Can infinite quantities exist? A philosophical approach


Ini­tially at­tracted to Less Wrong by Eliezer Yud­kowsky’s in­tel­lec­tual bold­ness in his “in­finite-sets athe­ism,” I’ve waited pa­tiently to dis­cover its ra­tio­nale. Some­times it’s said that our “in­tu­itions” speak for in­finity or against, but how could one, in a Kah­ne­man-ap­pro­pri­ate man­ner, ar­rive at in­tu­itions about whether the cos­mos is in­finite? In­tu­itions about in­finite sets might arise from an anal­y­sis of the con­cept of ac­tu­ally re­al­ized in­fini­ties. This is a dis­tinc­tively philo­soph­i­cal form of anal­y­sis and one some­what alien to Less Wrong, but it may be the only way to gain pur­chase on this ne­glected ques­tion. I’m by no means cer­tain of my rea­son­ing; I cer­tainly don’t think I’ve set­tled the is­sue. But for rea­sons I dis­cuss in this skele­tal ar­gu­ment, the con­cep­tual—as op­posed to the sci­en­tific or math­e­mat­i­cal—anal­y­sis of “ac­tu­ally re­al­ized in­fini­ties” has been largely avoided, and I hope to help be­gin a nec­es­sary dis­cus­sion.

1. The ac­tu­al­ity of in­finity is a paramount meta­phys­i­cal is­sue.

Some ma­jor is­sues in sci­ence and philos­o­phy de­mand tak­ing a po­si­tion on whether there can be an in­finite num­ber of things or an in­finite amount of some­thing. In­finity’s most ob­vi­ous sci­en­tific rele­vance is to cos­mol­ogy, where the ques­tion of whether the uni­verse is finite or in­finite looms large. But in­fini­ties are in­voked in var­i­ous phys­i­cal the­o­ries, and they seem of­ten to oc­cur in du­bi­ous the­o­ries. In quan­tum me­chan­ics, an (un­countable) in­finity of wor­lds is in­voked by the “many wor­lds in­ter­pre­ta­tion,” and an­thropic ex­pla­na­tions of­ten in­voke an ac­tual in­finity of uni­verses, which may them­selves be in­finite. Th­ese ap­pli­ca­tions make real in­finite sets a paramount meta­phys­i­cal prob­lem—if it in­deed is meta­phys­i­cal—but the or­tho­dox view is that, be­ing em­piri­cal, it isn’t meta­phys­i­cal at all. To view in­finity as a purely em­piri­cal mat­ter is the mod­ern view; we’ve learned not to place ex­ces­sive weight on purely con­cep­tual rea­son­ing, but whether con­cep­tual rea­son­ing can defini­tively set­tle the mat­ter differs from whether the mat­ter is fun­da­men­tally con­cep­tual.

Two de­vel­op­ments have dis­cour­aged the meta­phys­i­cal ex­plo­ra­tion of ac­tu­ally ex­ist­ing in­fini­ties: the math­e­mat­i­cal anal­y­sis of in­finity and the proffer of crank ar­gu­ments against in­finity in the ser­vice of ret­ro­grade causes. Although some marginal schools of math­e­mat­ics re­ject Can­tor’s in­ves­ti­ga­tion of trans­finite num­bers, I will as­sume the con­cept of in­finity it­self is con­sis­tent. My anal­y­sis per­tains not to the con­cept of in­finity as such but to the ac­tual re­al­iza­tion of in­finity. Ac­tual in­finity’s main de­trac­tor is a Chris­tian fun­da­men­tal­ist crank named William Lane Craig, whose cri­tique of in­finity, serv­ing the­ist first-cause ar­gu­ments, has made in­finity elimi­na­tivism in­tel­lec­tu­ally dis­rep­utable. Craig’s ar­gu­ments merely ap­peal to the strangeness of in­finity’s man­i­fes­ta­tions, not to the in­co­her­ence of its re­al­iza­tion. The stan­dard ar­gu­ments against in­finity, which pre­date Can­tor, have been well-re­futed, and I leave the math­e­mat­i­cal cri­tique of in­finity to the math­e­mat­i­ci­ans, who are mostly satis­fied. (See Gra­ham Oppy, Philo­soph­i­cal per­spec­tives on in­finity (2006).)

2. The prin­ci­ple of the iden­tity of in­dis­t­in­guish­ables ap­plies to physics and to sets, not to ev­ery­thing con­ceiv­able.

My novel ar­gu­ments are based on a re­vi­sion of a meta­phys­i­cal prin­ci­ple called the iden­tity of in­dis­t­in­guish­ables, which holds that two sep­a­rate things can’t have ex­actly the same prop­er­ties. Things are con­sti­tuted by their prop­er­ties; if two things have ex­actly the same prop­er­ties, noth­ing re­mains to make them differ­ent from one an­other. Phys­i­cal ob­jects do seem to con­form to the iden­tity of in­dis­t­in­guish­ables be­cause phys­i­cal ob­jects are in­di­vi­d­u­ated by their po­si­tions in space and time, which are prop­er­ties, but this is a phys­i­cal rather than a meta­phys­i­cal prin­ci­ple. Con­cep­tu­ally, brute dis­t­in­guisha­bil­ity, that is differ­ing from all other things sim­ply in be­ing differ­ent, is a prop­erty, al­though it pro­vides us with no ba­sis for iden­ti­fy­ing one thing and not an­other. There may be no way to use such a prop­erty in any phys­i­cal the­ory, we may never learn of such a prop­erty and thus never have rea­son to be­lieve it in­stan­ti­ated, but the prop­erty seems con­cep­tu­ally pos­si­ble.

But the iden­tity of in­dis­t­in­guish­ables does ap­ply to sets: in­dis­t­in­guish­able sets are iden­ti­cal. Prop­er­ties de­ter­mine sets, so you can’t define a proper sub­set of brutely dis­t­in­guish­able things.

3. Ar­gu­ments against ac­tu­ally ex­ist­ing in­finite sets.

A. Ar­gu­ment based on brute dis­t­in­guisha­bil­ity.

To show that the ex­is­tence of an ac­tu­ally ex­ist­ing in­finite set leads to con­tra­dic­tion, as­sume the ex­is­tence of an in­finite set of brutely dis­t­in­guish­able points. Now an­other point pops into ex­is­tence. The former and lat­ter sets are in­dis­t­in­guish­able, yet they aren’t iden­ti­cal. The pro­viso that the points them­selves are in­dis­t­in­guish­able al­lows the sets to be differ­ent yet in­dis­t­in­guish­able when they’re in­finite, prov­ing they can’t be in­finite.

B. Ar­gu­ment based on prob­a­bil­ity as limit­ing rel­a­tive fre­quency.

The pre­vi­ous ar­gu­ment de­pends on the co­her­ence of brute dis­t­in­guisha­bil­ity. The fol­low­ing prob­a­bil­ity ar­gu­ment de­pends on differ­ent in­tu­itions. Prob­a­bil­ities can be treated as ideal­iza­tions at in­finite limits. If you toss a coin, it will land heads roughly 50% of the time, and it gets closer to ex­actly 50% as the num­ber of tosses “ap­proaches in­finity.” But if there can ac­tu­ally be an in­finite num­ber of tosses, con­tra­dic­tion arises. Con­sider the pos­si­bil­ity that in an in­finite uni­verse or an in­finite num­ber of uni­verses, in­finitely many coin tosses ac­tu­ally oc­cur. The fre­quency of heads and of tails is then in­finite, so the rel­a­tive fre­quency is un­defined. Fur­ther­more, the fre­quency of rol­ling a 1 on a die also equals the fre­quency of rol­ling 2 – 6: both are (countably) in­finite. But if in­finite quan­tities ex­ist, then rel­a­tive fre­quency should equal prob­a­bil­ity. There­fore, in­finite quan­tities don’t ex­ist.

4. The nonex­is­tence of ac­tu­ally re­al­ized in­finite sets and the prin­ci­ple of the iden­tity of in­dis­t­in­guish­able sets to­gether im­ply the Gold model of the cos­mos.

Be­fore ap­ply­ing the con­clu­sion that ac­tu­ally re­al­ized in­fini­ties can’t ex­ist to­gether with the prin­ci­ple of the iden­tity of in­dis­t­in­guish­ables to a fun­da­men­tal prob­lem of cos­mol­ogy, caveats are in or­der. The ar­gu­ment uses only the most gen­eral and well-es­tab­lished phys­i­cal con­clu­sions and is oblivi­ous to phys­i­cal de­tail, and not be­ing com­pe­tent in physics, I must ab­stain even from as­sess­ing the weight the philo­soph­i­cal anal­y­sis that fol­lows should carry; it may be very slight. While the cos­molog­i­cal model I pro­pose isn’t origi­nal, the ar­gu­ment is origi­nal and as far as I can tell, novel. I am not propos­ing a phys­i­cal the­ory as much as sug­gest­ing meta­phys­i­cal con­sid­er­a­tions that might bear on physics, whereas it is for physi­cists to say how weighty these con­sid­er­a­tions are in light of ac­tual phys­i­cal data and the­ory.

The cos­molog­i­cal the­ory is the Gold model of the uni­verse, once fa­vored by Albert Ein­stein, ac­cord­ing to which the uni­verse un­der­goes a per­pet­ual ex­pan­sion, con­trac­tion, and re-ex­pan­sion. I as­sume a de­ter­minis­tic uni­verse, such that cy­cles are ex­actly iden­ti­cal: any con­trac­tion is thus in­dis­t­in­guish­able from any other, and any ex­pan­sion is in­dis­t­in­guish­able from any other. Since there is no room in physics for brute dis­t­in­guisha­bil­ity, they are iden­ti­cal be­cause no com­mon spa­tio-tem­po­ral frame­work al­lows their dis­tinc­tion. Thus, al­though the ex­pan­sion and con­trac­tion pro­cess is per­pet­ual and eter­nal, it is also finite; in fact, its num­ber is unity.

The Gold uni­verse—alone, with the pos­si­ble ex­cep­tion of the Hawk­ing uni­verse—avoids the dilemma of the re­al­iza­tion of in­finite sets or origi­na­tion ex nihilo.