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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#241

Post by simonh »

hamilyon2 wrote: post #186
Synaesthesia wrote: post #170

Earlier quoted context omitted.

The density of the sun is very low, like much lower than the atmosphere (throughout most of it, anyway,) But yes it has an enormous volume.

Given vastness of sun, it's age and existence of extremophiles, I would be surprised if there is no life there.

Life requires a stable environment in which persistent structures can be maintained over long periods of time. High temperatures are inimical to that. This is why we live in a part of the universe where temperatures rarely rise much above a few hundred degrees Kelvin. Above that most complex chemical structures break down.

Re: Conterintuitive facts in mathematics, CS, and physics

#242

Post by chriswarbo »

GPerson wrote: post #48

I didn’t see this one listed, and thought it was pretty cool when I studied it in a course a few years ago: https://en.m.wikipedia.org/wiki/Skolem's_paradox “ Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says…

I like to think of this as a game, with one player choosing the axioms and the other choosing a model. If the first player picks a (countable) set of axioms, the second player can always respond with a countable model. Likewise, if the second player picks a countable model, the first player can always extend the axioms in a consistent way, to rule out that model. This can alternate back-and-forth forever.

Uncountability is hence a 'leaky abstraction': something we want to investigate and study in general terms, even though particular occurances might have some loophole/edge-case.

I think about infinity and infinitesimals in a similar way, like iterative processes (e.g. the natural numbers arise from a process that increments; calculus arises from iteratively shrinking 'dx', e.g. by halving; etc.). Combining/interleaving such processes is tricky, so it's often more convenient to take their limits individually and manipulate those as objects; that's justified if those manipulations could potentially be implemented by some interleaving, but can otherwise result in paradoxes (e.g. Thomson's lamp)

Re: Conterintuitive facts in mathematics, CS, and physics

#243

Post by chriswarbo »

Ar-Curunir wrote: post #168
dfdz wrote: post #76

Earlier quoted context omitted.

I agree, I really don’t like this one either. There are many things in math that are counterintuitive, but the idea of a homomorphism is not one of them in my opinion. Once someone explains the idea, and provides a few examples it is very natural. I also don’t like the text explaining zero knowledge proof. It needs the phrase “practically speaking” somewhere or “for practical purposes” since it’s not true in a strict…

What do you mean by the line about zkps? We have perfectly-hiding proofs that reveal no information about the secret information, no matter how powerful the adversary is.

Yes, but they're not proofs in the mathematical sense, since there's always an (exponentially-shrinking) chance that the answers were only correct due to coincidence.

Re: Conterintuitive facts in mathematics, CS, and physics

#244

Post by Viliam1234 »

H8crilA wrote: post #81
GPerson wrote: post #48

I didn’t see this one listed, and thought it was pretty cool when I studied it in a course a few years ago: https://en.m.wikipedia.org/wiki/Skolem's_paradox “ Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says…

Just to make this more concrete: 1. There is a countable model of real numbers. 2. There even is a countable model of the entire set theory.

> There is a countable model of real numbers.

What exactly happens when you try to apply Cantor's diagonal argument to this model?

I guess that at some step, you get an answer like "outside of the model, yes this exists, but inside the model the answer is no", but I would like to see it precisely, how exactly the in-model reasoning diverges from the outside-model reasoning.

Re: Conterintuitive facts in mathematics, CS, and physics

#245

Post by mmmmmbop »

implements wrote: post #202
rawling wrote: post #190

Earlier quoted context omitted.

> Both the stellar day and the sidereal day are shorter than the mean solar day by about 3 minutes 56 seconds. This is a result of the Earth turning 1 additional rotation, relative to the celestial reference frame, as it orbits the Sun (so 366.25 rotations/y). https://en.wikipedia.org/wiki/Earth's_rotation

So we would need 367 unique date identifiers … but we’ve only got 366 (Feb 29th being the non-annual one). I get I may be being unintelligent, but isn’t the author confusing the rolling coin paradox with an obscure astronomical reference system and coming up with a ‘mistakenly technically correct’ result that doesn’t match experienced reality?

Imagine you're standing on a set point on the surface on the outer coin, e.g. the one touching the inner coin. As the outer coin rotates around the inner coin, your experienced reality will be that you see n-1 rotations.

In the example of the outer coin having 1/3 the radius of the inner coin, as the outer coin rolls around the inner coin 4 times, you would actually only touch the inner coin 3 times.

Re: Conterintuitive facts in mathematics, CS, and physics

#246

Post by ColinWright »

CJefferson wrote: post #182
ncmncm wrote: post #150

Number 19 is a clinker. Banach-Tarski applies only to objects in real-number space, but there are no such objects. For that to work, objects have to be infinitely divisible, but all of our objects are made out of atoms. Real-numbered space is a good enough approximation to our experience that we hardly ever encounter a model failure like this one.

I agree. Also, until you get super technical, it isn't really any different to "if you take the natural numbers, and split them into odd and even, you get two copies of the natural numbers".

I disagree ... the "two copies of the natural numbers" is sorta fine, except that they're more "spread out" so it's not at all surprising.

The surprising thing about BT is that the "pieces" are "moved around" ... there's no expansion or contraction.

Yes, the natural number thing helps to understand that simply counting things doesn't help, but the "rigid motion" aspect of BT takes it further.

Re: Conterintuitive facts in mathematics, CS, and physics

#247

Post by st_goliath »

> 19. Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

While you're at it, you can completely turn that sphere inside out without creating any holes or creases[1].

[1] https://www.youtube.com/watch?v=wO61D9x6lNY&t=92s

Re: Conterintuitive facts in mathematics, CS, and physics

#248

Post by dfdz »

chriswarbo wrote: post #243
Ar-Curunir wrote: post #168

Earlier quoted context omitted.

What do you mean by the line about zkps? We have perfectly-hiding proofs that reveal no information about the secret information, no matter how powerful the adversary is.

Yes, but they're not proofs in the mathematical sense, since there's always an (exponentially-shrinking) chance that the answers were only correct due to coincidence.

Exactly, practically it makes no different that the method could be fooled with a very tiny probability, but when making these counterintuitive statements I think it is important to be precise.

Ideally the reader should fully understand the statement and still feel amazed, rather than doubting the statement for a valid reason: perfect zero knowledge proof systems (which do not fail sometimes) are impossible and a reader would be right to think so

Re: Conterintuitive facts in mathematics, CS, and physics

#249

Post by FabHK »

Aerroon wrote: post #200
6gvONxR4sf7o wrote: post #11

> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood This one, on the other…

#16 is something video games taught me, particularly Path of Exile. In PoE resistance are a flat multiplier to incoming damage. Eg monster does 100 damage per attack and you have 60% resistance then you take 40 damage. The interesting thing is that the higher your resistances the more effective each additional percentage point of resistance is. Let's say a monster does 100 damage per attack. If you have 0% resistance…

Interestingly, the same logic applies to vaccination rates. Going from 0% to 5% vaccination has no impact on the course of the pandemic (except for those few vaccinated people, of course). Going from 75% to 80% has a much larger impact, and could stop the pandemic in its track (depending on R_0, and many other real-world complications of course).

(And the reason is just the same: what matters is the remainder.)

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