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.”


“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.”


“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.”

Med­i­ta­tion for next time:

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

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