# 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.

[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.

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[1] See The Use­ful Idea of Truth.

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

[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.

• Eliezer pre­sents a strong defence of the cor­re­spon­dence the­ory? Well, for some val­ues of “strong”. He puts for­ward the best pos­si­ble ex­am­ple of cor­re­spon­dence work­ing clearly and ob­vi­ously, and leaves the reader with the im­pres­sion that it works as well in all cases. In fact, the CToT is not uni­ver­sally ac­cepted be­cause there are a num­ber of cases where it is hard to ap­ply. One of them is maths’n’logic, the os­ten­si­ble sub­ject of your post­ing,

I would have thought that the out­stand­ing prob­lem with the cor­re­spon­dence the­ory of truth in re­la­tion to maths is: what do true math­e­mat­i­cal state­ments cor­re­spond to? Ie, what is the on­tol­ogy of maths? You seem to offer only the two sen­tences:-

“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” The first is rather vague. Truth preser­va­tion is a prop­erty of con­nected chains of state­ments. Such ar­gue­ments are valid when and only when they are truth pre­serv­ing, be­cause that is how val­idity is defined in this con­text. The con­clu­sion of a math­e­mat­i­cal ar­gu­ment is true when it’s premises are true AND when it is valid (AKA truth pre­serv­ing). Val­idity is not a vague syn­onym for truth: the ex­tra con­di­tion about the truth of the premises is im­por­tant.

Are math­e­mat­i­cal state­ments about all pos­si­ble causal fabrics? Is “causal fabric” a mean­ingful term? Choose one.

If a causal fabric is a par­tic­u­lar kind of math­e­mat­i­cal struc­ture, a di­rected acyclic graph, for in­stance, then it isn’t the only pos­si­ble topic of math­e­mat­ics, it’s too nar­row a ter­ri­tory, …maths can be about cyclic or undi­rected graphs, for in­stance.

On the other hand, if the phrase “causal fabric” doesn’t con­strain any­thing, then what is the ter­ri­tory...what does it do...and how you tell it is there? Un­der the stan­dard cor­re­spon­dence of truth as ap­plied to em­piri­cal claims, a claim is true if a cor­re­spond­ing piece of ter­ri­tory ex­ists, and false if doesn’t. But how can a piece of the math­e­mat­i­cal ter­ri­tory go miss­ing?

Math­e­mat­i­cal state­ments are proven by pre­sent­ing de­duc­tions from premises, ul­ti­mately from in­tu­itively ap­peal­ing ax­ioms. We can speak of the set of proven and prove­able the­o­rems as a ter­ri­tory, but that es­tab­lishes only a su­perfi­cial re­sem­blance to cor­re­spon­dence: ex­am­ined in de­tail, the math­e­mat­i­cal map-ter­ri­tory re­la­tion­ship works in re­verse. It is the ex­is­tence of a lump of phys­i­cal ter­ri­tory that proves the truth of the cor­re­spond­ing claim; whereas the truth of a math­e­mat­i­cal claim is proved non em­piri­cally, and the idea that it cor­re­sponds to some lump of metaphor­i­cal math­space is con­jured up sub­se­quently.

Is that too dis­mis­sive of math­e­mat­i­cal re­al­ism? The re­al­ist can in­sist that the­o­rems aren’t true un­less they cor­re­spond to some­thing in Pla­to­nia...even if a proof has been pub­lished and ac­cepted. But the idea that math­e­mat­i­ci­ans, de­spite all their efforts, are es­sen­tially in the dark about math­e­mat­i­call truth us quite a bul­let to bite. The re­al­ist can re­spond that math­e­mat­i­ci­ans are guided by some sort of con­tact be­tween their brains and the non phys­i­cal realm of Pla­to­nia, but that is not a claim a phys­i­cal­ist should sub­scribe to.

So , the in­tended con­clu­sion is that no math­e­mat­i­cal state­ment is made true by the ter­ri­tory, be­cause there is no suit­able ter­ri­tory to do so. Of course, the Law of the Ex­cluded Mid­dle, and the Prin­ci­ple of Non Con­tra­dic­tion are true in the sys­tems that em­ploy them, be­cause they are ax­ioms of the sys­tems that em­ploy them, and ax­ioms are true by stipu­la­tion.

I agree with the ex­am­ples you pre­sent to the effect that we need to pick and choose be­tween log­i­cal sys­tems ac­cord­ing to the do­main. I dis­agree with the con­clu­sion that an aban­don­ment of truth is nec­es­sary...or pos­si­ble.

To as­sert P is equiv­a­lent to as­sert­ing “P is true” (the defla­tion­ary the­ory in re­verse). That is still true if P is of the form “so and so works”. Prag­ma­tism is not or­thog­o­nal to, or tran­scen­dent of, truth. Prag­ma­tists need to be con­cerned about what truly works.

Two peo­ple might dis­agree be­cause they are run­ning on the same episte­mol­ogy, but have a differ­ent im­pres­sion of the ev­i­dence ap­ply­ing within that episte­mol­ogy. Or they might dis­agree about the episte­mol­ogy it­self. That can still ap­ply where they are dis­agree­ing about what works. So adopt­ing prag­ma­tism doesn’t make ob­ject level con­cerns about truth van­ish, and it doesnt make meta level con­cerns , episte­mol­ogy , van­ish ei­ther.

Math­e­mat­i­cal the­o­rems aren’t true uni­vo­cally, by cor­re­spon­dence to a sin­gle ter­ri­tory, but they are true by stipu­la­tion, where they can be proven. Univo­cal truth is wrong, and prag­ma­tism, as an al­ter­na­tive to truth, is wrong. What is right is con­tex­tual truth.

Every­one finds the PoNC per­sua­sive, yet many peo­ple be­lieve con­tra­dic­tory things...in a sense. What sense?

Con­sider:

A. Sher­lock Holmes lives at 221b Baker Street.

B. Sher­lock Holmes never lived, he’s a fic­tional char­ac­ter.

Most peo­ple would re­gard both of them as true … in differ­ent con­texts, the fic­tional and the real life. But some­one who be­lieved two con­tra­dic­tory propo­si­tions in the same con­text re­ally would be ir­ra­tional.

• That was a won­der­ful com­ment. I hope you don’t mind if I fo­cus on the last part in par­tic­u­lar. If you’d rather I ad­dressed more I can ac­com­mo­date that, al­though most of that will be sig­nal­ling agree­ment.

To as­sert P is equiv­a­lent to as­sert­ing “P is true” (the defla­tion­ary the­ory in re­verse). That is still true if P is of the form “so and so works”. Prag­ma­tism is not or­thog­o­nal to, or tran­scen­dent of, truth. Prag­ma­tists need to be con­cerned about what truly works.

I’ll note a few things in re­ply to this:

• I’m fine with some con­cep­tual over­lap be­tween my pro­posed episte­mol­ogy and other episte­molo­gies and vague memes.

• You might want to analyse state­ments “P” as mean­ing/​be­ing equiv­a­lent to “P is true,” but I am not go­ing to in­clude any ex­pli­ca­tion of “true” in my episte­mol­ogy for that anal­y­sis to an­chor it­self to.

• Con­tin­u­ing the above, part of what I am do­ing is taboo­ing “truth,” to see if we can for­mu­late an episte­mol­ogy-like frame­work with­out it.

• What “truly works” is more of a feel­ing or a pro­clivity than a propo­si­tion, un­til of course an agent de­vel­ops a model of what works and why.

What is right is con­tex­tual truth.

I agree with you here ab­solutely, mod­ulo vo­cab­u­lary. I would rather say that no sin­gle frame­work is uni­ver­sally ap­pro­pri­ate (prob­lem of in­duc­tion) and that de­vel­op­ing differ­ent tools for differ­ent con­texts is shrewd. But what I just said is more of a model in­spired by my episte­mol­ogy than part of the episte­mol­ogy it­self.

• You might want to analyse state­ments “P” as mean­ing/​be­ing equiv­a­lent to “P is true,” but I am not go­ing to in­clude any ex­pli­ca­tion of “true” in my episte­mol­ogy for that anal­y­sis to an­chor it­self to

Analysing P as “P is true” isn’t some pe­cu­liar­ity of mine: in less for­mal terms, to as­sert some­thing is to as­sert it as true. To put for­ward claims, and per­suade oth­ers that they should be­lieve them is to play a truth game...truth is what one should be­lieve,

So your episte­mol­ogy can’t dis­pense with truth, but offers no anal­y­sis of truth, How use­ful is that?

Con­tin­u­ing the above, part of what I am do­ing is taboo­ing “truth,” to see if we can for­mu­late an episte­mol­ogy-like frame­work with­out it.

Ta­boo­ing truth, or taboo­ing “truth”? It is al­most always pos­si­ble to stop us­ing a word, but con­tinue refer­ring to the con­cept by syn­ony­mous words or phrases. Do­ing with­out the con­cept is harder....do­ing with­out the use, the em­ploy­ment us harder still.

What “truly works” is more of a feel­ing or a pro­clivity than a propo­si­tion, un­til of course an agent de­vel­ops amodel of what works and why.

Noth­ing works just be­cause some­one feels it does. The truth of some­thing truly work­ing us given by the ter­ri­tory.

What is right is con­tex­tual truth.

I agree with you here ab­solutely, mod­ulo vo­cab­u­lary. I would rather say that no sin­gle frame­work is uni­ver­sally ap­pro­pri­ate (prob­lem of in­duc­tion)

Con­tex­tual truth is com­pat­i­ble with no truth?

• Un­for­tu­nately, not all con­struc­tivists or di­alethe­ists (as pro­po­nents of para­con­sis­tent logic are called) would agree with my view­point en­tirely.

I can­not form any idea of what your view­point is, al­though I’m fa­mil­iar with most of the log­i­cal top­ics you refer­enced. You always stop short of stat­ing it, all the way up to the last sen­tence of the last foot­note.

More gen­er­ally, and this is ad­dressed to ev­ery­one writ­ing on a com­plex sub­ject to an au­di­ence of di­verse and un­known back­grounds, try writ­ing your ma­te­rial back­wards. Start at the end, the con­clu­sion, and work back from that to the rea­sons for the con­clu­sion, and the rea­sons for the rea­sons, and so on. Stop be­fore you think you should and con­tinue clar­ifi­ca­tion in the com­ments as the need is re­vealed by the ques­tions.

• I’ve added an ad­den­dum that I hope will make things clearer.

• The cause of me be­liev­ing math is not “it’s true in ev­ery pos­si­ble case”, be­cause I can’t di­rectly ob­serve that. Nor is it “have been ap­plied suc­cess­fully in many cases so far”.

Ba­si­cally it’s “maths says it’s true” where maths is an in­ter­lock­ing sys­tem of many sub­sys­tems. MANY of these have been ap­plied suc­cess­fully in many cases so far. Many of them ren­der con­sid­er­ing them not true pointless, in the sense all my rea­son­ing and senses are in­valid if they don’t hold so I might as well give up and save com­put­ing time by con­di­tion­ing on them be­ing true. Some of them are in im­plicit in ev­ery sin­gle frame of my in­put stream. Many of them are used by my cog­ni­tion, and if I con­sis­tently didn’t con­di­tion on them be­ing true I’d have been un­able to read your post or write this re­ply. Many of them are di­rectly im­ple­mented in phys­i­cal sys­tems around me, which would cease to func­tion if they failed to hold in even one of the billions and billions of uses. Most im­por­tantly, many of them claim that sev­eral of the oth­ers must always be true of they them­selves are not, and while gödelian stuff means this can’t QUITE form a perfect loop in the strongest sense, the fact re­mains that if any of them fell ALL the oth­ers would fol­low like a house of cards; you cant have one of them with­out ALL the oth­ers.

You might try to imag­ine an uni­verse with­out math. And there are some pieces of math that might be iso­lated and in some sense work with­out the oth­ers. But there is a HUGE core of things that cant work with­out each other, nor with­out all those out­ly­ing pieces, at all even slightly. So your uni­verse couldn’t have ge­om­e­try, com­pu­ta­tion, dis­crete ob­jects that can be moved be­tween “piles”, any­thing re­sem­bling fluid dy­nam­ics, etc. Not much of an uni­verse, nor much sen­si­cal imag­in­abil­ity, AND it would be ne­ces­sity be pos­si­ble to simu­late in an uni­verse that does have all the maths so in some sense it still wouldn’t be “break­ing” the laws.

• Many of them ren­der con­sid­er­ing them not true pointless, in the sense all my rea­son­ing and senses are in­valid if they don’t hold so I might as well give up and save com­put­ing time by con­di­tion­ing on them be­ing true.

I call these sorts of mod­els sticky, in the sense that they are per­va­sive in our per­cep­tion and cat­e­gori­sa­tion. Sitcky cat­e­gories are the sort of thing that we have a hard time not tak­ing liter­ally. I haven’t gone into any of this yet, of course, but I like it when com­ments an­ti­ci­pate ideas and con­tinue trains of thought.

Maybe a short run-long run model would be good to illus­trate this stick­i­ness. In the short run, per­cep­tion is fixed; this also fixes cer­tain cat­e­gories, and the “de­gree of stick­i­ness” that differ­ent cat­e­gories have. For ex­am­ple, chair is re­mark­ably hard to get rid of, whereas “cor­pus­cle” isn’t quite as sticky. In the long run, when per­cep­tion is free, no cat­e­gory needs to be sticky. At least, not un­less we come up with a more re­stric­tive model of pos­si­ble per­cep­tions. I don’t think that such a re­stric­tive model would be ap­pro­pri­ate in a back­ground episte­mol­ogy. That’s some­thing that agents will de­velop for them­selves based on their needs and per­cep­tual ex­pe­rience.

Many of them are di­rectly im­ple­mented in phys­i­cal sys­tems

Differ­ent math­e­mat­i­cal mod­els of hu­man per­cep­tual ex­pe­rience might be perfectly suit­able for the same pur­pose.. Physics should be the clear­est ex­am­ple, since we have un­der­gone many differ­ent changes of math­e­mat­i­cal mod­els, and are cur­rently ex­pe­rienc­ing a plu­ral­ity of the­o­ries with differ­ent math­e­mat­ics in cos­mol­ogy. The differ­ences be­tween clas­si­cal me­chan­ics and quan­tum me­chan­ics should in par­tic­u­lar show this nicely: differ­ent for­mal­isms, but very good mod­els of a large class of ex­pe­riences.

you cant have one of them with­out ALL the oth­ers.

I think you slightly un­der­es­ti­mate the ver­sa­tility of math­e­mat­i­ci­ans in mak­ing their sys­tems work de­spite malfunc­tions. For in­stance, even if ZFC were proved in­con­sis­tent (as Ed­ward Nel­son hopes to do), we would not have to aban­don it as a foun­da­tion. Set the­o­rists would just do some ho­cus pocus in­volv­ing or­di­nals, and voila! all would be well. And there are sev­eral al­ter­na­tive for­mu­la­tions of ar­ith­metic, anal­y­sis, topol­ogy, etc. which are all ad­e­quate for most pur­poses.

You might try to imag­ine an uni­verse with­out math.

In the case of some math, this is easy to do. In other cases it is not. This is be­cause we don’t ex­pe­rience the freefloat­ing per­cep­tual long term, not be­cause cer­tain mod­els are nec­es­sary for all pos­si­ble agents and per­cep­tual con­tent.

• I don’t mean just sticky mod­els. The con­cepts I’m talk­ing about are things like “prob­a­bil­ity”, “truth”, “goal”, “If-then”, “per­sis­tent ob­jects”, etc. Believ­ing that a the­ory is true that says “true” is not a thing the­o­ries can be is ob­vi­ously silly. Believ­ing that there is no such things as de­ci­sion­mak­ing and that you’re a frac­tion of a sec­ond old and will cease to be within an­other frac­tion of a sec­ond might be philo­soph­i­cally more defen­si­ble, but con­di­tion­ing on it not be­ing true can never have bad con­se­quences while it has a chance of hav­ing good ones.

I were talk­ing about phys­i­cal sys­tems, not phys­i­cal laws. Com­put­ers, liv­ing cells, atoms, the fluid dy­nam­ics of the air… “Ap­plied suc­cess­fully in many cases”, where “many” is “billions of times ev­ery sec­ond”

Then ZFC is not one of those cores ones, just one of the periph­eral ones. I’m talk­ing ones like set the­ory as a whole, or ar­ith­metic, or Tur­ing ma­chines.

• Believ­ing that a the­ory is true that says “true” is not a thing the­o­ries can be is ob­vi­ously silly.

Oh okay. This is a two-part mi­s­un­der­stand­ing.

I’m not say­ing that the­o­ries can’t be true, I’m just not talk­ing about this truth thing in my meta-model. I’m perfectly a-okay with mod­els of truth pop­ping up wher­ever they might be handy, but I want to taboo the in­tu­itive no­tion and re­fuse to ex­pli­cate it. In­stead I’ll rely on other con­cepts to do much of the work we give to truth, and see what hap­pens. And if there’s work that they can’t do, I want to eval­u­ate whether it’s im­por­tant to in­clude in the meta-model or not.

I’m also not say­ing that my the­ory is true. At least, not when I’m talk­ing from within the the­ory. Per­haps I’ll find cer­tain facets of the cor­re­spon­dence the­ory use­ful for ex­plain­ing things or con­vinc­ing oth­ers, in which case I might claim it’s true. My episte­mol­ogy is just as much a model as any­thing else, of course; I’m de­vel­op­ing it with cer­tain goals in mind.

I were talk­ing about phys­i­cal sys­tems, not phys­i­cal laws. Com­put­ers, liv­ing cells, atoms, the fluid dy­nam­ics of the air… “Ap­plied suc­cess­fully in many cases”, where “many” is “billions of times ev­ery sec­ond”

The math we use to model com­pu­ta­tion is a model and a tool just as much as com­put­ers are tools; there’s noth­ing weird (at least from my point of view) about mod­els be­ing used to con­struct other tools. Liv­ing cells can be mod­eled suc­cess­fully with math, you’re right; but that again is just a model. And atoms are definitely the­o­ret­i­cal con­structs used to model ex­pe­riences, the per­sua­sive images of balls or clouds they con­jure notwith­stand­ing. Some­thing similar can be said about fluid dy­nam­ics.

I don’t mean any of this to be­lit­tle mod­els, of course, or make them seem whim­si­cal. Models are worth tak­ing se­ri­ously, even if I don’t think they should be taken liter­ally.

Then ZFC is not one of those cores ones, just one of the periph­eral ones. I’m talk­ing ones like set the­ory as a whole, or ar­ith­metic, or Tur­ing ma­chines.

The best ex­am­ple in the three is definitely ar­ith­metic; the other two aren’t con­vinc­ing. Math was done with­out set the­ory for ages, and be­sides we have other foun­da­tions available for mod­ern math that can be for­mu­lated en­tirely with­out talk­ing about sets. Tur­ing ma­chines can be re­placed with log­i­cal sys­tems like the lambda calcu­lus, or with other ma­chine mod­els like reg­ister ma­chines.

Arith­metic is more com­pel­ling, be­cause it’s very sticky. It’s hard not to take it liter­ally, and it’s hard to imag­ine things with­out it. This is be­cause some of the ideas it con­sti­tutes are at the core cluster of our cat­e­gories, i.e. they’re very sticky. But could you imag­ine that some agent might a) have goals that never re­quire ar­ith­meti­cal con­cepts, and b) that there could be mod­els that are non-ar­ith­meti­cal that could be used to­ward some of the same goals for which we use ar­ith­metic? I can imag­ine … vi­su­al­ise, ac­tu­ally, both, al­though I would have a very hard time trans­lat­ing my vi­sual into text with­out go­ing very meta first, or else writ­ing a ridicu­lously long post.

• Hmm, maybe I need to re­veal my episte­mol­ogy an­other step to­wards the bot­tom. Two things seem rele­vant here.

I think you you SHOULD take your best model liter­ally if you live in a hu­man brain, since it can never get com­pletely stuck re­quiring in­finite ev­i­dence due to it’s ar­chi­tec­ture, but does have limited com­pu­ta­tion and doubt can both con­fuse it and dam­age mo­ti­va­tion. The few down­sides there are can be fixed with in­junc­tions and heuris­tics.

Se­condly, you seem to be go­ing with fuzzy in­tu­itions or di­rect sen­sory ex­pe­rience as the most fun­da­men­tal. At my core is in­stead that I care about stuff, and that my out­put might de­ter­mine that stuff. The FIRST thing that hap­pens is con­di­tion­ing on that my de­ci­sions mat­ter, and then I start up­dat­ing on the in­put stream of a par­tic­u­lar in­stance/​im­ple­men­ta­tion of my­self. My work­ing defi­ni­tion of “real” is “stuff I might care about”.

My point wasn’t that the phys­i­cal sys­tems can be mod­eled BY math, but that they them­selves model math. Fur­ther, that if the math wasn’t True, then it wouldn’t be able to model the phys­i­cal sys­tems.

With the math sys­tems as well you seem to be com­ing from the op­po­site di­rec­tion. Set the­ory is a for­mal sys­tem, ar­ith­metic can model it us­ing gödel num­ber­ing, and you can’t pre­vent that or have it give differ­ent re­sults with­out break­ing ar­ith­metic en­tirely. Like­wise, set the­ory can model ar­ith­metic. It’s a pack­age deal. Lambda calcu­lus and reg­ister ma­chines are also mem­bers of that list of mu­tual mod­el­ing. I think even ba­sic ge­om­e­try can be made sort of Tur­ing com­plete some­how. Any im­ple­men­ta­tion of any of them must by ne­ces­sity model all of them, ex­actly as they are.

You can model an agent that doesn’t need the con­cepts, but it must be a very sim­ple agent with very sim­ple goals in a very sim­ple en­vi­ron­ment. To sim­ple to be rec­og­niz­able as agentlike by hu­mans.

• Could any­body provide a con­cise sum­mary of the ideas men­tioned above? I am try­ing to read ev­ery­thing but it is un­clear to me what point the au­thor is mak­ing, and how he/​she ar­rives at that con­clu­sion.

• I’ve added an ad­den­dum. If read­ing that doesn’t help, let me know and I’ll sum­marise it for you in an­other way.

• What are cases where mak­ing a de­ci­sion to model a prob­lem in a way where you rely on in­tu­ition­is­tic logic has high util­ity in the sense that it pro­duces a model that does good real world pre­dic­tions?

• In­tu­ition­is­tic logic can be in­ter­preted as the logic of finite ver­ifi­ca­tion.

Truth in in­tu­ition­is­tic logic is just prov­abil­ity. If you as­sert A, it means you have a proof of A. If you as­sert ¬A then you have a proof that A im­plies a con­tra­dic­tion. If you as­sert A ⇒B then you can pro­duce a proof of B from A. If you as­sert A ∨ B then you have a proof of at least one of A or B. Note that the law of ex­cluded mid­dle fails here be­cause we aren’t al­low­ing sen­tences A ∨ ¬A where you have no proof of A or that A im­plies a con­tra­dic­tion.

In all cases, the as­ser­tion of a for­mula must cor­re­spond to a proof, proofs be­ing (of course) finite. Us­ing this idea of finite ver­ifi­ca­tion is a nice way to de­velop topol­ogy for com­puter sci­ence and for­mal episte­mol­ogy (see Topol­ogy via Logic by Steven Vick­ers). Com­puter sci­ence is con­cerned with ver­ifi­ca­tion as proofs and pro­grammes (and the Curry-Howard iso­mor­phism comes in handy there), and for­mal episte­mol­ogy is con­cerned with ver­ifi­ca­tion as ob­ser­va­tions and sci­en­tific mod­el­ing.

That isn’t ex­actly a spe­cific ex­am­ple, but a class of ex­am­ples. Re­search on this is cur­rently very ac­tive.

• In­tu­ition­is­tic logic can be in­ter­preted as the logic of finite ver­ifi­ca­tion.

I don’t care of how it can be in­ter­preted but whether it’s use­ful. I asked for a prac­ti­cal ex­am­ple. Some­thing use­ful for guid­ing real world ac­tions. Maybe an ap­pli­ca­tion in biol­ogy or physics.

If you want to be “prag­matic” then it makes sense to look at whether your philos­o­phy ac­tu­ally is ap­pli­ca­ble to real world prob­lems.

• I think there’s a bit of a mi­s­un­der­stand­ing go­ing on here, though, be­cause I am perfectly okay with peo­ple us­ing clas­si­cal logic if they like. Clas­si­cal logic is a great way to model cir­cuits, for ex­am­ple, and it pro­vides some nice rea­son­ing heuris­tics.There’s noth­ing in my po­si­tion that com­mits us to aban­don­ing it en­tirely in favour of in­tu­ition­is­tic logic.

In­tu­ition­is­tic logic is ap­pli­ca­ble to at least three real-world prob­lems: for­mu­lat­ing foun­da­tions for math, ver­ify­ing pro­grammes, and com­put­er­ised the­o­rem-prov­ing. The last two in par­tic­u­lar will have ap­pli­ca­tions in ev­ery­thing from cli­mate mod­el­ing to pop­u­la­tion ge­net­ics to quan­tum field the­ory.

As it hap­pens, math­e­mat­i­cian An­drej Bauer wrote a much bet­ter defence of do­ing physics with in­tu­ition­is­tic logic than I could have: http://​​math.an­drej.com/​​2008/​​08/​​13/​​in­tu­ition­is­tic-math­e­mat­ics-for-physics/​​

• If one wants to un­der­stand an ab­stract prin­ci­ple it’s very use­ful to illus­trate the prin­ci­ple with con­crete prac­ti­cal ex­am­ples.

I don’t think that any­one on LW think that we shouldn’t a few con­struc­tivist math­e­mat­i­ci­ans around to do their job and make a few proof that ad­vance math­e­mat­ics. I don’t re­ally care about how the proofs of the math I use are de­rived pro­vided I can trust them.

If you call for a core change in episte­mol­ogy it sounds like you want more than that. To me it’s not clear what that more hap­pens to be. In case you don’t know the lo­cal LW defi­ni­tion of ra­tio­nal­ity is : “Be­hav­ing in a way that’s likely to make you win.”

• If you call for a core change in episte­mol­ogy it sounds like you want more than that. To me it’s not clear what that more hap­pens to be.

I’m go­ing to have to do some strate­gic re­view on what ex­actly I’m not be­ing clear about and what I need to say to make it clear.

In case you don’t know the lo­cal LW defi­ni­tion of ra­tio­nal­ity is : “Be­hav­ing in a way that’s likely to make you win.”

Yes, I share that defi­ni­tion, but that’s only the LW defi­ni­tion of in­stru­men­tal ra­tio­nal­ity; epistemic ra­tio­nal­ity on the other hand is mak­ing your map more ac­cu­rately re­flect the ter­ri­tory. Part of what I want is to scrap that and judge epistemic mat­ters in­stru­men­tally, like I said in the con­clu­sion and ad­den­dum.

Still, it’s clear I haven’t said quite enough. You men­tioned ex­am­ples, and that’s kind of what this post was in­tended to be: an ex­am­ple of ap­ply­ing the sort of rea­son­ing I want to a prob­lem, and con­trast­ing it with epistemic ra­tio­nal­ity rea­son­ing.

Part of the prob­lem with gen­er­at­ing a whole bunch of spe­cific ex­am­ples is that it wouldn’t help illus­trate the change much. I’m not say­ing that sci­ence as it’s prac­tised in gen­eral needs to be rad­i­cally changed. Mostly things would con­tinue as nor­mal, with a few ex­cep­tions (like the­o­ret­i­cal physics, but I’m go­ing to have to let that par­tic­u­lar ex­am­ple stew for a while be­fore I voice it out­side of pri­vate dis­cus­sions).

The main tar­get of my change is the way we con­cep­tu­al­ise sci­ence. Lots of episte­molog­i­cal work fo­cuses on ideal­ised car­i­ca­tures that are too pre­scrip­tive and poorly re­flect how we man­aged to achieve what we did in sci­ence. And I think that hav­ing a bet­ter philos­o­phy of sci­ence will make think­ing about some prob­lems in ex­is­ten­tial risk, par­tic­u­larly FAI, eas­ier.

• You men­tioned ex­am­ples, and that’s kind of what this post was in­tended to be: an ex­am­ple of ap­ply­ing the sort of rea­son­ing I want to a prob­lem, and con­trast­ing it with epistemic ra­tio­nal­ity rea­son­ing.

Ap­ply­ing it to what prob­lem? (If you mean the physics posts you linked to, I need more time to di­gest it fully)

The main tar­get of my change is the way we con­cep­tu­al­ise sci­ence.

No­body ac­tu­ally con­cep­tu­al­ises sci­ence as be­ing about de­riv­ing from think­ing “pink is my fa­vor­ity color and it isn’t” → “causal­ity doesn’t work”.

Lots of episte­molog­i­cal work fo­cuses on ideal­ised car­i­ca­tures that are too pre­scrip­tive and poorly re­flect how we man­aged to achieve what we did in sci­ence.

Then pick on of those car­i­ca­tures and analyse in de­tail how your episte­molog­i­cal leads to differ­ent think­ing about the is­sue.

And I think that hav­ing a bet­ter philos­o­phy of sci­ence will make think­ing about some prob­lems in ex­is­ten­tial risk, par­tic­u­larly FAI, eas­ier.

Yes, ob­vi­ously hav­ing a bet­ter philos­o­phy of sci­ence would be good.

• Ap­ply­ing it to what prob­lem? (If you mean the physics posts you linked to, I need more time to di­gest it fully)

No, not that com­ment, I mean the ini­tial post. The prob­lem is han­dling math­e­mat­i­cal sys­tems in an episte­mol­ogy. A lot of episte­molo­gies have a hard time with that be­cause of on­tolog­i­cal is­sues.

No­body ac­tu­ally con­cep­tu­al­ises sci­ence as be­ing about de­riv­ing from think­ing “pink is my fa­vor­ity color and it isn’t” → “causal­ity doesn’t work”.

No, but many peo­ple hold the view that you can talk about valid state­ments as con­strain­ing on­tolog­i­cal pos­si­bil­ities. This is in­clud­ing Eliezer of 2012. If you read the High Ad­vance Episte­mol­ogy posts on math, he does rea­son about the par­tic­u­lar log­i­cal laws con­strain­ing how the physics of time and space work in our uni­verse. And the view is very old, go­ing back to be­fore Aris­to­tle, through Leib­niz to the pre­sent.

• No, not that com­ment, I mean the ini­tial post. The prob­lem is han­dling math­e­mat­i­cal sys­tems in an episte­mol­ogy.

Han­dling math­e­mat­i­cal sys­tems in an episte­mol­ogy is in my idea a topic but not a spe­cific prob­lem. In that topic there are prob­a­bly a bunch of prac­ti­cal prob­lem but.

Molec­u­lar biol­ogy is a sub­ject. Pre­dict­ing pro­tein fold­ing re­sults is a spe­cific prob­lem.

If we look at FAI, writ­ing a bot that performs well in the pris­oner dilemma tour­na­ments that are about ver­ify­ing source code of the other bots is a spe­cific prob­lem.

The are also prob­lems in the daily busi­ness of do­ing sci­ence where the sci­en­tist has to de­cide be­tween mul­ti­ple pos­si­ble courses of ac­tion.