# MrMind

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• Two of my fa­vorite cat­e­gories show that they re­ally are ev­ery­where: the free cat­e­gory on any graph and the presheaves of gamma.

The first: take any di­rected graph, un­fo­cus your eyes and in­stead of ar­rows con­sider paths. That is a cat­e­gory!

The sec­ond: take any finite graph. Take sets and func­tions that re­al­ize this graph. This is a cat­e­gory, more­over you can make it dag­ger-com­pact, so you can do quan­tum me­chan­ics with it. Take as the finite graph gamma, which is just two ver­tex with two ar­rows be­tween them. Sets and func­tions that re­al­ize this graph are… any graph! So, CT al­lows you to do quan­tum me­chan­ics with graphs.

Amaz­ing!

• 9 Oct 2019 13:18 UTC
2 points

Lambda calcu­lus is though the in­ter­nal lan­guage of a very com­mon kind of cat­e­gory, so, in a sense, cat­e­gory the­ory al­lows lambda calcu­lus to do com­pu­ta­tions not only with func­tions, but also sets, topolog­i­cal spaces, man­i­folds, etc.

• While I share your en­thu­si­asm to­ward cat­e­gories, I find sus­pi­cious the claim that CT is the cor­rect frame­work from which to un­der­stand ra­tio­nal­ity. Around here, it’s mainly equated with Bayesian Prob­a­bil­ity, and the cat­e­go­rial grasp of prob­a­bil­ity or even mea­sure is less than im­pres­sive. The most in­ter­est­ing fact I’ve been able to dig up is that the Giry monad is the co­den­sity monad of the in­clu­sion of con­vex spaces into mea­sure spaces, hardly an illu­mi­nat­ing fact (ba­si­cally a con­voluted way of say­ing that prob­a­bil­ities are the most gen­eral ways of form­ing con­vex com­bi­na­tions out of mea­sures).

I’ve searched and searched for cat­e­go­rial an­swers or hints about the prob­lem of ex­tend­ing prob­a­bil­ities to other kinds of logic (or even sim­ply ex­tend­ing it to clas­si­cal pred­i­cate logic), but so far I’ve had no luck.

# Odds are not easier

21 Aug 2019 8:34 UTC
7 points
• I’d like to point out a source of con­fu­sion around Oc­cam’s Ra­zor that I see you’re fal­ling for, dis­pel­ling it will make things clearer: “you should not mul­ti­pli­cate en­tities with­out ne­ces­si­ties!”. This means that Oc­cam’s Ra­zor helps de­cide be­tween com­pet­ing the­o­ries if and only if they have the same ex­pla­na­tion and pre­dic­tive power. But in the his­tory of sci­ence, it was al­most never the case that com­pet­ing the­o­ries had the same power. Maybe it hap­pened a cou­ple of times (epicy­cles, the Copen­hagen in­ter­pre­ta­tion), but in all other in­stances a the­ory was se­lected not be­cause it was sim­pler, but be­cause it was much more pow­er­ful.

Con­trary to pop­u­lar mis­con­cep­tion, Oc­cam’s ra­zor gets to be used very, very rarely.

We do have, any­way, a for­mal­iza­tion of that prin­ci­ple in al­gorith­mic in­for­ma­tion the­ory: Solomonoff in­duc­tion. A agent that, to pre­dict the out­come of a se­quence, places the high­est prob­a­bil­ities in the short­est com­pat­i­ble pro­grams, will even­tu­ally out­perform ev­ery other class of pre­dic­tor. The catch here is the word ‘even­tu­ally’: in ev­ery mea­sure of com­plex­ity, there’s a con­stant that offset the val­ues due to the defi­ni­tion of the refer­ence uni­ver­sal Tur­ing ma­chine. Differ­ent refer­ences will in­di­cate differ­ent com­plex­ities for the same first pro­grams, but all mea­sure will con­verge af­ter a finite amount.

This is also why I think that the prob­lem ex­plain­ing thun­ders with “Thor vs clouds” is such a poor ex­am­ple of Oc­cam’s ra­zor: Solomonoff in­duc­tion is a for­mal­iza­tion of Oc­cam ra­zor for the­o­ries, not ex­pla­na­tions. Due to the afore­men­tioned con­stant, you can­not have ab­solutely sim­pler model of a finite se­quence of event. There’s no such a thing, it will always de­pend on the com­plex­ity of the start­ing Tur­ing ma­chine. How­ever, you can have even­tu­ally sim­pler mod­els of in­finite se­quence of events (in­finite se­quence pre­dic­tor are equiv­a­lent to pro­grams). In that case, the nat­u­ral event pro­gram will pre­vail be­cause it will al­low to con­trol bet­ter the out­comes.

• I ar­rived at the same con­clu­sion when I tried to make sense of the Me­taethics Se­quence. My sum­mary of Eliezer’s writ­ings is: “moral­ity is a bunch of men­tal com­pu­ta­tions shared be­tween most hu­man be­ings”. Mo­ral­ity thus grew out of our evolu­tive his­tory, and it should not be sur­pris­ing that in ex­treme situ­a­tions it might be in­co­her­ent or mal­adap­tive.

Only if you be­lieve that moral­ity should be like sys­tem­atic and uni­ver­sal and co­her­ent, then you can say that ex­treme ex­am­ples are un­cov­er­ing some­thing in­ter­est­ing about peo­ples’ moral­ity.

Other­wise, ex­treme situ­a­tions are as in­ter­est­ing as say­ing that peo­ple can­not men­tally fac­tor long num­bers.

• First of all, the com­mu­nity around LW2.0 can only be loosely as­so­ci­ated to a move­ment: I don’t think there’s any­one that ex­plic­itly en­dorses *ev­ery* tech­nique or the­ory ap­peared here. LW is not CFAR, is not the Align­ment fo­rum, etc. So I would cau­tion against en­tic­ing some­one into LW by say­ing that the com­mu­nity sup­ports this or that tech­nique.

The main ad­van­tage of ra­tio­nal­ity, in its pre­sent stage, is defen­sive: if you’re as­piring to be ra­tio­nal, you wouldn’t waste time at­tend­ing re­li­gious gath­er­ings that you de­spise; you wouldn’t waste money buy­ing in­effec­tive treat­ments (sugar pills, crys­tals, etc.); you wouldn’t waste re­sources fol­low­ing peo­ple that mis­take fic­tion for facts. At the mo­ment, ra­tio­nal­ity is just a very good filter for ev­ery product, knowl­edge and praxis that so­ciety pre­sents to you (hint: 99% of those things is crap).

On the other hand, what you can or should do with all the re­sources you’re not wast­ing, is some­thing ra­tio­nal­ity can­not an­swer in full to­day. Me­taethics and akra­sia are, af­ter all, the great­est un­solved prob­lems of our com­mu­nity.

There were no­to­ri­ous at­tempts (e.g. Tor­ture vs Dust specks or the Basilisk), but noth­ing has emerged with the clar­ity and effec­tive­ness of Bayesian rea­son­ing. Effec­tive Altru­ism and MIRI are per­haps the most fa­mous ex­am­ples of try­ing to solve the most press­ing prob­lems. A defini­tive frame­work though still eludes us.

• In Fo­er­ster’s pa­per, he links the in­crease in pro­duc­tivity lin­early with the in­crease in pop­u­la­tion. But Scott has also pro­posed that the rate of in­no­va­tion is slow­ing down, due to a log­a­r­ith­mic in­crease of pro­duc­tivity from pop­u­la­tion. So maybe Fo­er­ster’s model is still valid, and 1960 is only the year where we ex­hausted the al­most lin­ear part of progress (the “low hang­ing fruits”).

Per­haps nowa­days we com­bine the ex­po­nen­tial growth of pop­u­la­tion from pop­u­la­tion with the log­a­r­ith­mic in­crease in pro­duc­tivity, to get the lin­ear eco­nomic growth we see.

• Alge­braic topol­ogy is the dis­ci­pline that stud­ies ge­ome­tries by as­so­ci­at­ing them with alge­braic ob­jects (usu­ally, groups or vec­tor spaces) and ob­serv­ing how chang­ing the un­der­ly­ing space af­fects the re­lated alge­bras. In 1941, two math­e­mat­i­ci­ans work­ing in that field sought to gen­er­al­ize a the­o­rem that they dis­cov­ered, and needed to show that their solu­tion was still valid for a larger class of spaces, ob­tained by “nat­u­ral” trans­for­ma­tions. Nat­u­ral, at that point, was a term lack­ing a pre­cise defi­ni­tion, and only meant some­thing like “avoid­ing ar­bi­trary choices”, in the same way a vec­tor space is nat­u­rally iso­mor­phic to its dou­ble dual, while it’s iso­mor­phic to its dual only through the choice of a ba­sis.

The need to make pre­cise the no­tion of nat­u­ral­ity for alge­braic topol­ogy led them to the defi­ni­tion of nat­u­ral trans­for­ma­tion, which in turn re­quired the no­tion of func­tor which in turn re­quired the no­tion of cat­e­gory.

This an­swers ques­tions 1 and 2: cat­e­gory the­ory was born to give a pre­cise defi­ni­tion of nat­u­ral­ity, and was sought to gen­er­al­ize the “uni­ver­sal co­effi­cient the­o­rem” to a larger class of spaces.

This story is told with a lot of de­tails in the first para­graphs of Riehl’s won­der­ful “Cat­e­gory the­ory in con­text”.

To an­swer n° 3, though, even if cat­e­gory the­ory was rapidly ex­pand­ing dur­ing the ’50s and the ‘60s, it was only with the work of Law­vere (who I con­sider a ge­nius on par with Gödel) in the ’70s that it be­came a foun­da­tional dis­ci­pline: guided by his in­tu­itions, cat­e­gory the­ory be­came the unify­ing lan­guage for ev­ery branch of math­e­mat­ics, from ge­om­e­try to com­pu­ta­tion to logic to alge­bras. Ba­si­cally, it showed how the va­ri­ety of math­e­mat­i­cal dis­ci­plines are just differ­ent ways to say the same thing.

• Is it re­ally quite differ­ent, be­sides halo effect? It strongly de­pends on the de­tail, though if the two say the ex­act same thing, how are things differ­ent?

• The con­cept of “fake frame­work”, elu­ci­dated in the origi­nal post, to me it seems one of a model of re­al­ity that hides some com­plex­ity, some­times even to the point of be­ing very wrong, but that is nonethe­less use­ful be­cause it makes some other com­plex area man­age­able.

On the other hand, when I read the quotes you pre­sented, I see a rich tapestry of metaphors and jar­gon, of which the pro­po­nent him­self says that they can be wrong… but I fail com­pletely to see what part of re­al­ity they make man­age­able. Th­ese frame­works seems to just add com­plex­ity to com­plex­ity, with­out any real lev­er­age over re­al­ity. This makes those frame­works draw nearer fic­tion, rather than use­ful but sim­plified mod­els.

For ex­am­ple, if there’s no post-ra­tio­nal stage of de­vel­ope­ment, what use is the ad­vice of not con­fus­ing it with a pre-ra­tio­nal stage of de­vel­ope­ment? If En­light­en­ment is not a thing, what use is the ex­or­ta­tion to come up with a chronolog­i­cally ro­bust defi­ni­tion of the same?

This to me is the most strik­ing differ­ence be­tween “In­te­gral spiritu­al­ity” and say a road map. With the road map, you know ex­actly what is hid­den and why, and it’s ev­i­dent how to use it. With Wilber’s frame­work, it seems ex­actly the op­po­site.

Maybe this is due to of my un­fa­mil­iar­ity with that ma­te­rial… so some­one who has effec­tively found out some­thing use­ful out of that model can chime in and tell their ex­pe­rience, and I will stand cor­rected.

• I’m sorry, but you can­not re­ally learn any­thing from one ex­am­ple. I’m happy that your par­ents are far­ing well in their mar­riage, but if they didn’t would you have learned the same thing?

I’ve con­sulted a few statis­tics on ar­ranged mar­riage, and they all are:

• underpowered

• show­ing no sig­nifica­tive differ­ence be­tween au­tonomous and ar­ranged marriages

The lat­ter part is some­what sur­pris­ing for a Westerner, but given what you say, the same should be said for an In­dian com­ing from your back­ground.

The only con­clu­sion I can draw fairly con­clu­sively is that, for a long term re­la­tion­ship, the way or the why it started doesn’t re­ally mat­ter.

• Are you fa­mil­iar with the con­cept of fold/​un­fold? Folds are func­tions that con­sume struc­tures and pro­duce val­ues, while un­folds do the op­po­site. The com­po­si­tion of an un­fold plus a fold is called a hy­lo­mor­phism, of which the fac­to­rial is a perfect ex­am­ple: the un­fold cre­ates a list from 1 to n, the fold mul­ti­plies to­gether the en­tire list. Your sec­tion on the “two-fold re­cur­sion” is a perfect de­scrip­tion of a hy­lo­mor­phism: you take a goal, un­fold it into a plan com­posed of a list of micro-steps, then you fold it by ex­e­cut­ing each one of the micro-steps in or­der.

• Luke already wrote that there are at least four fac­tors that feed mo­ti­va­tion, and the ex­pec­ta­tion of suc­cess is only one of them. No amount of ex­pec­tancy can in­cre­ment drive if other fac­tors are lack­ing, and as Eliezer no­tice, it’s not sane to ex­pect only one fac­tor to be 10x the oth­ers so that it alone pow­ers the en­g­ine.

What Eliezer is ask­ing is ba­si­call if any­one has solved the ba­sic co­or­di­na­tion prob­lem of mankind, and I think he knows very well that the an­swer to his ques­tion is no. Also, be­cause we are op­er­at­ing in a rel­a­tively small mindspace (hu­mans’ sys­tem 1), the fact that no one solved that prob­lem in hun­dreds of thou­sands of years of co­op­er­a­tion points strongly to­ward the fact that such a solu­tion doesn’t ex­ist.

• Re: the third point, I think it’s im­por­tant to differ­en­ti­ate be­tween and , where is the true pre­dic­tion, that is what ac­tu­ally hap­pens when an agent performs the ac­tion .

is sim­ply the out­come the agent is aiming at, while is the out­come the agent even­tu­ally gets. So maybe it’s more in­ter­est­ing a mea­sure of similar­ity in , from which you can com­pare the two.

• Let’s say that is the set of available ac­tions and is the set of con­se­quences. is then the set of pre­dic­tions, where a sin­gle pre­dic­tion as­so­ci­ates to ev­ery pos­si­ble ac­tion a con­se­quence. is then a choice op­er­a­tor, that se­lects for each pre­dic­tion an ac­tion to take.

What we have seen so far:

• There’s no ‘gen­eral’ or ‘nat­u­ral’ choice op­er­a­tor, that is, ev­ery choice op­er­a­tor must be based on at least a par­tial knowl­edge of the do­main or the codomain;

• Un­less the pos­si­ble con­se­quences are triv­ial, a choice op­er­a­tor will choose the same ac­tion for many differ­ent pre­dic­tions, that is a choice op­er­a­tor only uses cer­tain fea­ture of the pre­dic­tions’ space and is in­differ­ent to any­thing else [1];

• A choice op­er­a­tor defines nat­u­rally a ‘preferred out­come’ op­er­a­tor, which is sim­ply the pre­dicted out­come of the cho­sen ac­tion, and is defined by ‘sand­wich­ing’ the choice op­er­a­tor be­tween two pre­dic­tions. I just thought in­ter­leave is a bet­ter name than sand­wich. It’s of type .

[1] To show this, let be a par­ti­tion of and let be the equiv­alence re­la­tion uniquely gen­er­ated by the par­ti­tion. Then

# You’re never wrong in­ject­ing com­plex­ity, but rarely you’re right

3 Oct 2018 14:20 UTC
38 points
• I won­der if there are any plau­si­ble ex­am­ples of this type where the con­straints don’t look like or­der­ing on B and search on A.

Yes, as I shown in my post, such op­er­a­tors must know at least an el­e­ment of one of the do­mains of the func­tion. If it knows at least an el­e­ment of A, a con­stant func­tion on that el­e­ment has the right type. Un­for­tu­nately, it’s not much in­ter­est­ing.

• It’s in­ter­est­ing to no­tice that there’s noth­ing with that type on hoogle (Haskell lan­guage search en­g­ine), so it’s not the type of any com­mon util­ity.

On the other hand, you can still say quite a bit on func­tions of that type, draw­ing from type and set the­ory.

First, let’s name a generic func­tion with that type . It’s pos­si­ble to show that k can­not be para­met­ric in both types. If it were, would be valid, which is ab­surd ( has an el­e­ment!). It’ also pos­si­ble to show that if k is not para­met­ric in one type, it must have ac­cess to at least an el­e­ment of that type (think about and ).

A sim­ple car­di­nal­ity ar­gu­ment also shows that k must be many-to-one (that is, non in­jec­tive): un­less B is 1 (the one el­e­ment type),

There is an in­ter­est­ing op­er­a­tor that uses k, which I call in­ter­leave:

Triv­ially,

It’s in­ter­est­ing be­cause par­tially ap­ply­ing in­ter­leave to some k has the type , which is the type of con­tinu­a­tions, and I sus­pect that this is what un­der­lies the com­mon us­age of such op­er­a­tors.