Conterintuitive facts in mathematics, CS, and physics • Hacker News Board

Re: Conterintuitive facts in mathematics, CS, and physics

#101

Post by p1necone » Wed, Oct 06, 2021, 2:07 AM UTC

p1necone wrote: post #91

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…

You're being overly pedantic (and I would argue actually incorrect). Kilograms and pounds are both referred to as "weight" in general conversation and nobody is going to be confused by this. Go to any supermarket in a country that uses the metric system, potatoes will be sold by the kilogram - it's the natural way to phrase this outside of America. In a physics context the definition of kilogram might be specifically…

Although the more I think about this the more I think the difference between technical and colloquial is actually that "weight" in colloquial use refers to mass, because force is not commonly relevant.

Re: Conterintuitive facts in mathematics, CS, and physics

#102

Post by anandoza » Wed, Oct 06, 2021, 2:12 AM UTC

umvi wrote: post #99

pessimizer wrote: post #61

Earlier quoted context omitted.

> > 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. Comes up a lot lately because of vaccine effectiveness e.g. 95% is twice as effective as 90%.

Wait, how is 95% effective twice as effective as 90%?

It halves your risk.

Re: Conterintuitive facts in mathematics, CS, and physics

#103

Post by duxup » Wed, Oct 06, 2021, 2:14 AM UTC

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.

Re: Conterintuitive facts in mathematics, CS, and physics

#104

Post by Kranar » Wed, Oct 06, 2021, 2:15 AM UTC

dwohnitmok wrote: post #12

29 is not correct as stated and falls prey to logical errors. Hamkins presents a formal take on it here: https://mathoverflow.net/questions/44102/is-the-analysis-as-... His conclusion (which I agree with) is > The claims made in both in your question and the Wikipedia page on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable…

This is very interesting but I think it relies heavily on interpretation.

For example there exists models of ZFC where all "real numbers" are definable, but said model does not include all the actual real numbers, it excludes any number that is not definable in ZFC. The issue is that the term "real number" is overloaded. In the formal sense it may refer only to numbers that are members of a model in which undefinable numbers are excluded. 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.

This actual set of real numbers does indeed contain members that are not definable in ZFC or any formal system, the issue is that there is no way to formalize this actual definition.

This is similar to what another poster mentioned about Skolem's paradox:

https://news.ycombinator.com/item?id=28767108

Re: Conterintuitive facts in mathematics, CS, and physics

#105

Post by londgine » Wed, Oct 06, 2021, 2:18 AM UTC

adenadel wrote: post #98

londgine wrote: post #86

> There are as many whole positive numbers as all fractions According to a specific non-colloquial definition of "as many".

Hm, I'm not so sure that just because the definition of a bijection is technical that it is not intuitive. I'll start an enumeration of the rationals 1 1/3 2 1/4 3 1/5 ... If you can prove that you can do this, is that really so non-colloquial? It is certainly what we mean by "as many" for all finite sets, so what is wrong with doing this for infinitely many sets?

If every whole positive number is a fraction, but not every fraction is a whole positive number, then colloquially, I wouldn't define them as having "as many" elements as each other. Now, if you want to say they have the same cardinality (and you define cardinality as existing a bijection), then I would agree fully.

Re: Conterintuitive facts in mathematics, CS, and physics

#106

Post by unholiness » Wed, Oct 06, 2021, 2:18 AM UTC

londgine wrote: post #86

> There are as many whole positive numbers as all fractions According to a specific non-colloquial definition of "as many".

My sister and I used to figure out who had more candy at halloween by lining up the pieces next to each other. The concept of bijection might be more intuitive than counting itself.

Re: Conterintuitive facts in mathematics, CS, and physics

#107

Post by analog31 » Wed, Oct 06, 2021, 2:20 AM UTC

>>> 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood

This is certainly counterintuitive, given that one-in-a-billion and 8 per month have different units of measure.

Re: Conterintuitive facts in mathematics, CS, and physics

#108

Post by bo1024 » Wed, Oct 06, 2021, 2:24 AM UTC

tgb wrote: post #83

bo1024 wrote: post #10

Great list! > 33. "...if you flip fair coins to generate n-dimensional vectors (heads => 1, tails => -1) then the probability they're linearly independent is at least 1-(1/2 + o(n))^n. I.e., they're very very likely independent! Counterintuitive facts about high dimensional geometry could get their own list. A side-1 cube in n dimensions has volume 1 of course, but a diameter-1 sphere inside it has volume approaching…

Only sort of true. It doesn't make sense to compare n dimensional volume to n+1 dimensional volumes, so the limit of the volume of an n-sphere isn't meaningful. The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising. In particular, it's equally valid and frankly nicer to define the unit n-sphere to be volume 1 rather than the unit cube. Do th…

> The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising.

This is why I start by recalling that the volume of the n-cube is always one, as the frame of reference. But I think people still find it surprising. Hard to tell, because...

> I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.

Hard to tell, because I don't find any of these facts surprising anymore -- would guess you're in the same boat!

Another good one is how you can fit exp(n) "almost-orthognal vectors" on the n-sphere.

Re: Conterintuitive facts in mathematics, CS, and physics

#109

Post by unholiness » Wed, Oct 06, 2021, 2:27 AM UTC

kristjansson wrote: post #40

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.

That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - i…

I don't think that argument holds water. Imagine rolling it on the inside of the circle – the center of the circle traces a path with only twice its radius, yet it still rolls 4 times around.

Re: Conterintuitive facts in mathematics, CS, and physics

#110

Post by machinelearning » Wed, Oct 06, 2021, 2:30 AM UTC

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.

The way to make sense of this is not to think about the water weight but the solid matter. By changing the proportion of water from 99% to 98%, you're also doubling the proportion of solid mass from 1% to 2%.