to define its se­man­tics.

So if I’m ques­tion­ing ev­ery­thing it seems I should at least be open to the idea of there be­ing no sin­gle model of the in­te­gers which can be said to be “right” in a non-ar­bi­trary way. I’d want to ques­tion first or­der logic too, but it’s hard to come up with a weaker (or differ­ent) sys­tem that’s both rigor­ous and ac­tu­ally use­ful for any­thing.

I’ve re­al­ized one thing though (based on this con­ver­sa­tion) - if the uni­verse is in­finite, defin­ing the in­te­gers in terms of the real world isn’t ob­vi­ously the right thing to do, as the real world may be fol­low­ing one of the non­stan­dard mod­els of the in­te­gers. Up­dat­ing in fa­vor of mean­ingful­ness not be­ing de­pen­dent on in­fini­tude of uni­verse.

• I’m a lit­tle con­fused as to which of two po­si­tions this is ad­vo­cat­ing:

1. Num­bers are real, se­ri­ous things, but the way that we pick them out is by hav­ing a cat­e­gor­i­cal set of ax­ioms. They’re in­ter­est­ing to talk about be­cause lots of things in the world be­have like them (to some de­gree).

2. Math­e­mat­i­cal talk is ac­tu­ally talk about what fol­lows from cer­tain ax­ioms. This is in­ter­est­ing to talk about be­cause lots of things obey the ax­ioms and so ex­hibit the the­o­rems (to some de­gree).

I read it as (1), with a side or­der of (2). Math­e­mat­i­cal talk is also about what fol­lows from cer­tain ax­ioms. The ax­ioms nail it down so that math­e­mat­i­ci­ans can be sure what other math­e­mat­i­ci­ans are talk­ing about.

Both of these have some prob­lems. The first one re­quires you to have weird, non-phys­i­cal num­bery-things.

Not weird, non-phys­i­cal num­bery-things, just non-phys­i­cal num­bery-things. If they seem weird, maybe it’s be­cause we only no­ticed them a few thou­sand years ago.

Not only this, but they’re a spe­cial ex­cep­tion to the the­ory of refer­ence that’s been de­vel­oped so far, in that you can re­fer to them with­out hav­ing a causal con­nec­tion.

No more than a mag­netic field is a spe­cial ex­cep­tion to the the­ory of elas­tic­ity. It’s just a phe­nomenon that is not de­scribed by that the­ory.

• But EY in­sists that maths does come un­der cor­re­spon­dence/​refer­ence!

“to delineate two differ­ent kinds of cor­re­spon­dence within cor­re­spon­dence the­o­ries of truth.”″

• Do we need a pro­cess for figur­ing out which ob­jects are likely to be­have like num­bers? And as good Bayesi­ans, for figur­ing out how likely that is?

• Er, yes? I mean it’s not like we’re born know­ing that cars be­have like in­te­gers and out­let elec­tric­ity doesn’t, since nei­ther of those things ex­isted an­ces­trally.

• I’m pretty sure that we’re born know­ing cars and car­like ob­jects be­have like in­te­gers.

• I think our eyes (or vi­sual cor­tex) knows that cer­tain things (up to 3 or 4 of them) be­have like in­te­gers since it both­ers to count them au­to­mat­i­cally.

• Wait, what? We may not be born know­ing what cars and elec­tric­ity are, but I would be sur­prised if we weren’t born with an abil­ity (or the ca­pac­ity to de­velop an abil­ity) to par­ti­tion our model of a car-con­tain­ing sec­tion of uni­verse into dis­crete “car” ob­jects, while not be­ing able to do the same for “elec­tric cur­rent” ob­jects.

• The an­ces­tral en­vi­ron­ment in­cluded peo­ple (who be­have like in­te­gers over mod­er­ate time spans) and wa­ter (which doesn’t be­have like in­te­gers)..

The bet­ter ques­tion would have been “how do peo­ple iden­tify ob­jects which be­have like in­te­gers?”.

• The bet­ter ques­tion would have been “how do peo­ple iden­tify ob­jects which be­have like in­te­gers?”.

The same way we iden­tify ob­jects which satisfy any other pred­i­cate? We de­ter­mine whether or not some­thing is a cat by com­par­ing it to our knowl­edge of what cats are like. We de­ter­mine whether or not some­thing is dan­ger­ous by com­par­ing it to our knowl­edge of what dan­ger­ous things are like.

Why do you ask this ques­tion speci­fi­cally of the in­te­gers? Is there some­thing spe­cial about them?

• Water does be­have like very large in­te­gers.

• So does elec­tric­ity. (And it does so ex­actly, whereas wa­ter con­tains differ­ent iso­topes of hy­dro­gen and oxy­gen...)

Any­way, I seem to re­call see­ing a Wikipe­dia ar­ti­cle about some ob­scure lan­guage where the word for ‘wa­ter’ is gram­mat­i­cally plu­ral, and think­ing ‘who knows if they’ve coined a back­formed sin­gu­lar for “wa­ter molecule”, at least in­for­mally or joc­u­larly’.

(Note also that nat­u­ral lan­guages don’t seem to have fixed rules for whether nouns like “rice” or “oats”—i.e. col­lec­tions of small ob­jects you could count but you would never nor­mally bother to—are mass nouns or plu­ral nouns.)

• If you’re go­ing to in­sist that differ­ent iso­topes dis­rupt the whole num­ber qual­ity of wa­ter, then frac­tional-charge quasi­par­ti­cles would like a word with your alle­ga­tion that elec­tric­ity can be com­pletely and ex­actly mod­eled us­ing in­te­gers.

• How do you de­ter­mine whether a phys­i­cal pro­cess “be­haves like in­te­gers”? The sec­ond-or­der ax­iom of in­duc­tion sounds com­pli­cated, I can­not eas­ily check that it’s satis­fied by ap­ples. If you use some sort of Bayesian rea­son­ing to figure out which ax­ioms work on ap­ples, can you de­scribe it in more de­tail?

• I don’t have an an­swer to the spe­cific ques­tion, only to the class of ques­tions. To ap­proach un­der­stand­ing this, we need to dis­t­in­guish be­tween re­al­ity and what points to re­al­ity, i.e, sym­bols. Our skill as hu­mans is in the ma­nipu­la­tion of sym­bols, as a kind of simu­la­tion of re­al­ity, with greater or lesser work­a­bil­ity for pre­dic­tion, based in prior ob­ser­va­tion, of new ob­ser­va­tions.

“Ap­ples” refers, in­ter­nally, to a set of re­sponses we cre­ated through our ex­pe­rience. We re­spond to re­al­ity as an “ap­ple” or as a “set of ap­ples,” only out of our his­tory. It’s ar­bi­trary. Count­ing, and thus “be­hav­ior like in­te­gers” ap­plies to the sim­plified, ar­bi­trary con­structs we call “ap­ples.” Real­ity is not di­vided into sep­a­rate ob­jects, but we have or­ga­nized our per­cep­tions into named ob­jects.

Ex­am­ples. If an “ap­ple” is a unique dis­crim­inable ob­ject, say all ap­ples have had a unique code ap­plied to them, then what can be counted is the codes. In­te­ger be­hav­ior is a be­hav­ior of codes.

Unique ap­plies can be picked up one at a time, be­ing trans­ferred to one bas­ket or an­other. How­ever, real ap­ples are not a con­stant. Ap­ples grow and ap­ples rot. Is a pile of rot­ten ap­ple an “ap­ple”? Is an ap­ple seed an ap­ple? Th­ese are ques­tions with no “true” an­swer, rather we choose an­swers. We end up with a bi­nary state for each pos­si­ble ob­ject: “yes, ap­ple,” or “no, not ap­ple.” We can count these states, they ex­ist in our mind.

If “ap­ple” refers to a va­ri­ety, we may have Mac­in­tosh, Fuji, Golden deli­cious, etc.

So I have a bas­ket with two ap­ples in it. That is, five pieces of fruit that are Mac­in­tosh and three that are Fuji.

I have an­other bas­ket with two ap­ples in it. That is, one Fuji and one Golden Deli­cious.

I put them all into one bas­ket. How many ap­ples are in the bas­ket? 2 + 2 = 3.

The ques­tion about in­te­ger be­hav­ior is about how cat­e­gories have been as­sem­bled. If “ap­ple” refers to an in­di­vi­d­ual piece of in­tact fruit, we can pick it up, move it around, and it re­mains the same ob­ject, it’s unique and there is no other the same in the uni­verse, and it be­longs to a class of ob­jects that is, again, unique as a class, the class is countable and classes will dis­play in­te­ger be­hav­ior.

That’s as far as I’ve got­ten with this. “In­te­ger be­hav­ior” is not a prop­erty of re­al­ity, per se, but of our per­cep­tions of re­al­ity.

• Well, it comes from the fact that ap­ples in a bowl is Ex­clu­sively just that, as ver­ified by your Bayesian rea­son­ing. There are no other “chains” of suc­ces­sors (shadow ap­ples? I can’t even imag­ine a good metaphor).

So, now you in fact have that bowl of ap­ples nar­rowed down to {0, S0, SS0, SSS0, …} which is iso­mor­phic to the nat­u­ral num­bers, so all other nat­u­ral num­ber prop­er­ties will be re­flected there.

• For thou­sands of years, math­e­mat­i­ci­ans tried prov­ing the par­allel pos­tu­late from Eu­clid’s four other pos­tu­lates, even though there are fairly sim­ple coun­terex­am­ples which show such a proof to be im­pos­si­ble. I sus­pect that at least part of the rea­son for this de­lay is a failure to ap­pre­ci­ate this post’s point : that a “straight line”, like a “num­ber” has to be defined/​speci­fied by a set of ax­ioms, and that a great cir­cle is in fact a “straight line” as far as the first four of Eu­clid’s pos­tu­lates are con­cerned.

• That’s not cor­rect. Ellip­tic ge­om­e­try fails to satisfy some of the other pos­tu­lates, de­pend­ing on how they are phrased. I’m not too fa­mil­iar with the stan­dard ways of mak­ing Eu­clid’s pos­tu­lates rigor­ous, but if you’re look­ing at Hilbert’s ax­ioms in­stead, then el­lip­tic ge­om­e­try fails to satisfy O3 (the third or­der ax­iom): if three points A, B, C are on a line, then any of the points is be­tween the other two. Pos­si­bly some other ax­ioms are vi­o­lated as well.

Notably, el­lip­tic ge­om­e­try does not con­tain any par­allel lines, while it is a the­o­rem of neu­tral ge­om­e­try that par­allel lines do in fact ex­ist.

Hyper­bolic ge­om­e­try was ac­tu­ally nec­es­sary to prove the in­de­pen­dence of Eu­clid’s fifth pos­tu­late, and few would call it a “fairly sim­ple coun­terex­am­ple”.

I agree that in­tro­duc­ing el­lip­tic ge­om­e­try (and other sim­ple ex­am­ples like the Fano plane) ear­lier on in his­tory would have made the dis­cus­sion of Eu­clid’s fifth pos­tu­late much more co­her­ent much sooner.

• Awe­some, I was look­ing for a good ex­pla­na­tion of the Peano ax­ioms!

About six months ago I had a se­ries of ar­gu­ments with my house­mate, who’s been do­ing a philos­o­phy de­gree at a Catholic uni­ver­sity. He ar­gued that I should leave the door open for some way other than ob­ser­va­tion to gather knowl­edge, be­cause we had things like maths giv­ing us knowl­edge in this other way, which meant we couldn’t as­sume we’d come up with some other other way to dis­cover, say, eth­i­cal or aes­thetic truths.

I couldn’t con­vince him that all we could do in ethics was rea­son from ax­ioms, be­cause he didn’t un­der­stand that maths was just rea­son­ing from ax­ioms—and I didn’t ac­tu­ally un­der­stand the Peano ax­ioms, so I couldn’t ex­plain them.

So, thanks for the post.

• Why does 2 + 2 come out the same way each time? Never mind the ques­tion of why the laws of physics are sta­ble—why is logic sta­ble? Of course I can’t imag­ine it be­ing any other way, but that’s not an ex­pla­na­tion.

My short an­swer is “be­cause we live in a causal uni­verse”.

To ex­pand on that:

Logic is a pro­cess that has been speci­fi­cally de­signed to be sta­ble. Any pro­cess that has gone through a de­sign speci­fi­cally in­tended to make it sta­ble, and re­fined for sta­bil­ity over gen­er­a­tions, is go­ing to have a higher prob­a­bil­ity of be­ing sta­ble. Logic, in short, is more likely than any­thing else in the uni­verse to be sta­ble.

So then the ques­tion is not why logic speci­fi­cally is sta­ble—that is by de­sign—but rather whether it is pos­si­ble for any­thing in the uni­verse to be sta­ble. And there is one thing that does ap­pear to be sta­ble; that if you have the same cause, then you will have the same effect. That the uni­verse is (at least mostly) causal. It is that causal­ity that gives logic its sta­bil­ity, as far as I can see.

• “Be­cause if you had an­other sep­a­rated chain, you could have a prop­erty P that was true all along the 0-chain, but false along the sep­a­rated chain. And then P would be true of 0, true of the suc­ces­sor of any num­ber of which it was true, and not true of all num­bers.”

But the ax­iom schema of in­duc­tion does not com­pletely ex­clude non­stan­dard num­bers. Sure if I prove some prop­erty P for P(0) and for all n, P(n) ⇒ P(n+1) then for all n, P(n); then I have ex­cluded the pos­si­bil­ity of some non­stan­dard num­ber “n” for which not P(n) but there are some prop­er­ties which can­not be proved true or false in Peano Arith­metic and there­fore whose truth hood can be al­tered by the pres­ence of non­stan­dard num­bers.

Can you give me a prop­erty P which is true along the zero-chain but nec­es­sar­ily false along a sep­a­rated chain that is in­finitely long in both di­rec­tions? I do not be­lieve this is pos­si­ble but I may be mis­taken.

• Can you give me a prop­erty P which is true along the zero-chain but nec­es­sar­ily false along a sep­a­rated chain that is in­finitely long in both di­rec­tions?

Pn(x) is “x is the nth suc­ces­sor of 0” (the 0th suc­ces­sor of a num­ber is it­self). P(x) is “there ex­ists some n such that Pn(x)”.

• I don’t see how you would define Pn(x) in the lan­guage of PA.

Let’s say we used some­thing like this:

``````Pn(x) iff ((0 + n) = x)
``````

Let’s look at the defi­ni­tion of +, a func­tion sym­bol that our model is al­lowed to define:

``````a + 0 = a
a + S(b) = S(a + b)
``````

“x + 0 = x” should work perfectly fine for non­stan­dard num­bers.

So go­ing back to P(x):

“there ex­ists some n such that ((0 + n) = x)”

for a non­stan­dard num­ber x, does there ex­ist some num­ber n such that ((0+n) = x)? Yup, the non­stan­dard num­ber x! Set n=x.

Oh, but when you said nth suc­ces­sor you meant n had to be stan­dard? Well, that’s the whole prob­lem isn’t it!

• But any non­stan­dard num­ber is not an nth suc­ces­sor of 0 for any n, even non­stan­dard n (what­ever that would mean). So your rephras­ing doesn’t mean the same thing, in­tu­itively—P is, in­tu­itively, “x is reach­able from 0 us­ing the suc­ces­sor func­tion”.

Couldn’t you say:

• P0: x = 0

• PS0: x = S0

• PSS0: x = SS0

and so on, defin­ing a set of prop­er­ties (we can con­struct these in­duc­tively, and so there is no Pn for non­stan­dard n), and say P(x) is “x satis­fies one such prop­erty”?

• An in­finite num­ber of ax­ioms like in an ax­iom schema doesn’t re­ally hurt any­thing, but you can’t have in­finitely long sin­gle ax­ioms.

``````∀x((x = 0) ∨ (x = S0) ∨ (x = SS0) ∨ (x = SSS0) ∨ …)
``````

is not an op­tion. And nei­ther is the ax­iom set

``````P0(x) iff x = 0
PS0(x) iff x = S0
PSS0(x) iff x = SS0
…
∀x(P0(x) ∨ PS0(x) ∨ PS0(x) ∨ PS0(x) ∨ …)
``````

We could in­stead try the axioms

``````P(0, x) iff x = 0
P(S0, x) iff x = S0
P(SS0, x) iff x = SS0
…
∀x(∃n(P(n, x)))
``````

but then again we have the prob­lem of n be­ing a non­stan­dard num­ber.

• What is n?

• It’s non-strictly-math­e­mat­i­cal short­hand for this.

• Not sure if I un­der­stand the point of your ar­gu­ment.

Are you say­ing that in re­al­ity ev­ery prop­erty P has ac­tu­ally three out­comes: true, false, un­de­cid­able? And that those always de­cid­able, like e.g. “P(n) <-> (n = 2)” can­not be true for all nat­u­ral num­bers, while those which can be true for all nat­u­ral num­bers, but mostly false oth­er­wise, are always un­de­cid­able for… some other val­ues?

Can you give me a prop­erty P which is true along the zero-chain but nec­es­sar­ily false along a sep­a­rated chain that is in­finitely long in both di­rec­tions? I do not be­lieve this is pos­si­ble but I may be mis­taken.

I don’t know.

Let’s sup­pose that for any spe­cific value V in the sep­a­rated chain it is pos­si­ble to make such prop­erty PV. For ex­am­ple “PV(x) <-> (x <> V)”. And let’s sup­pose that it is not pos­si­ble to make one such prop­erty for all val­ues in all sep­a­rated chains, ex­cept by say­ing some­thing like “P(x) <-> there is no such PV which would be true for all num­bers in the first chain and false for x”.

What would that prove? Would it con­tra­dict the ar­ti­cle? How speci­fi­cally?

• Are you say­ing that in re­al­ity ev­ery prop­erty P has ac­tu­ally three out­comes: true, false, un­de­cid­able?

By Godel’s in­com­plete­ness the­o­rem yes, un­less your the­ory of ar­ith­metic has a non-re­cur­sively enu­mer­able set of ax­ioms or is in­con­sis­tent.

And that those always de­cid­able, like e.g. “P(n) <-> (n = 2)” can­not be true for all nat­u­ral num­bers, while those which can be true for all nat­u­ral num­bers, but mostly false oth­er­wise, are always un­de­cid­able for… some other val­ues?

I’m hav­ing trou­ble un­der­stand­ing this sen­tence but I think I know what you are ask­ing about.

There are some prop­er­ties P(x) which are true for ev­ery x in the 0 chain, how­ever, Peano Arith­metic does not in­clude all these P(x) as the­o­rems. If PA doesn’t in­clude P(x) as a the­o­rem, then it is in­de­pen­dent of PA whether there ex­ist non­stan­dard el­e­ments for which P(x) is false.

Let’s sup­pose that for any spe­cific value V in the sep­a­rated chain it is pos­si­ble to make such prop­erty PV. What would that prove? Would it con­tra­dict the ar­ti­cle? How speci­fi­cally?

I think this is what I am say­ing I be­lieve to be im­pos­si­ble. You can’t just say “V is in the sep­a­rated chain”. V is a con­stant sym­bol. The model can as­sign con­stants to what­ever ob­ject in the do­main of dis­course it wants to un­less you add ax­ioms for­bid­ding it.

Hon­estly I am be­com­ing con­fused. I’m go­ing to take a break and think about all this for a bit.

• If our ax­iom set T is in­de­pen­dent of a prop­erty P about num­bers then by defi­ni­tion there is noth­ing in­con­sis­tent about the the­ory T1 = “T and P” and also noth­ing in­con­sis­tent about the the­ory T2= “T and not P”.

To say that they are not in­con­sis­tent is to say that they are satis­fi­able, that they have pos­si­ble mod­els. As T1 and T2 are in­con­sis­tent with each other, their mod­els are differ­ent.

The sin­gle zero-based chain of num­bers with­out non­stan­dard num­bers is a sin­gle model. There­fore, if there ex­ists a prop­erty about num­bers that is in­de­pen­dent of any the­ory of ar­ith­metic, that the­ory of ar­ith­metic does not log­i­cally ex­clude the pos­si­bil­ity of non­stan­dard el­e­ments.

By Godel’s in­com­plete­ness the­o­rems, a the­ory must have state­ments that are in­de­pen­dent from it un­less it is ei­ther in­con­sis­tent or has a non-re­cur­sively-enu­mer­able the­o­rem set.

Each in­stance of the ax­iom schema of in­duc­tion can be con­structed from a prop­erty. The set of prop­er­ties is re­cur­sively enu­mer­able, there­fore the set of in­stances of the ax­iom schema of in­duc­tion is re­cur­sively enu­mer­able.

Every the­o­rem of Peano Arith­metic must use a finite num­ber of ax­ioms in its proof. We can enu­mer­ate the the­o­rems of Peano Arith­metic by adding in­creas­ingly larger sub­sets of the in­finite set of in­stances of the ax­iom schema of in­duc­tion to our ax­iom set.

Since the the­ory of Peano Arith­metic has a re­cur­sively enu­mer­able set of the­o­rems it is ei­ther in­con­sis­tent or is in­de­pen­dent of some prop­erty and thus al­lows for the ex­is­tence of non­stan­dard el­e­ments.

• But the ax­iom schema of in­duc­tion does not com­pletely exclude

Eliezer isn’t us­ing an ax­iom schema, he’s us­ing an ax­iom of sec­ond or­der logic.

• I don’t see what the differ­ence is… They look very similar to me.

At some point you have to trans­late it into a (pos­si­bly in­finite) set of first-or­der ax­ioms or you wont be able to perform first-or­der re­s­olu­tion any­way.

• Can you give me a prop­erty P which is true along the zero-chain but nec­es­sar­ily false along a sep­a­rated chain that is in­finitely long in both di­rec­tions? I do not be­lieve this is pos­si­ble but I may be mis­taken.

For any num­ber n, n-n=0.

If you have a sep­a­rate chain that isn’t con­nected to zero, then this isn’t true.

How­ever this state­ment is pretty sim­ple and can be ex­pressed in first or­der logic. I have no idea why EY be­lieves that it re­quires sec­ond or­der logic to elimi­nate the pos­si­bil­ity of other chains that aren’t de­rived from zero.

• I love this in­quiry.

Num­bers do not ap­pear in re­al­ity, other than “men­tal re­al­ity.” 2+2=4 does not ap­pear out­side of the mind. Here is why:

To know that I have two ob­jects, I must ap­ply a pro­cess to my per­cep­tion of re­al­ity. I must rec­og­nize the ob­jects as dis­tinct, I must cat­e­go­rize them as “the same” in some way. And then I ap­ply an­other pro­cess, “count­ing.” That is ap­plied to my col­lected iden­ti­fi­ca­tions, not to re­al­ity it­self, which can just as eas­ily be seen as uni­tary, or sliced up in a prac­ti­cally in­finite num­ber of ways.

Num­ber, then, is a product of brain ac­tivity, and the ob­served prop­er­ties of num­bers are prop­er­ties of brain pro­cess. Some ex­am­ples.

I put two ap­ples in a bowl. I put two more ap­ples in the bowl. How many ap­ples are now in the bowl?

We may eas­ily say “four,” be­cause most of the time this pre­dic­tion holds. How­ever, it’s a mix­ing bowl, used as a blender, and what I have now is a bowl of ap­ple­sauce. How many ap­ples are in the bowl? I can’t count them! I put four ap­ples in, and none come out! Or some smaller num­ber than four. Or a greater num­ber (If I add some earth, air, fire, and wa­ter, and wait a lit­tle while....)

Ap­ples are com­plex ob­jects. How about it’s two deu­terium molecules? (Two deuterons each, with two elec­trons, elec­tron­i­cally bound.) How about the bowl is very small, con­fin­ing the molecules, re­duc­ing their free­dom of move­ment, and their rel­a­tive mo­men­tum is, tran­siently, close to zero?

How many deuterons? Ini­tially, four, but … it’s been calcu­lated that af­ter a cou­ple of fem­tosec­onds, there are none, there is one ex­cited atom of Beryl­lium-8, which promptly de­cays into two he­lium nu­clei and a lot of en­ergy. In the­ory. It’s only been calcu­lated, it’s not been proven, it merely is a pos­si­ble ex­pla­na­tion for cer­tain ob­served phe­nom­ena. Heh!

The point here: the iden­tity of an ob­ject, the defi­ni­tion of “one,” is ar­bi­trary, a tool, a de­vice for or­ga­niz­ing our ex­pe­rience of re­al­ity. What if it’s two red ap­ples and two green ap­ples? They don’t taste the same and they don’t look the same, at least not en­tirely the same. What we are count­ing is the iden­ti­fied ob­ject, “ap­ple.” Not what ex­ists in re­al­ity. Real­ity ex­ists, not “ap­ples,” ex­cept in our ex­pe­rience, largely as a product of lan­guage.

The prop­er­ties of num­bers, so uni­ver­sally rec­og­nized, fol­low from the tools we evolved for pre­dict­ing be­hav­ior, they are cer­tainly not ab­solutes in them­selves.

Hah! “Cer­tainly.” That, with “be­lieve” is a word that sets off alarms.

• The fact that one ap­ple added to one ap­ple in­vari­ably gives two ap­ples....

It’s al­most a tau­tol­ogy. What we have is an iter­ated iden­ti­fi­ca­tion. There are two ob­jects that are named “ap­ple,” they are iden­ti­cal in iden­ti­fi­ca­tion, but sep­a­rate and dis­tinct. This ap­pears in time. I’m count­ing my iden­ti­fi­ca­tions. The uni­ver­sal­ity of 1+1 = 2 is a product of a sin­gle brain de­sign. For an elephant, the same “prob­lem” might be “food plus food equals food.”

• Ba­si­cally, you’re say­ing that for an elephant, ap­ples be­have like clouds, be­cause the elephant has a con­cept of ap­ple that is like our con­cept of cloud. (I hope real elephants aren’t this dumb). I like this a lot, it clar­ifies what I felt was miss­ing from the cloud anal­ogy.

Hav­ing it ex­plic­itly stated is helpful. It leads to the in­sight that at bot­tom, out­side of di­rectly use­ful con­cepts and into pure on­tol­ogy/​episte­mol­ogy, there are no iso­lated in­di­vi­d­ual in­te­gers. There is only rel­a­tive mag­ni­tude on a broad con­tinuum. This makes ap­proach­ing QM much sim­pler.

• Mmmm. This is all pro­jected onto elephants, but maybe some­thing like what you say. I was just point­ing to a pos­si­ble al­ter­nate pro­cess­ing mode. An elephant might well rec­og­nize quan­tity, but prob­a­bly not through count­ing, which re­quires lan­guage. Quan­tity might be rec­og­nized di­rectly, by vi­sual com­par­i­son, for ex­am­ple. Big­ger pile/​smaller pile. More at­trac­tion vs. less at­trac­tion, there­fore move­ment to­ward big­ger pile. Or smell.

• I can’t figure out why you’re get­ting down­votes though.

1. I’m do­ing some­thing right.

2. I’m do­ing some­thing wrong.

3. I write too much.

4. I don’t ex­plain well enough.

5. It’s Thurs­day.

6. I have a strange name.

7. I’m Mus­lim.

8. I’m sen­si­ble.

9. I’m not.

10. It means noth­ing, which also means noth­ing.

11. Some­thing else.

Thanks, chaos­mo­sis, that was a nice thing to say. ….

• (So far I’ve down­voted many of your com­ments that con­tained what I be­lieve to be con­fused/​mys­ti­cal think­ing, du­bi­ous state­ments of un­clear mean­ing that I ex­pect can’t be made clear by un­pack­ing (what­ever their po­etic qual­ities may be); also, for similar rea­sons, some con­ver­sa­tions that I didn’t like tak­ing place, mostly with chaos­mo­sis, where I down­voted both sides.)

• Thanks, Vladimir. From where does “what I be­lieve” and what “I ex­pect” come? What is the source of “I didn’t like”?

Would you be more spe­cific? It could be helpful. (Some­where if not here?)

I “re­tracted” the list post be­cause it had three net down­votes and to see what “re­tract” ac­com­plishes here, and be­cause I’m will­ing to re­tract any in­effec­tive com­mu­ni­ca­tion, “right” and “wrong” have al­most noth­ing to do with it. It was still a nice thing for chaos­mo­sis to say.

• You do write un­usu­ally long com­ments and it’s slightly ir­ri­tat­ing (al­though I have not down­voted you so far).

• Yeah, thanks, Ali­corn. I’ve been “con­ferenc­ing”—as we used to call it in the 80s—for a long time, and I know the prob­lem. I ac­tu­ally love the up/​down vot­ing sys­tem here. I gives me some fairly fast feed­back as to how I’m oc­cur­ring to oth­ers. I’m pri­mar­ily here to learn, and learn­ing to com­mu­ni­cate effec­tively in a new con­text has always brought re­wards to me.

Ah, one more thing I’ll risk adding here. This is a Yud­kowsky thread and dis­cussing my post­ing may be se­ri­ously off-topic. I need to pay more at­ten­tion to con­text.

• I need to pay more at­ten­tion to con­text.

LessWrong is like di­gres­sion cen­tral. Some­one will make a post talk­ing about evolu­tion­ary psy­chol­ogy, and they’ll men­tion bow and ar­rows in an ex­am­ple, and then some­one else will re­spond with a study about how bow and ar­eas weren’t used un­til X date, and then a de­bate will hap­pen, and then it will go meta, and then, etc.

• I down­voted this one. HAHAHAHA. Chaotic neu­tral, my friend as­so­ci­ate.

In se­ri­ous­ness it was lengthy and not su­per hu­morous. Also, you’re Mus­lim.

• 2 Nov 2012 1:33 UTC
0 points
Parent

Would you ar­gue, then, that aliens or AIs might not dis­cover the fact that 1 + 1 = 2, or even con­sider it a fact at all?

• Okay, I don’ t have to spec­u­late or ar­gue. I’m an alien, and I don’t con­sider it a “fact,” un­less fact is defined to in­clude the con­se­quences of lan­guage. I.e, as an alien, I can see your pro­cess, and, within your pro­cess, I see that “1 + 1 = 2″ is gen­er­ally use­ful to your sur­vival. That I’ll ac­cept as a fact. How­ever, if you be­lieve that 1 + 1 = 2 is a “fact,” such that 1 + 1 <> 2 is nec­es­sar­ily “false,” I think you might be un­nec­es­sar­ily limited, harm­ing long-term sur­vival.

It’s also use­ful to my sur­vival, nor­mally. Some­times not. Some­times 1 + 1 = 1, or 1 + 1 = 0, work bet­ter. I’m not kid­ding.

The AI worth think­ing about is one which is greater than hu­man, so that a hu­man can rec­og­nize the limi­ta­tion of fixed ar­ith­metic in­di­cates to me that a su­per-hu­man AI would be able to do that or more.

• 2 Nov 2012 1:32 UTC
−1 points
Parent

It seems to me like there are at least two pos­si­ble defi­ni­tions of “re­al­ity”. One is “the set of all true state­ments” (such as “the sky is blue”, “salt dis­solves in wa­ter”, and “2 + 2 = 4”), and the other is “the set of all ‘as­pects of the world’”—things like “in world W, there is an elec­tron at point p and time t”, whose truth could, in prin­ci­ple, be varied in­de­pen­dently of all other “as­pects of the world”.

The choice of defi­ni­tion seems more or less ar­bi­trary.

• Okay, I’ll ex­plore this. “Real­ity” is in­de­pen­dent of state­ments and lan­guage. An “as­pect” is a point of view, or a thing seen from some point of view.

To be fair, how­ever, I think Real­ity is not sus­cep­ti­ble to or­di­nary defi­ni­tion, it can only be pointed to, hinted at. Defi­ni­tions are in­deed ar­bi­trary, re­al­ity is not.

Or, stated an­other way, re­al­ity may be defined as that of which there is only one, that is not owned by us, but owns us, that ex­isted when we were not, that will con­tinue to ex­ist when we are no longer. In­di­vi­d­u­ally or all of us or the en­tire uni­verse.

On the other hand, re­al­ity does not ex­ist in the same way as we ex­ist or that things ex­ist. Every state­ment that at­tempts to cap­ture re­al­ity, at least so far as I’ve seen, fails.

Real­ity is as­sumed to ex­ist in ap­ply­ing the sci­en­tific method. We trust that there is a sin­gle re­al­ity, not many.

Un­der­stand­ings of re­al­ity are many. Opinions are many. Sys­tems of lan­guage are many..

Real­ity is nei­ther true nor false, rather we cre­ate true and false as re­la­tion­ships we in­vent be­tween state­ments and re­al­ity. Th­ese in­ven­tions are not ac­tu­ally true or false; they are use­ful or not-use­ful. A “truth” can be very use­ful fora time and turn out to be limit­ing as the scope of ap­pli­ca­tion of a state­ment is widened.

• Hu­mans need fan­tasy to be hu­man.

“Tooth fairies? Hog­fathers? Lit­tle—”

Yes. As prac­tice. You have to start out learn­ing to be­lieve the lit­tle lies.

“So we can be­lieve the big ones?”

Yes. Jus­tice. Mercy. Duty. That sort of thing.

“They’re not the same at all!”

You think so? Then take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of jus­tice, one molecule of mercy.

• Su­san and Death, in Hog­father by Terry Pratchett

So far we’ve talked about two kinds of mean­ingful­ness and two ways that sen­tences can re­fer; a way of com­par­ing to phys­i­cal things found by fol­low­ing pinned-down causal links, and log­i­cal refer­ence by com­par­i­son to mod­els pinned-down by ax­ioms. Is there any­thing else that can be mean­ingfully talked about? Where would you find jus­tice, or mercy?

• It so hap­pens that the three “big lies” death men­tions are all re­lated to moral­ity/​ethics, which is a hard ques­tion. But let me take the con­ver­sa­tion and change it a bit:

“So we can be­lieve the big ones?”

Yes. Anger. Hap­piness. Pain. That sort of thing.

“They’re not the same at all!”

You think so? Then take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of hap­piness, one molecule of pain.

In this ver­sion, the fi­nal ar­gu­ment is still cor­rect—if I take the uni­verse and grind it down to a sieve, I will not be able to say “woo! that car­bon atom is an atom of hap­piness”. Since the penul­ti­mate ques­tion of this med­i­ta­tion was “Is there any­thing else”, at least I can an­swer that ques­tion.

Clearly, we want to talk about hap­piness for many rea­sons—even if we do not value hap­piness in it­self (for our­selves or oth­ers), pre­dict­ing what will make hu­mans happy is use­ful to know stuff about the world. There­fore, it is use­ful to find a way that al­lows us to talk about hap­piness. Hap­piness, though, is com­pli­cated, so let us put it aside for a minute to pon­der some­thing sim­pler: a so­lar sys­tem. I will sim­plify here, a so­lar sys­tem is one star and a bunch of planets ro­tat­ing around it. Though so­lar sys­tems effect each other through grav­ity or ra­di­a­tion, most of the effects of the rel­a­tive mo­tions in­side a so­lar sys­tem comes from in­side it­self, and this pat­tern re­peats it­self through­out the galaxy. Much like hap­piness, be­ing able to talk about so­lar sys­tems is use­ful—though I do not par­tic­u­larly value so­lar sys­tems in and of them­selves, it’s use­ful to have a con­cept of “a so­lar sys­tem”, which de­scribes things with com­mon­al­ities, and al­lows me to gen­er­al­ize.

If I grind the uni­verse, I can­not find an atom that is a so­lar sys­tem atom—grind­ing the uni­verse down de­stroys the “so­lar sys­tem” use­ful pat­tern. For bounded minds, hav­ing these pat­terns leads to good pre­dic­tive strength with­out hav­ing to figure out each and ev­ery atom in the so­lar sys­tem.

In essence, hap­piness is no differ­ent than so­lar sys­tem—both are crude words to de­scribe com­mon pat­terns. It’s just that hap­piness is a fea­ture of minds (mostly hu­man minds, but we talk about how dogs or lizards are happy, some­times, and it’s not sur­pris­ing—those minds are re­lated al­gorithms). I can­not say where ev­ery atom is in the case of a hu­man be­ing happy, but some atom con­figu­ra­tions are happy hu­mans, and some are not.

So: at the very least, hap­piness and so­lar sys­tems are part of the causal net­work of things. They de­scribe pat­terns that in­fluence other pat­terns.

Mercy is eas­ier than jus­tice and duty. Mercy is a spe­cific con­figu­ra­tion of atoms be­hav­ing a hu­man in a spe­cific way—even though the hu­man feels they are en­ti­tled to cause an­other hu­man hurt (“feel­ing en­ti­tled” is a set of spe­cific hu­man-mind-con­figu­ra­tions, re­gard­less of whether “en­ti­tle­ment” ac­tu­ally ex­ists), but does not do so (for spe­cific rea­sons, etc. etc.). In short, mercy de­scribes spe­cific pat­terns of atoms, and is part of causal net­works.

Duty and jus­tice—I ad­mit that I’m not sure what my re­duc­tion­ist metaethics are, and so it’s not ob­vi­ous what they mean in the causal net­work.

• We could make it even eas­ier :P

You say that a tiger has stripes, but I looked at some tiger atoms and didn’t see any stripes.

The harder ques­tion is what is a valid way of figur­ing out the im­por­tant prop­er­ties of the sys­tem.

• (Note: this is my first post. I may be wrong, and if so am cu­ri­ous as to how. Any­way, I figure it’s high time that my be­liefs stick their neck out. I ex­pect this will hurt, and apol­o­gize now should I later re­spond poorly.)

This may be the an­swer to a differ­ent ques­tion, but...

I play lots of role-play­ing games. Role-play­ing games are like make-be­lieve; events in them ex­ist in a shared counter-fac­tual space (in the play­ers’ imag­i­na­tion). Make-be­lieve has a prob­lem: if two peo­ple imag­ine differ­ent things, who is right? (This tends to end with a bunch of kids ar­gu­ing about whether the fic­tional T-Rex is al­ive or dead).

Role-play­ing games solve this prob­lem by hand­ing au­thor­ity over var­i­ous facets of the game to differ­ent things. The pro­tag­o­nists are con­trol­led by their re­spec­tive play­ers, the re­sults of choices by dice and rules, and most of the fic­tional world by the Game Master.*

So, in a role-play­ing game, when you ask what is true[RPG], you should di­rect that ques­tion to the ap­pro­pri­ate au­thor­ity. Ba­si­cally, truth[RPG] is ac­tu­ally canon (in the fan­dom sense; TV Trope’s page is good, but comes with the usual where-did-my-evening-go caveats).

Similarly, if we ask “where did Luke Sky­walker go to preschool?”, we’re ask­ing a ques­tion about canon.

That said, even canon needs to be in­ter­nally con­sis­tent. If some­one with au­thor­ity were to claim that Ta­tooine has no preschools, then we can con­clude that Luke Sky­walker didn’t go to preschool. If an au­thor­ity claims two in­con­sis­tent things, we can con­clude that the au­thor­ity is wrong (namely, in the math­e­mat­i­cal sense the canon wouldn’t match any pos­si­ble model).

I’ve long felt that ideas like moral­ity and liberty are a va­ri­ety of canon.

Speci­fi­cally, you can have au­thor­i­ties (a re­li­gion or philoso­pher tel­ling you stuff), and those au­thor­i­ties can be prov­ably wrong (be­cause they said some­thing in­con­sis­tent), but these ideas ex­ists in a kind of shared imag­i­nary space. Also, peo­ple can dis­agree with the canon and make up their own ideas.

Now, that space is still in­formed by re­al­ity. Even in fic­tion, we ex­pect grav­ity to drop off as the square of dis­tance, and we ex­pect solid ob­jects to be un­able to pass through each other.** With ideas, we can state that they are non­sen­si­cal (or, at min­i­mum, not use­ful) if they refers to real things which don’t ex­ist. A map of moral­ity is a map of a non-real thing, but moral­ity must in­ter­face with re­al­ity to be use­ful, so any­where the in­ter­face doesn’t line up with re­al­ity, moral­ity (or its map) is wrong.

*This is one pos­si­ble break­down. There are many oth­ers.

**In most games/​sto­ries, any­way. At first glance I’d ex­pect moral­ity to be bet­ter bound to re­al­ity, but I sup­pose there’s been plenty of peo­ple who’s moral sys­tem boiled down to “don’t do any­thing Ma’at would dis­ap­prove of”, backed up with con­cepts like the literal weight of sin (vs. the weight of a feather).

• I was go­ing to say that yes, I think there is an­other kind of thing that can be mean­ingfully talked about, and “jus­tice” and “mercy” and “duty” have some­thing to do with that sort of thing, but a more pro­to­typ­i­cal ex­am­ple would be “This court has ju­ris­dic­tion”. Espe­cially if many ex­perts were of the opinion that it didn’t, but the judge dis­agreed, but the su­pe­rior court re­versed her, and now the supreme court has de­cided to hear the case.

But then I re­al­ized that there was some­thing differ­ent about that kind of “truth”: I would not want an AI to as­sign a prob­a­bil­ity to the propo­si­tion The court did, in fact, have ju­ris­dic­tion (nor to, oh, It is the duty of any elected offi­cial to tell the pub­lic if they learn about a case of cor­rup­tion, say). I think so­cial con­struc­tions can tech­ni­cally be mean­ingfully talked about among hu­mans, and they are im­por­tant as hell if you want to un­der­stand hu­man com­mu­ni­ca­tion and be­hav­ior, but I guess on re­flec­tion I think that the fact that I would want an AI to rea­son in terms of more ba­sic facts is a hint that if we are dis­cussing episte­mol­ogy, if we’re dis­cussing what sorts of thin­gies we can know about and how we can know about them, rather than dis­cussing par­tic­u­lar prop­er­ties of the par­tic­u­larly in­ter­est­ing thin­gies called hu­mans, then it might be best to say that “The judge wrote in her de­ci­sion that the court had ju­ris­dic­tion” is a mean­ingful state­ment in the sense un­der con­sid­er­a­tion, but “The court had ju­ris­dic­tion” is not.

• The state­ment that the world is just is a lie. There ex­ist pos­si­ble wor­lds that are just—for in­stance, these wor­lds would not have chil­dren kid­napped and forced to kill—and ours is not one of them.

Thus, jus­tice is a mean­ingful con­cept. Jus­tice is a con­cept defined in terms of the world (pinned-down causal links) and also ir­re­ducibly nor­ma­tive state­ments. Nor­ma­tive state­ments do not re­fer to “the world”. They are use­ful be­cause we can log­i­cally de­duce im­per­a­tives from them. “If X is just, then do X.” is cor­rect, that is:

Do the right thing.

• I am not en­tirely sure how you ar­rived at the con­clu­sion that jus­tice is a mean­ingful con­cept. I am also un­clear on how you know the state­ment “If X is just, then do X” is cor­rect. Could you elab­o­rate fur­ther?

In gen­eral, I don’t think it is a suffi­cient test for the mean­ingful­ness of a prop­erty to say “I can imag­ine a uni­verse which has/​lacks this prop­erty, un­like our uni­verse, there­fore it is mean­ingful.”

• the state­ment “If X is just, then do X”

That’s an in­struc­tion, not a state­ment.

• I did not in­tend to ex­plain how i ar­rived at this con­clu­sion. I’m just stat­ing my an­swer to the ques­tion.

Do you think the state­ment “If X is just, then do X” is wrong?

• Like army1987 notes, it is an in­struc­tion and not a state­ment. Con­sid­er­ing that, I think “if X is just, then do X” is a good im­per­a­tive to live by, as­sum­ing some good defi­ni­tion of jus­tice. I don’t think I would de­scribe it as “wrong” or “cor­rect” at this point.

• OK. Ex­actly what you call it is unim­por­tant.

What mat­ters is that it gives jus­tice mean­ing.

• It may be in­com­plete. Do you have a place for Mercy?

• The rea­son I’m not mak­ing dis­tinc­tions among differ­ent moral words, though such dis­tinc­tions ex­ist in lan­guage, is that it seems the only new prob­lem cre­ated by these moral words is un­der­stand­ing moral­ity. Once you un­der­stand right and wrong, just and un­just can be defined just like you define reg­u­lar words, even if some­thing can be just but im­moral.

• In gen­eral, I don’t think it is a suffi­cient test for the mean­ingful­ness of a prop­erty to say “I can imag­ine a uni­verse which has/​lacks this prop­erty, un­like our uni­verse, there­fore it is mean­ingful.”

Um, math­e­mat­ics.

• I can’t imag­ine a uni­verse with­out math­e­mat­ics, yet I think math­e­mat­ics is mean­ingful. Doesn’t this mean the test is not suffi­cient to de­ter­mine the mean­ingful­ness of a prop­erty?

Is there some es­tab­lished think­ing on al­ter­nate uni­verses with­out math­e­mat­ics? My failure to imag­ine such uni­verses is hardly con­clu­sive.

• Sorry, mis­read what you wrote in the grand par­ent. I agree with you.

• I would find them un­der the cat­e­gory of pat­terns.

A neu­ral net­work is very good at recog­nis­ing pat­terns; and hu­man brains run on a neu­ral net­work ar­chi­tec­ture. Given a few ex­am­ples of what a word does or does not mean, we can quickly recog­nise the pat­tern and fit it into our vo­cab­u­lary. (Ap­par­ently, this can be used in lan­guage classes; the teacher will point to a va­ri­ety of ob­jects, in­di­cat­ing whether they are or are not vrugte, for ex­am­ple; and it won’t take that many ex­am­ples be­fore the stu­dent un­der­stands that vrugte means fruit but not veg­eta­bles).

Jus­tice and mercy are not pat­terns of ob­jects, but rather pat­terns of ac­tion. The man kil­led his en­emy, but has a wife and chil­dren to sup­port; send­ing him to Death Row might be just, but let­ting him have some way of earn­ing money while im­pris­oned might be mer­ciful. Similarly, happy, sad, and an­gry are emo­tional pat­terns; a per­son acts in this way when happy, and acts in that way when sad.

• They ex­ist in the same sense that num­bers ex­ist, or that mean­ingful ex­is­tence ex­ists, or that mean­ingful­ness ex­ists.

Once you grind the uni­verse into pow­der, none of those things ex­ists any­more.

• I’ve thought about this for a while, and I feel like you can re­place “Fan­tasy” and “Lies” with “Pat­terns” in that di­alogue, and have it make sense, and it also ap­pears to be an an­swer to your ques­tions. That be­ing said, it also feels like a sort of a cached thought, even though I’ve thought about it for a while. How­ever, I can’t think of a bet­ter way to ex­press it and all of the other thoughts I had ap­peared to be sig­nifi­cantly lower cal­iber and less clear.

Con­sid­er­ing that, I should then ask “Why isn’t ‘Pat­terns’ the an­swer?′

• “Jus­tice” and “mercy” can be found by look­ing at peo­ple, and in par­tic­u­lar how peo­ple treat each other. They’re phys­i­cal things, al­though they’re re­ally com­pli­cated kinds of phys­i­cal things.

• In par­tic­u­lar, the kind of thing that is de­stroyed when you grind it down into pow­der.

Hu­mans need fan­tasy to be hu­man.

“Tooth fairies? Hog­fathers? Lit­tle—”

Yes. As prac­tice. You have to start out learn­ing to be­lieve the lit­tle lies.

“So we can be­lieve the big ones?”

Yes. Cars. Chairs. Bi­cy­cles. That sort of thing.

“They’re not the same at all!”

You think so? Then take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of car, one molecule of bi­cy­cle.

Su­san and Death, in Hog­father by Terry Pratchett

Same thing.

• Jus­tice, mercy, duty, etc are found by com­par­i­son to log­i­cal mod­els pinned down by ax­ioms. Get­ting the ax­ioms right is damn tough, but if we have a de­cent set we should be able to say “If Alex kills Bob un­der cir­cum­stances X, this is un­just.” We can say this the same way that we can say “Two ap­ples plus two ap­ples is four ap­ples.” I can’t find an atom of ad­di­tion in the uni­verse, and this doesn’t make me re­ject ad­di­tion.

Also, the wide­spread con­ver­gence of the­o­ries of jus­tice on some is­sues (eg. Rape is un­just.) sug­gests that the­o­ries of jus­tice are at­tempt­ing to use their ax­ioms to pin down some­thing that is already there. Mo­ral philoso­phers are more likely to say “My ax­ioms are lead­ing me to con­clude rape is a moral duty, where did I mess up?” than “My ax­ioms are lead­ing me to con­clude rape is a moral duty, there­fore it is.” This also sug­gests they are pin­ning down some­thing real with ax­ioms. If it was oth­er­wise, we would ex­pect the sec­ond con­clu­sion.

• “the­o­ries of jus­tice are at­tempt­ing to use their ax­ioms to pin down some­thing that is already there”

So in other words, duty, jus­tice, mercy—moral­ity words—are ba­si­cally log­i­cal trans­for­ma­tions that trans­form the state of the uni­verse (or a par­tic­u­lar cir­cum­stance) into an ought state­ment.

Just as we de­rive valid con­l­cu­sions from premises us­ing log­i­cal state­ments, we de­rive moral obli­ga­tions from premises us­ing moral state­ments.

The term ‘util­ity fun­cion’ seems less novel now (novel as in, a de­par­ture from tra­di­tional ethics).

• Not quite. They don’t go all the way to com­plet­ing an ought state­ment, as this doesn’t solve the Is/​Ought di­chotomy. They are log­i­cal trans­for­ma­tions that make ap­ply­ing our val­ues to the uni­verse much eas­ier.

“X is un­just” doesn’t quite cre­ate an ought state­ment of “Don’t do X”. If I place value on jus­tice, that state­ment helps me eval­u­ate X. I may de­cide that some other con­sid­er­a­tion trumps jus­tice. I may de­cide to steal bread to feed my starv­ing fam­ily, even if I view the theft as un­just.

• This is my view.

• In peo­ple’s brains, and in pa­pers writ­ten by philos­o­phy stu­dents.

• The map is not the ter­ri­tory. We dis­cuss re­al­ity on many lev­els, but there is only one un­der­ly­ing level. Jus­tice, duty and the like are ab­strac­tions; we use the same sym­bol in mul­ti­ple places to define cer­tain pat­terns. You don’t get two iden­ti­cal ‘hap­pinesses’, like you get two iden­ti­cal atoms. It’s use­ful for us though, to talk about this ab­strac­tion at the macro level and not the micro, and it’s mean­ingful, given that we’re as­sum­ing the same ax­ioms. I think, stuff that causes other stuff is re­al­ity, and if we as­sume cer­tain ax­ioms that cor­re­spond to re­al­ity, any new truth­ful state­ments and con­cepts de­duced are mean­ingful be­cause they also cor­re­spond to re­al­ity. Every­thing there is cov­ered. Things that ex­ists, and things we think ex­ists.

• Math­e­mat­ics is a sys­tem for build­ing ab­stract state­ments that can be mapped to re­al­ity. The ax­ioms of a math­e­mat­i­cal (or other ax­io­matic) model define the con­di­tions that a sys­tem (such as a pair of ap­ples in the real uni­verse) must satisfy in or­der for the ab­stract model to be ap­pli­ca­ble as well as pro­vid­ing a schema for map­ping the ab­stract model to the con­crete sys­tem.

There are other kinds of ab­strac­tions we could mean­ingfully talk about and they need not be defined as pre­cisely as an ax­io­matic model like math­e­mat­ics. An ab­stract model could be defined as a re­la­tion­ship be­tween ab­stract ideas that can be mapped to a con­crete sys­tem by pin­ning down each of its con­stituent ab­strac­tions to a con­crete mem­ber of the sys­tem.

An ab­stract model may be pre­dic­tive, mean­ing it has an if-then struc­ture: if some re­la­tion be­tween ab­stract mem­bers holds then the model pre­dicts that some other re­la­tion will also hold. Such a pre­dic­tive model may be true or false for any given con­crete sys­tem that it is ap­plied to. The stan­dard we ex­pect of a math­e­mat­i­cal model is that it is valid (true for all con­crete sys­tems that it can be ap­plied to), yet an ab­stract model need not meet so high a stan­dard for it to be use­ful. We can imag­ine much fuzzier ab­stract mod­els that are true only some of the time but can be use­ful by pro­vid­ing gen­eral-pur­pose rules that al­low us to in­fer in­for­ma­tion about the ac­tual state of a con­crete sys­tem that matches the crite­ria of the model. If we know the prob­a­bil­ity of an ab­stract pre­dic­tive model be­ing cor­rect we can use it wher­ever it is ap­pli­ca­ble to in­form the con­struc­tion of causal mod­els. If we con­sider causal mod­els to op­er­ate in the realm of first or­der logic where we can quan­tify over and de­scribe re­la­tion­ships be­tween the ba­sic units of cause and effect in our uni­verse, an ab­stract model lives in the realm of higher or­der logic and can de­scribe the re­la­tion­ships be­tween causal re­la­tions and lower or­der ab­stract mod­els.

An ab­stract model need not be pre­dic­tive to be use­ful. It may be defined to be ap­pli­ca­ble only where the en­tire re­la­tion it de­scribes holds. In this case it sim­ply acts as a reusable sym­bol that is use­ful for rep­re­sent­ing a model of a con­crete sys­tem more com­pactly as in the way a func­tion in a com­puter pro­gram fac­tors out reusable logic, or a word in hu­man lan­guage fac­tors out a reusable ab­stract idea.

Jus­tice and Mercy are both fuzzy ab­stract mod­els. To the ex­tent that peo­ple agree on their defi­ni­tions they are mean­ingful for com­mu­ni­cat­ing a par­tic­u­lar re­la­tion­ship be­tween pinned-down ab­strac­tions. For ex­am­ple, Jus­tice may be defined (sim­plis­ti­cally) as de­scribing a re­la­tion­ship be­tween hu­man deeds and sub­se­quent events such that deeds la­bel­led ‘bad’ re­sult in pun­ish­ment and deeds la­bel­led ‘good’ re­sult in re­ward. The par­tic­u­lar deed and sub­se­quent event as well as the defi­ni­tions of good, bad, pun­ish­ment and re­ward are all com­po­nent ab­strac­tions of the ab­stract model called Jus­tice which must be pinned down in a con­crete sys­tem in or­der for the con­cept of Jus­tice to be ap­plied in that sys­tem.

Jus­tice may also be used as a pre­dic­tive model if you for­mu­late it as a pre­dic­tion from a good/​bad deed to a fu­ture re­ward/​pun­ish­ment event (or vice versa) and it would be use­ful for con­struct­ing a causal model of any par­tic­u­lar con­crete sys­tem to the ex­tent that this pre­dicted re­la­tion­ship matches the ac­tual un­der­ly­ing na­ture of that sys­tem.

Note: none of this is based on any for­mal study of logic out­side of this Episte­mol­ogy se­quence so some of the ter­minol­ogy in this post was in­vented by me just now.

• Right, re­sponse to the med­i­ta­tion:

It gets rather difficult talk­ing about hu­man men­tal con­structs, let’s be­gin by ask­ing my­self where would I find jus­tice/​mercy; al­most im­me­di­ately (which means that I need to do some more think­ing) I find that I think of hu­man emo­tional con­structs as a side effect of so­ciety and it’s group mind­set,

You think so? Then take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of jus­tice, one molecule of mercy.

• Su­san and Death, in Hog­father by Terry Pratchett

I would find that by grind­ing down the uni­verse to it’s com­po­nent molecules would com­pletely fail to find any num­ber of things that hu­man­ity finds im­por­tant; Hu­man­ity for one, to me ra­tio­nal­ism is, above all the study of the uni­verse and what it con­tains. And yet when it comes to most psy­cholog­i­cal phe­nomenon the mod­els start to break down, does this mean that a more re­fined model would be equally un­able to de­scribe the phe­nomenon, not nec­es­sar­ily. Be­cause as ra­tio­nal­ists one of our key teach­ings is that we can ob­serve some­thing by study­ing it’s causes and effects; jus­tice and mercy ex­ists in­so­far as we as hu­mans can com­pre­hend their na­ture. They ex­ist be­cause we can de­ter­mine the differ­ences be­tween a uni­verse where they ex­ist and the ones where they don’t ex­ist.

-re­posted in the right section

• ...Then take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of jus­tice, one molecule of mercy.

Then take the uni­verse and grind it down to the finest pow­der and sieve it through the finest sieve and then show me one atom of tem­per­a­ture, one molecule of pres­sure.

• So far we’ve talked about two kinds of mean­ingful­ness and two ways that sen­tences can re­fer; a way of com­par­ing to phys­i­cal things found by fol­low­ing pinned-down causal links, and log­i­cal refer­ence by com­par­i­son to mod­els pinned-down by ax­ioms. Is there any­thing else that can be mean­ingfully talked about? Where would you find jus­tice, or mercy?

You find them in­side coun­ter­fac­tual state­ments about the re­ac­tions of an im­plied hy­po­thet­i­cal rep­re­sen­ta­tive hu­man, judg­ing un­der un­der im­plied hy­po­thet­i­cal cir­cum­stances in which they have ac­cess to all rele­vant knowl­edge. There is clearly jus­tice if a wide va­ri­ety of these hy­po­thet­i­cal hu­mans agree that there is, un­der a wide va­ri­ety of these hy­po­thet­i­cal cir­cum­stances; there is clearly not jus­tice if they agree that there is not. If the hy­po­thet­i­cal peo­ple dis­agree with each other, then the defi­ni­tion fails.

Talk­ing about things like jus­tice, mercy and duty is mean­ingful, but the mean­ings are in­ter­me­di­ated by big, com­plex webs of ab­strac­tions which hu­mans keep in their brains, and the al­gorithms peo­ple use to ma­nipu­late those webs. They’re un­am­bigu­ous only to the ex­tent to which peo­ple suc­cess­fully keep those webs in sync with each other. In prac­tice, our ab­strac­tions mainly work by com­bin­ing bags of weak clas­sifiers and fea­ture-weighted similar­ity to pos­i­tive and nega­tive ex­am­ples. This works bet­ter for cases that are similar to the train­ing set, worse for cases that are novel and weird, and bet­ter for sim­pler ab­strac­tions and ab­strac­tions built on sim­pler con­stituents.

• Why couldn’t the hy­po­thet­i­cal om­ni­scient peo­ple in­side the veil of ig­no­rance de­cide that jus­tice doesn’t ex­ist? Or if they could, how does that para­graph go to­wards an­swer­ing the med­i­ta­tion? What dis­t­in­guishes them from the hy­po­thet­i­cal death who looks through ev­ery­thing in the uni­verse to try to find mercy? Aren’t you beg­ging the ques­tion here?

• 2 Nov 2012 2:34 UTC
−1 points
Parent

Jus­tice—The qual­ity of be­ing “just” or “fair”. Hu­mans call a situ­a­tion fair when ev­ery­one in­volved is happy af­ter­wards, with­out hav­ing had their de­sires forcibly thwarted (e.g. be­ing strapped into a chair and hooked into a mor­phine drip) along the way.

Mercy—Com­pas­sion­ate or kindly for­bear­ance shown to­ward an offen­der, an en­emy, or other per­son in one’s power. Hu­mans choose to en­gage in ac­tions char­ac­ter­ized this way on a daily ba­sis.

Duty—Some­thing that one is ex­pected or re­quired to do by moral or le­gal obli­ga­tion. Le­gal du­ties cer­tainly ex­ist; Earth is not an an­ar­chy.

Jus­tice, mercy, and duty are only words. The im­por­tant ques­tion to ask is whether or not they are use­ful. I cer­tainly think they are; I use each of those words at least once a week. Once the sym­bols have been re­placed by sub­stance, it is clear that we should not be look­ing for those things in sin­gle atoms, but very large col­lec­tions of them we call “hu­mans”, or slightly smaller (but still very large) col­lec­tions we call “hu­man brains”.

And as far as we know, atoms are not ar­ranged in con­figu­ra­tions that have the prop­er­ties we as­cribe to the tooth fairy.

• This de­pends a lot on your defi­ni­tion of mean­ingful­ness. Jus­tice and mercy are sub­jec­tive val­ues and not pre­dic­tive or de­scrip­tive state­ments about re­al­ity. But in my opinion sub­jec­tive val­ues are mean­ingful, in fact they’re mean­ing it­self and the only rea­son I con­sider de­scrip­tive state­ments about re­al­ity to be mean­ingful is that they help me achieve sub­jec­tive val­ues. I be­lieve that sub­jec­tive val­ues are ob­jec­tively valuable, or that the con­cept of ob­jec­tive value would make no sense, whichever you pre­fer. Changes in my be­liefs can­not change my fun­da­men­tal val­ues and my fun­da­men­tal val­ues are mo­ti­va­tional in a way that be­liefs are not, so I con­sider the fun­da­men­tal val­ues to be of prior im­por­tance.

RE: Sta­bil­ity of logic. Logic might not be sta­ble or it might change later, we don’t have any way of know­ing and the ques­tion isn’t use­ful and be­liev­ing in logic makes me happy and gives me re­wards.

• the con­cept of ob­jec­tive value would make no sense,

if your moral val­ues aren’t ob­jec­tive, why would any­one else be be­holden to them? And how could they be moral if they don’t reg­u­late oth­ers’ be­havi­our?

Logic might not be sta­ble or it might change later, we don’t have any way of know­ing

Why would it change, ab­sent our chang­ing the ax­ioms? Do you think it is part of the uni­verse?

• if your moral val­ues aren’t ob­jec­tive, why would any­one else be be­holden to them? And how could they be moral if they don’t reg­u­late oth­ers’ be­havi­our?

To the first ques­tion: Pos­si­bly be­cause your moral val­ues arose from a pro­cess that was al­most ex­actly the same for other in­di­vi­d­u­als, and such it’s rea­son­able to in­fer that their moral val­ues might be rather similar than com­pletely alien?

To the sec­ond: “And how could they be (blank?) if they don’t reg­u­late oth­ers’ be­havi­our?”, by which I mean, what do you mean by “moral”? What makes a value a “moral” value or not in this con­text?

I’m not sure why it should be nec­es­sary for a moral value to reg­u­late be­havi­our across in­di­vi­d­u­als in or­der to be valid.

• Pos­si­bly be­cause your moral val­ues arose from a pro­cess that was al­most ex­actly the same for other in­di­vi­d­u­als,

Why de­scribe them as sub­je­cive when they are in­ter­sub­jec­tive?

I’m not sure why it should be nec­es­sary for a moral value to reg­u­late be­havi­our across in­di­vi­d­u­als in or­der to be valid.

It would be nec­es­sary for them to be moral val­ues and not some­thing else, like aes­thetic val­ues. Be­cause moral­ity is largely to reg­u­late in­ter­ac­tions be­tween in­di­vi­d­u­als. That’s its job. Aes­thet­ics is there to make things beau­tiful, logic is there to work things out...

• moral­ity is largely to reg­u­late in­ter­ac­tions be­tween in­di­vi­d­u­als. That’s its job.

I don’t want to get into a dis­cus­sion of this, but if there’s an es­say-length-or-less ex­pla­na­tion you can point to some­where of why I ought to be­lieve this, I’d be in­ter­ested.

• I dont see that “moral­ity is largely to reg­u­late in­ter­ac­tions be­tween in­di­vi­d­u­als” is con­tentious. Did you have an­other job in mind for it?

• Well, since you ask: iden­ti­fy­ing right ac­tions.

But, as I say, I don’t want to get into a dis­cus­sion of this.

I cer­tainly agree with you that if there ex­ists some thing whose pur­pose is to reg­u­late in­ter­ac­tions be­tween in­di­vi­d­u­als, then it’s im­por­tant that that thing be com­pel­ling to all (or at least most) of the in­di­vi­d­u­als whose in­ter­ac­tions it is in­tended to reg­u­late.

• Well, since you ask: iden­ti­fy­ing right ac­tions.

Is that an end in it­self?

I cer­tainly agree with you that if there ex­ists some thing whose pur­pose is to reg­u­late in­ter­ac­tions be­tween in­di­vi­d­u­als, then it’s im­por­tant that that thing be com­pel­ling to all (or at least most) of the in­di­vi­d­u­als whose in­ter­ac­tions it is in­tended to reg­u­late.

Well, the law com­pells those who ar­ent com­pel­led by ex­hor­ta­tion. But laws need justii­ca­tion.

• Is that an end in it­self?

Not for me, no.

Is reg­u­lat­ing in­ter­ac­tions be­tween in­di­vi­d­u­als an end in it­self?

• Do you think it is pointless? Do you think it is a pre­lude to some­thing else>?

• I think iden­ti­fy­ing right ac­tions can be, among other things, a pre­lude to act­ing rightly.

Is reg­u­lat­ing in­ter­ac­tions be­tween in­di­vi­d­u­als an end in it­self?

• Is that an end in it­self?

What does that con­cept even mean? Are you ask­ing if there’s a moral obli­ga­tion to im­prove one’s own un­der­stand­ing of moral­ity?

Well, the law com­pells those who ar­ent com­pel­led by ex­hor­ta­tion. But laws need justii­ca­tion.

The jus­tifi­ca­tion for laws can be a com­bi­na­tion of prag­ma­tism and the val­ues of the ma­jor­ity.

• Does it serve a pur­pose by it­self? Judg­ing ac­tions to be right or wrong is ususally the pre­lude to had­nig out praise and blame, re­ward and pun­ish­ment.

The jus­tifi­ca­tion for laws can be a com­bi­na­tion of prag­ma­tism and the val­ues of the ma­jor­ity.

if the val­ues of the ma­jor­ity ar­ent jus­tified, how does thast jus­tify laws?

• Judg­ing ac­tions to be right or wrong is ususally the pre­lude to had­nig out praise and blame, re­ward and pun­ish­ment.

Also, some­times it’s a pre­lude to act­ing rightly and not act­ing wrongly.

• Nope. An agent with­out a value sys­tem would have no pur­pose in cre­at­ing a moral sys­tem. An agent with one might find it in­trin­si­cally valuable, but I per­son­ally don’t. I do find it in­stru­men­tally valuable.

Laws are jus­tified be­cause sub­jec­tive de­sires are in­her­ently jus­tified be­cause they’re in­her­ently mo­ti­va­tional. Many peo­ple re­verse the bur­den of proof, but in the real world it’s your logic that has to jus­tify it­self to your val­ues rather than your val­ues that have to jus­tify them­selves to your logic. That’s the way we’re de­signed and there’s no get­ting around it. I pre­fer it that way and that’s its own jus­tifi­ca­tion. Ab­stract lies which make me happy are bet­ter than truths that make me sad be­cause the con­cept of bet­ter it­self man­dates that it be so.

• Clar­ifi­ca­tion: From the per­spec­tive of a minor­ity, the laws are un­jus­tified. Or, they’re jus­tified, but still un­de­sir­able. I’m not sure which. Jus­tifi­ca­tion is an awk­ward paradigm to work within be­cause you haven’t proven that the con­cept makes sense and you haven’t pre­cisely defined the con­cept.

• Jus­tifi­ca­tion is an awk­ward paradigm to work within be­cause you haven’t proven that the con­cept makes sense.

Proof is a strong form of jus­tifi­ca­tion. If i don;t have jus­tifi­ca­tion, you don″t have proof.

• From the per­spec­tive of a minor­ity, the laws are un­jus­tified.

Why would the ma­jor­ity re­gard them as jus­tifed just be­cause they hap­pen to have them?

• They’re jus­tified in that there are no ar­gu­ments which ad­e­quately re­fute them, and that they’re mo­ti­vated to take those ac­tions. There are no ar­gu­ments which can re­fute one’s mo­ti­va­tions be­cause facts can only in­fluence val­ues via val­ues. Mo­ti­va­tions are what de­ter­mine ac­tions taken, not facts. That is why perfectly ra­tio­nal agents with iden­ti­cal knowl­edge but differ­ent val­ues would re­spond differ­ently to cer­tain data. If a babykil­ler learned about a baby they would eat it, if I learned about a baby I would give it a hug.

In terms of fram­ing, it might help you un­der­stand my per­spec­tive if you try not to think of it in terms of past atroc­i­ties. Think in terms of some­thing more neu­tral. The ma­jor­ity wants to make a gi­ant statue out of pur­ple bub­blegum, but the minor­ity wants to make a statue out of blue cot­ton candy, for ex­am­ple.

• I need a defi­ni­tion of jus­tified be­fore I go on

Well, it’s ike proof, but weaker.

They’re jus­tified in that there are no ar­gu­ments which ad­e­quately re­fute them, and that they’re mo­ti­vated to have them.

Lack of coun­ter­ar­gu­ment is not jus­tifi­ca­tion, nor is mo­ti­va­tion from some pos­si­ble ir­raitonal source.

The ma­jor­ity wants to make a gi­ant statue out of pur­ple bub­blegum, but the minor­ity wants to make a statue out of blue cot­ton candy, for ex­am­ple.

Or the ma­jor­ity want to shoot all left handed peo­ple, for ex­am­ple. Ma­jor­ity ver­dict isn’t even close to moral jus­tifi­ca­tion.

• Lack of coun­ter­ar­gu­ment is not jus­tifi­ca­tion, nor is mo­ti­va­tion from some pos­si­ble ir­raitonal source.

In util­i­tar­ian terms, mo­ti­va­tion is not “I’m mo­ti­vated to­day!”. The util­i­tar­ian mean­ing of mo­ti­va­tion is that a pro­gram which dis­plays “Hello World!” on a com­puter screen has for (ex­clu­sive) mo­ti­va­tion to do the ex­act pro­cess which makes it dis­play those words. The mo­ti­va­tion of this pro­gram is im­per­a­tive and ridicu­lously sim­ple and very blunt—it’s the pat­tern we’ve built into the com­puter to do cer­tain things when it gets cer­tain elec­tronic in­puts.

Mo­ti­va­tions are those core things which liter­ally cause ac­tions, whether it’s a sim­ple re­flex built into the ner­vous sys­tem which always causes some jolt of move­ment when­ever a cer­tain thing hap­pens (such as be­ing hit on the knee) or a very com­plex value sys­tem send­ing in­terfer­ing sig­nals within trillions of cells caus­ing a gi­ant an­i­mal to move one way or an­other de­pend­ing on the re­sult­ing out­come.

• I know.

• Mo­ti­va­tion is the only thing that causes ac­tions, it’s the only thing that it makes sense to talk about in refer­ence to pre­scrip­tive state­ments. Why do you define mo­ti­va­tion as ir­ra­tional? At worst, it should be ar­ra­tional. Even then, I see mo­ti­va­tion as its own jus­tifi­ca­tion and in­deed the ul­ti­mate source of all jus­tifi­ca­tions for be­lief in truth, etc. Un­til you can solve ev­ery para­dox ever, you need to ei­ther em­brace nihilism or em­brace sub­jec­tive value as the foun­da­tion of jus­tifi­ca­tion.

The ma­jor­ity ver­dict isn’t moral jus­tifi­ca­tion be­cause moral­ity is sub­jec­tive. But for peo­ple within the ma­jor­ity, their de­ci­sion makes sense. If I were in the com­mu­nity, I would do what they do. I be­lieve that it would be morally right for me to do so. Values are the only source of moral­ity that there is.

• Mo­ti­va­tion is the only thing that causes ac­tions, it’s the only thing that it makes sense to talk about in refer­ence to pre­scrip­tive state­ments.

That doesn’t fol­low. If it is the only thing that causes ac­tions, then it is rele­vant to why, as a mat­ter of fact, peo­ple do what they do—but that is de­scrip­tion, not pre­scrip­tion. Pre­scrip­tion re­quires ex­tra in­gre­di­ents.

Why do you define mo­ti­va­tion as ir­ra­tional?

I said that as a mat­ter fof fact it is not nec­es­sar­ily ra­tio­nal. My grounds are that you cna’t always ex­plain you mo­ti­va­tions on a ra­tioanl ba­sis.

Even then, I see mo­ti­va­tion as its own jus­tifi­ca­tion and in­deed the ul­ti­mate source of all jus­tifi­ca­tions for be­lief in truth, etc.

It may be the source of car­ing about truth and ra­tio­nal­ity. That does not mke it the source of truth and ra­tio­nal­ity.

Un­til you can solve ev­ery para­dox ever, you need to ei­ther em­brace nihilism or em­brace sub­jec­tive value as the foun­da­tion of jus­tifi­ca­tion.

That doens’t fol­low. I could em­brace non-eval­u­a­tive in­tu­tions, for in­stance.

The ma­jor­ity ver­dict isn’t moral jus­tifi­ca­tion be­cause moral­ity is sub­jec­tive.

Sub­jec­tive moral­ity can­not jus­tify laws tha pply to evey­body.

But for peo­ple within the ma­jor­ity, their de­ci­sion makes sense.

It may make sense as a set of per­sonal prefer­ences, but that doens’t jus­tify it be­ing bind­ing on oth­ers.

If I were in the com­mu­nity, I would do what they do.

Then you would have col­luded with atroc­i­ties in other his­tor­i­cal so­cieties.

Values are the only source of moral­ity that there is.

In­di­vi­d­ual val­ues do not sum to group moral­ity.

• That doesn’t fol­low. If it is the only thing that causes ac­tions, then it is rele­vant to why, as a mat­ter of fact, peo­ple do what they do—but that is de­scrip­tion, not pre­scrip­tion. Pre­scrip­tion re­quires ex­tra in­gre­di­ents.

In that case, pre­scrip­tion is im­pos­si­ble. Your sys­tem can’t han­dle the is-ought prob­lem.

I said that as a mat­ter fof fact it is not nec­es­sar­ily ra­tio­nal. My grounds are that you cna’t always ex­plain you mo­ti­va­tions on a ra­tioanl ba­sis.

Values are ra­tio­nal­ity neu­tral. If you don’t view mo­ti­va­tions and val­ues as iden­ti­cal, ex­plain why?

That doens’t fol­low. I could em­brace non-eval­u­a­tive in­tu­tions, for in­stance.

Th­ese in­tu­itions are vi­o­lated by para­doxes such as the prob­lem of in­duc­tion or the fact that log­i­cal jus­tifi­ca­tion is in­finitely re­gres­sive (tur­tles all the way down). Your choice is nihilism or an ar­bi­trary start­ing point, but logic isn’t a valid op­tion.

Sub­jec­tive moral­ity can­not jus­tify laws tha pply to evey­body.

Sure. Tech­ni­cally this is false if ev­ery­one is the same or very similar but I’ll let that slide. Why does this in­val­i­date sub­jec­tive moral­ity?

It may make sense as a set of per­sonal prefer­ences, but that doens’t jus­tify it be­ing bind­ing on oth­ers.

Why would I be mo­ti­vated by some­one else’s prefer­ences? The only thing rele­vant to my de­ci­sion is me and my prefer­ences. The fact that this de­ci­sion effects other peo­ple is ir­rele­vant, liter­ally ev­ery de­ci­sion effects other peo­ple.

Then you would have col­luded with atroc­i­ties in other his­tor­i­cal so­cieties.

I value hu­man life, you are wrong.

In­di­vi­d­ual val­ues do not sum to group moral­ity.

Group moral­ity does not ex­ist.

• In that case, pre­scrip­tion is im­pos­si­ble. Your sys­tem can’t han­dle the is-ought prob­lem.

Some­thing is not im­pos­si­ble just be­cause it re­quires ex­tra in­gre­di­ents.

Values are ra­tio­nal­ity neu­tral. If you don’t view mo­ti­va­tions and val­ues as iden­ti­cal, ex­plain why?

I don’t care about the differ­ence be­tween ir­ra­tional and ara­tional, they’re both non-ra­tio­nal.

Th­ese in­tu­itions are vi­o­lated by para­doxes such as the prob­lem of in­duc­tion or the fact that log­i­cal jus­tifi­ca­tion is in­finitely re­gres­sive (tur­tles all the way down).

Ground­ing out in an in­tu­ition­that can’t be jus­tified is no worse than ground­ing out in a value that can’t be jus­tified.

Your choice is nihilism or an ar­bi­trary start­ing point, but logic isn’t a valid op­tion.

You are (try­ing to) use logic right now, How come it works for you?

Why does this in­val­i­date sub­jec­tive moral­ity?

Be­cause moral­ity needs to be able to tell peo­ple why they should not always act on their first-oder im­pulses.

Why would I be mo­ti­vated by some­one else’s prefer­ences?

I didn’t say you should.If you have moral­ity as a higher or­der prefer­nce, you can be per­suaded to over­ride some of your first ort­der prefer­ences. In favour of moral­ity. Which is not sub­jec­tiv,e and there­fore not just some­one else;s val­ues.

The only thing rele­vant to my de­ci­sion is me and my prefer­ences.

You’ve ad­mit­ted that prefer­nces can in­clude em­pa­thy. They can in­clude re­spect for uni­ver­sal­is­able moral prin­ci­ples too. “My prefer­nces” does not have to equate to “self­ish prefer­neces”

The fact that this de­ci­sion effects other peo­ple is ir­rele­vant, liter­ally ev­ery de­ci­sion effects other peo­ple.

How does choos­ing vanilla over chocale chip af­fect other peo­ple?

I value hu­man life, you are wrong.

You need to make up your mind whether you value hu­man life more or less than go­ing along with the ma­jor­ity.

Group moral­ity does not exist

That claim needs jus­tifi­ca­tion.

• Some­thing is not im­pos­si­ble just be­cause it re­quires ex­tra in­gre­di­ents.

How do you gen­er­ate moral prin­ci­ples that con­flict with de­sire? How do you jus­tify moral prin­ci­ples that don’t spring from de­sire? Why would any­one adopt these moral prin­ci­ples or care what they have to say? How do you over­come the is-ought gap?

Give me a spe­cific ex­am­ple of an ob­jec­tive sys­tem that you think is valid and that over­comes the is-ought gap.

Be­cause moral­ity needs to be able to tell peo­ple why they should not always act on their first-oder im­pulses.

Mine can do that. Some im­pulses con­tra­dict other val­ues. Some val­ues out­weigh oth­ers. Some­times you make sac­ri­fices now for later gains.

I don’t know why you be­lieve moral­ity needs to be able to re­strict im­pulses, ei­ther. Mo­ral­ity is a guide to ac­tion. If that guide to ac­tion is iden­ti­cal to your in­her­ent first-or­der im­pulses, all the bet­ter for you.

I didn’t say you should.If you have moral­ity as a higher or­der prefer­nce, you can be per­suaded to over­ride some of your first ort­der prefer­ences. In favour of moral­ity. Which is not sub­jec­tiv,e and there­fore not just some­one else;s val­ues.

Let me rephrase. How can you gen­er­ate mo­ti­va­tional force from ab­stract prin­ci­ples? Why does moral­ity mat­ter if it has noth­ing to do with our val­ues?

You’ve ad­mit­ted that prefer­nces can in­clude em­pa­thy. They can in­clude re­spect for uni­ver­sal­is­able moral prin­ci­ples too. “My prefer­nces” does not have to equate to “self­ish prefer­neces”

Your prefer­ences might in­clude this, yes. I think that would be a weird thing to have built in your prefer­ences and that you should con­sider self-mod­ify­ing it out. Re­gard­less, that would be jus­tify­ing a be­lief in a uni­ver­sal­is­able moral prin­ci­ple through sub­jec­tive prin­ci­ples. You’re try­ing to jus­tify that be­lief through noth­ing but logic, be­cause that is the only way you can char­ac­ter­ize your sys­tem as truly ob­jec­tive.

How does choos­ing vanilla over chocale chip af­fect other peo­ple?

There are less vanilla chips for other peo­ple. It effects your diet which effects the way you will be­have. It will in­crease your hap­piness if you value vanilla chips more than choco­late ones. If some­one val­ues your hap­piness, they will be happy you ate vanilla chips. If some­one hates when you’re happy, they will be sad.

You need to make up your mind whether you value hu­man life more or less than go­ing along with the ma­jor­ity.

I don’t value go­ing along with the ma­jor­ity in and of it­self. If I’m a mem­ber of the ma­jor­ity and I have cer­tain val­ues then I would act on those val­ues, but my sta­tus as a mem­ber of the ma­jor­ity wouldn’t be rele­vant to moral­ity.

That claim needs jus­tifi­ca­tion.

Sure. Pain and plea­sure and value are the roots of moral­ity. They ex­ist only in in­ter­nal ex­pe­riences. My pain and your plea­sure are not in­ter­change­able be­cause there is no big Calcu­lat­ing util­ity god in the sky to ag­gre­gate the con­tent of our ex­pe­riences. Ex­pe­rience is always in­di­vi­d­ual and in­ter­nal and value can’t ex­ist out­side of ex­pe­rience and moral­ity can’t ex­ist out­side of value. The parts of your brain that make you value cer­tain ex­pe­riences are not con­nected to the parts of my brain that make me value cer­tain ex­pe­riences, which means the fact that your ex­pe­riences aren’t mine is suffi­cient to re­fute the idea that your ex­pe­riences would or should some­how mo­ti­vate me in and of them­selves.

• How do you gen­er­ate moral prin­ci­ples that con­flict with de­sire?

Did you no­tice my refer­ences to “firist or­der” and “higher or­der”?

How do you over­come the is-ought gap?

By us­ing ra­tio­nal-should as an in­ter­me­di­ate.

Mine can do that. Some im­pulses con­tra­dict other val­ues. Some val­ues out­weigh oth­ers. Some­times you make sac­ri­fices now for later gains.

Some­times you need to fol­low im­per­sonal, uni­ver­sali­able,...maybe even ob­jec­tive...moral rea­son­ing?

I don’t know why you be­lieve moral­ity needs to be able to re­strict im­pulses,

i don’t know why you think “do what thou wilt” is mor­laity. It would be like hav­ing a sys­tem of logic that can prove any claim.

ei­ther. Mo­ral­ity is a guide to ac­tion. If that guide to ac­tion is iden­ti­cal to your in­her­ent first-or­der im­pulses, all the bet­ter for you.

“All the bet­ter for me” does not mean “op­ti­mal moral­ity”. The job of logic is not to prove ev­ery­thing I hap­pen to be­lieve, and the job of moral­ity is not to con­firm all my im­pulses.

Let me rephrase. How can you gen­er­ate mo­ti­va­tional force from ab­stract prin­ci­ples?

Some peo­ple value rea­son, and the rest have value sys­tems tweaked by the threat of pun­ish­ment.

Why does moral­ity mat­ter if it has noth­ing to do with our val­ues?

You think no one val­ues moral­ity?

Your prefer­ences might in­clude this, yes. I think that would be a weird thing to have built in your prefer­ences and that you should con­sider self-mod­ify­ing it out.

What’s weird? Em­pa­thy? Mo­ral­ity? Ra­tioan­lity?

You’re try­ing to jus­tify that be­lief through noth­ing but logic, be­cause that is the only way you can char­ac­ter­ize your sys­tem as truly ob­jec­tive.

You say that like its a bad thing.

There are less vanilla chips for other people

Not nec­es­sar­ily. There might be a sur­plus.

But if you want to say that ev­ery­thing effects oth­ers, albeit to a ti y ex­tent, then it fol­lows that ev­ery­thing is a tiny bit moral.

I don’t value go­ing along with the ma­jor­ity in and of it­self.

You pre­vi­ouly made some state­ments that sounded a lot like that.

Sure. Pain and plea­sure and value are the roots of moral­ity.

That state­ment needs some jus­tifi­ca­tion. Is it bet­ter to do good things vol­un­tar­ily, or be­cause you are forced to?

Ex­pe­rience is always in­di­vi­d­ual and in­ter­nal and value can’t ex­ist out­side of ex­pe­rience and moral­ity can’t ex­ist out­side of value.

OK, I though ti was some­thing like that. The things is that sub­jects can have val­ues which are in­her­ently in­ter­per­sonal and even ob­jec­tive...things like em­pa­thy and ra­tio­nal­ity. So “value held by a sub­ject” does not im­ply “self­ish value”.

The parts of your brain that make you value cer­tain ex­pe­riences are not con­nected to the parts of my brain that make me value cer­tain ex­pe­riences, which means the fact that your ex­pe­riences aren’t mine is suffi­cient to re­fute the idea that your ex­pe­riences would or should some­how mo­ti­vate me in and of them­selves.

Yet agian, ob­jec­tive moral­ity is not a case of one sub­ject be­ing mo­ti­vated by an­other sub­jects val­ues. Ob­jec­tivity is not achieved by swap­ping sub­jects.

• Did you no­tice my refer­ences to “firist or­der” and “higher or­der”?

This is a black box. Ex­plain what they mean and how you gen­er­ate the con­nec­tion be­tween the two.

By us­ing ra­tio­nal-should as an in­ter­me­di­ate.

You claim that a ra­tio­nal-should ex­ists. Prove it.

Some­times you need to fol­low im­per­sonal, uni­ver­sali­able,...maybe even ob­jec­tive...moral rea­son­ing?

Us­ing ob­jec­tive prin­ci­ples as a tool to eval­u­ate trade­offs be­tween sub­jec­tive val­ues is not the same as us­ing ob­jec­tive prin­ci­ples to pro­duce moral truths.

i don’t know why you think “do what thou wilt” is mor­laity. It would be like hav­ing a sys­tem of logic that can prove any claim.

That’s not my defi­ni­tion of moral­ity, it’s the con­clu­sion I end up with. Your anal­ogy doesn’t seem valid to me be­cause I don’t con­clude that all moral claims are equal but that all de­sires are good. Re­press­ing de­sires or failing to achieve de­sires is bad. Ad­di­tion­ally, its clear to me why a log­i­cal sys­tem that proves ev­ery­thing is good is bad, but why would a moral sys­tem that did the same be in­valid?

“All the bet­ter for me” does not mean “op­ti­mal moral­ity”. The job of logic is not to prove ev­ery­thing I hap­pen to be­lieve, and the job of moral­ity is not to con­firm all my im­pulses.

I agree. I didn’t claim ei­ther of those things. Mo­ral­ity doesn’t have a job out­side of dis­t­in­guish­ing be­tween right and wrong.

What’s weird? Em­pa­thy? Mo­ral­ity? Ra­tioan­lity?

The idea that all prin­ci­ples you act upon must be uni­ver­sal­iz­able. It’s bad be­cause in­di­vi­d­u­als are differ­ent and should act differ­ently. The prin­ci­ple I defend is a uni­ver­sal­iz­able one, that in­di­vi­d­u­als should do what they want. The differ­ence be­tween mine and yours is that mine is broad and all peo­ple are happy when its ap­plied to their case, but yours is nar­row and ex­clu­sive and ego­cen­tric be­cause it ne­glects differ­ences in in­di­vi­d­ual val­ues, or holds those differ­ences to be morally ir­rele­vant.

Not nec­es­sar­ily. There might be a sur­plus.

But if you want to say that ev­ery­thing effects oth­ers, albeit to a ti y ex­tent, then it fol­lows that ev­ery­thing is a tiny bit moral.

Sub­trac­tion, have you heard of it?

Some things are neu­tral even though they effect oth­ers.

That state­ment needs some jus­tifi­ca­tion. Is it bet­ter to do good things vol­un­tar­ily, or be­cause you are forced to?

Vol­un­tar­ily, be­cause that means you’re act­ing on your val­ues.

OK, I though ti was some­thing like that. The things is that sub­jects can have val­ues which are in­her­ently in­ter­per­sonal and even ob­jec­tive...things like em­pa­thy and ra­tio­nal­ity. So “value held by a sub­ject” does not im­ply “self­ish value”.

If I val­ued ra­tio­nal­ity, why would that re­sult in spe­cific moral de­crees? Value held by a sub­ject doesn’t im­ply self­ish value, but it does im­ply that the val­ues of oth­ers are only rele­vant to my moral­ity in­so­far as I em­pathize with those oth­ers.

Yet agian, ob­jec­tive moral­ity is not a case of one sub­ject be­ing mo­ti­vated by an­other sub­jects val­ues. Ob­jec­tivity is not achieved by swap­ping sub­jects.

“Ob­jec­tivity” in ethics is achieved by aban­don­ing in­di­vi­d­ual val­ues and be­liefs and try­ing to pro­duce state­ments which would be val­ued and be­lieved by ev­ery­one. That’s stupid be­cause we can never es­cape the lo­cus of the self and be­cause moral­ity emerges from in­ter­nal pro­cesses and ne­glect­ing those in­ter­nal pro­cesses means that there is zero foun­da­tion for any sort of moral­ity. I’m say­ing that moral­ity is only ac­cessible in­ter­nally, and that the things which pro­duce moral­ity are in­ter­nal sub­jec­tive be­liefs.

If you con­tinue to dis­agree, I sug­gest we start over. Let me know and I’ll post an ar­gu­ment that I used last year in de­bate. I feel like start­ing over would clar­ify things a lot be­cause we’re get­ting mud­dled down in a line-by-line back-and-forth hy­per­spe­cific con­ver­sa­tion here.

• This is a black box. Ex­plain what [first or­der and higher or­der] mean and how you gen­er­ate the con­nec­tion be­tween the two.

Usual mean­ing in this type of di­s­uc­s­sion.

You claim that a ra­tio­nal-should ex­ists. Prove it.

If I can prove any­thing to you, you are already run­ning on ra­tio­nal_should.

Us­ing ob­jec­tive prin­ci­ples as a tool to eval­u­ate trade­offs be­tween sub­jec­tive val­ues is not the same as us­ing ob­jec­tive prin­ci­ples to pro­duce moral truths.

Why not?

That’s not my defi­ni­tion of moral­ity, it’s the con­clu­sion I end up with.

That doens’t help. It;s not moral­ity whether it’s as­sumed or con­cluded.

The idea that all prin­ci­ples you act upon must be [is weird]

It’s bad be­cause in­di­vi­d­u­als are differ­ent and should act differ­ently.

In­di­vi­d­u­als are differ­ent and would act differntly. You are ar­gu­ing as though peo­ple should never do anythng un­less it is morally obli­gated, as though moral rules are all en­com­pass­ing. I never said that. Mo­ral­ity does not need to de­tem­ine evey ac­tion any more than civil law does.

The prin­ci­ple I defend is a uni­ver­sal­iz­able one, that in­di­vi­d­u­als should do what they want.

That isn’t uni­ver­sal­is­able be­cause you don;t want to be mur­dered. The cor­rect form is “in­di­vi­d­u­als should do what they want un­less it harms an­other”.

The differ­ence be­tween mine and yours is that mine is broad and all peo­ple are happy when its ap­plied to their case,

We don’t have it. if peo­ple wanted your prin­ci­ple, they would abol­ish all laws.

but yours is nar­row and ex­clu­sive and ego­cen­tric

!!!

If I val­ued ra­tio­nal­ity, why would that re­sult in spe­cific moral de­crees?

Look at ex­am­ples of peo­ple ar­gu­ing about moral­ity.

ETA: Bet­ter re­strict that to liber­als.

There’s plenty about, even on this site.

Value held by a sub­ject doesn’t im­ply self­ish value, but it does im­ply that the val­ues of oth­ers are only rele­vant to my moral­ity in­so­far as I em­pathize with those oth­ers.

Nope. Ra­tion­al­ity too.

“Ob­jec­tivity” in ethics is achieved by aban­don­ing in­di­vi­d­ual val­ues and be­liefs

Of course not. It is a perfectly ac­cept­able prin­ci­ple that peo­ple should be al­lowed to re­al­ise their val­ues so long as they do not harm oth­ers. Where do you ge these ideas?

and try­ing to pro­duce state­ments which would be val­ued and be­lieved by ev­ery­one.

Just ev­ery­one ra­tio­nal. The po­lice are there for a reason

That’s stupid be­cause we can never es­cape the lo­cus of the self and be­cause moral­ity emerges from in­ter­nal processes

Yet again: we can in­ter­nally value what is ob­jec­tive and im­par­tial. “In me” doesn’t im­ply “for me”.

and ne­glect­ing those in­ter­nal pro­cesses means that there is zero foun­da­tion for any sort of moral­ity.

I’m say­ing that moral­ity is only ac­cessible in­ter­nally, and that the things which pro­duce moral­ity are in­ter­nal sub­jec­tive be­liefs.

Yet again: “In me” doesn’t im­ply “for me”.

If you con­tinue to dis­agree, I sug­gest we start over. Let me know and I’ll post an ar­gu­ment that I used last year in de­bate. I feel like start­ing over would clar­ify things a lot be­cause we’re get­ting mud­dled down in a line-by-line back-and-forth hy­per­spe­cific con­ver­sa­tion here.

If you like.

• Laws are jus­tified be­cause sub­jec­tive de­sires are in­her­ently jus­tified be­cause they’re in­her­ently mo­ti­va­tional.

What you need to justfy is im­pris­on­ing some­one for offend­ing against val­ues they don’t nec­es­sar­ily sub­sribe to. That you are mo­ti­vated by your val­ues, and the crim­i­nal by theirs, doens’t give you the right to jail them.

• I ac­tu­ally see that as counter-in­tu­itive.

“Mo­ral­ity” is in­deed be­ing used to reg­u­late in­di­vi­d­u­als by some in­di­vi­d­u­als or groups. When I think of moral­ity, how­ever, I think “greater to­tal util­ity over mul­ti­ple agents, whose value sys­tems (util­ity func­tions) may vary”. Mo­ral­ity seems largely about tak­ing ac­tions and mak­ing de­ci­sions which achieve greater util­ity.

• I do this, ex­cept I only use my own util­ity and not other agents. For me, out­side of em­pa­thy, I have no more rea­son to help other peo­ple achieve their val­ues than I do to help the Babyeaters eat ba­bies. The util­ity func­tions of oth­ers don’t in­her­ently con­nect to my mo­ti­va­tional states, and graft­ing the val­ues of oth­ers onto my de­ci­sion calcu­lus seems weird.

I think most peo­ple be­come util­i­tar­i­ans in­stead of ego­ists be­cause they em­pathize with other peo­ple, while never see­ing the fact that to the ex­tent that this em­pa­thy moves them it is their own value and within their own util­ity func­tion. They then build the ab­stract moral the­ory of util­i­tar­i­anism to for­mal­ize their in­tu­itions about this, but be­cause they’ve over­looked the ego­ist in­ter­me­di­ary step the model is slightly off and some­times leads to con­clu­sions which con­tra­dict ego­ist im­pulses or ego­ist con­clu­sions.

• Or they adopt ul­ti­tar­i­an­sim, or some other non-sub­jec­tive sys­tem, be­cause they value hav­ing a moral sys­tem that can ap­ply to, per­suade, and jus­tify it­self to oth­ers. (Or in short: they value hav­ing a moral sys­tem).

• In my view there’s a differ­ence be­tween hav­ing a moral sys­tem (defined as some­thing that tells you what is right and what is wrong) and hav­ing a sys­tem that you use to jus­tify your­self to oth­ers. That differ­ence gen­er­ally isn’t rele­vant be­cause hu­mans tend to em­pathize with each other and hu­mans have a very close cluster of val­ues so there are lots of com­mon in­ter­ests.

• My com­puter won’t load the web­site be­cause it’s ap­par­ently hav­ing is­sues with flash, can you please sum­ma­rize? If you’re just mak­ing a dis­tinc­tion be­tween your­self and your be­liefs, sure, I’ll con­cede that. I was a bit sloppy with my ter­minol­ogy there.

• Its not “My be­liefs” ei­ther.” jus­tifi­ca­tion is the rea­son why some­one (prop­erly) holds the be­lief, the ex­pla­na­tion as to why the be­lief is a true one, or an ac­count of how one knows what one knows.”

• Okay. I think I’ve ex­plained the jus­tifi­ca­tion then. Spe­cific moral sys­tems aren’t nec­es­sar­ily in­ter­change­able from per­son to per­son, but they can still be ex­plained and jus­tified in a gen­eral sense. “My val­ues tell me X, there­fore X is moral” is the form of jus­tifi­ca­tion that I’ve been defend­ing.

• Yet again, you run into the prob­lem that you need it to be wrong for other peo­ple to mur­der you, which you can’t jus­tify with your val­ues alone.

• No I don’t. I need to be stronger than the peo­ple who want to mur­der me, or to live in a so­ciety that de­ters mur­der. If some­one wants to mur­der me, it’s prob­a­bly not the best strat­egy to start try­ing to con­vince them that they’re be­ing im­moral.

You’re mak­ing an ar­gu­men­tum ad con­se­quen­tum. You don’t de­cide metaeth­i­cal is­sues by de­cid­ing what kind of moral­ity it would be ideal to have and then work­ing back­wards. Just be­cause you don’t like the type of sys­tem that moral­ity leads to over­all doesn’t mean that you’re jus­tified in ig­nor­ing other moral ar­gu­ments.

The benefit of my sys­tem is that it’s right for me to mur­der peo­ple if I want to mur­der them. This means I can do things like self defense or kil­ling Nazis and pe­dophiles with min­i­mal moral dam­age. This isn’t a rea­son to sup­port my sys­tem, but it is kind of neat.

• No I don’t. I need to be stronger than the peo­ple who want to mur­der me,

That’s giv­ing up on moral­ity not defend­ing sub­jec­tive moral­ity.

or to live in a so­ciety that de­ters mur­der.

Same prob­lem. That’s ei­ther group moral­ity or non moral­ity.

If some­one wants to mur­der me, it’s prob­a­bly not the best strat­egy to start try­ing to con­vince them that they’re be­ing im­moral.

I didn;t say it was the best prac­ti­cal strat­egy. The moral an the prac­ti­cal are differnt things. I am say­ing that for moral­ity to be what it is, it needs to offer rea­sons for peo­ple to not act on some of their first or­der val­ues. That moral­ity is not le­gal­ity or brue force or a a magic spell is not rele­vant.

You’re mak­ing an ar­gu­men­tum ad con­se­quen­tum. You don’t de­cide metaeth­i­cal is­sues by de­cid­ing what kind of moral­ity it would be ideal to have and then work­ing back­wards.

I am start­ing wth what kind of moral­ity it would be ad­e­quate to have. If you can’t bang in a nail with it, it isn’t a ham­mer.

Just be­cause you don’t like the type of sys­tem that moral­ity leads to overall

Where on eath did I say that?

The benefit of my sys­tem is that it’s right for me to mur­der peo­ple if I want to mur­der them.

That’s not a benefit, be­cause mur­der is just the sort of thing mor­laity is sup­posed to con­demn.. Ham­mers are for nails, not screws, and moral­ity is not for “i can do what­ever I want re­gard­less”.

This means I can do things like self defense

Jus­tifi­able self defense is not mur­der. You seem to have con­fused eth­i­cal ob­jec­tiv ism (moral­ity is not just per­sonal prefer­ence) with eth­i­cal ab­solutism (moral prin­ci­ples have no ex­cep­tions). Read yer wikipe­dia!

• That’s giv­ing up on moral­ity not defend­ing sub­jec­tive moral­ity.

Mo­ral­ity is a guide for your own ac­tions, not a guide for get­ting peo­ple to do what you want.

Same prob­lem. That’s ei­ther group moral­ity or non moral­ity.

Ra­tional self in­ter­ested in­di­vi­d­u­als de­cide to cre­ate a po­lice force.

Ar­gu­men­tum ad con­se­quen­tums are still in­valid.

I didn;t say it was the best prac­ti­cal strat­egy. The moral an the prac­ti­cal are differnt things. I am say­ing that for moral­ity to be what it is, it needs to offer rea­sons for peo­ple to not act on some of their first or­der val­ues. That moral­ity is not le­gal­ity or brue force or a a magic spell is not rele­vant.

Sure, but moral­ity needs to have mo­ti­va­tional force or its use­less and stupid. Why should I care? Why should the bur­glar? If you’re go­ing to keep in­sist­ing that moral­ity is what’s pre­vent­ing peo­ple from do­ing evil things, you need to ex­plain how your ac­count­ing of moral­ity over­rules in­her­ent mo­ti­va­tion and de­sire, and why its jus­tified in do­ing that.

I am start­ing wth what kind of moral­ity it would be ad­e­quate to have. If you can’t bang in a nail with it, it isn’t a ham­mer.

This is not how metaethics works. You don’t get to start with a pre­defined no­tion of ad­e­quate. That’s the op­po­site of ob­jec­tivity. By ne­glect­ing metaethics, you’re defend­ing a model that’s just as sub­jec­tive as mine, ex­cept that you don’t ac­knowl­edge that and you seek to vil­ify those who don’t share your prefer­ences.

Where on eath did I say that?

You’re ar­gu­ing that sub­jec­tive moral­ity can’t be right be­cause it would lead to con­clu­sions you find un­de­sir­able, like ran­dom mur­ders.

That’s not a benefit, be­cause mur­der is just the sort of thing mor­laity is sup­posed to con­demn.. Ham­mers are for nails, not screws, and moral­ity is not for “i can do what­ever I want re­gard­less”.

Stop mud­dling the de­bate with un­jus­tified as­sump­tions about what moral­ity is for. If you want to talk about some­thing else, fine. My defi­ni­tion of moral­ity is that moral­ity is what tells in­di­vi­d­u­als what they should and should not do. That’s all I in­tend to talk about.

You’ve con­ceded nu­mer­ous things in this con­ver­sa­tion, also. I’m done ar­gu­ing with you be­cause you’re ig­nor­ing any point that you find in­con­ve­nient to your po­si­tion and be­cause you haven’t shown that you’re ra­tio­nal enough to es­cape your dogma.

• Mo­ral­ity is a guide for your own ac­tions,

No, it is largely about reg­u­lat­ing in­ter­ac­tions such as rape theft and mur­der.

not a guide for get­ting peo­ple to do what you want.

I never said moral­ity is to make oth­ers do what I want. That is per­sis­tent straw man on your part

Ra­tional self in­ter­ested in­di­vi­d­u­als de­cide to cre­ate a po­lice force.

So?

Ar­gu­men­tum ad con­se­quen­tums are still in­valid.

“It’s not a ham­mer if it can’t bang in nail” isn’t in­valid.

Sure, but moral­ity needs to have mo­ti­va­tional force or its use­less and stupid. Why should I care?

If your ar ra­tio­nal you will care abotu raiton­al­ity based moral­ity. If you are not...what are you do­ing on LW?

Why should the bur­glar? If you’re go­ing to keep in­sist­ing that moral­ity is what’s pre­vent­ing peo­ple from do­ing evil things, you need to ex­plain how your ac­count­ing of moral­ity over­rules in­her­ent mo­ti­va­tion and de­sire, and why its jus­tified in do­ing that.

The mo­ti­va­tion to be ra­tio­nal is a mo­ti­va­tion. I didn’t say non-mo­ti­va­tions over­ride mo­ti­va­tions. Higher or­der and lower or­der, re­mem­ber.

This is not how metaethics works. You don’t get to start with a pre­defined no­tion of ad­e­quate.

Why not? I can see apri­ori what would make a ham­mer ad­e­quate.

You’re ar­gu­ing that sub­jec­tive moral­ity can’t be right be­cause it would lead to con­clu­sions you find un­de­sir­able, like ran­dom mur­ders.

Con­clu­sions that just about anyne would find un­der­sir­able. Ob­jec­tion to ran­dom mur­der is not some weird pecadillo of mine.

Stop mud­dling the de­bate with un­jus­tified as­sump­tions about what moral­ity is for.

Cal­ling some­thing un­jus­tified doens’t prove an­ty­hing.

My defi­ni­tion of moral­ity is that moral­ity is what tells in­di­vi­d­u­als what they should and should not do.

What’s the differnce? If you should not do a mur­der (your defin­tiion), then a po­ten­tial in­ter­ac­tion has been reg­u­lated (my ver­sion).;

You’ve con­ceded nu­mer­ous things in this con­ver­sa­tion, also. I’m done ar­gu­ing with you be­cause you’re ig­nor­ing any point that you find in­con­ve­nient to your po­si­tion

and be­cause you haven’t shown that you’re ra­tio­nal enough to es­cape your dogma.

What dogma?

• No, it is largely about reg­u­lat­ing in­ter­ac­tions such as rape theft and mur­der.

This is a sub­set of my pos­si­ble in­di­vi­d­ual ac­tions. Every in­ter­ac­tion is an ac­tion.

Mo­ral­ity is not poli­ti­cal, which is what you’re mak­ing it into. Mo­ral­ity is about right and wrong, and that’s all.

I never said moral­ity is to make oth­ers do what I want. That is per­sis­tent straw man on your part

You’re us­ing moral­ity for more than in­di­vi­d­ual ac­tions. There­fore, you’re us­ing it for other peo­ple’s ac­tions, for per­suad­ing them to do what you want to do. Other­wise, your at­tempt to dis­t­in­guish your view from mine fails.

“It’s not a ham­mer if it can’t bang in nail” isn’t in­valid.

Then you’re us­ing a differ­ent defi­ni­tion of moral­ity which has more con­straints than my defi­ni­tion. My defi­ni­tion is that moral­ity is any­thing that tells an in­di­vi­d­ual which ac­tions should or should not be taken, and that no other re­quire­ments are nec­es­sary for moral­ity to ex­ist. If your con­cep­tion of moral­ity guides in­di­vi­d­ual ac­tions as well, but also has ad­di­tional re­quire­ments, I’m con­tend­ing that your ad­di­tional re­quire­ments have no valid meta­phys­i­cal foun­da­tion.

The mo­ti­va­tion to be ra­tio­nal is a mo­ti­va­tion. I didn’t say non-mo­ti­va­tions over­ride mo­ti­va­tions. Higher or­der and lower or­der, re­mem­ber.

Ra­tion­al­ity is not a mo­ti­va­tion, it is value-neu­tral.

Why not? I can see apri­ori what would make a ham­mer ad­e­quate.

What moral sys­tem do you defend? How does ra­tio­nal­ity re­sult in moral prin­ci­ples? Can you give me an ex­am­ple?

Con­clu­sions that just about anyne would find un­der­sir­able. Ob­jec­tion to ran­dom mur­der is not some weird pecadillo of mine.

Not rele­vant. Peo­ple are stupid. Ar­gu­men­tum ad con­se­quen­tums are log­i­cally in­valid. Use Wikipe­dia if you doubt this.

Cal­ling some­thing un­jus­tified doens’t prove an­ty­hing.

If your as­sump­tions were jus­tified, I missed it. Please jus­tify them for me.

What’s the differnce? If you should not do a mur­der (your defin­tiion), then a po­ten­tial in­ter­ac­tion has been reg­u­lated (my ver­sion).;

Our defi­ni­tions over­lap in some in­stances but aren’t iden­ti­cal. You add con­straints, such as the idea that any moral sys­tem which jus­tifies mur­der is not a valid moral sys­tem. Yours is also nar­rower than mine be­cause mine holds that moral­ity ex­ists even in the con­text of wholly iso­lated in­di­vi­d­u­als, whereas yours says moral­ity is about in­ter­per­sonal in­ter­ac­tions.

I was mis­taken be­cause I hadn’t seen your other com­ment. I read the com­ments out of or­der. My apolo­gies.

What dogma?

You’re ar­gu­ing from defi­ni­tions in­stead of show­ing the rea­son­ing pro­cess which starts with ra­tio­nal prin­ci­ples and ends up with moral prin­ci­ples.

• Every in­ter­ac­tion is an ac­tion.

It is not rati­nal to de­cide ac­tions which are in­te­ac­tions on the prefer­neces of one party alone.

Mo­ral­ity is not political

Weren’t you say­ing that the ma­jor­ity de­cide what is moral?

You’re us­ing moral­ity for more than in­di­vi­d­ual ac­tions.

Arent you?

There­fore, you’re us­ing it for other peo­ple’s ac­tions, for per­suad­ing them to do what you want to do.

Every­body is us­ing it for their and ev­ery­body el­ses ac­tions. I play no cen­tral role.

If your con­cep­tion of moral­ity guides in­di­vi­d­ual ac­tions as well, but also has ad­di­tional re­quire­ments, I’m con­tend­ing that your ad­di­tional re­quire­ments have no valid meta­phys­i­cal foun­da­tion.

That de­pends on whether or not your “in­di­vi­d­ual ac­tions” in­lcude in­terac­i­tons. if they do, the in­ter­ests of the other par­ties need to be taken into ac­count.

Ra­tion­al­ity is not a mo­ti­va­tion, it is value-neu­tral.

How does any­one end up raitional if no-one is mo­ti­vated to be? Are you quite sure you haven’t confused

“ra­tio­nal­ity is value neu­tral be­cause if you don’t get any val­ues out of it your don’t put into it”

with

“No one would ever value ra­tio­nal­ity”

I don’t have to jus­tify com­mon defin­tions.

What moral sys­tem do you defend?

Where did I say I was defend­ing one? I said sub­jec­tivism doen’t work.

Ar­gu­men­tum ad con­se­quen­tums are log­i­cally in­valid.

You can­not log­i­cally con­clude that some­thing ex­ists in ob­jec­tive re­al­ity be­cause you like its con­se­quences. But moral­ity doens’t ex­ist in ob­jec­tive re­al­ity. it is a hu­man cre­ation, and hu­mans are en­ti­tled to re­ject ver­sions of ti that don’t work be­caue they dont work.

If your as­sump­tions were jus­tified, I missed it. Please jus­tify them for me.

The bur­den is on you to ex­plain how your defi­ni­tion “moral­ity is about right and wrong” is differ­ent from mine: “moral­ity is about the re­qua­tion of con­duct”.

Our defi­ni­tions over­lap in some in­stances but aren’t iden­ti­cal. You add con­straints, such as the idea that any moral sys­tem which jus­tifies mur­der is not a valid moral sys­tem.

It ob­viiusly isn’t. If our defi­ni­tions differ, mine is right.

Yours is also nar­rower than mine be­cause mine holds that moral­ity ex­ists even in the con­text of wholly iso­lated in­di­vi­d­u­als, whereas yours says moral­ity is about in­ter­per­sonal in­ter­ac­tions.

I said “largely”.

You’re ar­gu­ing from definitions

You say that like its a bad thing.

in­stead of show­ing the rea­son­ing pro­cess which starts with ra­tio­nal prin­ci­ples and ends up with moral prin­ci­ples.

Why would I need to do that to show that sub­jec­tivism is wrong?

• I don’t want to spend any more time on this. I’m done.

• Your us­age of the words “sub­jec­tive” and “ob­jec­tive” is con­fus­ing.

Utili­tar­i­anism doesn’t for­bid that each in­di­vi­d­ual per­son (agent) have differ­ent things they value (util­ity func­tions). As such, there is no uni­ver­sal spe­cific sim­ple rule that can ap­ply to all pos­si­ble agents to max­i­mize “moral­ity” (to­tal sum util­ity).

It is “ob­jec­tive” in the sense that if you know all the util­ity func­tions, and try to achieve the max­i­mum pos­si­ble to­tal util­ity, this is the best thing to do from an ex­ter­nal stand­point. It is also “ob­jec­tive” in the sense that when your own util­ity is max­i­mized, that is the best pos­si­ble thing that you could have, re­gard­less of what­ever any­one might think about it.

How­ever, it is also “sub­jec­tive” in the sense that each in­di­vi­d­ual can have their own util­ity func­tion, and it can be what­ever you could imag­ine. There are no re­stric­tions in util­i­tar­i­anism it­self. My util­ity is not your util­ity, un­less your util­ity func­tion has a com­po­nent that val­ues my util­ity and you have full knowl­edge of my util­ity (or even if you don’t, but that’s a the­o­ret­i­cal nit­pick).

Utili­tar­i­anism alone doesn’t ap­ply to, per­suade, or jus­tify any ac­tion that af­fects val­ues to any­one else. It can be abused as such, but that’s not what it’s there for, AFAIK.

• I think spe­cific ap­pli­ca­tions of util­i­tar­i­anism might say that mod­ify­ing the val­ues of your­self or of oth­ers would be benefi­cial even in terms of your cur­rent util­ity func­tion.

• Yeah.

When things start get­ting in­ter­est­ing is when not only are some val­ues im­ple­mented as vari­able-weight within the func­tion, but the func­tions them­selves be­come part of the calcu­la­tion, and util­ity func­tions be­come mod­u­lar and par­tially re­cur­sive.

I’m cur­rently con­vinced that there’s at least one (per­haps well-hid­den) such re­cur­sive mod­ule of util­ity-for-util­ity-func­tions cur­rently built into the hu­man brain, and that clever hack­ing of this mod­ule might be very benefi­cial in the long run.

• Utili­tar­i­anism alone doesn’t ap­ply to, per­suade, or jus­tify any ac­tion that af­fects val­ues to any­one else. It can be abused as such, but that’s not what it’s there for,

Are you say­ing that no form of util­i­tar­i­an­sim will ever con­clude fhat one per­son should sac­ri­fice some value for the benefit of the many?

• No form of the offi­cial the­ory in the pa­pers I read, at the very least.

Many ap­pli­ca­tions or im­ple­men­ta­tions of util­i­tar­i­anism or util­i­tar­ian (-like) sys­tems do, how­ever, en­force rules that if one agent’s weighed util­ity loss im­proves the to­tal weighed util­ity of mul­ti­ple other agents by a sig­nifi­cant mar­gin, that is what is right to do. The mar­gin’s size and spe­cific num­bers and un­cer­tainty val­ues will vary by sys­tem.

I’ve never seen a sys­tem that would en­force such rules with­out a weigh­ing func­tion for the util­ities of some kind to cor­rect for limited in­for­ma­tion and un­cer­tainty and diminish­ing-re­turns-like prob­lems.

• No form of the offi­cial the­ory in the pa­pers I read, at the very least.

Many ap­pli­ca­tions or im­ple­men­ta­tions of util­i­tar­i­anism or util­i­tar­ian (-like) sys­tems do, how­ever, en­force rules that if one agent’s weighed util­ity loss im­proves the to­tal weighed util­ity of mul­ti­ple other agents by a sig­nifi­cant mar­gin, that is what is right to do. The mar­gin’s size and spe­cific num­bers and un­cer­tainty val­ues will vary by sys­tem.

It seems to me that these two para­graphs con­tara­dict each other. Do you think the “he should” means some­thing differ­ent to “it is right for him to do so”?

• No, they don’t have any ma­jor differ­ences in util­i­tar­ian sys­tems.

It seems I was con­fused when try­ing to an­swer your ques­tion. Utili­tar­i­anism can be seen as an ab­stract sys­tem of rules to com­pute stuff.

Cer­tain ways to ap­ply those rules to com­pute stuff are also called util­i­tar­i­anism, in­clud­ing the philos­o­phy that the max­i­mum to­tal util­ity of a pop­u­la­tion should pre­clude over the util­ity of one in­di­vi­d­ual.

If util­i­tar­i­anism is sim­ply the set of rules you use to com­pute which things are best for one sin­gle purely self­ish agent, then no, noth­ing con­cludes that the agent should sac­ri­fice any­thing. If you ad­here to the clas­si­cal philos­o­phy re­lated to those rules, then yes, any hu­man will con­clude what I’ve said in that sec­ond para­graph in the grand­par­ent (or some­thing similar). This lat­ter (the philos­o­phy) is his­tor­i­cally what ap­peared first, and is also what’s ex­posed on wikipe­dia’s page on util­i­tar­i­anism.

• If util­i­tar­i­anism is sim­ply the set of rules you use to com­pute which things are best for one sin­gle purely self­ish agent,

Isn’t that de­ci­sion the­ory?

• I share this view. When I ap­pear to forfeit some util­ity in fa­vor of some­one else, it’s be­cause I’m ac­tu­ally max­i­miz­ing my own util­ity by de­riv­ing some from the knowl­edge that I’m im­prov­ing the util­ity of other agents.

Other agents’s util­ity func­tions and val­ues are not di­rectly val­ued, at least not among hu­mans. Some (most?) of us just do in­di­rectly value im­prov­ing the value and util­ity of other agents, ei­ther as an in­stru­men­tal step or a ter­mi­nal value. Be­cause of this, I be­lieve most peo­ple who have/​pro­fess the be­lief of an “in­nate good­ness of hu­man­ity” are mind-pro­ject­ing their own value-of-oth­ers’-util­ity.

Whether this is a true value ac­tu­ally shared by all hu­mans is un­known to me. It is pos­si­ble that those who ap­pear not to have this value are sim­ply bro­ken in some tem­po­ral, en­vi­ron­ment-based man­ner. It’s also pos­si­ble that this is a purely en­vi­ron­ment-learned value that be­comes “ter­mi­nal” in the pro­cess of be­ing trained into the brain’s re­ward cen­ters due to its in­stru­men­tal value in many situ­a­tions.

• Be­cause moral­ity is largely to reg­u­late in­ter­ac­tions be­tween in­di­vi­d­u­als. That’s its job.

You are an­thro­po­mor­phiz­ing con­cepts. Mo­ral­ity is a hu­man ar­ti­fact, and ar­ti­facts have no more pur­pose than nat­u­ral ob­jects.

Mo­ral­ity is a use­ful tool to reg­u­late in­ter­ac­tions be­tween in­di­vi­d­u­als. There are efforts to make it a bet­ter tool for that pur­pose. That does not mean that moral­ity should be used to reg­u­late in­ter­ac­tions.

• You are an­thro­po­mor­phiz­ing con­cepts. Mo­ral­ity is a hu­man ar­ti­fact, and ar­ti­facts have no more pur­pose than nat­u­ral ob­jects.

Hu­man ar­ti­facts are gen­er­ally cre­ated to do jobs, eg hammers

Mo­ral­ity is a use­ful tool

Tool. Like i said.

That does not mean that moral­ity should be used to reg­u­late in­ter­ac­tions.

Does that mean you have a bet­ter tool in mind, or that in­ter­ac­tion don’t need reg­u­la­tion?

• If I put a ham­mer un­der a table to keep the table from wob­bling, am I us­ing a tool or not? If the ham­mer is the only ob­ject within range that is the right size for the table, and there is no task which re­quires a weighted lever, is the ham­mer in­tended to bal­ance the table sim­ply by virtue of be­ing the best tool for the job?

Fit-for-task is a differ­ent qual­ity than pur­pose. Ham­mers are use­ful tools to drive nails, but poor tools for de­ter­min­ing what nails should be driven. There are many nails that should not be driven, de­spite the pres­ence of ham­mers.

• If I put a ham­mer un­der a table to keep the table from wob­bling, am I us­ing a tool or not?

f you can’t bang in nails with it, it isnt a ham­mer. What you can do with it isn’t rele­vant.

There are many nails that should not be driven, de­spite the pres­ence of ham­mers.

???

So we can judge things morally wrong, be­cause we have a tool to do the job, but we shouldn’t in many cases, be­cause...? (And what kind of “shouldn’t” is that?)

• If you can’t bang in nails with it, it isnt a ham­mer. What you can do with it isn’t rele­vant.

By that, the ab­sence of nails makes the weighted lever not a ham­mer. I think that ham­mer­ness is in­trin­sic and not based on the pres­ence of nails; like­wise moral­ity can ex­ist when there is only one ac­tive moral agent.

• The metaphor was that you could, in prin­ci­ple, drive nails liter­ally ev­ery­where you can see, in­clud­ing in your brain. Will you agree that one should not drive nails liter­ally ev­ery­where, but only in se­lect lo­ca­tions, us­ing the right type of nail for the right lo­ca­tion? If you don’t, this part of the con­ver­sa­tion is not sal­vage­able.

• What is that sup­posed to be anal­gous to? If you have a work­able sys­tem of ethics, then it doens’t make judg­ments willy nilly, any­more than a work­able sys­tem of logic al­lows quodli­bet.

• The metaphor was that you could, in prin­ci­ple, make rules and laws for liter­ally any pos­si­ble ac­tion, in­clud­ing liv­ing. Will you agree that one should not make fixed rules for liter­ally all ac­tions, but only for se­lect high-nega­tive-im­pact ones, us­ing the right type of rule for the right ac­tion?

(Edited for ex­plicit anal­ogy.)

Ba­si­cally, it’s not be­cause you have a moral­ity (ham­mer) that hap­pens to be con­ve­nient for mak­ing laws and rules of in­ter­ac­tions (bal­anc­ing the table) that moral­ity is nec­es­sar­ily the best and in­tended tool for mak­ing rules and that moral­ity it­self tells you what you should make laws about or that you even should make laws in the first place.

• Mo­ral rules and le­gal laws aren’t the same thing. Modern soc­i­ties don’t leg­is­late against adultery, al­though they may con­sider it against the moral rules.

If you are go­ing to over­ride a moral rule, (ie nei­ther pun­ish nor even dis­aprove of) an ac­tion, what would you over­ride it in favour of? What would count more?

• I would re­fuse to al­low moral judge­ment on things which lie out­side of the realm of ap­pro­pri­ate moral­ity. Modern so­cieties don’t leg­is­late against adultery be­cause con­sen­sual sex is amoral. Us­ing moral guidelines to de­ter­mine which peo­ple are al­lowed to have con­sen­sual sex is like us­ing a ham­mer to open a win­dow.

• Oh, that was your con­cern. I has no bear­ing on what I was say­ing.

• I don’t see where I’ve im­plied that one would over­ride a moral rule. What I’m say­ing is that most cur­rent moral sys­tems are not good enough to even make ra­tio­nal rules about some types of ac­tions in the first place, and that in the long run we would re­gret do­ing so af­ter do­ing some metaethics.

Uncer­tainty and the lack of re­li­a­bil­ity of our own minds and de­ci­sion sys­tems are key points of the above.

• Why de­scribe them as sub­je­cive when they are in­ter­sub­jec­tive?

Be­cause they’re not writ­ten on a stone tablet handed down to Hu­man­ity from God the Holy Creator, or de­rived some other ver­ifi­able, falsifi­able and phys­i­cal fact of the uni­verse in­de­pen­dent of hu­mans? And be­cause there are pos­si­ble vari­a­tions within the value sys­tems, rather than them be­ing perfectly uniform and iden­ti­cal across the en­tire species?

I have warn­ing lights that there’s an ar­gu­ment about defi­ni­tions here.

• Be­cause they’re not writ­ten on a stone tablet handed down to Hu­man­ity from God the Holy Creator, or de­rived some other ver­ifi­able, falsifi­able and phys­i­cal fact of the uni­verse in­de­pen­dent of hu­mans?

That would make them not-ob­jec­tive. Sub­jec­tive and in­ter­sub­jec­tive re­main as op­tions.

And be­cause there are pos­si­ble vari­a­tions within the value sys­tems, rather than them be­ing perfectly uniform and iden­ti­cal across the en­tire species?

Then, again, why would any­one else be be­holden to my val­ues?

• Be­cause valu­ing oth­ers’ sub­jec­tive val­ues, or act­ing as if one did, is of­ten a win­ning strat­egy in game-the­o­retic terms.

If one posits that by work­ing to­gether we can achieve an utopia where each in­di­vi­d­ual’s val­ues are max­i­mized, and that to work to­gether effi­ciently we need to at least act ac­cord­ing to a model that would as­sign util­ity to oth­ers’ val­ues, would it not fol­low that it’s in ev­ery­one’s best in­ter­ests for ev­ery­one to build and fol­low such mod­els?

The free-loader prob­lem is an ob­vi­ous down­side of the above sim­plifi­ca­tion, but that and other is­sues don’t seem to be part of the pre­sent dis­cus­sion.

• Be­cause valu­ing oth­ers’ sub­jec­tive val­ues, or act­ing as if one did, is of­ten a win­ning strat­egy in game-the­o­retic terms.

That doesn’t make them be­holden—obli­gated. They can opt not to play that game. They can opt not to vvalue win­ning.

If one posits that by work­ing to­gether we can achieve an utopia where each in­di­vi­d­ual’s val­ues are max­i­mized, and that to work to­gether effi­ciently we need to at least act ac­cord­ing to a model that would as­sign util­ity to oth­ers’ val­ues, would it not fol­low that it’s in ev­ery­one’s best in­ter­ests for ev­ery­one to build and fol­low such mod­els?

Only if they achieve satis­fac­tion for in­di­vi­d­u­als bet­ter than their be­hav­ing self­ishly. A utopia that is bet­ter on av­erae or in to­tal need not be bet­ter for ev­ery­one in­di­vi­d­u­ally.

• Could you taboo “be­holden” in that first? I’m not sure the “feel­ing of moral duty borned from guilt” I as­so­ci­ate with the word “obli­gated” is quite what you have in mind.

They can opt not to play that game. They can opt not to value win­ning.

Within con­text, you can­not opt to not value win­ning. If you wanted to “not win”, and the preferred course of ac­tion is to “not win”, this merely means that you had a hid­den func­tion that as­signed greater util­ity to a lower ap­par­ent util­ity within the game.

In other words, you just didn’t truly value what you thought you val­ued, but some other thing in­stead, and you end up hav­ing in fact won at your ob­jec­tive of not win­ning that sub-game within your over­ar­ch­ing game of opt­ing to play the game or not (the de­ci­sion to opt to play the game or not is it­self a sep­a­rate higher-tier game, which you have won by de­cid­ing to not-win the lower-tier game).

A utopia which pur­ports to max­i­mize util­ity for each in­di­vi­d­ual but fails to op­ti­mize for higher-tier or meta util­ities and val­ues is not truly max­i­miz­ing util­ity, which vi­o­lates the premises.

(sorry if I’m ar­gu­ing a bit by defi­ni­tion with the utopia thing, but my premise was that the utopia brings each in­di­vi­d­ual agent’s util­ity to its max­i­mum pos­si­ble value if there ex­ists a max­i­mum for that agent’s func­tion)

• I wouldn’t let my val­ues be changed if do­ing so would thwart my cur­rent val­ues. I think you’re con­tend­ing that the utopia would satisfy my cur­rent val­ues bet­ter than the sta­tus quo would, though.

In that case, I would only re­sist the utopia if I had a de­on­tic pro­hi­bi­tion against chang­ing my val­ues (I don’t have very strong ones but I think they’re in here some­where and for some things). You would call this a hid­den util­ity func­tion, I don’t think that ad­e­quately mod­els the idea that hu­mans are satis­ficers and not perfect util­i­tar­i­ans. Deon­tol­ogy is some­times a way of iden­ti­fy­ing satis­fic­ing con­di­tions for hu­man be­hav­ior, in that sense I think it can be a much stronger ar­gu­ment.

Even sup­pos­ing that we were perfect util­i­tar­i­ans, if I placed more value on main­tain­ing my cur­rent val­ues than I do on any­thing else, I would still re­ject mod­ify­ing my­self and mov­ing to­wards your utopia.

• Do you think the utopia is fea­si­ble?

• Naw. But even if it was, if I placed value on main­tain­ing my cur­rent val­ues to a high de­gree, I wouldn’t mod­ify.

• Within con­text, you can­not opt to not value win­ning. If you wanted to “not win”, and the preferred course of ac­tion is to “not win”, this merely means that you had a hid­den func­tion that as­signed greater util­ity to a lower ap­par­ent util­ity within the game.

Games emerge where peo­ple have things other peo­ple value. If some­one doens’t value those sorts of things, they are not go­ing to game-play.

A utopia which pur­ports to max­i­mize util­ity for each in­di­vi­d­ual but fails to op­ti­mize for higher-tier or meta util­ities and val­ues is not truly max­i­miz­ing util­ity, which vi­o­lates the premises.

I don’t see where higher-tier func­tions come in.

You are as­sum­ign that a utopia will max­imise ev­ery­ones value in­diivi­d­u­ally AND that val­ues di­verge. That’s a tall or­der.

• “The ax­ioms aren’t things you’re ar­bi­trar­ily mak­ing up, or as­sum­ing for con­ve­nience-of-proof, about some pre-ex­is­tent thing called num­bers. You need ax­ioms to pin down a math­e­mat­i­cal uni­verse be­fore you can talk about it in the first place. The ax­ioms are pin­ning down what the heck this ‘NUM-burz’ sound means in the first place—that your mouth is talk­ing about 0, 1, 2, 3, and so on.”

Ok NOW I fi­nally get the whole Peano ar­ith­metic thing. …Took me long enough. Thanks kindly, un­usu­ally-fast-think­ing math­e­mat­i­cian!

• The bound­ary be­tween phys­i­cal causal­ity and log­i­cal or math­e­mat­i­cal im­pli­ca­tion doesn’t always seem to be clearcut. Take two ex­am­ples.

(1) The product of two and an in­te­ger is an even in­te­ger. So if I dou­ble an in­te­ger I will find that the re­sult is even. The first state­ment is clearly a time­less math­e­mat­i­cal im­pli­ca­tion. But by re­cast­ing the equa­tion as a pro­ce­dure I in­tro­duce both an im­plied sep­a­ra­tion in time be­tween ac­tion and out­come, and an im­plied phys­i­cal em­bod­i­ment that could be sub­ject to er­ror or in­ter­rup­tion. Thus the truth of the sec­ond for­mu­la­tion strictly de­pends on both a math­e­mat­i­cal fact and phys­i­cal facts.

(2) The end­point of a phys­i­cal pro­cess is causally re­lated to the ini­tial con­di­tions by the phys­i­cal laws gov­ern­ing the pro­cess. The sen­si­tivity of the end­point to the ini­tial con­di­tions is a quite sep­a­rate phys­i­cal fact, but re­quires no new phys­i­cal laws: it is a math­e­mat­i­cal im­pli­ca­tion of the phys­i­cal laws already noted. Again, the re­la­tion­ship de­pends on both phys­i­cal and math­e­mat­i­cal truths.

Is there a rec­og­nized name for such hy­brid cases? They could per­haps be de­scribed as “quasi-causal” re­la­tion­ships.

• I love this post, and will be recom­mend­ing it.

Speak­ing as a non-math­e­mat­i­cian I think I would have tried to ex­press ‘there’s only one chain’ by say­ing some­thing like ‘all num­bers can be reached by a finite amount of repeti­ti­tions of con­sid­er­ing the suc­ces­sor of a num­ber you’ve already con­sid­ered, start­ing from zero’.

• We can try to write that down as “For all x, there is an n such that x = S(S(...S(0)...)) re­peated n times.”

The two prob­lems that we run into here are: first, that re­peat­ing S n times isn’t some­thing we know how to do in first-or­der logic: we have to say that there ex­ists a se­quence of rep­e­ti­tions, which re­quires quan­tify­ing over a set. Se­cond, it’s not clear what sort of thing “n” is. It’s a num­ber, ob­vi­ously, but we haven’t pinned down what we think num­bers are yet, and this state­ment be­comes awk­ward if n is an el­e­ment of some other chain that we’re try­ing to say doesn’t ex­ist.

• re­peat­ing S n times isn’t some­thing we know how to do in first-or­der logic

Why not? Re­peat­ing S n times is just ad­di­tion, and ad­di­tion is defined in the peano first or­der logic ax­ioms. I just took these from my text­book:

∀y.plus(0,y,y)

∀x.∀y.∀z.(plus(x,y,z) ⇒ plus(s(x),y,s(z)))

∀x.∀y.∀z.∀w.(plus(x,y,z) ∧ ¬same(z,w) ⇒ ¬plus(x,y,w))

I’ve also seen ad­di­tion defined re­cur­sively some­how, so each step it sub­tracted 1 from the sec­ond num­ber and added 1 to the first num­ber, un­til the sec­ond num­ber was equal to zero. Some­thing like this:

∀x.∀y.∀z.∀w.(plus(x,y,z) ⇒ plus(s(x),w,z) ∧ same(s(w),y))

From this you could define sub­trac­tion in a similar way, and then state that all num­bers sub­tracted from them­selves, must equal 0. This would rule out non­stan­dard num­bers.

• From this you could define sub­trac­tion in a similar way, and then state that all num­bers sub­tracted from them­selves, must equal 0. This would rule out non­stan­dard num­bers.

That will not rule out non­stan­dard mod­els of the first-or­der Peano ax­ioms. If a sub­trac­tion pred­i­cate is defined by:

∀x. sub(x,0,x)

∀x.∀y.∀z. sub(x,y,z) ⇒ sub(s(x),s(y),z)

then you don’t need to add that all num­bers sub­tracted from them­selves, must equal 0. ∀x.sub(x,x,0) is already a the­o­rem, which can be proved al­most im­me­di­ately from those ax­ioms and the first-or­der in­duc­tion schema. Be­ing a the­o­rem, it is true in all mod­els. Every non­stan­dard el­e­ment of a non­stan­dard model, sub­tracted from it­self, gives 0.

It may seem odd that a state­ment proved by in­duc­tion is nec­es­sar­ily true even of those el­e­ments of a non-stan­dard model that, in our men­tal pic­ture of them, can­not be reached by count­ing up­wards from zero, but the in­duc­tion ax­iom scheme ex­plic­itly says just that: if P(0) and ∀x.(P(x) ⇒ P(s(x))) then ∀x.P(x). The con­clu­sion is not limited to stan­dard val­ues of x, be­cause the lan­guage can­not dis­t­in­guish stan­dard from non-stan­dard val­ues.

• If you already have an ax­iom of in­duc­tion then you’ve already ruled out non­stan­dard num­bers and this isn’t an is­sue. I was try­ing to show that with­out the sec­ond or­der logic ax­iom of in­duc­tion, you can rule out non­stan­dard num­bers.

The re­cur­sive sub­tract pred­i­cate will never reach zero on a non­stan­dard num­ber, there­fore it can not be true that n-n=0.

• If you already have an ax­iom of in­duc­tion then you’ve already ruled out non­stan­dard num­bers and this isn’t an is­sue. I was try­ing to show that with­out the sec­ond or­der logic ax­iom of in­duc­tion, you can rule out non­stan­dard num­bers.

Without sec­ond-or­der logic, you can­not rule out non­stan­dard num­bers. As Epicte­tus men­tioned, the Lowen­heim-Skolem The­o­rem im­plies that if there is a model of first-or­der Peano ar­ith­metic, there are mod­els of all in­finite car­di­nal­ities.

You have to dis­t­in­guish the ax­ioms from the mean­ings one in­tu­itively at­taches to them. We have an in­tu­itive idea of the nat­u­ral num­bers, and the Peano ax­ioms (in­clud­ing the in­duc­tion schema) seem to be true of them. How­ever, ZFC set the­ory (for ex­am­ple) prov­ably con­tains mod­els of those ax­ioms other than the nat­u­ral num­bers of our in­tu­ition.

The in­duc­tion schema seems to for­mal­ise our no­tion that ev­ery nat­u­ral num­ber is reach­able by count­ing up from zero. But look more closely and you can in­tu­itively read it like this: if you can prove that P is true of ev­ery num­ber you can reach by count­ing, then P is true of ev­ery num­ber (even those you can’t reach by count­ing, if there are any).

The pred­i­cate “is a stan­dard num­ber” would be a coun­terex­am­ple to that, but the in­duc­tion schema is as­serted only for for­mu­las P ex­press­ible in the lan­guage of Peano ar­ith­metic. Given the ex­is­tence of non-stan­dard mod­els, the fact that “is a stan­dard num­ber” does not satisfy the in­duc­tion schema demon­strates that it is not defin­able in the lan­guage.

The sub­trac­tion pred­i­cate prov­ably satis­fies ∀n. n-n = 0. There­fore ev­ery model of the Peano ax­ioms satis­fies that—it would not be a model if it did not.

(Tech­ni­cal re­mark: I should not have added “sub” as a new sym­bol, which cre­ates a differ­ent lan­guage, an ex­ten­sion of Peano ar­ith­metic. In­stead, “sub(x,y,z) should be in­tro­duced as a met­al­in­guis­tic ab­bre­vi­a­tion for y+z=x, which is a for­mula of Peano ar­ith­metic. One can still prove ∀x. sub(x,x,0), and with­out even us­ing in­duc­tion. Ex­pand­ing the ab­bre­vi­a­tion gives x+0 = x, which is one of the ax­ioms, e.g. as listed here.)

• I re­fer you to the Lowen­heim-Skolem The­o­rem:

Every (countable) first-or­der the­ory that has an in­finite model, has a model of size k for ev­ery in­finite car­di­nal k. You can­not use first-or­der logic to ex­clude non-stan­dard num­bers un­less you want to aban­don in­finite mod­els al­to­gether.

• Re­peat­ing S n times is not ad­di­tion: ad­di­tion is the thing defined by those ax­ioms, no more, and no less. You can prove the state­ments:

∀x. plus(x, 1, S(x))

∀x. plus(x, 2, S(S(x)))

∀x. plus(x, 3, S(S(S(x))))

and so on, but you can’t write “∀x. plus(x, n, S(S(...n...S(x))))” be­cause that doesn’t make any sense. Nei­ther can you prove “For ev­ery x, x+n is reached from x by ap­ply­ing S to x some num­ber of times” be­cause we don’t have a way to say that for­mally.

From out­side the Peano Ax­ioms, where we have our own no­tion of “num­ber”, we can say that “Ad­ding N to x is the same as tak­ing the suc­ces­sor of x N times”, where N is a real-hon­est-to-god-nat­u­ral-num­ber. But even from the out­side of the Peano Ax­ioms, we can­not con­vince the Peano Ax­ioms that there is no num­ber called “pi”. If pi hap­pens to ex­ist in our model, then all the val­ues …, pi-2, pi-1, pi, pi+1, pi+2, … ex­ist, and to­gether they can be used to satisfy any the­o­rem about the nat­u­ral num­bers you con­coct. (For in­stance, sub(pi, pi, 0) is a true state­ment about sub­trac­tion, so the state­ment “∀x. sub(x, x, 0)” can be proven but does not rule out pi.)

• “For ev­ery x, x+n is reached from x by ap­ply­ing S to x some num­ber of times” be­cause we don’t have a way to say that for­mally.

But that’s what I’m try­ing to say. To say n num­ber of times, you start with n and sub­tract 1 each time un­til it equals zero. So for ad­di­tion, 2+3 is equal to 3+2, is equal to 4+1, is equal to 5+0. For sub­trac­tion you do the op­po­site and sub­tract one from the left num­ber too each time.

If no num­ber of sub­tract 1′s cause it to equal 0, then it can’t be a num­ber.

• I know that’s what you’re try­ing to say be­cause I would like to be able to say that, too. But here’s the prob­lems we run into.

1. Try writ­ing down “For all x, some num­ber of sub­tract 1′s cause it to equal 0”. We can write the “∀x. ∃y. F(x,y) = 0″ but in place of F(x,y) we want “y iter­a­tions of sub­tract 1′s from x”. This is not some­thing we could write down in first-or­der logic.

2. We could write down sub(x,y,0) (in your no­ta­tion) in place of F(x,y)=0 on the grounds that it ought to mean the same thing as “y iter­a­tions of sub­tract 1′s from x cause it to equal 0”. Un­for­tu­nately, it doesn’t ac­tu­ally mean that be­cause even in the model where pi is a num­ber, the re­sult­ing ax­iom “∀x. ∃y. sub(x,y,0)” is true. If x=pi, we just set y=pi as well.

The best you can do is to add an in­finitely long ax­iom “x=0 or x = S(0) or x = S(S(0)) or x = S(S(S(0))) or …”

• I think I’m start­ing to get it. That there is no prop­erty that a nat­u­ral num­ber could be defined as hav­ing, that a in­finite chain couldn’t also satisfy in the­ory.

That’s re­ally dis­ap­point­ing. I took a course on logic and the most in­spiring mo­ment was when the pro­fes­sor wrote down the ax­ioms of peano ar­ith­mitic. They are more or less for­mal­iza­tions of all the stuff we learned about num­bers in grade school. It was cool that you could just write down what you are talk­ing about for­mally and use pure logic to prove any the­o­rem with them. It’s sad that it’s so limited you can’t even ex­press num­bers.

• “Why does 2+2 come out the same way each time?”

Thoughts that seem rele­vant:

1. Ad­di­tion is well defined, that is if x=x’ and y=y’ then x+y = x’+y’. Not ev­ery com­putable trans­for­ma­tion has this prop­erty. Con­sider the non-well-defined func­tion <+> on frac­tions given by a/​b <+> c/​d = (a+c)/​(b+d) We know that 39 = 13 and 25 = 410 but 719 != 38.

2. We have the Church-Rosser The­o­rem http://​​en.wikipe­dia.org/​​wiki/​​Church%E2%80%93Rosser_the­o­rem as a sort of guaran­tee (in the lambda calcu­lus) that if I com­pute one way and you com­pute an­other, then we can even­tu­ally reach com­mon ground.

3. If we con­sider “a logic” to be a set of rules for ma­nipu­laing strings, then we can come up with some ax­ioms for clas­si­cal logic that char­ac­ter­ize it uniquely. That is to say, we can log­i­cally pin­point clas­si­cal logic (say, with the ax­ioms of boolean alge­bra) just like we can we can log­i­cally pin­point the nat­u­ral num­bers (with the peano ax­ioms).

• 2 Nov 2012 1:25 UTC
3 points
Parent

I’d say that your “non-well-defined func­tion on frac­tions” isn’t ac­tu­ally a func­tion on frac­tions at all; it’s a func­tion on frac­tional ex­pres­sions that fails to define a func­tion on frac­tions.

• Fair enough. We could have “num­ber ex­pres­sions” which de­note the same num­ber, like “ssss0“, “4”, “2+2”, “2*2”. Then the ques­tion of well-defined­ness is whether our method of com­put­ing ad­di­tion gives the same re­sult for each of these differ­ent num­ber ex­pres­sions.

• Be­cause you can prove once and for all that in any pro­cess which be­haves like in­te­gers, 2 thin­gies + 2 thin­gies = 4 thin­gies.

I ex­pected at this point the math­e­mat­i­cian to spell out the con­nec­tion to the ear­lier dis­cus­sion of defin­ing ad­di­tion ab­stractly—“for ev­ery re­la­tion R that works ex­actly like ad­di­tion...”

• x + Sy = Sz -

That looks a bit odd.

• I think the idea is that one speaker got cut off by the other af­ter hav­ing said “x+Sy=Sz”.

• 8 Dec 2012 1:47 UTC
2 points
Parent

If this were Wikipe­dia, some­one would write a rant about the im­por­tance of us­ing ty­po­graph­i­cally cor­rect char­ac­ters for the hy­phen, the minus sign, the en dash, and the em dash ( - − – and — re­spec­tively).

• Yeah, I un­der­stood that af­ter about 10 sec­onds of con­fu­sion, which seems un­nec­es­sary.

• I’m new here, so watch your toes...

As has been men­tioned or al­luded to, the un­der­ly­ing premise may well be flawed. By con­sid­er­able ex­trap­o­la­tion, I in­fer that the un­stated in­tent is to find a re­li­able method for com­pre­hend­ing math­e­mat­ics, start­ing with nat­u­ral num­bers, such that an al­gorithm can be cre­ated that con­sis­tently ar­rives at the most ra­tio­nal an­swer, or set of an­swers, to any prob­lem.

Every­one read­ing this has had more than a lit­tle train­ing in math­e­mat­ics. Per­mit me to digress to en­sure ev­ery­one re­calls a few facts that may not be suffi­ciently ap­pre­ci­ated. Our gen­eral ed­u­ca­tion is the only sub­stan­tive differ­ence be­tween Homo Sapi­ens to­day and Homo Sapi­ens 200,000 years ago.

With each gen­er­a­tion the early ed­u­ca­tion of our offspring in­cludes in­creas­ingly so­phis­ti­cated con­cepts. Th­ese are in­ter­nal­ized as re­li­able, even if the un­der­ly­ing rea­sons have been treated very lightly or not at all. Our abil­ity to use and record ab­stract sym­bols ap­peared at about the same time as farm­ing. The con­cept that “1” stood for a sin­gle ob­ject and “2″ rep­re­sented the con­cept of two ob­jects was es­tab­lish along with a host of other con­cep­tual con­structs. Through the en­su­ing mil­len­nia we now have an ad­vanced sym­bol­ogy that en­ables us to con­tem­plate very com­plex prob­lems.

The di­gres­sion is to point out that very com­plex con­cepts, such as hu­man logic, re­quire a com­plex sym­bol­ogy. I strug­gle with un­der­stand­ing how con­tem­plat­ing a sim­ple ar­tifi­cially con­strained prob­lem about nat­u­ral num­bers helps me to un­der­stand how to think ra­tio­nally or ad­vance the state of the art. The ex­am­ple and hu­man ra­tio­nal­ity are two very differ­ent classes of prob­lem. Hope­fully some­one can en­lighten me.

There are some very in­ter­est­ing base al­ter­na­tives that seem to me to be bet­ter suited to a dis­cus­sion of hu­man ra­tio­nal­ity. Ex­am­in­ing the shape of the Pareto front gen­er­ated by PIBEA (Prospect Indi­ca­tor Based Evolu­tion­ary Al­gorithm for Mul­tiob­jec­tive Op­ti­miza­tion Prob­lems) runs for var­i­ous real-world vari­ables would fa­cil­i­tate dis­cus­sions around how each of us weights each vari­able and what con­di­tional vari­ables change the weight (e.g., ur­gency).

Again, I in­tend no offense. I am seek­ing un­der­stand­ing. Bear in mind that my back­ground is in ap­pli­ca­tion of ad­vanced al­gorithms in real-world situ­a­tions.

• Due to all this talk about logic I’ve de­cided to take a lit­tle closer look at Goedel’s the­o­rems and re­lated is­sues, and found this nice LW post that did a re­ally good job dis­pel­ling con­fu­sion about com­plete­ness, in­com­plete­ness, SOL se­man­tics etc.: Com­plete­ness, in­com­plete­ness, and what it all means: first ver­sus sec­ond or­der logic

If there’s any­thing else along these lines to be found here on LW—or for that mat­ter, any­where, I’m all ears.

• 1 Nov 2012 17:31 UTC
0 points

Never mind the ques­tion of why the laws of physics are sta­ble—why is logic sta­ble? Of course I can’t imag­ine it be­ing any other way, but that’s not an ex­pla­na­tion.

A hid­den med­i­ta­tion, me­thinks.

• try pon­der­ing this one. Why does 2 + 2 come out the same way each time? Never mind the ques­tion of why the laws of physics are sta­ble—why is logic sta­ble? Of course I can’t imag­ine it be­ing any other way, but that’s not an ex­pla­na­tion.

Do you have an an­swer which will be re­vealed in a later post?

• Every num­ber has a suc­ces­sor. If two num­bers have the same suc­ces­sor, they are the same num­ber. There’s a num­ber 0, which is the only num­ber that is not the suc­ces­sor of any other num­ber. And ev­ery prop­erty true at 0, and for which P(Sx) is true when­ever P(x) is true, is true of all num­bers. In com­bi­na­tion, those premises nar­row down a sin­gle model in math­e­mat­i­cal space, up to iso­mor­phism. If you show me two mod­els match­ing these re­quire­ments, I can perfectly map the ob­jects and suc­ces­sor re­la­tions in them

The prop­erty “is the only num­ber which is not the suc­ces­sor of any num­ber” man­i­festly is false for ev­ery Sx.

There is a num­ber ′ (spo­ken “prime”). The suces­sor of ′ is ‘. ’ and ′ are the same num­ber.

There is a num­ber A. Every prop­erty which is true of 0, and for which P(Sx) is true when­ever P(x) is true, is true of A. The suc­ces­sor of A is B. The suc­ces­sor of B is C. The suc­ces­sor of C is A.

Both of these can be elimi­nated by adding a prop­erty P1: EDIT for cor­rect­ness: It is true of a num­ber y that if Sx=y, then y≠x; It is fur­ther true of the num­ber Sy that if Sx=y, Sy≠x. &etc But P1 was not re­quired in your de­scrip­tion of num­bers.

There is also an in­finite se­ries, … −3,-2,-1,o,1,2,3… which also shares all of the prop­er­ties zero for which P(Sx) is true when­ever P(x) is true.

I can’t eas­ily find a way to ex­clude any of the in­finite chains us­ing the ax­ioms de­scribed here.

• But isn’t P1 re­quired by “And ev­ery prop­erty true at 0, and for which P(Sx) is true when­ever P(x) is true, is true of all num­bers.”?

• No, Sx=x is not pro­hibited by that.

I also phrased the meta-prop­erty wrong; I meant to al­low it to ex­plode from zero and will edit to clar­ify.

• 2 Nov 2012 4:09 UTC
3 points
Parent

Ac­tu­ally, it is pro­hibited. Let P(x) be the prop­erty x ≠ S(x). I will now demon­strate that ev­ery nat­u­ral num­ber x has the prop­erty P.

0 ≠ S(0), be­cause 0 is not the suc­ces­sor of any nat­u­ral num­ber (in­clud­ing it­self).

Sup­pose, for an ar­bi­trary nat­u­ral num­ber, k, we know that k ≠ S(k). Sup­pose now that S(k) = S(S(k)). Since k and S(k) have the same suc­ces­sor, we know that k = S(k). But this con­tra­dicts our origi­nal as­sump­tion.

So, for any nat­u­ral num­ber k, we know that (k ≠ S(k)) im­plies (S(k) ≠ S(S(k))), or, to rephrase, if P(k) is true, P(S(k)) must be true as well.

Every prop­erty true at 0, and for which P(Sx) is true when­ever P(x) is true, is true of all num­bers.

So, all mod­els satis­fy­ing Eliezer’s ax­ioms (ac­tu­ally, Peano’s) do in­deed have the prop­erty x ≠ S(x), no mat­ter which nat­u­ral num­ber x you are con­sid­er­ing.

Edits: Paren­the­ses, gram­mar.

• Beat me to the punch! And with clearer for­mat­ting, too.

• That breaks the sin­gle chain; what pro­hibits the finite loop, or the in­finite chain

Con­sider the in­finite set of prop­er­ties of the form x ≠ Sn(x)… and ev­ery finite chain is also bro­ken, through a similar method.

How­ever, I don’t see what prop­erty is true of zero and all of its suc­ces­sors that can­not be true of an in­finite chain, ev­ery mem­ber of which is a suc­ces­sor to a differ­ent num­ber.

I would also ask if it is any differ­ent to make the change “IFF two num­bers have the same suc­ces­sor, they are the same num­ber.”

• Let the prop­erty P be, “is a stan­dard num­ber”.

Then P would be true of 0, true along the suc­ces­sor-chain of 0, and false at a sep­a­rated in­finite suc­ces­sor-chain.

Thus P would be a prop­erty that was true of 0 and true of the suc­ces­sor of ev­ery ob­ject of which it was true, yet not true of all num­bers.

This would con­tra­dict the sec­ond-or­der ax­iom, so this must not be a model of sec­ond-or­der PA.

• And why aren’t −1, ev­ery pre­de­ces­sors of −1, and ev­ery suc­ces­sor of −1 a stan­dard num­ber? Or are we sim­ply us­ing sec­ond-or­der logic to de­clare in a round­about way that there is ex­actly one se­ries of num­bers?

Let us pos­tu­late a pre­de­ces­sor func­tion: Iff a num­ber x is a suc­ces­sor of an­other num­ber, then the pre­de­ces­sor of x is the num­ber which has x as a suc­ces­sor. The pre­de­ces­sor func­tion re­tains ev­ery prop­erty that is defined to be re­tained by the suc­ces­sor func­tion. The al­ter­nate chain C’ has o: o has EVERY prop­erty of 0 (in­clud­ing null prop­er­ties) ex­cept that it is the suc­ces­sor to -a, and the suc­ces­sor to o is a. Those two prop­er­ties are not true of S(x) given that they are true of x. ev­ery suc­ces­sor of o in the chain meets the defi­ni­tion of num­ber; I can’t find a prop­erty that is not true of -a and the pre­de­ces­sors of a but that is true for ev­ery nat­u­ral num­ber.

• `P(k) := R(k, 0)`

`R(k, n) := (k = n) ∨ R(k, Sn)`

P(0) and P(k) ⇒ P(Sk) can be eas­ily proved, so

`P(k) for all k`

Or some­thing like that.

• In that case, P(o) is true, and P(k)->P(Pk) is equally prov­able.

• No...?

The above ba­si­cally says that `P(k)` is “is within the suc­ces­sor chain of 0”. Note that the base case is `k = 0`, not `k = o`. Any­way, the point is, since such a prop­erty is pos­si­ble (in­clud­ing only the ob­jects that are some n-suc­ces­sor of 0), the ax­iom of in­duc­tion im­plies that num­bers that fol­low 0 are the only num­bers.

ETA: Read­ing your par­ent post again, the prob­lem is it’s im­pos­si­ble to have an `o` that has ev­ery prop­erty `0` does. As a demon­stra­tion, `Z(k) := (k = 0)` is a valid prop­erty. It’s true only of `0`. `R(k, 0)` is similarly a prop­erty that is only true of `0`, or `SS..0`.

• I’m hav­ing a hard time pars­ing what you are try­ing to say here.

That breaks the sin­gle chain; what pro­hibits the finite loop, or the in­finite chain?

A finite loop of size one is pro­hibited by ∀x (x ≠ S(x)); this is prov­able from the Peano ax­ioms (and thus holds true in all mod­els of the Peano ax­ioms), a finite loop of size two is pro­hibited by ∀x (x ≠ S(S(x))); this is prov­able from the Peano ax­ioms, a finite loop of size three is pro­hibited by ∀x (x ≠ S(S(S(x)))); this is prov­able from the Peano ax­ioms, and so on.

I don’t see what prop­erty is true of zero and all of its suc­ces­sors that can­not be true of an in­finite chain, ev­ery mem­ber of which is a suc­ces­sor to a differ­ent num­ber.

Con­sider the chains 0, 1, 2, 3, … and 0|, 1|, 2|, 3|, …. Let P(n) be “n is a num­ber with­out a ‘|’.” This is true for ev­ery num­ber in the first chain (“zero and all of its suc­ces­sors”), and not true in the other chain. Thus the set of num­bers in ei­ther of those two chains is not a model of the Peano ax­ioms. This is ex­actly what Eliezer said, but I thought it might be helpful for you to see it vi­su­ally.

Some­thing you should note is that prop­er­ties are fun­da­men­tally ex­ten­sion­ally defined in math­e­mat­ics. So Q(n) := “n is a nat­u­ral num­ber less than 3” is ac­tu­ally, fun­da­men­tally, the col­lec­tion of num­bers {0,1,2}. It does not need a de­scrip­tion. Math is struc­turally based. The names we give num­bers and prop­er­ties don’t mat­ter; it is the re­la­tion­ships be­tween them that we care about. So if we define the same rules about the “mag­i­cally change” op­er­a­tion and the start­ing num­ber “fairy” that we do about the “suc­ces­sor” op­er­a­tion and the start­ing num­ber “zero” in the Peano ax­ioms, you are refer­ring to the same struc­ture. Read the be­gin­ning of Truly Part of You.

As for the proof of this fact, it’s un­for­tu­nately not triv­ial. A math­e­mat­i­cal logic pro­fes­sor at your lo­cal col­lege could prob­a­bly ex­plain it to you. Also, see this page. The proof is in­com­plete, but you could check the refer­ence there, if you’re cu­ri­ous.

I would also ask if it is any differ­ent to make the change “IFF two num­bers have the same suc­ces­sor, they are the same num­ber.”

“Only if two num­bers have the same suc­ces­sor are they the same num­ber.” This is a tau­tol­ogy in logic. If two num­bers are re­ally the same, ev­ery­thing about them must be the same, in­clud­ing their suc­ces­sors (and it is kind of wrong to say “two” in the first place). The other di­rec­tion of im­pli­ca­tion is non-triv­ial, and thus it must be in­cluded in the Peano ax­ioms.

• That proof only ap­plies to well or­dered sets. The set [… -a, o, a, …] has no least el­e­ment, and is there­fore not well or­dered.

• The in­te­gers with that or­der­ing are not a model of the Peano ax­ioms! (By the way, the in­te­gers can be well-or­dered: 0, −1, 1, −2, 2, …)

Read the Wikipe­dia ar­ti­cle on the Peano ax­ioms up through the first para­graph in the mod­els sec­tion. Do you dis­agree with the ar­ti­cle’s state­ment that “Any two mod­els of the Peano ax­ioms are iso­mor­phic.”? If so, why? It has been proven by at least Richard Dedekind and Seth Warner; it also seems in­tu­itively true to me. Given that, I’d need strong ev­i­dence to change my po­si­tion.

If you don’t dis­agree with that, what is it, speci­fi­cally, that Eliezer said in this post that you dis­agree with? I had a lot of trou­ble un­der­stand­ing your origi­nal top-level com­ment, but it seemed like you had a con­trar­ian tone. If I mis­in­ter­preted your origi­nal com­ment, for­give me; I have no crit­i­cism of it ex­cept for that it was not clearly worded. But I think Eliezer’s post was very well-writ­ten and use­ful.

• Speci­fi­cally: I don’t think that the listed con­straints are suffi­cient to uniquely de­scribe ex­actly the set nat­u­ral num­bers. If the prop­erty ‘Is ei­ther zero or in the suc­ces­sion chain of zero’ is al­lowed as a prop­erty of a num­ber, then the round­about de­scrip­tion “ev­ery prop­erty true at 0, and for which P(Sx) is true when­ever P(x) is true, is true of all num­bers” is log­i­cally iden­ti­cal with “There are no chains that do not start with zero.”

Why not sim­ply state that di­rectly as part of your defi­ni­tion, since it is the in­tent? It can even be done in first-or­der logic with a lit­tle bit of work some­what above my level.

EDIT: I hadn’t con­sid­ered that ev­ery finite and countably in­finite set could be well-or­dered by fiat. It seems coun­ter­in­tu­itive that any num­ber could be defined to be the least el­e­ment, but there’s noth­ing in defi­ni­tion of well-or­dered that pre­vents se­lect­ing an ar­bi­trary least el­e­ment.

• I think we’re in agree­ment with each other. The in­te­gers are not well-or­dered by ‘<’ as ‘<’ is tra­di­tion­ally in­ter­preted; they are well-or­dered by other, differ­ent re­la­tions (that can be for­mal­ized in logic); see the Wikipe­dia ar­ti­cle on well-or­der­ing.

The rea­son that we don’t start out with “There are no chains that do not start with zero.” is, I spec­u­late, at least two-fold:

1. In Peano’s origi­nal ax­ioms, he used the other for­mu­la­tion. So there is a prece­dent.

2. Peano’s for­mu­la­tion can be ex­pressed eas­ily in first-or­der Peano ar­ith­metic. Peano’s for­mu­la­tion de­scribes what is go­ing on within the sys­tem; whereas “there are no chains that do not start with zero” is dis­cussing the struc­ture of the sys­tem from the out­side. They do come out equiv­a­lent (I think) in sec­ond-or­der logic, but Peano’s for­mu­la­tion is the one that is eas­ily ex­pressed in first-or­der logic.

• “All nat­u­ral num­bers can be gen­er­ated by iter­at­ing the suc­ces­sor func­tion on zero.” “The small­est set k which in­cludes zero and the suc­ces­sor of ev­ery mem­ber of it­self is the set of nat­u­ral num­bers.”

I think that both of those for­mu­la­tions can be phrased in first-or­der logic...

• Prop­er­ties are sets of num­bers, so with­out get­ting into tech­ni­cal­ities, you need sec­ond-or­der logic to talk about the small­est set such that what­ever (since you need to quan­tify over all can­di­date sets).

Similarly, to say that you can get x by iter­at­ing the suc­ces­sor func­tion on zero re­quires sec­ond-or­der logic. First-or­der logic isn’t even suffi­cient to define ad­di­tion with­out adding ax­ioms for what ad­di­tion does.

• If you use set the­ory, then yes. Usu­ally, how­ever, math­e­mat­i­ci­ans don’t want to have to worry about things like the ax­iom of reg­u­lar­ity when all they wanted to talk about in the first place was the nat­u­ral num­bers!

• You can’t talk about what the nat­u­ral num­bers are and are not with­out some form of set the­ory.

“0 is the only num­ber which is not the suc­ces­sor of any num­ber” re­quires set the­ory to be mean­ingful.

• You can’t talk about what the nat­u­ral num­bers are and are not with­out some form of set the­ory.

But you can talk about some of the prop­er­ties they have, and quite of­ten that is all we care about.

Also, the stronger your sys­tem is, the more likely it is that your for­mu­la­tion is in­con­sis­tent (and if the sys­tem is in­con­sis­tent, you’re definitely not de­scribing any­thing mean­ingful). I’m much more con­fi­dent that first-or­der Peano ar­ith­metic is con­sis­tent than I am that first-or­der ZFC set the­ory is con­sis­tent.

• No. You can rephrase that as: “Every nat­u­ral num­ber is ei­ther 0 or the suc­ces­sor of some num­ber”.

• What does “Every x” mean in the ab­sence of set the­ory?

• En­joy A Prob­lem Course in Math­e­mat­i­cal Logic. Read Defi­ni­tion 6.4, Defi­ni­tion 6.5, and Defi­ni­tion 6.6 (Edit: They are on PDF pages 47-50, book pages 35-38.). It means that, within each model of the ax­ioms, it is the case that ev­ery ob­ject in the model has the speci­fied prop­erty. The nat­u­ral num­bers hap­pen to be a model of first-or­der Peano ar­ith­metic.

Let me ask you what “ev­ery x” means in first-or­der ZFC set the­ory. An­swer care­fully—it has a countable model.

• I think that this is the sort of case in which it is use­ful to do some hand-wav­ing to in­di­cate that you’ve re­al­ized that your rea­son­ing was wrong but that you have ad­di­tional rea­son­ing to back up your con­clu­sion, as oth­er­wise it can ap­pear that you’ve re­al­ized your rea­son­ing was wrong but want to stick to the con­clu­sion any­way, and there­fore need to come up with new rea­son­ing to sup­port it.

Any­way, con­sider the fol­low­ing: let Pn(x) mean “x is the nth suc­ces­sor of 0” (where the 0th suc­ces­sor of a num­ber is it­self). Then, by in­duc­tion, ev­ery num­ber x has the prop­erty “there ex­ists some n such that Pn(x)”.

I don’t think that change has an effect, you’re just adding “if two num­bers are the same num­ber, they have the same suc­ces­sor”, right? Which is already true.

• Is zero the ze­roth suc­ces­sor of zero, by that prop­erty? Is that com­pat­i­ble with zero not be­ing a suc­ces­sor of any num­ber?

• As I said, the “ze­roth suc­ces­sor” of a num­ber is it­self. That is, zero is the re­sult of ap­ply­ing the suc­ces­sor func­tion to it­self zero times. You have to ap­ply a func­tion at least once in or­der to have ap­plied the func­tion (and thus ob­tained a re­sult of ap­ply­ing the func­tion, e.g., calcu­lated a suc­ces­sor).

If you don’t like the term, you can think of it this way:

• P0(x): x = 0

• P1(x): x = S0

• P2(x): x = SS0

and so forth.

• S0 != 0

Sup­pose Sx != x. Then, if SSx = Sx, then Sx is the suc­ces­sor of both x and Sx, so x=Sx. This is false by as­sump­tion, so SSx != Sx.

Thus, by “And ev­ery prop­erty true at 0, and for which P(Sx) is true when­ever P(x) is true, is true of all num­bers.”, for no num­ber x is Sx=x.

Is there some­thing wrong with this rea­son­ing?

• EY is talk­ing from a po­si­tion of faith that in­finite model the­ory and sec­ond-or­der logic are good and rea­son­able things.

It is pos­si­ble to in­stead start from a po­si­tion of doubt that the in­finite model the­ory and sec­ond or­der logic are good and rea­son­able things (based on my mem­ory of hav­ing stud­ied in col­lege whether model the­ory and sec­ond or­der logic can be for­mal­ized within Zer­melo-Frankel set the­ory, and what the first-or­der-ness of Zer­melo-Frankel has to do with it.).

We might be fine with a proof-the­o­retic ap­proach, which starts with the same ideas “zero is a num­ber”, “the suc­ces­sor of a num­ber is a differ­ent num­ber”, but then goes to a proof-the­o­retic rule of in­duc­tion some­thing like “I’d be happy to say ‘All num­bers have such-and-such prop­erty’ if there were a proof that zero has that prop­erty and an­other also proof that if a num­ber has that prop­erty, then its suc­ces­sor also has that prop­erty.”

We don’t need to talk about mod­els at all—in par­tic­u­lar we don’t need to talk about in­finite mod­els.

Se­cond-or­der ar­ith­metic is suffi­cient to get what EY wants (a nice pretty model uni­verse) but I have two ob­jec­tions. First it is too strong—of­ten the first suffi­cient ham­mer that you find in math­e­mat­ics is rarely the one you should end up us­ing. Se­cond, the goal of a nice pretty model uni­verse pre­sumes a stance of faith in (in­finite) model the­ory, but the in­finite model the­ory is not for­mal­ized. If you do for­mal­ize it then your for­mal­iza­tion will have al­ter­na­tive “un­de­sired” in­ter­pre­ta­tions (by Lowen­heim-Skolem).

• EY is talk­ing from a po­si­tion of faith that in­finite model the­ory and sec­ond-or­der logic are good and rea­son­able things.

I think this is a fal­lacy of gray. Math­e­mat­i­ci­ans have been us­ing in­finite model the­ory and sec­ond-or­der logic for a while, now; this is strong ev­i­dence that they are good and rea­son­able.

Edit: Link for­mat­ting, sorry. I wish there was a way to pre­view com­ments be­fore sub­mit­ting....

• Se­cond-or­der logic is not part of stan­dard, main­stream math­e­mat­ics. It is part of a field that you might call math­e­mat­i­cal logic or “foun­da­tions of math­e­mat­ics”. Foun­da­tions of a build­ing are rele­vant to the strength of a build­ing, so the name im­plies that foun­da­tions of math­e­mat­ics are rele­vant to the strength of main­stream math­e­mat­ics. A more ac­cu­rate anal­ogy would be the re­la­tion­ship be­tween physics and philos­o­phy of physics—dis­cov­er­ies in episte­mol­ogy and philos­o­phy of sci­ence are more of­ten driven by physics than the other way around, and the field “philos­o­phy of physics” is a back­wa­ter by com­par­i­son.

As is prob­a­bly ev­i­dent, I think the good, solid math­e­mat­i­cal logic is in­tu­ition­ist and con­struc­tive and higher-or­der and based on proof the­ory first and model the­ory only sec­ond. It is easy to analo­gize from their names to a straight line be­tween first-or­der, sec­ond-or­der, and higher-or­der logic, but in fact they’re not in a straight line at all. First-or­der logic is main­stream math­e­mat­ics, sec­ond-or­der logic is math­e­mat­i­cal logic fla­vored with faith in the re­al­ity of in­finite mod­els and set the­ory, and higher-or­der logic is math­e­mat­i­cal logic that is (usu­ally) con­struc­tive and proof-the­o­retic and built with an aware­ness of com­puter sci­ence.

• Your view is not main­stream.

• To an ex­tent, but I think it’s ob­vi­ous that most math­e­mat­i­ci­ans couldn’t care less whether or not their the­o­rems are ex­press­ible in sec­ond-or­der logic.

• Yes, be­cause most math­e­mat­i­ci­ans just take SOL at face value. If you be­lieve in SOL and use the cor­re­spond­ing English lan­guage in your proofs—i.e., you as­sume there’s only one field of real num­bers and you can talk about it—then of course it doesn’t mat­ter to you whether or not your the­o­rem hap­pens to re­quire SOL taken at face value, just like it doesn’t mat­ter to you whether your proof uses ~~P->P as a log­i­cal ax­iom. Only those who dis­trust SOL would try to avoid proofs that use it. That most math­e­mat­i­ci­ans don’t care is pre­cisely how we know that dis­be­lief in SOL is not a main­stream value. :)

• The stan­dard story is that ev­ery­thing math­e­mat­i­ci­ans prove is to be in­ter­preted as a state­ment in the lan­guage of ZFC, with ZFC it­self be­ing in­ter­preted in first-or­der logic. (With a side-or­der of angst­ing about how to talk about e.g. “all” vec­tor spaces, since there isn’t a set con­tain­ing all of them—IMO there are var­i­ous good ways of re­solv­ing this, but the stan­dard story con­sid­ers it a prob­lem; cer­tainly in so far as SOL pro­vides an an­swer to these con­cerns at all, it’s not “the one” an­swer that ev­ery­body is ob­vi­ously im­plic­itly us­ing.) So when they say that there’s only one field of real num­bers, this is sup­posed to mean that you can for­mal­ize the field ax­ioms as a ZFC pred­i­cate about sets, and then prove in ZFC that be­tween any two sets satis­fy­ing this pred­i­cate, there is an iso­mor­phism. The fact that the se­man­tics of first-or­der logic don’t pin down a unique model of ZFC isn’t seen as con­flict­ing with this; the math­e­mat­i­cian’s state­ment that there is only one com­plete or­dered field (up to iso­mor­phism) is sup­posed to desugar to a for­mal state­ment of ZFC, or more pre­cisely to the meta-as­ser­tion that this for­mal state­ment can be proven from the ZFC ax­ioms. Math­e­mat­i­cal prac­tice seems to me more in line with this story than with yours, e.g. math­e­mat­i­ci­ans find noth­ing strange about in­tro­duc­ing the re­als through ax­ioms and then talk about a “neigh­bour­hood ba­sis” as some­thing that as­signs to each real num­ber a set of sets of real num­bers—you’d need fourth-or­der logic if you wanted to talk about neigh­bour­hood bases as ob­jects with­out hav­ing some kind of set the­ory in the back­ground. And peo­ple who don’t seem to care a fig about logic will use Zorn’s lemma when they want to prove some­thing that uses choice, which seems quite rooted in set the­ory.

Now I do think that math­e­mat­i­ci­ans think of the ob­jects they’re dis­cussing as more “real” than the stan­dard story wants them to, and just us­ing SOL in­stead of FOL as the se­man­tics in which we in­ter­pret the ZFC ax­ioms would be a good way to, um, tell a bet­ter story—I re­ally like your post and it has con­vinced me of the use­ful­ness of SOL—but I think if we’re sim­ply try­ing to de­scribe how math­e­mat­i­ci­ans re­ally think about what they’re do­ing, it’s fairer to say that they’re just tak­ing set the­ory at face value—not think­ing of set the­ory as some­thing that has ax­ioms that you for­mal­ize in some logic, but see­ing it as as fun­da­men­tal as logic it­self, more or less.

• Um, I think when an or­di­nary math­e­mat­i­cian says that there’s only one com­plete or­dered field up to iso­mor­phism, they do not mean, “In any given model of ZFC, of which there are many, there’s only one or­dered field com­plete with re­spect to the pred­i­cates for which sets ex­ist in that model.” We could ask some nor­mal math­e­mat­i­ci­ans what they mean to test this. We could also prove the iso­mor­phism us­ing logic that talked about all pred­i­cates, and ask them if they thought that was a fair proof (with­out call­ing at­ten­tion to the quan­tifi­ca­tion over pred­i­cates).

Tak­ing set the­ory at face value is tak­ing SOL at face value—SOL is of­ten seen as im­port­ing set the­ory into logic, which is why math­e­mat­i­ci­ans who care about it all are some­times sus­pi­cious of it.

• Um, I think when an or­di­nary math­e­mat­i­cian says that there’s only one com­plete or­dered field up to iso­mor­phism, they do not mean, “In any given model of ZFC, of which there are many, there’s only one or­dered field com­plete with re­spect to the pred­i­cates for which sets ex­ist in that model.” We could ask some nor­mal math­e­mat­i­ci­ans what they mean to test this.

The stan­dard story, as I un­der­stand it, is claiming that mod­els don’t even en­ter into it; the or­di­nary math­e­mat­i­cian is sup­posed to be say­ing only that the cor­re­spond­ing state­ment can be proven in ZFC. Of course, that story is ac­tu­ally told by lo­gi­ci­ans, not by peo­ple who learned about mod­els in their one logic course and then promptly for­got about them af­ter the exam. As I said, I don’t agree with the stan­dard story as a fair char­ac­ter­i­za­tion of what math­e­mat­i­ci­ans are do­ing who don’t care about logic. (Though I do think it’s a co­her­ent story about what the in­for­mal math­e­mat­i­cal English is sup­posed to mean.)

Tak­ing set the­ory at face value is tak­ing SOL at face value—SOL is of­ten seen as im­port­ing set the­ory into logic, which is why math­e­mat­i­ci­ans who care about it all are some­times sus­pi­cious of it.

Is it a fair-rephras­ing of your point that what nor­mal math­e­mat­i­ci­ans do re­quires the same or­der of on­tolog­i­cal com­mit­ment as the stan­dard (non-Henkin) se­man­tics of SOL, since if you take SOL as prim­i­tive and in­ter­pret the ZFC ax­ioms in it, that will give you the cor­rect pow­er­set of the re­als, and if you take set the­ory as prim­i­tive and for­mal­ize the se­man­tics of SOL in it, you will get the cor­rect col­lec­tion of stan­dard mod­els? ’Cause I agree with that (and I see the value of SOL as a par­tic­u­larly sim­ple way of mak­ing that on­tolog­i­cal com­mit­ment, com­pared to say ZFC). My point was that math­e­mat­i­cal English maps much more di­rectly to ZFC than it does to SOL (there’s still cod­ing to be done, but much less of it when you start from ZFC than when you start from SOL); e.g., you ear­lier said that “[o]nly those who dis­trust SOL would try to avoid proofs that use it”, and you can’t re­ally use on­tolog­i­cal com­mit­ments in proofs, what you can ac­tu­ally use is no­tions like “for all prop­er­ties of real num­bers”, and many no­tions peo­ple ac­tu­ally use are ones more di­rectly pre­sent in ZFC than SOL, like my ex­am­ple of quan­tify­ing over the neigh­bour­hood bases (map­pings from re­als to sets of sets of re­als).

• So to take your ex­am­ple of real num­bers—if some­one didn’t want to use SOL, they would still prove the same the­o­rems, they would just end up prov­ing that they are true for any Archimedean com­plete to­tally or­dered field. In gen­eral, I think most math­e­mat­ics (i.e. math­e­mat­ics out­side set the­ory and logic) is ro­bust with re­spect to foun­da­tions: rarely is it the case that a change in ax­ioms makes a proof in­valid, it just means you’re talk­ing about some­thing slightly differ­ent. The idea of the proof is still pre­served.

• I agree with this state­ment—and yet you did not con­tra­dict my state­ment that sec­ond or­der logic is also not part of main­stream math­e­mat­ics.

A topol­o­gist might care about man­i­folds or home­o­mor­phisms—they do not care about foun­da­tions of math­e­mat­ics—and it is not the case that only one foun­da­tion of math­e­mat­ics can sup­port topol­ogy. The weaker foun­da­tion is prefer­able.

• The last sen­tence is not ob­vi­ous at all. The goal of math­e­mat­ics is not to be cor­rect a lot. The goal of math­e­mat­ics is to pro­mote hu­man un­der­stand­ing. Strong ax­ioms help with that by sim­plify­ing rea­son­ing.

• If you as­sume A and de­rive B you have not proven B but rather A im­plies B. If you can in­stead as­sume a weaker ax­iom Aprime, and still de­rive B, then you have proven Aprime im­plies B, which is stronger be­cause it will be ap­pli­ca­ble in more cir­cum­stances.

• In what “cir­cum­stances” are man­i­folds and home­o­mor­phisms use­ful?

• If you were writ­ing soft­ware for some­thing in­tended to tra­verse the In­ter­plane­tary trans­fer net­work then you would prob­a­bly use charts and at­lases and tran­si­tion func­tions, and you would study (sym­plec­tic) man­i­folds and home­o­mor­phisms in or­der to un­der­stand those more-ap­plied con­cepts.

If an oth­er­wise use­ful the­o­rem as­sumes that the man­i­fold is ori­entable, then you need to show that your prac­ti­cal man­i­fold is ori­entable be­fore you can use it—and if it turns out not to be ori­entable, then you can’t use it at all. If in­stead you had an analo­gous the­o­rem that ap­plied to all man­i­folds, then you could use it im­me­di­ately.

• There’s a differ­ence be­tween as­sum­ing that a man­i­fold is ori­entable and as­sum­ing some­thing about set the­ory. The phase space is, of course, only ap­prox­i­mately a man­i­fold. On a very small level it’s—well, some­thing we’re not very sure of. But all the math you’ll be do­ing is an ap­prox­i­ma­tion of re­al­ity.

So some big macro­scopic fea­ture like ori­entabil­ity would be a prob­lem to as­sume. Ori­entabil­ity cor­re­sponds to some­thing in phys­i­cal re­al­ity, and some­thing that clearly mat­ters for your calcu­la­tion.

The ax­iom of choice or what­ever set-the­o­retic as­sump­tion cor­re­sponds to noth­ing in phys­i­cal re­al­ity. It doesn’t mat­ter if the the­o­rems you are us­ing are right for the situ­a­tion, be­cause they are ob­vi­ously all wrong, be­cause they are about sym­plec­tic dy­nam­ics on a man­i­fold, and physics isn’t ac­tu­ally sym­plec­tic dy­nam­ics on a man­i­fold! The only thing that mat­ters is how eas­ily you can find a good-enough ap­prox­i­ma­tion to re­al­ity. More foun­da­tional as­sump­tions make this eas­ier, and do not im­pede one’s ap­prox­i­ma­tion of re­al­ity.

Note that physi­cists fre­quently make ar­gu­ments that are just plain un­am­bigu­ously wrong from a math­e­mat­i­cal per­spec­tive.

• I un­der­stand your point—it’s akin to the Box quote “all mod­els are wrong but some are use­ful”—when choos­ing among (false) mod­els, choose the most use­ful one. How­ever, it is not the case that stronger as­sump­tions are more use­ful—of course stronger as­sump­tions make the task of prov­ing eas­ier, but the task as a whole in­cludes both prov­ing and also build­ing a sys­tem based on the the­o­rems proven.

My pri­mary point is that EY is im­ply­ing that sec­ond-or­der logic is nec­es­sary to work with the in­te­gers. Peo­ple work with the in­te­gers with­out us­ing sec­ond-or­der logic all the time. If he said that he is only in­tro­duc­ing sec­ond-or­der logic for con­ve­nience in prov­ing and there are cer­tainly other ways of do­ing it, and that some peo­ple (in­tu­ition­ists and fini­tists) think that in­tro­duc­ing sec­ond-or­der logic is a du­bi­ous move, I’d be happy.

My other claim that sec­ond-or­der logic is un­phys­i­cal and that its un­phys­i­cal­ity prob­a­bly does rip­ple out to make prac­ti­cal tasks more difficult, is a sec­ondary one. I’m happy to agree that this sec­ondary claim is not main­stream.

• My pri­mary point is ac­tu­ally that I don’t care if math is use­ful. Math is awe­some. This is ob­vi­ously an ex­tremely rare view­point, but very com­mon among.

But I do agree with that quote, more or less. I think that po­ten­tially some mod­els are true, but those mod­els are al­most cer­tainly less use­ful for most pur­poses than the crude and easy to work with ap­prox­i­ma­tions.

I agree that sec­ond-or­der logic is not nec­es­sary to work with the in­te­gers. Se­cond-or­der logic is nec­es­sary to work with the in­te­gers and only the in­te­gers, how­ever. Some­what prob­le­mat­i­cally, it’s not ac­tu­ally pos­si­ble to work with sec­ond-or­der logic.

What sort of prac­ti­cal tasks are you think­ing of?

• Well, it’s strong ev­i­dence that math­e­mat­i­ci­ans find these things use­ful for pub­lish­ing pa­pers.