# riceissa

Karma: 313 (LW), 11 (AF)
Page 1
• We can model suc­cess as a com­bi­na­tion of do­ing use­ful things and avoid­ing mak­ing mis­takes. As a par­tic­u­lar ex­am­ple, we can model in­tel­lec­tual suc­cess as a com­bi­na­tion of com­ing up with good ideas and avoid­ing bad ideas. I claim that ra­tio­nal­ity helps us avoid mis­takes and bad ideas, but doesn’t help much in gen­er­at­ing good ideas and use­ful work.

Eliezer Yud­kowsky has made similar points in e.g. “Un­teach­able Ex­cel­lence” (“much of the most im­por­tant in­for­ma­tion we can learn from his­tory is about how to not lose, rather than how to win”, “It’s eas­ier to avoid du­pli­cat­ing spec­tac­u­lar failures than to du­pli­cate spec­tac­u­lar suc­cesses. And it’s of­ten eas­ier to gen­er­al­ize failure be­tween do­mains.“) and “Teach­ing the Un­teach­able”.

• Thoughts on #10:

I am con­fused about this ex­er­cise. The stan­dard/​mod­ern proof of Gödel’s sec­ond in­com­plete­ness the­o­rem uses the Hilbert–Ber­nays–Löb deriv­abil­ity con­di­tions, which are stated as (a), (b), (c) in ex­er­cise #11. If the ex­er­cises are meant to be solved in se­quence, this seems to im­ply that #10 is solv­able with­out us­ing the deriv­abil­ity con­di­tions. I tried do­ing this for a while with­out get­ting any­where.

Maybe an­other way to state my con­fu­sion is that I’m pretty sure that up to ex­er­cise #10, noth­ing that dis­t­in­guishes Peano ar­ith­metic from Robin­son ar­ith­metic has been in­tro­duced (it is only with the in­tro­duc­tion of the deriv­abil­ity con­di­tions in #11 that this differ­ence be­comes ap­par­ent). It looks like there is a ver­sion of the sec­ond in­com­plete­ness the­o­rem for Robin­son ar­ith­metic, but the pa­per says “The proof is by the con­struc­tion of a non­stan­dard model in which this for­mula [i.e. for­mula ex­press­ing con­sis­tency] is false”, so I’m guess­ing this proof won’t work for Peano ar­ith­metic.

• My solu­tion for #12:

Sup­pose for the sake of con­tra­dic­tion that such a for­mula ex­ists. By the di­ag­o­nal lemma ap­plied to , there is some sen­tence such that, prov­ably, . By the sound­ness of our the­ory, in fact . But by the prop­erty for we also have , which means , a con­tra­dic­tion.

This seems to be the “se­man­tic” ver­sion of the the­o­rem, where the prop­erty for is stated out­side the sys­tem. There is also a “syn­tac­tic” ver­sion where the prop­erty for is stated within the sys­tem.

• At­tempted solu­tion and some thoughts on #9:

Define a for­mula tak­ing one free vari­able to be .

Now define to be . By the defi­ni­tion of we have .

We have

The first step fol­lows by the defi­ni­tion of , the sec­ond by the defi­ni­tion of , the third by the defi­ni­tion of , and the fourth by the prop­erty of men­tioned above. Since by the type sig­na­ture of , this com­pletes the proof.

Things I’m not sure about:

It’s a lit­tle un­clear to me what the no­ta­tion means. In par­tic­u­lar, I’ve as­sumed that takes as in­puts Gödel num­bers of for­mu­las rather than the for­mu­las them­selves. If takes as in­puts the for­mu­las them­selves, then I don’t think we can as­sume that the for­mula ex­ists with­out do­ing more ar­ith­me­ti­za­tion work (i.e. the equiv­a­lent of would need to know how to con­vert from the Gödel num­ber of a for­mula to the for­mula it­self).

If the bi­con­di­tional “” is a con­nec­tive in the logic it­self, then I think the same proof works but we would need to as­sume more about than is given in the prob­lem state­ment, namely that the the­ory we have can prove the sub­sti­tu­tion prop­erty of .

The as­sump­tion about the quan­tifier com­plex­ity of and was barely used. It was just given to us in the type sig­na­ture for , and the same proof would have worked with­out this as­sump­tion, so I am con­fused about why the prob­lem in­cludes this as­sump­tion.

• That link works, thanks!

• The link no longer works (I get “This pro­ject has not yet been moved into the new ver­sion of Over­leaf. You will need to log in and move it in or­der to con­tinue work­ing on it.“) Would you be will­ing to re-post it or move it so that it is visi­ble?

• Some other sources of ex­er­cises you might want to check out (that have solu­tions and that I have used at least partly):

• Mul­ti­ple choice quizzes (the ones re­lated to lin­ear alge­bra are de­ter­mi­nants, el­e­men­tary ma­tri­ces, in­ner product spaces, lin­ear alge­bra, lin­ear sys­tems, lin­ear trans­for­ma­tions, ma­tri­ces, and vec­tor spaces)

• Vipul Naik’s quizzes (dis­clo­sure: I am friends with Vipul and also do con­tract work for him)

Re­gard­ing Axler’s book (since it has been men­tioned in this thread): there are sev­eral “lev­els” of lin­ear alge­bra, and Axler’s book is at a higher level (em­pha­sis on ab­stract vec­tor spaces and co­or­di­nate-free ways of do­ing things) than the 3Blue1Brown videos (more con­crete, work­ing in ). Axler’s book also as­sumes that the reader has had ex­po­sure to the lower level ma­te­rial (e.g. he does not talk about row re­duc­tion and el­e­men­tary ma­tri­ces). So I’m not sure I would recom­mend it to some­one start­ing out try­ing to learn the ba­sics of lin­ear alge­bra.

Gra­tu­itous re­marks:

• I think differ­ent re­sources cov­er­ing ma­te­rial in a differ­ent or­der and us­ing differ­ent ter­minol­ogy is in some sense a fea­ture, not a bug, be­cause it al­lows one to look at the sub­ject from differ­ent per­spec­tives. For in­stance, the “done right” in Axler’s book comes from one such change in per­spec­tive.

• I find that learn­ing math­e­mat­ics well takes an un­in­tu­itively long time; it might be un­re­al­is­tic to ex­pect to learn the ma­te­rial well un­less one puts in a lot of effort.

• I think there is a case to be made for the im­por­tance of strug­gling in learn­ing (dis­clo­sure: I am the au­thor of the page).

• Is there (or will there be) a way to see a list of the lat­est posts, re­stricted to posts that are ques­tions? (I am won­der­ing about this both in the GraphQL API and in the site UI.)

• Can you say more about why ex­er­cise 17.6.3 is wrong?

If we define by then for dis­tinct we have

We also have since

In gen­eral, the deriva­tive is , which is con­tin­u­ous on .

• My solu­tion for #3:

Define by . We know that is con­tin­u­ous be­cause and the iden­tity map both are, and by the limit laws. Ap­ply­ing the in­ter­me­di­ate value the­o­rem (prob­lem #2) we see that there ex­ists such that . But this means , so we are done.

Coun­terex­am­ple for the open in­ter­val: con­sider defined by . First, we can ver­ify that if then , so in­deed maps to . To see that there is no fixed point, note that the only solu­tion to in is , which is not in . (We can also view this graph­i­cally by plot­ting both and and check­ing that they do not in­ter­sect in .)

• Here is my at­tempt, based on Hoagy’s proof.

Let be an in­te­ger. We are given that and . Now con­sider the points in the in­ter­val . By 1-D Sperner’s lemma, there are an odd num­ber of such that and (i.e. an odd num­ber of “seg­ments” that be­gin be­low zero and end up above zero). In par­tic­u­lar, is an even num­ber, so there must be at least one such num­ber . Choose the small­est and call this num­ber .

Now con­sider the se­quence . Since this se­quence takes val­ues in , it is bounded, and by the Bolzano–Weier­strass the­o­rem there must be some sub­se­quence that con­verges to some num­ber .

Con­sider the se­quences and . We have for each . By the limit laws, as . Since is con­tin­u­ous, we have and as . Thus and , show­ing that , as de­sired.

• 19 Nov 2018 1:29 UTC
11 points
AF
in reply to: Hoagy's comment

I’m hav­ing trou­ble un­der­stand­ing why we can’t just fix in your proof. Then at each iter­a­tion we bi­sect the in­ter­val, so we wouldn’t be us­ing the “full power” of the 1-D Sperner’s lemma (we would just be us­ing some­thing close to the base case).

Also if we are only given that is con­tin­u­ous, does it make sense to talk about the gra­di­ent?

• I had a similar thought while read­ing this post, but I’m not sure in­vok­ing causal­ity is nec­es­sary (hav­ing a di­rec­tion still seems nec­es­sary). Just in terms of propo­si­tional logic, I would ex­plain this post as fol­lows:

1. Ini­tially, one has the im­pli­ca­tion stored in one’s mind.

2. Some­one as­serts .

3. Now one’s mind (per­haps sub­con­sciously) does a modus po­nens, and ob­tains .

4. How­ever, is an un­de­sir­able be­lief, so one wants to deny it.

5. In­stead of re­ject­ing the im­pli­ca­tion , one adamantly de­nies .

The “buck­ets er­ror” is the im­pli­ca­tion , and “flinch­ing away” is the de­nial of . Flinch­ing away is about pro­tect­ing one’s episte­mol­ogy be­cause deny­ing is still bet­ter than ac­cept­ing . Of course, it would be best to re­ject the im­pli­ca­tion , but since one can’t do this (by as­sump­tion, one makes the buck­ets er­ror), it is prefer­able to “flinch away” from .

ETA (2019-02-01): It oc­curred to me that this is ba­si­cally the same thing as “one man’s modus po­nens is an­other man’s modus tol­lens” (see e.g. this post) but with some ex­tra emo­tional con­no­ta­tions.

• I was con­fused about this too, but now I think I have some idea of what’s go­ing on.

Nor­mally prob­a­bil­ity is defined for events, but ex­pected value is defined for ran­dom vari­ables, not events. What is hap­pen­ing in this post is that we are tak­ing the ex­pected value of events, by way of the con­di­tional ex­pected value of the ran­dom vari­able (con­di­tion­ing on the event). In sym­bols, if is some event in our sam­ple space, we are say­ing , where is some ran­dom vari­able (this ran­dom vari­able is sup­posed to be clear from the con­text, so it doesn’t ap­pear on the left hand side of the equa­tion).

Go­ing back to cousin_it’s lot­tery ex­am­ple, we can for­mal­ize this as fol­lows. The sam­ple space can be and the prob­a­bil­ity mea­sure is defined as and . The ran­dom vari­able rep­re­sents the lot­tery, and it is defined by and .

Now we can calcu­late. The ex­pected value of the lot­tery is:

The ex­pected value of win­ning is:

The “probu­til­ity” of win­ning is:

So in this case, the “probu­til­ity” of win­ning is the same as the ex­pected value of the lot­tery. How­ever, this is only the case be­cause the situ­a­tion is so sim­ple. In par­tic­u­lar, if was not equal to zero (while win­ning and los­ing re­mained ex­clu­sive events), then the two would have been differ­ent (the ex­pected value of the lot­tery would have changed while the “probu­til­ity” would have re­mained the same).

• I don’t see refer­ence num­ber 17 (“Per­sonal cor­re­spon­dence with Carl Shul­man”) used in the body of the post. What in­for­ma­tion from that refer­ence is used in the post?

• do we have any statis­tics about it?

For ses­sions and pageviews from Google An­a­lyt­ics, I wrote a post about it in April 2017. Since you men­tion scrap­ing, per­haps you mean some­thing like post and com­ment counts; if so, I’m not aware of any statis­tics about that.

Wei Dai has a web ser­vice to re­trieve all posts and com­ments of par­tic­u­lar users that I find use­ful (not sure if you will find it use­ful for gath­er­ing statis­tics, but I thought I would men­tion it just in case).

• Based on de­scrip­tions on the FHI web­site, it looks like Kyle Scott filled this role, from July 2015 to Septem­ber 2017.

Kyle brings over 5 years of op­er­a­tions ex­pe­rience to the Fu­ture of Hu­man­ity In­stu­tute. He keeps daily op­er­a­tions run­ning smoothly, and man­ages in­com­ing and out­go­ing re­quests for Prof. Nick Bostrom.

Strate­gi­cally, he works to im­prove the pro­cesses and ca­pac­ity of the office and free up the at­ten­tion and time of Prof. Nick Bostrom.

Kyle came to the Fu­ture of Hu­man­ity In­sti­tute from the Effec­tive Altru­ism move­ment, de­ter­min­ing that this job po­si­tion would be his most effec­tive con­tri­bu­tion to so­ciety. Learn more about Effec­tive Altru­ism here.

The page is still up but it doesn’t look like he holds the po­si­tion any­more.

He seems to be a pro­ject man­ager at BERI now:

Kyle man­ages var­i­ous pro­jects sup­port­ing BERI’s part­ner in­sti­tu­tions. He grad­u­ated Whit­man Col­lege with a B.A. in Philos­o­phy. He spent two years work­ing in ca­reer ser­vices and sub­se­quently moved to Oxford where he worked for 80,000 Hours, the Cen­tre for Effec­tive Altru­ism and most re­cently at the Fu­ture of Hu­man­ity In­sti­tute as Nick Bostrom’s Ex­ec­u­tive As­sis­tant.

On Novem­ber 13, 2017 FHI opened the po­si­tion for ap­pli­ca­tions.

ETA: Louis Franc­ini comes to the same con­clu­sion on Quora. (Con­text: I asked the ques­tion on Quora, figured out the an­swer, posted this com­ment, then Louis an­swered my ques­tion.)

• In some re­cent com­ments over at the Effec­tive Altru­ism Fo­rum you talk about anti-re­al­ism about con­scious­ness, say­ing in par­tic­u­lar “the case for ac­cept­ing anti-re­al­ism as the an­swer to the prob­lem of con­scious­ness seems pretty weak, at least as ex­plained by Brian”. I am won­der­ing if you could elab­o­rate more on this. Does the case for anti-re­al­ism about con­scious­ness seem weak be­cause of your gen­eral un­cer­tainty on ques­tions like this? Or is it more that you find the case for anti-re­al­ism speci­fi­cally weak, and you hold some con­trary po­si­tion?

I am es­pe­cially cu­ri­ous since I was un­der the im­pres­sion that many peo­ple on LessWrong hold es­sen­tially similar views.