# What useless things did you understand recently?

Please re­ply in the com­ments with things you un­der­stood re­cently. The only con­di­tion is that they have to be use­less in your daily life. For ex­am­ple, “I found this idea that defeats pro­cras­ti­na­tion” doesn’t count, be­cause it sounds use­ful and you might be de­luded about its truth. Whereas “I figured out how con­struc­tion cranes are con­structed” qual­ifies, be­cause you aren’t likely to use it and it will stay true to­mor­row.

I’ll start. To­day I un­der­stood how Heyt­ing alge­bras work as a model for in­tu­ition­is­tic logic. The main idea is that you rep­re­sent sen­tences as shapes. So you might have two sen­tences A and B shown as two cir­cles, then “A and B” is their in­ter­sec­tion, “A or B” is their union, etc. But “A im­plies B” doesn’t mean one cir­cle lies in­side the other, as you might think! In­stead it’s a shape too, con­sist­ing of all points that lie out­side A or in­side B (or both). There were some other de­tails about closed and open sets, but these didn’t cause a prob­lem for me, while “A im­plies B” made me stum­ble for some rea­son. I prob­a­bly won’t use Heyt­ing alge­bras for any­thing ever, but it was pretty fun to figure out.

PS: please don’t feel pres­sured to post some­thing su­per ad­vanced. It’s re­ally, hon­estly okay to post ba­sic things, like why a stream of tap wa­ter nar­rows as it falls, or why the sky is blue (though I don’t claim to un­der­stand that one :-))

• (This isn’t a thing I learned re­cently, it’s an an­swer to some­thing cousin_it said he didn’t un­der­stand. Though I would not be sur­prised [1] if in fact he already un­der­stands all this and what he’s not-un­der­stand­ing is some deeper more de­tailed thing that I don’t un­der­stand ei­ther.)

[1] Merely on the gen­eral grounds that cousin_it strikes me as a clever per­son who knows many things.

The sky is blue for the same rea­son as the sun is yel­low. The ac­tual light from the sun is white (a bet­ter way to say this: our idea of what counts as white is de­rived from the spec­trum of the sun), and as it passes through the at­mo­sphere some of it gets scat­tered in other di­rec­tions. So if you look at the sky but not di­rectly at the sun, you are nec­es­sar­ily see­ing scat­tered light; and if you look di­rectly at the sun, you are see­ing the sun’s light with the scat­tered light re­moved.

Shorter-wave­length light scat­ters more eas­ily than longer-wave­length light. You can do the ac­tual calcu­la­tions and find ex­actly how much more eas­ily—per­haps these de­tails are what cousin_it is say­ing he doesn’t un­der­stand—but qual­i­ta­tively it’s ob­vi­ous enough: a pho­ton gets scat­tered when it ex­cites one of the atom’s elec­trons, which af­ter a while re­turns to a lower-en­ergy state and re-ra­di­ates, and higher-en­ergy pho­tons do that more read­ily.

When the sun­light’s path through the at­mo­sphere to you is longer, at sun­rise or sun­set, more scat­ter­ing hap­pens, which is why the sun looks red­der then. More of the shorter-wave­length light is go­ing el­se­where.

• That’s the best ex­pla­na­tion of Rayleigh scat­ter­ing I’ve ever seen, thank you!

I guess the in­ter­est­ing ques­tions be­gin when you try to con­vert the ex­pla­na­tion to a pre­dic­tion, like “Mommy, was the sky always blue?” or “will it be blue in the fu­ture?” That re­quires know­ing a lot more things then just Rayleigh scat­ter­ing. My knowl­edge is just enough to tell me that I don’t have a clue. For ex­am­ple, even with just Rayleigh scat­ter­ing (ig­nor­ing all other fac­tors) the sky could also be vi­o­let (even shorter wave­length, right?) or or­ange (if the at­mo­sphere was thicker and most blue light got scat­tered into space). Then you get into things like the spec­trum of the Sun, the com­po­si­tion of the at­mo­sphere, the way wa­ter washes out dust, the fac­tors that pre­vent los­ing wa­ter to space, the role of the bio­sphere, etc. To an­swer these in­no­cent ques­tions it seems like you need to know liter­ally all sci­ences!

• As a mat­ter of fact, the ni­tro­gen makes sky blue, but the oxy­gen makes it green. Had been more oxy­gen than ni­tro­gen in our at­mo­sphere, they sky would have been green, all else equal.

You can also say, that this blue color is the color of 20000 K, on the Wein’s di­a­gram. Which is the tem­per­a­ture (ki­netic en­ergy) of the ni­tro­gen atom hit by an UV pho­ton of the ap­pro­pri­ate en­ergy to be ab­sorbed.

And our planet in fact loses wa­ter by the hy­dro­gen es­cap­ing. 50 kilo­gram per sec­ond.

Well, this I think I know with­out Googling, You may re­fine this by—Googling it.

• Is this ac­tu­ally true? Do you have a source? I have tried Googling for it.

My un­der­stand­ing is that the sky’s blue color was caused by Rayleigh scat­ter­ing. This scat­ter­ing is higher for shorter wave­lengths. There’s no broad peak in scat­ter­ing as­so­ci­ated with ni­tro­gen ab­sorp­tion lines (which I imag­ine would be very nar­row­band, rather than broad­band).

Wikipe­dia’s ar­ti­cle on Rayleigh scat­ting men­tions oxy­gen twice but makes no refer­ence to your the­ory.

https://​​en.wikipe­dia.org/​​wiki/​​Rayleigh_scattering

• That’s the best ex­pla­na­tion of Rayleigh scat­ter­ing I’ve seen, thank you!

The re­ally fun ques­tions be­gin when you try to con­vert ex­pla­na­tion to pre­dic­tion, like “was the sky always blue?” or “what color will it be in the fu­ture?” To an­swer these, you pretty much need to know all the sci­ences, from as­tro­physics to evolu­tion­ary his­tory. My ed­u­ca­tion is only enough to tell me that I don’t have a clue. Just look at other planets, they all have differ­ently col­ored skies due to differ­ent fac­tors, which could also af­fect Earth at other points in time.

• That’s the best ex­pla­na­tion of Rayleigh scat­ter­ing I’ve seen in a while, thank you!

The in­ter­est­ing ques­tions start at the next layer. For ex­am­ple, the same Rayleigh scat­ter­ing could also lead to a vi­o­let or pur­ple sky de­pend­ing on the com­po­si­tion of sun­light. Or it could lead to an or­ange sky if the at­mo­sphere was thicker and most of the blue got scat­tered into space. Or it could be all sorts of other col­ors due to at­mo­spheric gases or dust. At long timescales, all these fac­tors can change a lot. So if I try to con­vert the sim­ple ex­pla­na­tion into a pre­dic­tion—“Daddy, was the sky always blue? What color will it be in a billion years?”—my mind goes ev­ery­where at once.

• Sorry, I thought it was clini­cal enough of an illus­tra­tion not to need a warn­ing. It was in the mid­dle of a BBC News ar­ti­cle, af­ter all.

• For a long time it was odd to me that cacti have lots of spikes and big thorns. I sup­posed that the goal was to ward off big ru­mi­nants like cows, but that doesn’t re­ally make much sense, since the desert isn’t re­ally overflow­ing with big an­i­mals that eat a lot of plants.

It turns out that pro­tec­tion from preda­tors is only a sec­ondary goal. The main goal is pro­tec­tion from the en­vi­ron­ment. The spikes cap­ture and slow the air mov­ing around the plant, to pre­serve mois­ture and pro­tect against the heat.

• Hang on, I’m not sure I buy it. Why are they so thin, hard and sharp then? Some kind of fuzz or flat leaves would work bet­ter.

• About differ­ent trans­la­tions of the same thing (in the spe­cific case I have in mind, Lord of the Rings + Hob­bit trans­lated into Rus­sian and Ukrainian). Some of them go af­ter the in­tent & the pic­ture, and some of them go af­ter the di­rect mean­ing (not the Google-trans­late style, but the “say­ing as much us­ing as similar means as is har­mo­nious within the lan­guage” one). Thus, Владыка Элронд is not ex­actly лорд (lord) Elrond—“lord” (and лорд) are much more com­mon and for­mal than the strange, old владыка - but the lord­ship it does con­vey ex­ists out­side of time and the gen­eral struc­ture of con­tem­po­rary so­ciety.

And so we have lots of Tolk­ien, and you poor na­tive speak­ers don’t… :)

• The same holds for trans­la­tions from Rus­sian to English. For ex­am­ple, Con­stance Gar­nett’s trans­la­tion of The Brothers Kara­ma­zov is quite differ­ent from the Pe­vear/​Volokhon­sky trans­la­tion. It seemed to me that Dos­toyevsky’s dark hu­mor was bet­ter cap­tured in the Pe­vear/​Volokhon­sky trans­la­tion. The Pe­vear/​Volokhon­sky trans­la­tion was quite en­joy­able, IMO.

• Damn. Never thought I’d want to read D in English :) he’s quite formidable in the origi­nal.

(It’s a pity that I can’t find any­thing by Yuri Tynyanov in English; The death of am­bas­sador plenipo­ten­tiary (Смерть Вазир-Мухтара) with its odd and won­der­ful word us­age, styled some­what to Pushkin’s times it de­scribes, would be a gem… I re­ally thought it ex­isted in trans­la­tion.)

• Ama­zon lists a vol­ume con­tain­ing English trans­la­tions of two nov­el­las by Tynyanov—Lieu­tenant Kije and Young Vi­tushish­nokov. Are ei­ther of those good choices as in­tro­duc­tions to Tynyanov?

• Oh, so I missed it! I think any of these is ok. Just re­mem­ber to drop it in time, some peo­ple find him a bit heavy.

• I just or­dered the vol­ume con­tain­ing Lieu­tenant Kije and Young Vi­tushish­nokov. I’m in the mid­dle of a cou­ple of things already though, so I may not get started on Tynyanov right away. I’m look­ing for­ward to it though—thanks for the recom­men­da­tion!

Also—you are work­ing on a trans­la­tion, aren’t you? How’s that go­ing? And, is it a trans­la­tion into English?

• ...and Leonid An­dreyev’s The Black Masks. (I’m on the phone, so have not Googled it, sorry.) ‘The God our Lord placed the sword in my hands, and with death I pun­ished the mad Lorenzo, and yet he was a knight’...

• There’s quite a lot of An­dreyev’s work available in English. Some trans­la­tions are ap­par­ently in the pub­lic do­main as they are available for free on Ama­zon in ebook form. I don’t re­ally en­joy read­ing plays as a rule (The Black Masks is a play, I be­lieve), so I down­loaded the nov­ella The Seven Who Were Hanged. It’ll be a while be­fore I get around to read­ing it, as my read­ing list is fairly long (and get­ting longer, thanks to your great sug­ges­tions!).

Is The Seven Who Were Hanged a good in­tro­duc­tion to An­dreyev?

• (Was con­sid­er­ing two an­swers: “I am” and “It is”. “It is” seems to be fit­ting:)

It’s go­ing very slowly, be­cause I hate pro­pa­ganda, my offi­cial job is in the state of “wtf is the cen­tral office THINKING?!”, & akra­sia. Shouldn’t for­get akra­sia.

• Yeah. The Rus­sian trans­la­tion of LOTR by Mu­raviev and Kis­ti­akovsky is amaz­ing, eas­ily on par with the origi­nal, es­pe­cially the po­ems.

• But they added so much text!

But the po­ems, yeah. It was some­times im­pos­si­ble to imag­ine this was a trans­la­tion, af­ter all.

• I gained an un­der­stand­ing of how to in­ter­pret zen koans. It’s kinda fun and yields a very calm state of mind when play­ing with them in your head.

It might be use­ful but I didn’t re­ally go seek­ing this, I mostly stum­bled across it.

• I sus­pect that many koans in­clude puns or cul­tural refer­ences that stop work­ing af­ter trans­la­tion. Un­for­tu­nately, a “puz­zle you can only un­der­stand if you are a bud­dha” and a “puz­zle you can only un­der­stand if you are a bud­dha and fluent in Chi­nese and fa­mil­iar with cen­turies old cul­tural refer­ences” may seem quite similar from the out­side.

• There are a few like that but with some duct tape and a vague un­der­stand­ing of Chi­nese and monastery cul­ture you can get a glimpse of them. Also see­ing mul­ti­ple trans­la­tions can give you some clues.

• Okay. How do you do it?

• Will pub­lish in a few days, af­ter I pre­sent at my lo­cal dojo . No guaran­tee it works be­cause my de­scrip­tion is pos­si­bly rough and in per­son I can get feed­back on your un­der­stand­ing of the de­scrip­tion and say, okay try this ex­pla­na­tion in­stead…

• (Not very fa­mil­iar with math.)

The Heyt­ing-alge­braic defi­ni­tion of im­pli­ca­tion makes in­tu­itive sense to me, or at least af­ter you state your con­fu­sion. ‘One cir­cle lies in­side the other’ is like say­ing A is a sub­set of B, which is a state­ment that de­scribes a re­la­tion be­tween two sets, and not a state­ment that de­scribes a set, so we shouldn’t ex­pect that that men­tal image would cor­re­spond to a set. Fur­ther­more, the defi­ni­tion of im­pli­ca­tion you’ve given is very similar to the ma­te­rial im­pli­ca­tion rule; that we may sub­sti­tute ‘P im­plies Q’ with ‘not-P or Q’.

Also, I have per­son­ally been en­joy­ing your re­cent posts with few pre­req­ui­sites. (Seems to be a thing.)

• Thanks! I’m not an amaz­ing writer like Eliezer, but I en­joy be­ing on LW and I want other peo­ple to en­joy it as well.

The defi­ni­tion of im­pli­ca­tion is ac­tu­ally a bit more com­plex, you need to take the largest open sub­set of “not-P or Q”. Similarly, nega­tion isn’t just com­ple­ment, but the largest open sub­set of the com­ple­ment. That’s what makes the in­tu­ition­is­tic stuff work, oth­er­wise you get clas­si­cal logic as Alex said. But topol­ogy isn’t ev­ery­one’s cup of tea, so I left it out.

• That I were blessed with a won­der­ful favourite teacher and a crazy, but won­der­ful su­per­vi­sor in col­lege.

Be­cause when some­one of mine dies—be it a rel­a­tive or a dog—or gets di­ag­nosed with in­cur­able dis­ease, I go to ei­ther of them and we drink tea or just have a walk to­gether, and talk of ir­rele­vant things.

Only got it af­ter the fourth time, though...

• Why so many of our spices are toxic to other an­i­mals and in­sects.

• Why?

• Most of the in­ter­est­ing-tast­ing or psy­choac­tive chem­i­cals that plants make are there to ward off be­ing eaten or in­fected. Caf­feine and mint oil are among the plant in­sec­ti­cides, all sorts of other things are tox­ins to ver­te­brates. By virtue of be­ing megafauna we can tol­er­ate amounts of toxic stuff that will kill smaller or­ganisms by mix­ing them with other foods, and our par­tic­u­lar bio­chem­istry hap­pens to be par­tic­u­larly strong against some (and weak against oth­ers, just try to eat hem­lock). Stuff we can tol­er­ate but still has effects on us (caf­feine, cap­saicin) can be in­ter­est­ingly psy­choac­tive, stuff that doesn’t hurt us can be in­ter­est­ing to taste (mint, cin­na­mon, gar­lic), and there’s in­ter­est­ing cor­re­la­tions be­tween spice use and par­a­site load in food (that could be con­founded six ways to mars)...

• Ni­co­tine, too, is an in­sec­ti­cide.

• So you might have two sen­tences A and B shown as two cir­cles, then “A and B” is their in­ter­sec­tion, “A or B” is their union, etc. But “A im­plies B” doesn’t mean one cir­cle lies in­side the other, as you might think! In­stead it’s a shape too, con­sist­ing of all points that lie in­side B or out­side A (or both).

There’s noth­ing in­tu­ition­is­tic about this. You can do ex­actly the same thing with clas­si­cal logic, if you just for­get about the topolog­i­cal “other de­tails” that you al­luded to.

• Yeah I know. I’m only look­ing at it now be­cause in­tu­ition­is­tic logic can’t be re­duced to finite truth ta­bles like clas­si­cal logic, it re­ally needs these pic­tures. That’s kind of weird in it­self, but hard to ex­plain in a short post.

• I guess to­day I’m learn­ing about Heyt­ing alge­bras too.

I don’t think that cir­cle method works. “Not Not A” isn’t nec­es­sar­ily the same thing as “A” in a Heyt­ing alge­bra, though your method sug­gests that they are the same. You can try to fix this by adding or re­mov­ing the cir­cle bor­ders through nega­tion op­er­a­tions, but even that yields in­con­sis­tent re­sults. For ex­am­ple, if you add the bor­der on each nega­tion, “A or Not A” yields 1 un­der your method, though it should not in a Heyt­ing alge­bra. If you re­move the bor­der on each nega­tion “A is a sub­set of Not Not A” is false un­der your method, though it should yield true.

I think it’s eas­ier to think of Heyt­ing alge­bra in terms of func­tions and ar­gu­ments. “A im­plies B” is a func­tion that takes an ar­gu­ment of type A and pro­duces an ar­gu­ment of type B. 0 is null. “A and B” is the set of ar­gu­ments a,b where a is of type A and b is of type B. If null is in the ar­gu­ment list, then the whole ar­gu­ment list be­comes null. “Not A” is a func­tion that takes an ar­gu­ment of type A and pro­duces 0. “Not Not A” can be thought of in two ways: (1) it takes an ar­gu­ment of type Not A and pro­duces 0, or (2) it takes an ar­gu­ment of type [a func­tion that takes an ar­gu­ment of type A and pro­duces 0] and pro­duces 0.

If “(A and B and C and …) → 0” then “A → (B → (C → … → 0))”. If you’ve worked with pro­gram­ming lan­guages where lambda func­tions are com­mon, it’s like tak­ing a func­tion of 2 ar­gu­ments and turn­ing it into a func­tion of 1 ar­gu­ment by fix­ing one of the ar­gu­ments.

I don’t see it on the Wikipe­dia page, but I’d guess that “A or B” means “(Not B im­plies A) and (Not A im­plies B)”.

If you don’t already, I highly recom­mend study­ing cat­e­gory the­ory. Most ab­stract math­e­mat­i­cal con­cepts have sim­ple defi­ni­tions in cat­e­gory the­ory. The cat­e­gory the­o­retic defi­ni­tion of Heyt­ing alge­bras on Wikipe­dia con­sists of 6 lines, and it’s enough to un­der­stand all of the above ex­cept the Or re­la­tion.

• Yeah, I men­tioned the topol­ogy com­pli­ca­tions.

If you re­move the bor­der on each nega­tion “A is a sub­set of Not Not A” is false un­der your method, though it should yield true.

How so? I thought re­mov­ing the bor­der on each nega­tion was the right way. (Also you need to start out with no bor­der, ba­si­cally you should have open sets at each step.)

Lambda calcu­lus is in­deed a nice way to un­der­stand in­tu­ition­ism, that’s how I imag­ined it since for­ever :-) Also the con­nec­tion be­tween Peirce’s law and call/​cc is nice. And the way it pre­vents con­fluence is also re­ally nice. This stack­overflow ques­tion has prob­a­bly the best ex­pla­na­tion.

• How so? I thought re­mov­ing the bor­der on each nega­tion was the right way.

I gave an ex­am­ple of where re­mov­ing the bor­der gives the wrong re­sult. Are you ask­ing why “A is a sub­set of Not Not A” is true in a Heyt­ing alge­bra? I think the proof goes like this:

• (1) (a and not(a)) = 0

• (2) By #1, (a and not(a)) is a sub­set of 0

• (3) For all c,x,b, ((c and x) is a sub­set of b) = (c is a sub­set of (x im­plies b))

• (4) By #2 and #4, a is a sub­set of (not(a) im­plies 0)

• (5) For all c, not(c) = (c im­plies 0)

• (6) By #4 and #5, a is a sub­set of not(not(a))

Maybe your method is work­able when you in­ter­pret a Heyt­ing sub­set to be a topolog­i­cal su­per­set? Then 1 is the ini­tial (empty) set and 0 is the ter­mi­nal set. That doesn’t work with in­ter­sec­tions though. “A and Not A” must yield 0, but the in­ter­sec­tion of two non-ter­mi­nal sets can­not pos­si­bly yield a ter­mi­nal set. The union can though, so I guess that means you’d have to rep­re­sent And with a union. That still doesn’t work though be­cause “Not A and Not Not A” must yield 0 in a Heyt­ing alge­bra, but it’s miss­ing the bor­der of A in the topolog­i­cal method, so it again isn’t ter­mi­nal.

I don’t see how the topolog­i­cal method is work­able for this.

• In a topolog­i­cal space, defining

1. X ∨ Y as X ∪ Y

2. X ∧ Y as X ∩ Y

3. X → Y as Int( X^c ∪ Y )

4. ¬X as Int( X^c )

does yield a Heyt­ing alge­bra. This means that the un­der­stand­ing (but not the ex­pla­na­tion) of /​u/​cousin_it checks out: re­mov­ing the bor­der on each nega­tion is the “right way”.

No­tice that un­der this in­ter­pre­ta­tion X is always a sub­set of ¬¬X.:

1. Int(X^c) is a sub­set of X^c; by defi­ni­tion of Int(-).

2. Int(X^c)^c is a su­per­set of X^c^c = X; since tak­ing com­ple­ments re­verses con­tain­ment.

3. Int( Int(X^c)^c ) is a su­per­set of Int(X) = X; since Int(-) pre­serves con­tain­ment.

But Int( Int(X^c)^c ) is just ¬¬X. So X is always a sub­set of ¬¬X.

How­ever, in many cases ¬¬X is not a sub­set of X. For ex­am­ple, take the Eu­clidean plane with the usual topol­ogy, and let X be the plane with one point re­moved. Then ¬X = Int( X^c ) = ∅ is empty, so ¬¬X is the whole plane. But the whole plane is ob­vi­ously not a sub­set of the plane with one point re­moved.