Astray with the Truth: Logic and Math

LessWrong has one of the strongest and most com­pel­ling pre­sen­ta­tions of a cor­re­spon­dence the­ory of truth on the in­ter­net, but as I said in A Prag­matic Episte­mol­ogy, it has some defi­cien­cies. This post delves into one ex­am­ple: its treat­ment of math and logic. First, though, I’ll sum­marise the episte­mol­ogy of the se­quences (es­pe­cially as pre­sented in High Ad­vanced Episte­mol­ogy 101 for Begin­ners).

Truth is the cor­re­spon­dence be­tween be­liefs and re­al­ity, be­tween the map and the ter­ri­tory.[1] Real­ity is a causal fabric, a col­lec­tion of vari­ables (“stuff”) that in­ter­act with each other.[2] True be­liefs mir­ror re­al­ity in some way. If I be­lieve that most maps skew the rel­a­tive size of Elles­mere Is­land, it’s true when I com­pare ac­cu­rate mea­sure­ments of Elles­mere Is­land to ac­cu­rate mea­sure­ments of other places, and find that the differ­ences aren’t pre­served in the scal­ing of most maps. That is an ex­am­ple of a truth-con­di­tion, which is a re­al­ity that the be­lief can cor­re­spond to. My be­lief about world maps is true when that scal­ing doesn’t match up in re­al­ity. All mean­ingful be­liefs have truth-con­di­tions; they trace out paths in a causal fabric.[3] Another way to define truth, then, is that a be­lief is true when it traces a path which is found in the causal fabric the be­liever in­hab­its.

Beliefs come in many forms. You can have be­liefs about your ex­pe­riences past, pre­sent and fu­ture; about what you ought to do; and, rele­vant to our pur­poses, about ab­strac­tions like math­e­mat­i­cal ob­jects. Math­e­mat­i­cal state­ments are true when they are truth-pre­serv­ing, or valid. They’re also con­di­tional: they’re about all pos­si­ble causal fabrics rather than any one in par­tic­u­lar.[4] That is, when you take a true math­e­mat­i­cal state­ment and plug in any ac­cept­able in­puts,[5] you will end up with a true con­di­tional state­ment about the in­puts. Let’s illus­trate this with the dis­junc­tive syl­l­o­gism:

((A∨B) ∧ ¬A) ⇒ B

Let­ting A be “All pen­guins ski in De­cem­ber” and B be “Mar­ti­ans have been dec­i­mated,” this reads “If all pen­guins ski in De­cem­ber or Mar­ti­ans have been dec­i­mated, and some pen­guins don’t ski in De­cem­ber, then Mar­ti­ans have been dec­i­mated.” And if the hy­poth­e­sis ob­tains (if it’s true that (A∨B) ∧ ¬A), then the con­clu­sion (B) is claimed to fol­low.[6]

That’s it for re­view, now for the sub­stance.

Sum­mary. First, from ex­am­in­ing the truth-con­di­tions of be­liefs about val­idity, we see that our sense of what is ob­vi­ous plays a sus­pi­cious role in which state­ments we con­sider valid. Se­cond, a ma­jor failure mode in fol­low­ing ob­vi­ous­ness is that we sac­ri­fice other goals by sep­a­rat­ing the pur­suit of truth from other pur­suits. This ele­va­tion of the truth via the epistemic/​in­stru­men­tal ra­tio­nal­ity dis­tinc­tion pre­vents us from see­ing it as one in­stru­men­tal goal among many which may some­times be ir­rele­vant.


What are the truth-con­di­tions of a be­lief that a cer­tain log­i­cal form is valid or not?

A prop­erty of valid state­ments is be­ing able to plug any propo­si­tion you like into the propo­si­tional vari­ables of the state­ment with­out dis­turb­ing the out­come (the con­di­tional state­ment will still be true). Liter­ally any propo­si­tion; valid forms about ev­ery­thing that can be ar­tic­u­lated by means of propo­si­tions. So part of the truth-con­di­tions of a be­lief about val­idity is that if a sen­tence is valid, ev­ery­thing is a model of it. In that case, causal fabrics, which we in­ves­ti­gate by means of propo­si­tions,[7] can’t help but be con­strained by what is log­i­cally valid. We would never ex­pect to see some uni­verse where in­putting propo­si­tions into the dis­junc­tive syl­l­o­gism can out­put false with­out be­ing in er­ror. Call this the log­i­cal law view. This sug­gests that we could check a bunch of in­puts and uni­verses con­struc­tions un­til we feel satis­fied that the sen­tence will not fail to out­put true.

It hap­pens that sen­tences which peo­ple agree are valid are usu­ally sen­tences that peo­ple agree are ob­vi­ously true. There is some­thing about the struc­ture of our thought that makes us very will­ing to ac­cept their val­idity. Per­haps you might say that be­cause re­al­ity is con­strained by valid sen­tences, sapi­ent chunks of re­al­ity are go­ing to be pre­dis­posed to recog­nis­ing val­idity …

But what sep­a­rates that hy­poth­e­sis from this al­ter­na­tive: “valid sen­tences are rules that have been ap­plied suc­cess­fully in many cases so far”? That is, af­ter all, the very pro­cess that we use to check the truth-con­di­tions of our be­liefs about val­idity. We con­sider hy­po­thet­i­cal uni­verses and we ap­ply the rules in rea­son­ing. Why should we go fur­ther and claim that all pos­si­ble re­al­ities are con­strained by these rules? In the end we are very de­pen­dent on our in­tu­itions about what is ob­vi­ous, which might just as well be due to flaws in our thought as log­i­cal laws. And our in­sis­tence of cor­rect­ness is no ex­cuse. In that re­gard we may be no differ­ent than cer­tain ants that mis­take liv­ing mem­bers of the colony for dead when their body is cov­ered in a cer­tain pheromone:[8] prone to a re­ac­tion that is just as ob­vi­ously astray to other minds as it is ob­vi­ously right to us.

In light of that, I see no rea­son to be con­fi­dent that we can dis­t­in­guish be­tween suc­cess in our limited ap­pli­ca­tions and nec­es­sary con­straint on all pos­si­ble causal fabrics.

And de­spite what I said about “suc­cess so far,” there are clear cases where stick­ing to our strong in­tu­ition to take the log­i­cal law view leads us astray on goals apart from truth-seek­ing. I give two ex­am­ples where ob­ses­sive fo­cus on truth-seek­ing con­sumes valuable re­sources that could be used to­ward a host of other wor­thy goals.

The Law of Non-Con­tra­dic­tion. The is law is prob­a­bly the most ob­vi­ous thing in the world. A propo­si­tion can’t be truth and false, or ¬(P ∧ ¬P). If it were both, then you would have a model of any propo­si­tion you could dream of. This is an ex­tremely scary prospect if you hold the log­i­cal law view; it means that if you have a true con­tra­dic­tion, re­al­ity doesn’t have to make sense. Causal­ity and your ex­pec­ta­tions are mean­ingless. That is the prin­ci­ple of ex­plo­sion: (P ∧ ¬P) ⇒ Q, for ar­bi­trary Q. Sup­pose that pink is my favourite colour, and that it isn’t. Then pink is my favourite colour or causal­ity is mean­ingless. Ex­cept pink isn’t my favourite colour, so causal­ity is mean­ingless. Ex­cept it is, be­cause ei­ther pink is my favourite colour or causal­ity is mean­ingful, but pink isn’t. There­fore pix­ies by a similar ar­gu­ment.

Is (P ∧ ¬P) ⇒ Q valid? Most peo­ple think it is. If you hyp­no­tised me into for­get­ting that I find that sort of ques­tion sus­pect, I would agree. I can *feel* the pull to­ward as­sent­ing its val­idity. If ¬(P ∧ ¬P) is true it would be hard to say why not. But there are nonethe­less very good rea­sons for ditch­ing the law of non-con­tra­dic­tion and the prin­ci­ple of ex­plo­sion. De­spite its in­tu­itive truth and gen­eral ob­vi­ous­ness, it’s ex­tremely in­con­ve­nient. Solv­ing the prob­lem of the con­sis­tency of var­i­ous PA and ZFC, which are cen­tral to math­e­mat­ics, has proved very difficult. But of course part of the mo­ti­va­tion is that if there were an in­con­sis­tency, the prin­ci­ple of ex­plo­sion would ren­der the en­tire sys­tem use­less. This un­de­sir­able effect has led some to de­velop para­con­sis­tent log­ics which do not ex­plode with the dis­cov­ery of a con­tra­dic­tion.

Set­ting aside whether the law of non-con­tra­dic­tion is re­ally truly true and the prin­ci­ple of ex­plo­sion re­ally truly valid, wouldn’t we be bet­ter off with foun­da­tional sys­tems that don’t buckle over and die at the mer­est whiff of a con­tra­dic­tion? In any case, it would be nice to al­ter the de­bate so that the truth of these state­ments didn’t eclipse their util­ity to­ward other goals.

The Law of Ex­cluded Mid­dle. P∨¬P: if a propo­si­tion isn’t true, then it’s false; if it isn’t false, then it’s true. In terms of the LessWrong episte­mol­ogy, this means that a propo­si­tion ei­ther ob­tains in the causal fabric you’re em­bed­ded in, or it doesn’t. Like the pre­vi­ous ex­am­ple this has a strong in­tu­itive pull. If that pull is cor­rect, all sen­tences Q ⇒ (P∨¬P) must be valid since ev­ery­thing mod­els true sen­tences. And yet, though doubt­ing it can seem ridicu­lous, and though I would not doubt it on its own terms[9], there are very good rea­sons for us­ing sys­tems where it doesn’t hold.

The use of the law of ex­cluded mid­dle in proofs severely in­hibits the con­struc­tion of pro­grammes based on proofs. The bar­rier is that the law is used in ex­is­tence proofs, which show that some math­e­mat­i­cal ob­ject must ex­ist but give no method of con­struct­ing it.[10]

Re­mov­ing the law, on the other hand, gives us in­tu­ition­is­tic logic. Via a map­ping called the Curry-Howard iso­mor­phism all proofs in in­tu­ition­is­tic logic are trans­lat­able into pro­grammes in the lambda calcu­lus, and vice versa. The lambda calcu­lus it­self, as­sum­ing the Church-Tur­ing the­sis, gives us all effec­tively com­putable func­tions. This cre­ates a deep con­nec­tion be­tween proof the­ory in con­struc­tive math­e­mat­ics and com­putabil­ity the­ory, fa­cil­i­tat­ing au­to­matic the­o­rem prov­ing and proof ver­ifi­ca­tion and ren­der­ing ev­ery­thing we do more com­pu­ta­tion­ally tractable.

Even if we the above weren’t tempt­ing and we de­cided not to re­strict our­selves to con­struc­tive proofs, we would be stuck with in­tu­ition­is­tic logic. Just as clas­si­cal logic is as­so­ci­ated with Boolean alge­bras, in­tu­ition­is­tic logic is as­so­ci­ated with Heyt­ing alge­bras. And it hap­pens that the open set lat­tice of a topolog­i­cal space is a com­plete Heyt­ing alge­bra even in clas­si­cal topol­ogy.[11] This is closely re­lated to topos the­ory; the in­ter­nal logic of a topos is at least[12] in­tu­ition­is­tic. As I un­der­stand it, many topoi can be con­sid­ered as foun­da­tions for math­e­mat­ics,[13] and so again we see a clas­si­cal the­ory point­ing at con­struc­tivism sug­ges­tively. The moral of the story: in clas­si­cal math­e­mat­ics where the law of ex­cluded mid­dle holds, ob­jects in which it fails arise nat­u­rally.

Work in the foun­da­tions of math­e­mat­ics sug­gests that con­struc­tive math­e­mat­ics is at least worth look­ing into, set­ting aside whether the law of ex­cluded mid­dle is too ob­vi­ous to doubt. Let­ting its truth hold us back from in­ves­ti­gat­ing the mer­its of liv­ing with­out it crip­ples the ca­pa­bil­ities of our math­e­mat­i­cal pro­jects.


Un­for­tu­nately, not all con­struc­tivists or di­alethe­ists (as pro­po­nents of para­con­sis­tent logic are called) would agree how I framed the situ­a­tion. I have blamed the ten­dency to stick to dis­cus­sions of truth for our in­abil­ity to move for­ward in both cases, but they might blame the in­abil­ity of their op­po­nents to see that the laws in ques­tion are false. They might urge that if we take the suc­cess of these laws as ev­i­dence of their truth, then failures or short­com­ings should be ev­i­dence against them and we should sim­ply re­vise our views ac­cord­ingly.

That is how the prob­lem looks when we wear our epistemic ra­tio­nal­ity cap and fo­cus on the truth of sen­tences: we con­sider which ex­pe­riences could tip us off about which rules gov­ern causal fabrics, and we or­ganise our be­liefs about causal fabrics around them.

This fram­ing of the prob­lem is coun­ter­pro­duc­tive. So long as we are dis­cussing these ab­stract prin­ci­ples un­der the con­straints of our own minds,[14] I will find any dis­cus­sion of their truth or falsity highly sus­pect for the rea­sons high­lighted above. And be­yond that, the psy­cholog­i­cal pull to­ward the re­spec­tive po­si­tions is too force­ful for this mode of de­bate to make progress on rea­son­able timescales. In the in­ter­ests of ac­tu­ally achiev­ing some of our goals I favour drop­ping that de­bate en­tirely.

In­stead, we should put on our in­stru­men­tal ra­tio­nal­ity cap and con­sider whether these con­cepts are work­ing for us. We should think hard about what we want to achieve with our math­e­mat­i­cal sys­tems and tai­lor them to perform bet­ter in that re­gard. We should recog­nise when a path is moot and trace a differ­ent one.

When we wear our in­stru­men­tal ra­tio­nal­ity cap, math­e­mat­i­cal sys­tems are not at­tempts at cre­at­ing images of re­al­ity that we can use for other things if we like. They are tools that we use to achieve po­ten­tially any goal, and po­ten­tially none. If af­ter care­ful con­sid­er­a­tion we de­cide that cre­at­ing images of re­al­ity is a fruit­ful goal rel­a­tive to the other goals we can think of for our sys­tems, fine. But that should by no means be the de­fault, and if it weren’t math­e­mat­ics would be headed el­se­where.


ADDENDUM

[Added due to ex­pres­sions of con­fu­sion in the com­ments. I have also al­tered the origi­nal con­clu­sion above.]

I gave two broad weak­nesses in the LessWrong episte­mol­ogy with re­spect to math.

The first con­cerned its on­tolog­i­cal com­mit­ments. Think­ing of val­idity as a prop­erty of log­i­cal laws con­strain­ing causal fabrics is in­dis­t­in­guish­able in prac­ti­cal pur­poses from think­ing of val­idity as a prop­erty of sen­tences rel­a­tive to some ax­ioms or ac­cord­ing to strong in­tu­ition. Since our for­mu­la­tion and use of these sen­tences have been in fa­mil­iar con­di­tions, and since it is very difficult (per­haps im­pos­si­ble) to de­ter­mine whether their psy­cholog­i­cal weight is a bias, in­fer­ring any of them as log­i­cal laws above and be­yond their use­ful­ness as tools is spu­ri­ous.

The sec­ond con­cerned cases where the log­i­cal law view can hold us back from achiev­ing goals other than dis­cov­er­ing true things. The law of non-con­tra­dic­tion and the law of ex­cluded mid­dle are as old as they are ob­vi­ous, yet they pre­vent us from strength­en­ing our math­e­mat­i­cal sys­tems and mak­ing their use con­sid­er­ably eas­ier.

One di­ag­no­sis of this prob­lem might be that some­times it’s best to set our episte­mol­ogy aside in the in­ter­ests of prac­ti­cal pur­suits, that some­times our episte­mol­ogy isn’t rele­vant to our goals. Un­der this di­ag­no­sis, we can take the LessWrong episte­mol­ogy liter­ally and be­lieve it is true, but tem­porar­ily ig­nore it in or­der to solve cer­tain prob­lems. This is a step for­ward, but I would make a stronger di­ag­no­sis: we should have a back­ground episte­mol­ogy guided by in­stru­men­tal rea­son, in which the episte­mol­ogy of LessWrong and epistemic rea­son are tools that we can use if we find them con­ve­nient, but which we are not com­mit­ted to tak­ing liter­ally.

I pre­scribe an episte­mol­ogy that a) sees the­o­ries as no differ­ent from ham­mers, b) doesn’t take the con­tent of the­o­ries liter­ally, and c) lets in­stru­men­tal rea­son guide the de­ci­sion of which the­ory to adopt when. I claim that this is the best frame­work to use for achiev­ing our goals, and I call this a prag­matic episte­mol­ogy.

---

[1] See The Use­ful Idea of Truth.

[2] See The Fabric of Real Things and Stuff that Makes Stuff Hap­pen.

[3] See The Use­ful Idea of Truth and The Fabric of Real Things.

[4] See Proofs, Im­pli­ca­tions, and Models and Log­i­cal Pin­point­ing.

[5] Ac­cept­able in­puts be­ing given by the uni­verse of dis­course (also known as the uni­verse or the do­main of dis­course), which is dis­cussed on any text cov­er­ing the se­man­tics of clas­si­cal logic, or clas­si­cal model the­ory in gen­eral.

[6] A vi­sual ex­am­ple us­ing modus po­nens and cute cud­dly kit­tens is found in Proofs, Im­pli­ca­tions, and Models.

[7] See The Use­ful Idea of Truth.

[8] See this pa­per by biol­o­gist E O Wil­son.

[9] What I mean is that I would not claim that it “isn’t true,” which usu­ally makes the de­bate stag­nate.

[10] For con­crete­ness, read these ex­am­ples of non-con­struc­tive proofs.

[11] See here, para­graph two.

[12] Given cer­tain fur­ther re­stric­tions, a topos is Boolean and its in­ter­nal logic is clas­si­cal.

[13] This is an amus­ing and vague-as-ad­ver­tised sum­mary by John Baez.

[14] Com­mu­ni­ca­tion with very differ­ent agents might be a way to cir­cum­vent this. Re­ceiv­ing ad­vice from an AI, for in­stance. Still, I have rea­sons to find this fishy as well, which I will ex­plore in later posts.