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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#161

Post by nwallin »

> 17. At any given moment on the earth's surface, there exist 2 antipodal points (on exactly opposite sides of the earth) with the same temperature and barometric pressure: youtube.com/watch?v=cchIr1OXc8E

This is not necessarily true. They say a picture is worth 1000 words: https://media.deseretdigital.com/file/5894488349

Regardomg Gabriel's Horn and Banach-Tarski, the paradox is described as a trumpet, or a ball, made out of molecules, atoms, electrons, protons, neutrons, quarks-- but the mathematical proof is... not that. It's pretty common that intuition about objects made out of a finite number of parts breaks down when describing a construct with infinitely many parts.

Re: Conterintuitive facts in mathematics, CS, and physics

#162

Post by wodenokoto »

tshaddox wrote: post #136
wodenokoto wrote: post #32

Earlier quoted context omitted.

It’s also terribly phrased. > The vast majority of real numbers can't be described. But it is impossible to give a single example. If we accept the first sentence and assume the second refers to indescribable numbers, then isn’t it obvious that we have no examples of things we cannot describe? If the second sentence refers to real numbers in general I can give one example or two.

How could you give an example? The example you give would be a description of a number.

It’s only the vast majority that can’t be described.

So either it is claimed that it is counter intuitive that you can’t give an example of something you can’t describe. That is not counter intuitive- that is basically the definition of indescribable.

The other way the sentence can be read is that you can’t give an example of a real number. Of course you can. It’s only the vast majority of real numbers that can’t be described. There’s still infinitely many we can describe. 1 is a real number.

Re: Conterintuitive facts in mathematics, CS, and physics

#163

Post by rsj_hn »

Picard's Great Theorem would be one of my favorite counterintuitive facts:

As a holomorphic function approaches an essential singularity, it takes on every possible value (except at most one) infinitely often. An astonishing result.

https://en.wikipedia.org/wiki/Picard_theorem

And the nice thing is you can prove this with fairly elementary techniques (e.g. first year complex analysis is all you need).

Re: Conterintuitive facts in mathematics, CS, and physics

#164

Post by kergonath »

syncsynchalt wrote: post #31
lotsofpulp wrote: post #14

Earlier quoted context omitted.

> https://en.wikipedia.org/wiki/Potato_paradox The Wikipedia link above says: > Fred brings home 100 kg of potatoes, which (being purely mathematical potatoes) consist of 99% water. He then leaves them outside overnight so that they consist of 98% water. What is their new weight? The surprising answer is 50 kg. It annoys me when mass is used interchangeably with force (weight), so I went to the Wikipedia source, and…

> Wonder why the person that wrote the Wikipedia article changed it up when it is supposed to be a direct quote. They likely changed it from lb to kg because that would be more friendly to an international audience, without realizing that lb is a measure of weight and kg is a measure of mass. Therefore they didn't know to change "weight" to "mass".

I have been to the US quite regularly, and been living in the UK for a number of years, and I have never seen someone using pounds as a unit of force instead of a unit of mass meaning roughly 500g, give or take. Second meaning was about 1.20€.

FWIW, the pound is a proper unit of mass: https://en.m.wikipedia.org/wiki/Pound_(mass) .

Re: Conterintuitive facts in mathematics, CS, and physics

#165

Post by markc »

hodgesrm wrote: post #13

> 0% selected the right answer on this SAT question: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total? That's fun. I of course immediately selected 3 which means I could have a bright career in test preparation ahead of me.

>A rolls one trip around circle B

Without the diagram (which I didn't see until watching the video) this is ambiguous. In my visualization, the plane of the small circle (A) was perpendicular to the plane of the larger circle (B). (Think of circle B drawn on paper, while circle A is a coin on its edge)

With that interpretation of "rolls one trip around", 3 is indeed is the correct answer.

Re: Conterintuitive facts in mathematics, CS, and physics

#166

Post by silisili »

duxup wrote: post #103
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 like the money hall problem. I understand the answer, the answer makes sense to me. And yet but when I think of it how I think of it initially ... it still makes no sense.

Easiest for me if thinking only in fractions and percentages, and realizing that the dry mass is a constant.

The obvious example is 99g water, 1g dry mass. Knowing that 1g cannot change, what do we need water to be to equal 98%? 49g.

Re: Conterintuitive facts in mathematics, CS, and physics

#167

Post by Ar-Curunir »

charcircuit wrote: post #58

>It is possible to compute over encrypted data without access to the secret key I don't think this is counterintuitive for most people. The most basic encryption scheme that everyone knows is the Caesar cipher. It's easy to see that shifts of the cipher text will cause shifts in the plain text.

The interesting part is performing arbitrary computations over encrypted data.

Re: Conterintuitive facts in mathematics, CS, and physics

#168

Post by Ar-Curunir »

dfdz wrote: post #76
charcircuit wrote: post #58

>It is possible to compute over encrypted data without access to the secret key I don't think this is counterintuitive for most people. The most basic encryption scheme that everyone knows is the Caesar cipher. It's easy to see that shifts of the cipher text will cause shifts in the plain text.

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.

Re: Conterintuitive facts in mathematics, CS, and physics

#169

Post by dwohnitmok »

Kranar wrote: post #146
dwohnitmok wrote: post #143

Earlier quoted context omitted.

> In another sense the term "real number" refers to actual real numbers as we humans intend for them to exist but do not have a precise formal definition. Ah a Platonist in the flesh. Don't see many of you on HN. I don't think real numbers truly, objectively exist and think of them more as artifacts of human thought, but that's a deep deep rabbit hole. I'm kind of curious then, what do you believe the cardinality of…

I think I'm with you on that. Real numbers don't exist in an objective sense, I mean they exist in the same sense that an Escher painting of a hand drawing a hand exists, but they don't exist in the sense that a hand drawing a hand actually exists. When I was in high school I remember thinking that computers use the discrete to approximate the continuous and that it is the continuum that is real and the discrete that…

If you're sympathetic to the finitist cause, the idea that all mathematical objects are in fact definable is right up that alley. It's nice that this happens to line up acceptance of infinity, but finitism is basically entirely predicated on definability.

Re: Conterintuitive facts in mathematics, CS, and physics

#170

Post by Synaesthesia »

d_burfoot wrote: post #117

Surprising fact about the sun: it actually produces less heat per unit volume than a compost pile - this makes sense when you consider that fusion events are very rare. The reason the sun is so hot is that it has an enormous ratio of volume to surface area. Heat emission due to radiative cooling is proportional to the surface area, but heat creation due to fusion is proportional to the volume.

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.
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