The role of mathematical truths

Re­lated to: Math is sub­junc­tively ob­jec­tive, How to con­vince me that 2+2=3

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.