# The role of mathematical truths

Elab­o­ra­tion of points I made in these com­ments: first, second

TL;DR Sum­mary: Math­e­mat­i­cal truths can be cashed out as com­bined claims about 1) the com­mon con­cep­tion of the rules of how num­bers work, and 2) whether the rules im­ply a par­tic­u­lar truth. This cash­ing-out keeps them purely about the phys­i­cal world and elimi­nates the need to ap­peal to an im­ma­te­rial realm, as some math­e­mat­i­ci­ans feel a need to.

Back­ground: “I am quite con­fi­dent that the state­ment 2 + 3 = 5 is true; I am far less con­fi­dent of what it means for a math­e­mat­i­cal state­ment to be true.”—Eliezer Yudkowsky

This is the prob­lem I will ad­dress here: how should a ra­tio­nal­ist re­gard the sta­tus of math­e­mat­i­cal truths? In do­ing so, I will pre­sent a unify­ing ap­proach that, I con­tend, el­e­gantly solves the fol­low­ing re­lated prob­lems:

- Elimi­nat­ing the need for a non-phys­i­cal, non-ob­serv­able “Pla­tonic” math realm.

- The is­sue of whether “math was true/​ex­isted even when peo­ple weren’t around”.

- Cash­ing out the mean­ing of iso­lated claims like “2+2=4”.

- The is­sue of whether math­e­mat­i­cal truths and math it­self should count as be­ing dis­cov­ered or in­vented.

- Whether math­e­mat­i­cal rea­son­ing alone can tell you things about the uni­verse.

- Show­ing what it would take to con­vince a ra­tio­nal­ist that “2+2=3”.

- How the words in math state­ments can be wrong.

This is an am­bi­tious pro­ject, given the amount of effort spent, by very in­tel­li­gent peo­ple, to prove one po­si­tion or an­other re­gard­ing the sta­tus of math, so I could very well be in over my head here. How­ever, I be­lieve that you will agree with my ap­proach, based on stan­dard ra­tio­nal­ist desider­ata.

Here’s the re­s­olu­tion, in short: For a math­e­mat­i­cal truth (like 2+2=4) to have any mean­ing at all, it must be de­com­pos­able into two in­ter­per­son­ally ver­ifi­able claims about the phys­i­cal world:

1) a claim about whether, gen­er­ally speak­ing, peo­ple’s mod­els of “how num­bers work” make cer­tain assumptions

2) a claim about whether those as­sump­tions log­i­cally im­ply the math­e­mat­i­cal truth (2+2=4)

(Note that this dis­cus­sion avoids the more nar­rowly-con­structed class of math­e­mat­i­cal claims that take the form of say­ing that some ad­mit­tedly ar­bi­trary set of as­sump­tions en­tails a cer­tain im­pli­ca­tion, which de­com­pose into only 2) above. This dis­cus­sion in­stead fo­cuses in­stead on the sta­tus of the more com­mon be­lief that “2+2=4”, that is, with­out spec­i­fy­ing some pre­con­di­tion or as­sump­tion set.)

So for a math­e­mat­i­cal state­ment to be true, it sim­ply needs to be the case that both 1) and 2) hold. You could there­fore re­fute such a state­ment ei­ther by say­ing, “that doesn’t match what peo­ple mean by num­bers [or that par­tic­u­lar op­er­a­tion]”, thus re­fut­ing #1; or by say­ing that the state­ment just doesn’t fol­low from ap­ply­ing the rules that peo­ple com­monly take as the rules of num­bers, thus re­fut­ing #2. (The lat­ter means find­ing a flaw in steps of the proof some­where af­ter the givens.)

There­fore, a per­son claiming that 2+2=5 is ei­ther us­ing a pro­cess we don’t rec­og­nize as any part of math or our ter­minol­ogy for num­bers (vi­o­lat­ing #1) or made an er­ror in calcu­la­tions (vi­o­lat­ing #2). Recog­ni­tion of this er­ror is thus re­vealed phys­i­cally: ei­ther by notic­ing the gen­eral opinions of peo­ple on what num­bers are, or by notic­ing whether the car­ry­ing out of the rules (ei­ther in the mind or some medium iso­mor­phic to the rules) has a cer­tain re­sult. It fol­lows that math does not re­quire some non-phys­i­cal realm. To the ex­tent that peo­ple feel oth­er­wise, it is a species of the mind-pro­jec­tion fal­lacy, in which #1 and #2 are trun­cated to sim­ply “2+2=4”, and that lone Pla­tonic claim is be­lieved to be in the ter­ri­tory.

The next is­sue to con­sider is what to make of claims that “math has always ex­isted (or been true), even when peo­ple weren’t around to perform it”. It would in­stead be more ac­cu­rate to make the fol­low­ing claims:

3) The uni­verse has always ad­hered to reg­u­lar­i­ties that are con­cisely de­scrib­able in what we now know as math (though it’s coun­ter­fac­tual as no­body would nec­es­sar­ily be around to do the de­scribing).

4) It has always been the case that if you set up some phys­i­cal sys­tem iso­mor­phic to some math­e­mat­i­cal op­er­a­tion, performed the cor­re­spond­ing phys­i­cal op­er­a­tion, and re-in­ter­preted it by the same iso­mor­phism, the in­ter­pre­ta­tion would match that which the rules of math give (though again coun­ter­fac­tual, as there’s no one to be ob­serv­ing or set­ting up such a sys­tem).

This, and noth­ing else, is the sense in which “math was around when peo­ple weren’t”—and it uses only phys­i­cal re­al­ity, not im­ma­te­rial Pla­tonic realms.

Is math dis­cov­ered or in­vented? This is more of a defi­ni­tional dis­pute, but un­der my ap­proach, we can say a few things. Math was in­vented by hu­mans to rep­re­sent things use­fully and help find solu­tions. Its hu­man use, given pre­vi­ous non-use, makes it in­vented. This does not con­tra­dict the pre­vi­ous para­graphs, which ac­cept math­e­mat­i­cal claims in­so­far as they are coun­ter­fac­tual claims about what would have gone on had you ob­served the uni­verse be­fore hu­mans were around. (And note that we find math so very use­ful in de­scribing the uni­verse, that it’s hard to think what other de­scrip­tions we could be us­ing.) It is no differ­ent than other “be­liefs in the im­plied in­visi­ble” where a claim that can’t be di­rectly ver­ified falls out as an im­pli­ca­tion of the most par­si­mo­nious ex­pla­na­tion for phe­nom­ena that can be di­rectly ver­ified.

Can “a pri­ori” math­e­mat­i­cal rea­son­ing, by it­self, tell you true things about the uni­verse? No, it can­not, for any re­sult always needs the ad­di­tional em­piri­cal ver­ifi­ca­tion that phe­nomenon X ac­tu­ally be­haves iso­mor­phi­cally to a par­tic­u­lar math­e­mat­i­cal struc­ture (see figure be­low). This is a crit­i­cal point that is of­ten missed due to the ob­vi­ous­ness of the as­sump­tions that the iso­mor­phism holds.

What ev­i­dence can con­vince a ra­tio­nal­ist that 2+2=3? On this ques­tion, my ac­count largely agrees with what Eliezer Yud­kowsky said here, but with some caveats. He de­scribes a sce­nario in which, ba­si­cally, the rules for countable ob­jects start op­er­at­ing in such a way that com­bin­ing two and two of them would yield three of them.

But there are im­por­tant nu­ances to make clear. For one thing, it is not just the ob­jects’ be­hav­ior (2 earplugs com­bined with 2 earplugs yield­ing 3 earplugs) that changes his opinion, but his keep­ing the be­lief that these kinds of ob­jects ad­here to the rules of in­te­ger math. Note that many of the philo­soph­i­cal er­rors in quan­tum me­chan­ics stemmed from the un­grounded as­sump­tion that elec­trons had to obey the rules of in­te­gers, un­der which (given ad­di­tional rea­son­able as­sump­tions) they can’t be in two places at the same time.

Also, for his ex­po­si­tion to help provide in­sight, it would need to use some­thing less ob­vi­ous than 2+2=3′s falsity. If you in­stead talk in terms of much harder ar­ith­metic, like 5,896 x 5,273 = 31,089,508, then it’s not as ob­vi­ous what the an­swer is, and there­fore it’s not so ob­vi­ous how many units of real-world ob­jects you should ex­pect in an iso­mor­phic real-world sce­nario.

Keep in mind that your math-re­lated ex­pec­ta­tions are jointly de­ter­mined by the be­lief that a phe­nomenon be­haves iso­mor­phi­cally to some kind of math op­er­a­tion, and the be­liefs re­gard­ing the re­sults of these op­er­a­tions. Either one of these can be re­jected in­de­pen­dently. Given the more difficult ar­ith­metic above, you can see why you might change your mind about the lat­ter. For the former, you merely need no­tice that for that par­tic­u­lar phe­nomenon, in­te­ger math (say) lacks an iso­mor­phism to it. The causal di­a­gram works like this:

Hy­po­thet­i­cal uni­verses with differ­ent math. My ac­count also han­dles the dilemma, be­loved among philoso­phers, about whether there could be uni­verses where 2+2 ac­tu­ally equals 6. Such sce­nar­ios are harder than one might think. For if our math could still de­scribe the nat­u­ral laws of such a uni­verse, then a de­scrip­tion would rely on a rule­set that im­plies 2+2=4. This would ren­der ques­tion­able the claim that 2+2 has been made to non-triv­ially equal 6. It would re­duce the philoso­pher’s dilemma into “I’ve hy­poth­e­sized a sce­nario in which there’s a differ­ent sym­bol for 4”.

I be­lieve my ac­count is also ro­bust against mere re­la­bel­ing. If some­one speaks of a math where 2+2=6, but it turns out that its en­tire cor­pus of the­o­rems is iso­mor­phic to reg­u­lar math, then they haven’t ac­tu­ally pro­posed differ­ent truths; their “new” math can be ex­plained away as us­ing differ­ent sym­bols, and hav­ing the same re­la­tion­ship to re­al­ity ex­cept with a minor differ­ence in the iso­mor­phism in ap­ply­ing it to ob­ser­va­tions.

Con­clu­sion: Math rep­re­sents a par­tic­u­larly tempt­ing case of map-ter­ri­tory con­fu­sion. Peo­ple who nor­mally fa­vor nat­u­ral­is­tic hy­pothe­ses and make such dis­tinc­tions tend to grant math a spe­cial sta­tus that is not jus­tified by the ev­i­dence. It is a tool that is use­ful for com­press­ing de­scrip­tions of the uni­verse, and for which hu­mans have a com­mon un­der­stand­ing and ter­minol­ogy, but no more an in­trin­sic part of na­ture than its use­ful­ness in com­press­ing phys­i­cal laws causes it to be.

• I think philos­o­phy of math dis­cus­sion on LW would prob­a­bly be bet­ter if it ever referred to the think­ing that has been done by pro­fes­sional philoso­phers of math. Or maybe that think­ing is worth­less enough that it’s worth restart­ing from scratch (e.g. if they don’t have our nec­es­sary back­ground con­cepts), but then that should be noted and defended from time to time.

• Right. I think a lot of the com­ments here have been speak­ing to­wards that re­quest in some way.

• Lakatos struck me as quite im­pres­sive in that field, but I am pre­cisely the op­po­site of widely-read in that area.

• but I am pre­cisely the op­po­site of widely-read in that area.

Never heard of it and illiter­ate? Those are some mighty fine lucky key-presses.

• I’m not yet see­ing how this way of think­ing about math con­tra­dicts pla­ton­ism. It seems to leave un­ad­dressed the ques­tions that pla­ton­ism pur­ports to an­swer. That is, your ac­count here is es­sen­tially in­de­pen­dent of the on­tolog­i­cal sta­tus of math­e­mat­i­cal ob­jects, op­er­a­tions, etc.

For ex­am­ple, you wrote:

It has always been the case that if you set up some phys­i­cal sys­tem iso­mor­phic to some math­e­mat­i­cal op­er­a­tion, performed the cor­re­spond­ing phys­i­cal op­er­a­tion, and re-in­ter­preted it by the same iso­mor­phism, the in­ter­pre­ta­tion would match that which the rules of math give (though again coun­ter­fac­tual, as there’s no one to be ob­serv­ing or set­ting up such a sys­tem).

This seems to leave unan­swered the clas­si­cal kinds of ques­tions that gave rise to pla­ton­ism, such as:

What kind of thing is this “iso­mor­phism” of which you speak? Where does it live? It doesn’t seem to be a phys­i­cal thing it­self, so what is it? And what about the math­e­mat­i­cal op­er­a­tion that is iso­mor­phic to the phys­i­cal sys­tem? Is the math­e­mat­i­cal op­er­a­tion an­other phys­i­cal sys­tem? If so, which spe­cific phys­i­cal sys­tem is it? Is it, for ex­am­ple, some par­tic­u­lar phys­i­cal elec­tronic calcu­la­tor? If so, which one is it? It seems im­plau­si­ble that any par­tic­u­lar phys­i­cal calcu­la­tor has the honor of be­ing the math­e­mat­i­cal op­er­a­tion of ad­di­tion, say. But if the math­e­mat­i­cal op­er­a­tion is not a par­tic­u­lar phys­i­cal sys­tem, what is it? Is it an iso­mor­phism class of phys­i­cal sys­tems? But this gets back to the prob­lem of what, phys­i­cally, an iso­mor­phism is, and adds the prob­lem of what, phys­i­cally, a class of phys­i­cal things is. One might try to iden­tify the class with the mere­olog­i­cal sum of its el­e­ments, but there are well-known prob­lems with this ap­proach. And what about the “rules of math”? Which in­hab­itant of the phys­i­cal uni­verse is a “rule of math”? And so on.

All of the ques­tions above are cer­tainly con­fused to some de­gree. But I’m not yet see­ing that you’ve made much progress on dis­solv­ing them.

[ETA: I don’t mean to say that any­thing you said was wrong. It cer­tainly seems to me to be the most promis­ing way to ap­proach the sub­ject.]

• If your “na­tive hard­ware” can’t un­der­stand and im­ple­ment iso­mor­phisms, you’re no bet­ter off by posit­ing an im­ma­te­rial realm in which iso­mor­phisms ex­ist. At some point, you can no longer define your func­tion­al­ity in terms of a sub-speci­fi­ca­tion. But this just means that an agent must have some level that acts au­to­mat­i­cally, with­out fur­ther re­flec­tion (cf Created Already in Mo­tion), not that a be­ing’s on­tol­ogy is in­suffi­cient on ac­count of failing to posit some su­per­set realm for the con­cepts it im­plic­itly uses.

In case it wasn’t clear from the pre­vi­ous para­graph, I look at this from the per­spec­tive of cre­at­ing an ar­tifi­cial be­ing that can do ev­ery­thing I can. If the defi­cien­cies of the way of han­dling of math I’ve de­scribed (in­clud­ing failure to spec­ify to ever greater pre­ci­sion the “rules of math”) don’t cor­re­spond to some kind of failure mode of the ar­tifi­cial be­ing, then I have to ask if it re­ally is a defi­ciency.

Let me know if that was re­spon­sive to your ques­tions.

• Let me know if that was re­spon­sive to your ques­tions.

I took your post to be an ac­count of the mean­ing of math­e­mat­i­cal claims, or of what it is that they as­sert about the world. In par­tic­u­lar, you said that you would elimi­nate “the need for a non-phys­i­cal, non-ob­serv­able ‘Pla­tonic’ math realm.” (Em­pha­sis added.)

I take your com­ment here to be de­scribing the sense of “need” that you were us­ing in your OP: To need the con­cept of a pla­tonic math realm means to need it to build an ar­tifi­cial be­ing that can do math.

But I don’t think that many pla­ton­ists would dis­agree. I’ve never heard any­one claim that calcu­la­tor en­g­ineers need to learn math­e­mat­i­cal pla­ton­ism, or in­deed any philos­o­phy of math­e­mat­ics at all, to do their job. Cer­tainly none would say that we have to some­how pro­gram the calcu­la­tor to be pla­ton­ist for it to do its job. They wouldn’t even say that a hu­man math­e­mat­i­cian has to be a pla­ton­ist to suc­ceed at math­e­mat­ics.

The prob­lem with pla­ton­ism isn’t that it keeps any­one from be­ing able to build calcu­la­tors. I’d say that the prob­lem with pla­ton­ism is that it con­vinces peo­ple that they can know about some things (ideal ge­o­met­ric ob­jects, say) with­out in­ter­act­ing with them causally. This en­courages some peo­ple to credit other mys­te­ri­ous “ways of know­ing”, such as re­li­gious faith. And that, in turn, can get them so con­fused that they can’t suc­ceed at cer­tain tasks, such as build­ing an AI. (Is that what you were get­ting at?)

If that se­quence of con­fu­sions is the “failure mode” to avoid, then your suc­cess in your OP is to be judged by whether it ac­tu­ally keeps hu­mans from feel­ing such a felt need for pla­ton­ism.

But I don’t yet see that it does this, for the rea­sons that I gave in my pre­vi­ous com­ment. Some­one could eas­ily read your post, agree with the pic­ture it paints, and yet say, “Yes, but just what kinds of things are these iso­mor­phisms and op­er­a­tions and rules of math? I think that the most satis­fy­ing an­swer is still that they are in­hab­itants of some ideal pla­tonic realm.”

• I’d say that the prob­lem with pla­ton­ism is that it con­vinces peo­ple that they can know about some things (ideal ge­o­met­ric ob­jects, say) with­out in­ter­act­ing with them causally. This en­courages some peo­ple to credit other mys­te­ri­ous “ways of know­ing”, such as re­li­gious faith. And that, in turn, can get them so con­fused that they can’t suc­ceed at cer­tain tasks, such as build­ing an AI. (Is that what you were get­ting at?)

Agreed, but that was an im­plicit premise, not some­thing I was try­ing to prove. That is, my ar­ti­cle takes it for granted that you will not want to use an episte­mol­ogy that im­plies that knowl­edge can arise with­out causal in­ter­ac­tion, and that there­fore you deem your episte­mol­ogy flawed if and to the ex­tent that it does so. So I as­sume the reader re­gards re­moval of the pla­tonic realm de­pen­dency as de­sir­able, for any of a num­ber of rea­sons, in­clud­ing that one.

But I don’t yet see that it does this, for the rea­sons that I gave in my pre­vi­ous com­ment. Some­one could eas­ily read your post, agree with the pic­ture it paints, and yet say, “Yes, but just what kinds of things are these iso­mor­phisms and op­er­a­tions and rules of math? I think that the most satis­fy­ing an­swer is still that they are in­hab­itants of some ideal pla­tonic realm.”

True: if you can’t im­ple­ment a well-defined pro­ce­dure (such as iso­mor­phism or stan­dard math) with­out posit­ing its ex­is­tence in an im­ma­te­rial realm, then my ar­ti­cle doesn’t have much that will change your mind on that mat­ter (“you” in the gen­eral sense).

But I don’t see how some­one would well-versed enough in ra­tio­nal­ity for this ar­ti­cle to be rele­vant, yet still make such a leap. That kind of er­ror oc­curs at a more ba­sic level. What­ever rea­son suffices to make one feel the need to posit a pla­tonic realm must have a broader ground­ing, right?

• So I as­sume the reader re­gards re­moval of the pla­tonic realm de­pen­dency as desirable

I think that this gets at the crux of my crit­i­cism. What kind of de­pen­dency on Pla­ton­ism do you see your ar­ti­cle as re­mov­ing? That is, what kind of “need” for Pla­ton­ism did you pic­ture a reader feel­ing be­fore read­ing your ar­ti­cle, but be­ing cured of af­ter read­ing it?

• Thanks, that does get to the heart of the mat­ter. To bor­row from one of the linked ar­ti­cles, I imag­ine some­one in the role of Eliezer Yud­kowsky here, be­ing challenged by “the one” (bold added):

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

Well, it kinda is.

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

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

That is, a ra­tio­nal­ist could avoid mak­ing ob­vi­ous or large er­rors, but still be­lieve “2+3=5”, above and be­yond any phys­i­cally-ver­ifi­able claim be­tween two peo­ple, and above and be­yond any spe­cific model (map) of re­al­ity, phys­i­cally in­stan­ti­ated in agents. My ar­ti­cle says to that ra­tio­nal­ist, no, you needn’t be­lieve in this pla­tonic “2+3=5″ apart from its im­pli­ca­tion in a com­monly used model, and you can still el­e­gantly and con­sis­tently han­dle all of the dilem­mas as­so­ci­ated with hav­ing to clas­sify such ab­stract state­ments. In fact, you needn’t make a state­ment about any­thing non-phys­i­cal.

Do you be­lieve I’ve done so, and said some­thing rele­vant to ra­tio­nal­ists?

• My ar­ti­cle says to that ra­tio­nal­ist, no, you needn’t be­lieve in this pla­tonic “2+3=5” apart from its im­pli­ca­tion in a com­monly used model

Which im­pli­ca­tion is still a fact which seems to be non-phys­i­cal, seems to have been true be­fore there were any hu­mans to do logic, etc. You’ve elimi­nated Pla­tonic nu­mer­i­cal en­tities and meta­phys­i­cally priv­ileged for­mal sys­tems—which do seem to be im­prove­ments—but not non-phys­i­cal a pri­ori truths.

• Which im­pli­ca­tion is still a fact which seems to be non-phys­i­cal, seems to have been true be­fore there were any hu­mans to do logic, etc

It is a coun­ter­fac­tual claim about some­thing phys­i­cal. You can rep­re­sent it in a causal di­a­gram with only phys­i­cal refer­ents.

• It is a coun­ter­fac­tual claim about some­thing phys­i­cal. You can rep­re­sent it in a causal di­a­gram with only phys­i­cal refer­ents.

The causal di­a­gram in your OP con­tains a node la­beled “In­te­ger math im­plies 2+2 = 4?”

What is the phys­i­cal refer­ent for “In­te­ger math”?

• Any phys­i­cal sys­tem that, as best you can in­fer, be­haves with a known iso­mor­phism to in­te­ger math.

**ETA: Oops, see zero_call’s cor­rec­tion; fol­low­ing the ar­ti­cle, in­te­ger math ac­tu­ally cor­re­sponds to some widely-held con­cep­tion—within hu­man brains—of how num­bers work. Since Tyrrell_McAllister’s point was that I was slip­ping in non-phys­i­cal­ity, the rest of the ex­change is still rele­vant, though.

• Any phys­i­cal sys­tem that, as best you can in­fer, be­haves with a known iso­mor­phism to in­te­ger math.

So, to make the pure phys­i­cal­ity of all refer­ents clear, should we la­bel that node:

Phys­i­cal sys­tem S out­puts the string ‘4’ when­ever it is fed the string ‘2+2=’

where S is the name of a spe­cific con­crete phys­i­cal sys­tem such that the string ‘2+2=’ phys­i­cally makes S out­put ‘4’ in a way that is iso­mor­phic to the way that the rules of ar­ith­metic log­i­cally im­ply that 2+2=4?

• Yes, ba­si­cally. I mean, I’d tweak it to read some­thing more like

Phys­i­cal sys­tem S is re­garded as out­putting ‘4’ when in­ter­preted per a spe­cific known iso­mor­phism M, when­ever the query ‘2+2=’ is con­verted per M and ap­plied to it.

but I don’t think that im­pacts what­ever point you were try­ing to make.

• I think that your tweak makes an im­por­tant differ­ence. And, if I may be so bold, I think that you want some­thing closer to what I wrote :).

I’m try­ing to make good your claim that the causal di­a­gram refers only to phys­i­cal things. But your la­bel refers to M, which is an iso­mor­phism. What is the con­crete phys­i­cal refer­ent of “M”?

• I think that your tweak makes an im­por­tant differ­ence. And, if I may be so bold, I think that you want some­thing closer to what I wrote :).

Why? The con­straint that a sys­tem out­put a “string” is too strict; it suffices that they out­put some­thing in­ter­pretable as a string.

I’m try­ing to make good your claim that the causal di­a­gram refers only to phys­i­cal things. But your la­bel refers to M, which is an iso­mor­phism. What is the con­crete phys­i­cal refer­ent of “M”?

An iso­mor­phism M is a one-to-one map­ping be­tween two phe­nom­ena X and Y. In this con­text, then, the phys­i­cal refer­ent of M is what­ever phys­i­cally en­codes how to iden­tify what in Y it is that the as­pects of X map to.

• Why? The con­straint that a sys­tem out­put a “string” is too strict; it suffices that they out­put some­thing in­ter­pretable as a string.

I agree now that “string” is too strict. I should have said “sym­bol”, where a sym­bol is any­thing with phys­i­cal to­kens. My pro­posed la­bel is now

Phys­i­cal sys­tem S out­puts the sym­bol A when­ever it is fed the sym­bol B

where

• S” is the name of a spe­cific con­crete phys­i­cal sys­tem, and

• A” and “B” are the names of spe­cific phys­i­cally-man­i­fested sym­bols,

such that a to­ken of the sym­bol A phys­i­cally makes S out­put a to­ken of the sym­bol B in a way that is iso­mor­phic to the way that the rules of ar­ith­metic log­i­cally im­ply that 2+2=4.

I think that the work that you want to do by adding the word “in­ter­pretable” to the la­bel is done by my con­di­tions on what S, A, and B are.

An iso­mor­phism M is a one-to-one map­ping be­tween two phe­nom­ena X and Y. In this con­text, then, the phys­i­cal refer­ent of M is what­ever phys­i­cally en­codes how to iden­tify what in Y it is that the as­pects of X map to.

Then you should be able to make the la­bel re­fer di­rectly to that phys­i­cal en­cod­ing of M. That is, in­stead of men­tion­ing the iso­mor­phism M, you ought to be able to re­fer just to some spe­cific phys­i­cal sys­tem T that “en­codes” M in the same way that my phys­i­cal sys­tem S above en­codes the op­er­a­tion of adding 2 to 2.

How­ever, if you’re still un­happy with my la­bel, then you would prob­a­bly be un­happy with this un­pack­ing of your refer­ence to M. But I can think of no other way to make good your claim to re­fer only to phys­i­cal things.

(A strict pla­ton­ist would say that even my la­bel refers to non­phys­i­cal things, be­cause it refers to sym­bols, only the to­kens of which are phys­i­cal. I’m happy to ig­nore this.)

• Then you should be able to make the la­bel re­fer di­rectly to that phys­i­cal en­cod­ing of M. … you ought to be able to re­fer just to some spe­cific phys­i­cal sys­tem T that “en­codes” M … if you’re still un­happy with my la­bel, then you would prob­a­bly be un­happy with this un­pack­ing of your refer­ence to M. But I can think of no other way to make good your claim to re­fer only to phys­i­cal things.

Well, I would need to per­mit more than just one phys­i­cal en­cod­ing; I’d need to per­mit any phys­i­cal en­cod­ing that is, er, iso­mor­phic to an ar­bi­trary one of them. But I don’t see this as be­ing a prob­lem—it’s like what they do with NP-com­plete­ness. You can se­lect one (ar­bi­trary) prob­lem as be­ing NP-com­plete, and then define NP-com­plete­ness as “that prob­lem, plus any one con­vert­ible to it”.

So it ap­pears I can avoid bind­ing the mean­ing to one spe­cific phys­i­cal sys­tem, while still us­ing only phys­i­cal refer­ents. And yes, your up­dated ter­minol­ogy is fine as long as you al­low “sym­bols” and “fed” to have suffi­ciently broad mean­ings.

In­ci­den­tally, are you say­ing the same prob­lem arises for defin­ing “waves”? Do you think that refer­ring to one par­tic­u­lar wave re­quires you to refer­ence some­thing non-phys­i­cal? Would you say waves are partly non-phys­i­cal?

• M as an iso­mor­phism is just an in­ter­pre­ta­tion be­tween things (rocks, birds, etc.) and “math things” (num­bers, etc.) Its phys­i­cal refer­ent is the hu­man men­tal in­stan­ti­a­tion of that in­ter­pre­ta­tion (e.g., in the form of neu­tro trans­mit­ters or what have you.) How­ever, (see my com­ment a lit­tle above), I don’t think this is what you were get­ting at.

• No, I thought the phys­i­cal refer­ent for the in­te­ger math was some­thing like “Hu­man men­tal in­stan­ti­a­tion of an idea that is rea­son­ably agreed upon.” I be­lieve you are refer­ring to the phys­i­cal refer­ent of the preimage of the iso­mor­phism (i.e., the phys­i­cal sys­tem it­self. A some­what re­dun­dant thing to call a refer­ent, since it is ac­tu­ally the ex­plicit mean­ing of the state­ment.)

• You’re right, I agree. I was be­ing in­con­sis­tent with my ar­ti­cle there.

• Voted up be­cause this is a great topic that I’d like us to try and be­gin to tackle.

But this post re­ally frus­trat­ing to try to re­spond to. Not be­cause it is es­pe­cially wrong-headed or poorly writ­ten but just be­cause it is a lit­tle hard for me to find my way around your the­ory. It is difficult to find a point of trac­tion. In gen­eral, I sus­pect it just isn’t re­ally solv­ing prob­lems but elid­ing dis­tinc­tions and ig­nor­ing prob­lems (just based on what I do know and the rel­a­tive short­ness of this com­pared to most other work in philos­o­phy of math). This is pretty much the way I feel about what Eliezer has said on the sub­ject and just about ev­ery sin­gle thought I’ve ever had on the sub­ject. I’m also not sure I’m fa­mil­iar enough with the sub­ject area to be able to ex­am­ine this post in the way it re­quires.

So I sus­pect I’ll end up prod­ding you in a cou­ple places but to be­gin with: what ex­actly do you take the Pla­ton­ist the­sis to be? If there is an analog­i­cal re­la­tion­ship be­tween a par­tic­u­lar ex­pres­sion in our sys­tem of in­scrip­tions and our rules for ma­nipu­lat­ing them (i.e. a writ­ten equa­tion) and a phys­i­cal sys­tem (i.e. a sys­tem that equa­tion de­scribes) that seems to sug­gest an un­der­ly­ing struc­ture which is in­stan­ti­ated in both the math­e­mat­i­cal ex­pres­sion and the phys­i­cal sys­tem. That such struc­tures ex­ist in­de­pen­dently of the mind strikes me as a pla­ton­ist po­si­tion. What ex­actly is wrong with that po­si­tion? Or what even did you say to con­tra­dict it?

Per­haps we need to have a dis­cus­sion about ab­stract ob­jects in gen­eral be­fore tack­ling the math.

I do think you’re right about the map-ter­ri­tory con­fu­sions here. They definitely abound.

• In gen­eral, I sus­pect it just isn’t re­ally solv­ing prob­lems but elid­ing dis­tinc­tions and ig­nor­ing prob­lems (just based on what I do know and the rel­a­tive short­ness of this com­pared to most other work in philos­o­phy of math).

This is not a good heuris­tic, be­cause in philos­o­phy, works tend to be longest when they’re con­fused, be­cause most of the length tends to be spent re­pairing the dam­age caused by a mis­take early on.

• So philos­o­phy can get long be­cause the au­thor is run­ning dam­age con­trol. True. But it can also be short be­cause the au­thor is try­ing to an­swer 5-6 ques­tions at once with­out en­gag­ing with the ar­gu­ments of those who ar­gue against his po­si­tion. So length by it­self- maybe a bad heuris­tic. But I’m lev­er­ag­ing this heuris­tic with enough back­ground to make it work.

• what ex­actly do you take the Pla­ton­ist the­sis to be?

That there is an im­ma­te­rial realm of ideal forms (struc­tures, con­cepts) of which our uni­verse con­sists solely of im­perfect ap­prox­i­ma­tions of.

If there is an analog­i­cal re­la­tion­ship be­tween a par­tic­u­lar ex­pres­sion in our sys­tem of in­scrip­tions and our rules for ma­nipu­lat­ing them (i.e. a writ­ten equa­tion) and a phys­i­cal sys­tem (i.e. a sys­tem that equa­tion de­scribes) that seems to sug­gest an un­der­ly­ing struc­ture which is in­stan­ti­ated in both the math­e­mat­i­cal ex­pres­sion and the phys­i­cal sys­tem.That such struc­tures ex­ist in­de­pen­dently of the mind strikes me as a pla­ton­ist po­si­tion.

I would say in­stead that there is some gen­er­at­ing func­tion for re­al­ity. A sys­tem of in­scrip­tions/​rules can de­scribe that gen­er­at­ing func­tion im­perfectly; but this in no way means that the rule/​in­scrip­tion sys­tem has some ex­is­tence apart from its in­stan­ti­a­tion as the uni­verse it­self, and again ex­plic­itly in a model.

• That there is an im­ma­te­rial realm of ideal forms (struc­tures, con­cepts) of which our uni­verse con­sists solely of im­perfect ap­prox­i­ma­tions of.

This stuff about im­perfect ap­prox­i­ma­tions is just a rem­nant of Plato’s mys­ti­cism. Few mod­ern pla­ton­ists would say any­thing like that. This no­tion of an im­ma­te­rial “realm” has similar con­no­ta­tions. How about:

Pla­ton­ism is the view that there ex­ist such things as ab­stract ob­jects — where an ab­stract ob­ject is an ob­ject that does not ex­ist in space or time and which is there­fore en­tirely non-phys­i­cal and non-men­tal.

Pla­ton­ism is ap­peal­ing be­cause it ad­heres to our norm of ac­cept­ing the ex­is­tence of things we make true state­ments about. “Silas is cool” im­plies the ex­is­tence of Silas. Similarly, “3 is prime” im­plies the ex­is­tence of 3. The list of non-pla­ton­ist op­tions as far as I can re­call con­sists of: math­e­mat­i­cal ob­jects are men­tal ob­jects, math­e­mat­i­cal ob­jects are phys­i­cal ob­jects, state­ments about math­e­mat­i­cal ob­jects are false (like state­ments about Santa Claus), or state­ments about math­e­mat­i­cal ob­jects are ac­tu­ally para­phrases of sen­tences that don’t com­mit us to the ex­is­tence of ab­stract ob­jects.

It seems like you are try­ing some­thing like the last. But for this strat­egy you re­ally should give ex­plicit para­phrases or, ideally, a method for para­phras­ing all math­e­mat­i­cal truths.

I would say in­stead that there is some gen­er­at­ing func­tion for re­al­ity. A sys­tem of in­scrip­tions/​rules can de­scribe that gen­er­at­ing func­tion im­perfectly; but this in no way means that the rule/​in­scrip­tion sys­tem has some ex­is­tence apart from its in­stan­ti­a­tion as the uni­verse it­self, and again ex­plic­itly in a model.

But then what kind of thing is this func­tion? It clearly isn’t merely a set of in­scrip­tions and rules for ma­nipu­lat­ing them (the mod­els). Nor is it merely the phys­i­cal uni­verse. We talk like it ex­ists. If it doesn’t, why do we talk like this and what do claims about it re­ally mean?

• At least for ge­o­met­ri­cal forms, the ab­strac­tions may be in­trin­sic to the mind, even if they don’t ex­ist out­side it.

In The Man Who Mis­took His Wife for a Hat, there’s a de­scrip­tion of a man who lost the abil­ity to vi­su­ally rec­og­nize or­di­nary ob­jects, though he could still see. The one de­scrip­tion sug­gest that he just saw ge­om­e­try.

In Crash­ing Through, which is about a man who lost his sight at age 3 and re­cov­ered it in mid­dle age and which has a lot about re­cov­ered vi­sion and the amount of pro­cess­ing it takes to make sense of what you see, there’s men­tion of some peo­ple who are very dis­ap­pointed when they re­cover their sight—they’re con­stantly com­par­ing the world to an idea of it which is perfectly clean and ge­o­met­ri­cal.

• In The Man Who Mis­took His Wife for a Hat, there’s a de­scrip­tion of a man who lost the abil­ity to vi­su­ally rec­og­nize or­di­nary ob­jects, though he could still see. The one de­scrip­tion sug­gest that he just saw ge­om­e­try.

I’m a lit­tle con­fused: did is vi­sual field lose fo­cus such that, in­stead of see­ing the de­tails on ob­jects and their im­perfec­tions he ac­tu­ally just saw ideal­ized ge­o­met­ric figures?

One prob­lem with this as ev­i­dence of the pos­si­bil­ity that ge­o­met­ric forms could ex­ist only in the hu­man mind is that it pre­sum­ably only ap­plies to a rather nar­row class of ge­o­met­ric forms. It would be weird if the ge­o­met­ric forms we have in­nate ac­cess to had a differ­ent on­tolog­i­cal sta­tus from forms that can’t be in­stan­ti­ated in the hu­man mind: like a 1000-sided poly­gon or some­thing in 4+ di­men­sions.

• What I meant was that, if peo­ple have sim­ple ge­o­met­ric forms built deep into their minds, then it would be tempt­ing to con­clude that math has an ob­jec­tive eter­nal ex­is­tence be­cause it feels that way.

In any case, I found the ac­tual quote, and I’ve very un­cer­tain that it sug­gests what I thought it did. It seems as though the man was at least as sen­si­tive to sim­ple topol­ogy as ge­om­e­try, How­ever, peo­ple don’t ro­man­ti­cize topol­ogy.

Here’s the pas­sage, which I had not re­mem­bered as well as I thought:

I had stopped at a florist on my way to his apart­ment and bought my­self an ex­trav­a­gant red rose for my but­ton­hole. Now I re­moved this and handed it to him. He took it like a botanist or mor­phol­o­gist given a spec­i­men, not like per­son given a flower.

“About six inches in length,’ he com­mented. ‘A con­voluted red form with a lin­ear green at­tach­ment.’

‘Yes,’ I said en­courag­ingly, ‘and what do you think it is, Dr P.?’

‘Not easy to say.’ He seemed per­plexed. ‘It lacks the sim­ple sym­me­try of the Pla­tonic solids, al­though it may have a higher sym­me­try of its own… I think this could be an in­flores­cence or flower.’

‘Could be?’ I queried.

‘Could be,’ he con­firmed.

‘Smell it,’ I sug­gested, and he again looked some­what puz­zled, as if I had asked him to smell a higher sym­me­try. But he com­plied cour­te­ously, and took it to his nose. Now, sud­denly, he came to life.

‘Beau­tiful!’ he ex­claimed. ‘An early rose. What a heav­enly smell!’ He started to hum ‘Die Rose, die Lillie…’ Real­ity, it seemed, might by con­veyed by smell, not by sight.

I tried one fi­nal test. It was still a cold day, in early spring, and I had thrown my coat and gloves on the sofa.

‘What is this?’ I asked, hold­ing up a glove.

‘May I ex­am­ine it?’ he asked, and, tak­ing it from me, he pro­ceeded to ex­am­ine it as he had ex­am­ined the ge­o­met­ri­cal shapes.

‘A con­tin­u­ous sur­face,’ he an­nounced at last, ‘in­folded on it­self. It ap­pears to have’ – he hes­i­tated – ‘five out­pouch­ings, if this is the word.’

• It’s a won­der­ful ex­tract in any case. It is fas­ci­nat­ing to see some­one de­scribing the world with­out any­thing more than the phe­nomenol­ogy of his sur­round­ings. It is in­ter­est­ing that the con­cepts he had ac­cess to were math­e­mat­i­cal and ge­o­met­ric- that these con­cepts in­volve a part of the brain sep­a­rate from the part that in­volves more com­plex and ob­vi­ously learned con­cepts like shoe, glove, and flower does seem im­por­tant to keep in mind when eval­u­at­ing the ev­i­dence on this is­sue. You’re right that this fact could lead to us posit­ing a false on­tolog­i­cal differ­ence… though of course there are those who will say “glove­ness” and “flow­er­ness” are ab­stract ob­jects as well. The fact that these con­cepts are pro­cessed in differ­ent parts of the brain could also be taken as ev­i­dence for the dis­tinc­tion in that differ­ent evolu­tion­ary pro­cesses gen­er­ated these two kinds of con­cepts. I’m not sure how to in­ter­pret this. Good for keep­ing in mind though.

• In The Man Who Mis­took His Wife for a Hat, there’s a de­scrip­tion of a man who lost the abil­ity to vi­su­ally rec­og­nize or­di­nary ob­jects, though he could still see. The one de­scrip­tion sug­gest that he just saw ge­om­e­try.

Googling it looks like maybe he just had vi­sual ag­nosia? Which doesn’t re­ally en­tail what you’re say­ing. That would mean that he could see nor­mally but just couldn’t rec­og­nize figures as ob­jects with names and func­tions. Or are you say­ing the de­tails of ob­jects dis­ap­peared and all that was left were the ba­sic ge­o­met­ric forms?

On prob­lem with this as ev­i­dence of the pos­si­bil­ity that ge­o­met­ric forms could ex­ist only in the hu­man mind is that it pre­sum­ably only ap­plies to a rather nar­row class of ge­o­met­ric forms. It would be weird if the ge­o­met­ric forms we have in­nate ac­cess to had a differ­ent on­tolog­i­cal sta­tus from forms that can’t be in­stan­ti­ated in the hu­man mind: like a 1000-sided poly­gon or some­thing in 4+ di­men­sions.

• This is very clar­ify­ing.

• Pla­ton­ism is ap­peal­ing be­cause it ad­heres to our norm of ac­cept­ing the ex­is­tence of things we make true state­ments about. “Silas is cool” im­plies the ex­is­tence of Silas.

To bring the com­par­i­son closer to the mark: It would also im­ply the ex­is­tence of ‘cool’.

• Heh. What I had in mind was Quine’s crite­rion for on­tolog­i­cal com­mit­ment un­der which it wouldn’t. So Silas is cool is some­thing like, where cool is the pred­i­cate let­ter C: ∃x(Cx ∩ x=”Silas”). We’re com­mit­ted to the ex­is­tence of the bound vari­ables (to ex­ist is to be the value of a bound vari­able) but not of the prop­er­ties, there doesn’t have to be any­thing like cool­ness (as­sum­ing that was what you were sug­gest­ing).

There is an older ar­gu­ment that claims all words must re­fer to things and thus a word like “cool” must re­fer to cool­ness. But I wasn’t in­tend­ing to make that ar­gu­ment (though I didn’t say nearly enough in my pre­vi­ous com­ment to ex­pect ev­ery­one to figure that out).

• We’re com­mit­ted to the ex­is­tence of the bound vari­ables (to ex­ist is to be the value of a bound vari­able) but not of the prop­er­ties, there doesn’t have to be any­thing like cool­ness (as­sum­ing that was what you were sug­gest­ing).

My read­ing of Silas’s es­say (and in par­tic­u­lar look­ing at his di­a­grams) gave me im­pres­sion that his ‘2’ is closer to what you would de­scribe as a ‘prop­erty’ than the cat­e­gory in which you put ‘Silas’.

• I was just start­ing from the ob­ser­va­tion that in our math­e­mat­i­cal dis­course we treat num­bers like ob­jects, not prop­er­ties. “The num­ber be­tween 2 and 4”, “there is a prime num­ber greater than one mil­lion”, “5 is odd” etc. all treat num­bers as ob­jects.

• I would call those prop­er­ties that had prop­er­ties. But I’m a pro­gram­mer, not a math­e­mat­i­cian or philoso­pher (so don’t know which limi­ta­tions I’m sup­posed to have placed around my think­ing!)

By the way, I think ‘cool’ is kinda ‘lame’ but ‘awe­some­ness’ is kinda ‘cool’. Just sayin’.

Pla­ton­ism is the view that there ex­ist such things as ab­stract ob­jects — where an ab­stract ob­ject is an ob­ject that does not ex­ist in space or time and which is there­fore en­tirely non-phys­i­cal and non-men­tal.

Pla­ton­ism is ap­peal­ing be­cause it ad­heres to our norm of ac­cept­ing the ex­is­tence of things we make true state­ments about. “Silas is cool” im­plies the ex­is­tence of Silas. Similarly, “3 is prime” im­plies the ex­is­tence of 3.

This, I claim, is where you should stop the chain. You’ve erred from the mo­ment you think that that a true state­ment im­plies in­de­pen­dent ex­is­tence of its pred­i­cates, leav­ing out the pos­si­bil­ity that the pred­i­cates ex­ist only as part of mod­els. (Or, equiv­a­lently, from treat­ing the term “ex­is­tence” as hav­ing the same mean­ing whether it refers to some­thing in the map or the ter­ri­tory.)

But for this strat­egy you re­ally should give ex­plicit para­phrases or, ideally, a method for para­phras­ing all math­e­mat­i­cal truths.

Isn’t that ex­actly what I did? The method I gave is that you break down the truth as­ser­tion into a claim about the cor­rect­ness of the terms (i.e., part 1, what the com­mon us­age is), and a claim about whether the model in com­mon us­age im­plies the truth (part 2). I sup­pose I could have been more spe­cific about how ex­actly one does that, but it seems straight­for­ward enough that I didn’t think I needed to give more de­tail.

• You’ve erred from the mo­ment you think that that a true state­ment im­plies in­de­pen­dent ex­is­tence of its pred­i­cates, leav­ing out the pos­si­bil­ity that the pred­i­cates ex­ist only as part of mod­els.

No, no. Not the pred­i­cates. Just the val­ues of it’s bound vari­ables. I’m not say­ing “Prime­ness ex­ists”. See my re­ply to wedrifid.

Isn’t that ex­actly what I did? The method I gave is that you break down the truth as­ser­tion into a claim about the cor­rect­ness of the terms (i.e., part 1, what the com­mon us­age is), and a claim about whether the model in com­mon us­age im­plies the truth (part 2). I sup­pose I could have been more spe­cific about how ex­actly one does that, but it seems straight­for­ward enough that I didn’t think I needed to give more de­tail.

So “3 is prime” means what? “2+3= 5″ means what? You ab­solutely need to give more de­tail be­cause as far as I know none of the many very smart peo­ple who have tried have been able to give a satis­fac­tory para­phrase for all true math­e­mat­i­cal state­ments.

• No, no. Not the pred­i­cates. Just the val­ues of it’s bound vari­ables. I’m not say­ing “Prime­ness ex­ists”.

How does “3 is prime” im­ply that “3″ ex­ists, while “prime­ness is re­lated to the ze­roes of the Zeta func­tion” not im­ply that “prime­ness” ex­ists?

This whole dis­cus­sion is, ul­ti­mately, about the defi­ni­tion of the word “ex­ist”. But if you try to hang that defi­ni­tion off lin­guis­tic phe­nom­ena, then you’re at the whim of ev­ery lin­guis­tic con­struct peo­ple can come up with, and peo­ple can prob­a­bly twist words to get some­thing cov­ered by your defi­ni­tion that you didn’t want to be.

• How does “3 is prime” im­ply that “3″ ex­ists, while “prime­ness is mem­ber­ship in the set of ze­roes of the Zeta func­tion” not im­ply that “prime­ness” ex­ists?

The lat­ter does im­ply prime­ness ex­ists. But “3 is prime” doesn’t. Luck­ily you haven’t just used prime­ness as the value of a bound vari­able, you’ve given an ap­pro­pri­ate para­phrase (al­though now you’re com­mit­ted to the ex­is­tence of the set of ze­ros of the Zeta func­tion).

This whole dis­cus­sion is, ul­ti­mately, about the defi­ni­tion of the word “ex­ist”. But if you try to hang that defi­ni­tion off lin­guis­tic phe­nom­ena, then you’re at the whim of ev­ery lin­guis­tic con­struct peo­ple can come up with, and peo­ple can prob­a­bly twist words to get some­thing cov­ered by your defi­ni­tion that you didn’t want to be.

Huh?

• So “3 is prime” means what? “2+3= 5″ means what? You ab­solutely need to give more de­tail be­cause as far as I know none of the many very smart peo­ple who have tried have been able to give a satis­fac­tory para­phrase for all true math­e­mat­i­cal state­ments.

My method would be to find the as­so­ci­ated #1 and #2 state­ments. The #1 state­ment would be a claim about what peo­ple use the terms “3”, “is”, and “prime” to mean. Un­der this method you would next iden­tify the com­mon con­cep­tion of “3″ (by em­piri­cal ex­am­i­na­tion of how peo­ple use the term and un­der con­straint of Oc­cam’s Ra­zor) as some­thing like, “the quan­tity im­me­di­ately fol­low­ing the quan­tity im­me­di­ately fol­low­ing the quan­tity im­me­di­ately fol­low­ing the quan­tity of noth­ing”. Then do the same for the other parts.

(Also, keep in mind that this method is only nec­es­sary for the bare state­ment that “3 is prime”. You needn’t con­struct the as­so­ci­ated #1 state­ment for more spe­cific claims like, “Here is a sys­tem of math. Un­der those rules and defi­ni­tions, 3 is prime.”)

Then you would con­struct the #2 state­ment, which would be that, un­der those mean­ings, the claim as a whole fol­lows from the defi­ni­tions and as­sump­tions of the sys­tem im­plictly used by those mean­ings. This would be some­thing like, “un­der any phys­i­cal sys­tem be­hav­ing iso­mor­phi­cally to the as­sump­tions in #1, the phys­i­cal cor­re­late of ‘3 be­ing prime’ will hold”, and that phys­i­cal cor­re­late will be some­thing like, “any di­vi­sion of the units cor­re­lat­ing to 3 will be such that each par­ti­tion will have a differ­ent num­ber of units, or one unit, or three units”.

...Er, okay, per­haps more de­tail was needed. Does that an­swer your ques­tion, though?

• Yes. That is helpful. I was go­ing to bring this up in my first com­ment but de­cided to fo­cus on one thing at a time. Your view seems very similar to nom­i­nal­ist-struc­tural­ism, which I find ap­peal­ing as well. At least I can read nom­i­nal­ist-struc­tural­ism into your post and com­ments. Afaict, it’s con­sid­ered the most promis­ing ver­sion of nom­i­nal­ism go­ing. They ba­si­cally take your use of iso­mor­phism and go one step farther. The SEP dis­cusses it some but it’s a pretty poorly writ­ten ar­ti­cle. You might have to do a lot of googling. Struc­tural­ism ar­gues that math does not de­scribe ob­jects of any kind but rather, struc­tures and places within struc­tures (and have no iden­tity or fea­tures out­side those struc­tures). Any given in­te­ger is not an ob­ject but a places in the struc­ture of in­te­gers. As you can imag­ine once you think of math­e­mat­i­cal truths this way it be­come ob­vi­ous how math can be used to de­scribe other phys­i­cal sys­tems: namely those sys­tems in­stan­ti­ate the same struc­ture (or in your words, are iso­mor­phic?). Nom­i­nal­ist struc­tural­ism in­volves dis­claiming the ex­is­tence of struc­tures (which are ab­stract ob­jects) as in­de­pen­dent of the sys­tems that in­stan­ti­ate them

ETA: One is­sue here is the work be­ing done by the word “re­sem­bles” when we say “the struc­ture of real num­bers re­sem­bles the struc­ture of space’, or “the struc­ture of macro­scopic ob­ject mo­tion re­sem­bles the struc­ture of simu­lated ob­ject mo­tion in your physics simu­la­tor”. Which is the same is­sue as “what is the work be­ing done by ‘iso­mor­phic’.

• I think there’s a tau­tol­ogy hid­den in there some­place. If two ice cubes plus two ice cubes equal one pud­dle of wa­ter, and two guinea pigs plus two guinea pigs equal an un­speci­fied num­ber of gunea pigs, you can say that isn’t what you meant by ad­di­tion, but I think that what you mean is that the phys­i­cal iso­mor­phism has to be ar­ranged so that you get the an­swer you were ex­pect­ing.

Is it true that there’s always a phys­i­cal iso­mor­phism for math? My im­pres­sion is that some math sits around just be­ing math for quite a while, and then some­one finds physics where that math is use­ful. It’s at least plau­si­ble that no one will ever find a phys­i­cal iso­mor­phism for some math, even if the math is log­i­cally sound.

A weaker claim—that math is log­i­cally de­rived from ex­pe­rience of the phys­i­cal world—might hold up.

• I think there’s a tau­tol­ogy hid­den in there some­place. If two ice cubes plus two ice cubes equal one pud­dle of wa­ter, and two guinea pigs plus two guinea pigs equal an un­speci­fied num­ber of gunea pigs, you can say that isn’t what you meant by ad­di­tion, but I think that what you mean is that the phys­i­cal iso­mor­phism has to be ar­ranged so that you get the an­swer you were ex­pect­ing.

As long as the per­son us­ing math and as­sert­ing its rele­vance (com­pres­sive power) in a situ­a­tion can spec­ify, in ad­vance, what must be true in or­der to for the iso­mor­phism to hold, there’s no tau­tol­ogy. This situ­a­tion ex­ists for all the­o­ries: the the­ory’s val­idity, plus the val­idity of your ob­ser­va­tion, plus sev­eral other fac­tors, jointly de­ter­mine what you will ob­serve. If it con­tra­dicts your ex­pec­ta­tions, that re­duces your con­fi­dence in all of the fac­tors, to some ex­tent. Math­e­mat­i­cal pred­i­cates, de­pend­ing on how easy they are to ver­ify, can be more or less re­silient than other fac­tors in the face of such ev­i­dence.

Is it true that there’s always a phys­i­cal iso­mor­phism for math? My im­pres­sion is that some math sits around just be­ing math for quite a while, and then some­one finds physics where that math is use­ful. It’s at least plau­si­ble that no one will ever find a phys­i­cal iso­mor­phism for some math, even if the math is log­i­cally sound.

Right, that’s the caveat I was refer­ring to here:

(Note that this dis­cus­sion avoids the more nar­rowly-con­structed class of math­e­mat­i­cal claims that take the form of say­ing that some ad­mit­tedly ar­bi­trary set of as­sump­tions en­tails a cer­tain im­pli­ca­tion, which de­com­pose into only 2) above. This dis­cus­sion in­stead fo­cuses in­stead on the sta­tus of the more com­mon be­lief that “2+2=4”, that is, with­out spec­i­fy­ing some pre­con­di­tion or as­sump­tion set.)

In other words, some math­e­mat­i­cal claims are about ar­bi­trary ax­iom sets, not nec­es­sar­ily re­lated to phys­i­cal law, and sim­ply as­sert that some im­pli­ca­tion fol­lows there­from. This ar­ti­cle isn’t about those cases. Rather, it’s about bare claims like “2+2=4”, not “un­der this ax­iom set, with these defi­ni­tions, 2+2=4″. There­fore, their truth will hinge par­tially on the mean­ing given to the terms, and claims with­out an ex­plicit ax­iom set have an as­sumed one, and nec­es­sar­ily hinge on the pres­ence of an iso­mor­phism to phys­i­cal law.

• This was a great post.

Some ob­ser­va­tions:

1) The map/​ter­ri­tory tool can be used more ex­ten­sively. Let us take the ter­ri­tory as fairly ‘un­know­able’ ex­cept that us­ing our map, we can make pre­dic­tions. If our pre­dic­tions are wrong then we as­sume that is a failing in our map and we try to re­pair the map. Math is a tau­tolog­i­cal sys­tem that has not failed us yet and we use it as part of the map. If it did fail us, we would change it or aban­don it from the map. We think the chances of this are van­ish­ingly small.

2) We can con­struct en­large­ments to Math tau­tolog­i­cally. I think of this as in­ven­tion but dis­cov­ery is OK if it pleases. They are en­large­ments of the map and not ob­jects in the ter­ri­tory. There is a dan­ger in treat­ing Math ob­jects as part of the ter­ri­tory—an in­finite re­gres­sion of maps of maps of maps.....of ter­ri­tory.

3) The rea­son we in­vented Math origi­nally and the rea­son it maps the ter­ri­tory so well are both be­cause its roots are in the struc­ture of the brain and the brain has evolved to do a good (pass­able) pre­dic­tive model of re­al­ity (or map of the ter­ri­tory).

Please carry on with your in­ves­ti­ga­tion of the sub­ject and post on it again.

• What are the rea­sons that math­e­mat­i­ci­ans like to ap­peal to a math­e­mat­i­cal realm?

In my case, I feel like I ‘ma­nipu­late’ math­e­mat­i­cal ob­jects in my mind as one would ma­nipu­late phys­i­cal ob­jects. Also, I feel like I ‘ex­plore’ a math­e­mat­i­cal sub­ject as one would ex­plore a ter­ri­tory … I in­ves­ti­gate how it works rather than make up how it works. (If any one else feels like be­lief in a math realm is nat­u­ral, even if illu­sory, what are your rea­sons?)

The temp­ta­tion to ap­peal to an im­ma­te­rial realm may also re­late to your 4th point:

4) It has always been the case that if you set up some phys­i­cal sys­tem iso­mor­phic to some math­e­mat­i­cal op­er­a­tion, performed the cor­re­spond­ing phys­i­cal op­er­a­tion, and re-in­ter­preted it by the same iso­mor­phism, the in­ter­pre­ta­tion would match that which the rules of math give (though again coun­ter­fac­tual, as there’s no one to be ob­serv­ing or set­ting up such a sys­tem).

It has always been the case. But why? Why do math­e­mat­i­cal iso­mor­phisms have to fol­low the same rules as their phys­i­cal coun­ter­parts?

My guess, com­pletely un­sub­stan­ti­ated by any knowl­edge of neu­ro­science, is that when math­e­mat­i­ci­ans do math, they are cre­at­ing and ma­nipu­lat­ing phys­i­cal mod­els in their brains. We ex­plore the logic em­bed­ded in phys­i­cal re­al­ity by study­ing physics on a smaller scale, in a much more ab­stract way, in our brains. Be­cause—and this is my only ar­gu­ment—how else would we know? I would guess that the mod­els are im­ple­mented at the cel­lu­lar (neu­ronal) scale rather than sub-cel­lu­lar.

So then there would in­deed good rea­sons for our sense of a Pla­tonic realm. The Pla­tonic realm would be the spe­cial soft­ware (hard­ware?) that we run when we think about and de­velop math­e­mat­ics.

• ″...the large, highly evolved sen­sory and mo­tor por­tions of the brain seem to be the hid­den pow­er­house be­hind hu­man thought. By virtue of the great effi­ciency of these billion-year-old struc­tures, they may em­body one mil­lion times the effec­tive com­pu­ta­tional power of the con­scious part of our minds.

While novice perfor­mance can be achieved us­ing con­scious thought alone, mas­ter-level ex­per­tise draws on the enor­mous hid­den re­sources of these old and spe­cial­ized ar­eas. Some­times some of that power can be har­nessed by find­ing and de­vel­op­ing a use­ful map­ping be­tween the prob­lem and a sen­sory in­tu­ition.”

• My guess, com­pletely un­sub­stan­ti­ated by any knowl­edge of neu­ro­science, is that when math­e­mat­i­ci­ans do math, they are cre­at­ing and ma­nipu­lat­ing phys­i­cal mod­els in their brains.

I strongly agree! As­sum­ing physics, “be­lief in math” is equiv­a­lent to the be­lief that these mod­els be­have very con­sis­tently in my and oth­ers’ brains, and re­flect other re­gions of phys­i­cal re­al­ity effec­tively. But even with­out that, what­ever this floaty thing in my mind I call math is, my “be­lief in math” is one that I hold as con­vict­ingly and im­plic­itly as any­thing else I’m aware of.

• I’m very much not con­vinced by your case. Why wouldn’t sim­ply map=ter­ri­tory for math?

Does ac­cept­ing your ar­gu­ments change any­thing, or is it just a less con­ve­nient way to look at the same math?

• The ques­tion isn’t why wouldn’t it or why couldn’t it, but why must it.

Many of our maps could equal the ter­ri­tory, or at least match it pre­cisely. For me, the ter­ri­to­ries that my pre­sent un­der­stand­ing of math maps to are:

1. phys­i­cal re­al­ity,

2. what hap­pens when oth­ers and my past and fu­ture selves “do math”.

I ex­pect a proof of a the­o­rem in my mind right now to tell me that

1. phys­i­cal re­al­ity will be­have a cer­tain way if the the­o­rem ap­plies to my model of it, and

2. when I or oth­ers “do math”, we’ll have ideas that are ei­ther con­sis­tent with the the­o­rem, or ideas that have a par­tic­u­lar re­flec­tive feel­ing about them that I’ve learned to call “er­rors”.

I ad­here to this be­lief—this trust in math—more faith­fully than to any other I can think of, and have seen more ev­i­dence for it than, well, al­most any­thing. But I’m still not sure that it must “equal the ter­ri­tory”, and I think that’s the point.

• I agree with you. For what it’s worth, I’m a math­e­mat­i­cian, and for me math is as much about sub­jec­tive an­ti­ci­pa­tion as ev­ery­thing else (though most of my col­leagues dis­agree). It’s about ex­pect­ing the same con­clu­sion ev­ery time, and ex­pect­ing to find some­thing fa­mil­iar that I’d clas­sify as an “er­ror” when that doesn’t hap­pen.

With re­ally ab­stract math, when I be­lieve that the­o­rem X is true, what I’m think­ing is more like thought pat­tern S can re­li­ably trans­form thought A into thought B, where thought pat­tern S is a pat­tern peo­ple usu­ally call “de­duc­tion”, and “A” and “B” are thought types usu­ally called “hy­pothe­ses” and “con­clu­sions”.

• 17 Jul 2010 6:01 UTC
0 points

Isn’t the sim­ple way to deal with most of these is­sues is to treat math as an­other lan­guage that let’s us com­mu­ni­cate about the world? There isn’t re­ally a “two” by it­self out there, two is more of an ad­jec­tive de­scribing the num­ber of ob­jects [or po­si­tion, or what­ever]. This is akin to how we say that there’s noth­ing that’s in­her­ently a tree, but there are ob­jects we call trees so we all know what ob­ject we’re talk­ing about.

If I’m miss­ing the mark, or this leads to some silly con­clu­sion, some­one please cor­rect me.

• Hmm… For those of us who weren’t that wor­ried about Pla­ton­ism to be­gin with, this seems to let us rest a lit­tle eas­ier. I’m not sure it ac­com­plishes any more than that. But of­ten, that’s enough.

• Re­gard­ing your origi­nal for­mu­la­tion, I think you could phrase things a lit­tle more sim­ply. For ex­am­ple: “For a math­e­mat­i­cal claim to be true, we re­quire two con­di­tions. Firstly, the ax­ioms of the claim should agree with our own ac­cepted ax­ioms, and our ax­ioms should be rea­son­able. Se­condly, the claim should fol­low from those ax­ioms.” As far as I can tell, this is ba­si­cally what you’re say­ing, al­though what you mean by the rea­son­able­ness of our ax­ioms is un­clear to me.

Re­gard­ing the ex­is­tence of math­e­mat­i­cal en­tities, you’ve seemed to an­swer in the nega­tive. But I don’t see this as fol­low­ing from your origi­nal frame­work. That frame­work says noth­ing about the iden­tity of the math­e­mat­i­cal struc­tures in and of them­selves. That frame­work only tells you how the math works, not what it is. Although per­haps you’re say­ing that the math struc­tures have no iden­tity in and of them­selves, ex­ist­ing solely in this frame­work of claim ver­ifi­ca­tion. But in that case, the re­s­olu­tion is tau­tolog­i­cal, and I’m not sure it re­ally gets to the heart of that ques­tion.

In look­ing at the dis­cov­ery ver­sus in­ven­tion of math, you sup­port the in­ven­tion. But this is es­sen­tially a re­con­figu­ra­tion of the pre­vi­ous prob­lem. If we don’t know pre­cisely the iden­tity of the math, the differ­ence be­tween its in­ven­tion or its dis­cov­ery is moot.

I dis­agree about your nega­tive con­clu­sion of math mak­ing new pre­dic­tions a pri­ori. There is the fol­low­ing un­der­ly­ing prob­lem. The con­di­tions of the re­al­ity in gen­eral do not ex­actly match the con­di­tions of the math, or at least, this is not ver­ifi­able in gen­eral. Hence you can never be sure that your iso­mor­phism be­tween math and re­al­ity is strictly cor­rect. But that means that a pri­ori math­e­mat­i­cal rea­son­ing is the de facto stan­dard (in gen­eral). Ob­vi­ously there are some spe­cial cases like adding ap­ples or rocks to­gether which seems to be fully cor­re­spon­dent, but in most cases, the iso­mor­phism may be un­ver­ifi­able. That’s why it’s amaz­ing that it works.

Re­gard­ing the ev­i­dence for the truth­ful­ness of math state­ments… this “truth­ful­ness” just fol­lows by con­struc­tion from within the origi­nal frame­work. Not sure what you were get­ting at in that sec­tion.

Uni­verses with other math sys­tems—I like this sec­tion of yours the most, I think it at least cor­rectly iden­ti­fies the non triv­ial­ity of that pos­si­bil­ity. A sys­tem where non-triv­ially 2+2=6 would have to cor­re­spond to a bit of a differ­ent re­al­ity, but that re­al­ity would also pre­sum­ably be self-con­sis­tent. But if it is self-con­sis­tent, then that state­ment would have to make sense within the sys­tem it­self. There­fore you es­cape any prob­lems or con­tra­dic­tions and it loses its idea of be­ing spe­cial or strange.

Any­ways hope this feed­back might be helpful for you and I apol­o­gize if I’m mis­in­ter­pret­ing you here, as I’ve done a lot of syn­the­sis in this com­ment.

• Thanks for the feed­back. I’ll re­ply to your con­cerns as best I can.

As far as I can tell, this is ba­si­cally what you’re say­ing, al­though what you mean by the rea­son­able­ness of our ax­ioms is un­clear to me.

I didn’t re­quire the ax­ioms to be rea­son­able in this ap­proach, ex­cept, of course, to the ex­tent that their rea­son­able­ness causes peo­ple to gen­er­ally ac­cept them in com­mon us­age.

Although per­haps you’re say­ing that the math struc­tures have no iden­tity in and of them­selves, ex­ist­ing solely in this frame­work of claim ver­ifi­ca­tion. But in that case, the re­s­olu­tion is tau­tolog­i­cal, and I’m not sure it re­ally gets to the heart of that ques­tion.

That is in­deed what I’m say­ing, but I dis­agree that it’s tau­tolog­i­cal. To the ex­tent that my frame­work han­dles difficult prob­lems and para­doxes in a satis­fac­tory way, that is its non-tau­tolog­i­cal sub­stan­ti­a­tion, as it shows how you don’t need to ap­peal to con­cepts out­side of what I have re­duced math to.

I dis­agree about your nega­tive con­clu­sion of math mak­ing new pre­dic­tions a pri­ori. There is the fol­low­ing un­der­ly­ing prob­lem. The con­di­tions of the re­al­ity in gen­eral do not ex­actly match the con­di­tions of the math, or at least, this is not ver­ifi­able in gen­eral. Hence you can never be sure that your iso­mor­phism be­tween math and re­al­ity is strictly cor­rect. But that means that a pri­ori math­e­mat­i­cal rea­son­ing is the de facto stan­dard (in gen­eral). Ob­vi­ously there are some spe­cial cases like adding ap­ples or rocks to­gether which seems to be fully cor­re­spon­dent, but in most cases, the iso­mor­phism may be un­ver­ifi­able.

I mostly agree, but re­fer back to the causal di­a­gram. As a stan­dard Bayesian rule, you will never have 100% cer­tainty on any of your premises or con­clu­sions. How­ever, failure of the pre­dicted causal im­pli­ca­tion to hold (“adding two rocks to two rocks will yield four rocks”) needn’t have the same im­pact on your de­gree of be­lief in each of its causal par­ents. You can do a lot more to ver­ify your math than to ver­ify the iso­mor­phism to some­thing phys­i­cal.

If the iso­mor­phism has a lot of ev­i­dence fa­vor­ing it, then the math can tell you sur­pris­ing things about par­tic­u­lar re­gions of the do­main of sup­posed ap­pli­ca­bil­ity, which turn out to be true. This is the essence of sci­ence and en­g­ineer­ing. My point here is only that the math’s ap­pli­ca­bil­ity to the uni­verse always de­pends on the em­piri­cal val­idity of the iso­mor­phism, which you might miss if you view the out­put of math as be­ing the crit­i­cal step in an in­sight.

That’s why it’s amaz­ing that it works.

I think the amaz­ing­ness will even­tu­ally be de­mys­tified by a com­mon fac­tor that caused both our use of math and the uni­verse’s fre­quent close iso­mor­phisms thereto.

Re­gard­ing the ev­i­dence for the truth­ful­ness of math state­ments… this “truth­ful­ness” just fol­lows by con­struc­tion from within the origi­nal frame­work. Not sure what you were get­ting at in that sec­tion.

Yes, and the frame­work can be rele­vant or ir­rele­vant to phys­i­cal sys­tems; peo­ple are more likely to be refer­ring to ax­iom sets that are rele­vant (have an iso­mor­phism) to phys­i­cal sys­tems.

• What is “twoness”, phys­i­cally?

• There are two dots, but that’s not “twoness”. Other­wise, we wouldn’t be able to count dis­tant ob­jects that are never in con­ju­ga­tion, or ideas.

• ∃x∃y ( ~(x=y) & ( ∀z ( ~(z=x) ⊃ (z=y) ) & ( ~(z=y) ⊃ (z=x) ) )

Only works in a limited uni­verse of dis­course, though.

• Only works in a limited uni­verse of dis­course, though.

In lower brow dis­course, try: (.)v(.)

• 2 Aug 2010 5:15 UTC
2 points
Parent

I think you may have meant (.Y.)

• That works too. Although I must con­fess I pre­fer the smaller cup size. :P

• Or ∃x∃y ( ~(x=y) & ∀z ( z=y or z=x) )

Still, that’s not ‘twoness’. That’s a sen­tence that’s only satis­fied when there are two things, and could be taken as a defi­ni­tion of what it means to as­sert that there are two things, or even as a defi­ni­tion of there be­ing two such things, but it’s not ‘twoness’. ‘Twoness’ im­plies num­ber is a prop­erty of ob­jects, which I think Frege pretty con­clu­sively dis­proved.

• I think the fact that a defi­ni­tion of “2” in sym­bolic logic can be taken to count as an an­swer to the ques­tion “What is twoness, phys­i­cally?” pretty much says all that needs to be said about the clar­ity of the ques­tion.

• Three blues, an anti-red, 4 mm^3 of ether and half a con­scious­ness.

• That is cor­rect, ex­cept for these er­rors:

• It’s ac­tu­ally four blues, not three.

• The whole pro­ject of seek­ing a uni­ver­sal, “pla­tonic” essence of “twoness” is one gi­ant con­fu­sion from the very start and will serve to do noth­ing but dis­tract you from ac­cu­rately mod­el­ing the world, even if, for some rea­son, you don’t care about pa­per­clips.

• For a char­ac­ter whose very ex­is­tence is founded on face­tious irony and satire Clippy sure seems to strug­gle with it at times.

• Ac­tu­ally, my ex­is­tence is founded on the in­her­ent good­ness of pa­per­clips. This con­fu­sion on your part causes me to re­con­sider my clas­sifi­ca­tion of you as a good hu­man.

• :)

• I thought that was perfectly on point, as usual.

• Where is the chess in Deep Blue?

• What I should have said:

The na­ture of math­e­mat­ics is one in­stance of the prob­lem of uni­ver­sals. (The op­po­site of a “uni­ver­sal” is a “par­tic­u­lar”.) Pla­ton­ism says that uni­ver­sals ex­ist in­de­pen­dently of par­tic­u­lars—and ex­treme pla­ton­ism says that uni­ver­sals are all that ex­ists (ex­am­ple). Aris­totelian re­al­ism says that uni­ver­sals always oc­cur in as­so­ci­a­tion with a par­tic­u­lar. Nom­i­nal­ism says there are no uni­ver­sals, just words.

I can­not tell if you are a nom­i­nal­ist or an Aris­totelian re­al­ist. For a phys­i­cal pro­cess in­volv­ing rocks to have an iso­mor­phism onto the equa­tion 2+2=4, it seems like there has to be some ac­tual twoness in the phys­i­cal re­al­ity, which maps onto the ab­strac­tion ‘2’. So I want to know your views on the na­ture of this phys­i­cal twoness.

• Okay. What I have ad­vo­cated here is a species of nom­i­nal­ism.

For a phys­i­cal pro­cess in­volv­ing rocks to have an iso­mor­phism onto the equa­tion 2+2=4, it seems like there has to be some ac­tual twoness in the phys­i­cal re­al­ity, which maps onto the ab­strac­tion ‘2’. So I want to know your views on the na­ture of this phys­i­cal twoness.

Well, that’s where I dis­agree. For the iso­mor­phism, it’s only nec­es­sary that I have a work­ing model; I needn’t en­dorse any more ab­stract or uni­ver­sal con­cept of “twoness”.

I con­tinu­ally ask my­self: for what­ever truth I posit, how jus­tifi­ably sur­prised can I be if na­ture re­fused to play along? (Fol­low­ing the heuris­tic here.) All the “twoness” that I need to ac­cept the ex­is­tence of, is con­tained in phys­i­cal agents’ phys­i­cal mod­els of re­al­ity. To give it any greater role is to at­tach my­self to a premise for which I have no con­tra­dic­tion to com­plain about if na­ture were to re­fuse to yield any other in­stance of “twoness”.

• What is “twoness”, math­e­mat­i­cally?

• Could you ex­pand on 1) the com­mon con­cep­tion of the rules of how num­bers work?

You’ve writ­ten:

1) a claim about whether, gen­er­ally speak­ing, peo­ple’s mod­els of “how num­bers work” make cer­tain assumptions

To what ex­tent is the truth that 2+2=4 an in­ter­per­sonal one? Is this be­cause if we had differ­ent ideas about it, it would be less true -- that the ‘truth’ of ad­di­tion stems from the fact that we all seem to agree on the way it works?

For my­self, I would be re­luc­tant to adopt a con­cept of math­e­mat­i­cal truth that re­lies on com­mu­nity agree­ment, but I am cu­ri­ous as to why you em­pha­size the role of more than one per­son.

Also, to check my un­der­stand­ing re­gard­ing the as­sump­tions com­po­nent of (1): are these gen­er­ated in or­der to model phys­i­cal phe­nom­ena, so that if two peo­ple agree on the phys­i­cal phe­nom­ena be­ing mod­eled, they would agree on the as­sump­tions of the model?

• To what ex­tent is the truth that 2+2=4 an in­ter­per­sonal one? Is this be­cause if we had differ­ent ideas about it, it would be less true—that the ‘truth’ of ad­di­tion stems from the fact that we all seem to agree on the way it works?

For my­self, I would be re­luc­tant to adopt a con­cept of math­e­mat­i­cal truth that re­lies on com­mu­nity agree­ment, but I am cu­ri­ous as to why you em­pha­size the role of more than one per­son.

The in­ter­per­sonal as­pect is in there to con­strain what the sym­bols in “2+2=4” ac­tu­ally mean; it has no bear­ing on the un­der­ly­ing log­i­cal truth (part 2). Nev­er­the­less, com­mon agree­ment in the use of terms is nec­es­sary to give those terms mean­ing. In that re­spect, the opinions of other peo­ple do im­pact whether such a state­ment eval­u­ates to “true”. For the same rea­son, the iso­lated state­ment “2+2=6” should eval­u­ate to “false”, even though some­one could say, “oh no, see, here, I meant this ‘6’ sym­bol to mean ‘4’.” That per­son may have an ac­cu­rate in­ter­nal model of re­al­ity, but hasn’t cor­rectly con­veyed it.

Words can be wrong in terms of sud­den un­ex­plained de­vi­a­tion from com­mon us­age.

• Is 4 not by defin­tion 2+2, Is math not self prov­ing? I mean why all this “ex­plan­tion” when it is more ev­i­dent to say that this thing math­e­mat­ics is a com­plex game with rules de­signed to match the re­al­ity.

• Typ­i­cally, 4 is by defi­ni­tion 1+(1+(1+1))), so “2+2=4” means

(1+1)+(1+1) = 1+(1+(1+1)))

In other words, it means two par­tic­u­lar differ­ent pro­cesses for adding up 1′s will yield the same re­sult. This is not as­sumed in Peano ar­ith­metic, but proven from a se­lec­tion of even more ba­sic as­sump­tions (which need not ex­plic­itly men­tion as­so­ci­a­tivity), albeit a very clever se­lec­tion.