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cousin_it30

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.

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by cousin_it
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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.

Your turn!

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 :-))

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What useless things did you understand recently?
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[-]gjm100

(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.

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[-]cousin_it20

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!

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[-]Thomas00

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.

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[-]TedSanders60

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

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[-][anonymous]00

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.

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[-][anonymous]00

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.

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[-]Daniel_Burfoot100

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.

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[-]cousin_it30

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.

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[-]Daniel_Burfoot20

There are lots of cacti that are mostly hairy/fuzzy instead of pointy.

In terms of air flow protection purchased vs biological effort expended, I'm not sure a leaf is better than a spike.

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[-]Elo00

The fur/spike can also be at specific widths to block certain wavelengths of light

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[-]TedSanders00

Wavelengths of visible light are around ~500 nm. Even infrared is on the order of micrometers. I don't think the spikes that we're imagining are micrometers apart.

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[-]Elo00

https://www.ncbi.nlm.nih.gov/pubmed/18507791

Looks like photon density is a thing. I was under the impression that I read somewhere that spikes can intercept (harsh desert) light and make it less harmful to the plant

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[-]Elo00

No, they can be micrometers wide and long enough to cause disruption of waves

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[-]bogus10

Please do not link to NSFW tentacle-porn without warning!

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[-]Kaj_Sotala00

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.

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[-]Elo00

hardly.

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[-]MaryCh20

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... :)

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[-]g_pepper00

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.

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[-]MaryCh10

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.)

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[-]g_pepper00

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?

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[-]MaryCh00

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.

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[-]g_pepper10

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?

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[-]MaryCh00

...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'...

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[-]g_pepper00

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?

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[-]MaryCh00

...I'll have to read it myself first. Probably, yes:)

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[-]MaryCh00

(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.

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[-]cousin_it00

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

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[-]MaryCh00

But they added so much text!

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

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[-]Elo10

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.

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[-]Viliam10

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.

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[-]Elo00

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.

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[-]Brillyant00

Okay. How do you do it?

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[-]Elo00

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...

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[-]Gram_Stone10

(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.)

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[-]cousin_it30

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.

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[-]MaryCh00

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...

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[-][anonymous]00

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

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[-][anonymous]160

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)...

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[-]CronoDAS50

Nicotine, too, is an insecticide.

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[-]AlexMennen00

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.

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[-]cousin_it00

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.

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[-]sen00

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.

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[-]cousin_it00

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.

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[-]sen00

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.

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[-]TezlaKoil30

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.

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[-]sen10

I see. Thanks for the explanation.

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