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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#291

Post by mjburgess »

jerf wrote: post #284
mjburgess wrote: post #220

Earlier quoted context omitted.

Well this is a popular idea imported from comptuer science, but there's absolutely no evidence for it -- and plenty against. Eg., QM is only linear in infinitely-dimensional real-spaces, etc. Essentially of a physics uses real spaces indispensably. There is no evidence whatsoever that this is dispensible; other than the fever dreams of discrete mathematicians.

Remember, though: QM is wrong. Relativity also depends on continuous spaces, but it is also wrong. All the theories in physics that depend on continuous space are also wrong. By "wrong", I mean, we know they can't predict everything correctly. QM itself can't derive relativity. Relativity doesn't have QM in it, and break down at extremes like black holes. They're both very, very, very accurate in their domains, but p…

They're not "wrong" in tests of their real-valuedness though.

I'm somewhat confident there is an empirical test of real-valuedness in areas of physics which require infinite-valued spaces.

However, either way -- the positions of the other commenters was that *geometry* is somehow a dispensable approximation in physics!

This is an extremely radical claim with no evidence whatsoever. Rather some discrete mathematicians simply wish it were the case.

It is true that *maybe* (!) spacetime will turn out discrete, and likewise, Hilbert spaces, etc. -- and all continuous and infinite dimensional things will be discretised.

This however is a project without a single textbook. There is no such physics. There are no empirical predictions. There are no theories. This is a project within discrete mathematics.

Re: Conterintuitive facts in mathematics, CS, and physics

#292

Post by ParanoidMarvin »

ColinWright wrote: post #246
CJefferson wrote: post #182

Earlier quoted context omitted.

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.

What do you mean "spread out"? Aren't there the same amount of even numbers as natural numbers?, because they both are countable sets. https://en.wikipedia.org/wiki/Countable_set

Re: Conterintuitive facts in mathematics, CS, and physics

#293

Post by jerf »

mjburgess wrote: post #291
jerf wrote: post #284

Earlier quoted context omitted.

Remember, though: QM is wrong. Relativity also depends on continuous spaces, but it is also wrong. All the theories in physics that depend on continuous space are also wrong. By "wrong", I mean, we know they can't predict everything correctly. QM itself can't derive relativity. Relativity doesn't have QM in it, and break down at extremes like black holes. They're both very, very, very accurate in their domains, but p…

They're not "wrong" in tests of their real-valuedness though. I'm somewhat confident there is an empirical test of real-valuedness in areas of physics which require infinite-valued spaces. However, either way -- the positions of the other commenters was that *geometry* is somehow a dispensable approximation in physics! This is an extremely radical claim with no evidence whatsoever. Rather some discrete mathematicians…

"They're not "wrong" in tests of their real-valuedness though."

Yes, they are, or more accurate, they're not right enough for you to confidently assert the structure of space time at scales below the Planck scale. You are doing so on the basis of theories known to be broken at that scale. You are not entitled to use the theories that way.

Even the Planck scale being the limit is a mathematical number; I'm not sure we have concrete evidence of that size being the limit. I've seen a few proposed experiments that would measure at that resolution (such as certain predictions made by LQG about light traveling very long distances and different wavelengths traveling at very slightly different speeds) but I'm not aware of any that have panned out enough to have a solid result of any kind.

Re: Conterintuitive facts in mathematics, CS, and physics

#294

Post by raldi »

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.

Imagine Circle B is reduced to infinitesimal size, like rolling a quarter around a needle. It still makes one full revolution, even though the ratio of the circumferences is effectively infinite.

Re: Conterintuitive facts in mathematics, CS, and physics

#295

Post by raldi »

jcims wrote: post #216

I needed a half cup of something for a recipe and only found the 1/3 cup measure. Then it occurred to me that a third and a half (of a third) is equal to a half. So simple but somehow doesn't feel right.

3 measures = 1 cup

Now divide both sides of the equation by 2.

Re: Conterintuitive facts in mathematics, CS, and physics

#296

Post by ryathal »

dartharva wrote: post #264
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.

I'm confused, I also immediately came to 3 when I read this question. Is that wrong? What's the correct answer?

the answer is 4. The reason it's 4 is because distance traveled is relative to the center of the circle. A circle will travel it's circumference in a rotation, but the distance traveled isn't actually the circumference of the inner circle, because that isn't where the center of the circle is. It actually travels the sum of the two circles radii.

Re: Conterintuitive facts in mathematics, CS, and physics

#297

Post by TMWNN »

dartharva wrote: post #265
TMWNN wrote: post #141

Earlier quoted context omitted.

No. While two 12" pizzas have more pizza than one 18" pizza, it's possible for the two 12" pizzas to be a better value. If the total price of the two 12" pizzas is $7.20, for example, they are a better value than the 18" pizza once the price of the 18" pizza is greater than $8.10. More generally, the two 12" pizzas are a better value if the 18" pizza's price is higher than 112.5% of the two 12" pizzas' total price.

But that's almost never the case.

How do you know, unless you do the calculation?

Re: Conterintuitive facts in mathematics, CS, and physics

#298

Post by ColinWright »

ParanoidMarvin wrote: post #292
ColinWright wrote: post #246

Earlier quoted context omitted.

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.

What do you mean "spread out"? Aren't there the same amount of even numbers as natural numbers?, because they both are countable sets. https://en.wikipedia.org/wiki/Countable_set

>>> ... if you take the natural numbers, and split them into odd and even, you get two copies of the natural numbers ...

>> ... the "two copies of the natural numbers" is sorta fine, except that they're more "spread out" ...

> What do you mean "spread out"? Aren't there the same amount of even numbers as natural numbers?

Yes, there are the same number, but when you look at just the even numbers, they are each distance 2 from their neighbours, whereas the natural numbers are all distance 1 from their neighbours. So people are less surprised, because the even numbers are "spread out", they are less dense in any given area. To map the even numbers back onto the natural numbers you have to "compress" them.

But this is not the case with the Banach-Tarski Theorem. There is a set, A, and another set B, which is just A rotated around, and they are disjoint. So they have a union, C=AuB. But when you rotate C, you can get an exact copy of A. There's no squashing or spreading needed.

So we have A and B, with B=r(A), and A intersect B is empty. Then we have C=AuB. No problem here.

The challenge comes that there is a rotation, s, such that s(C)=A.

So even though C is made up of two copies of A, it's actually identical to A. So start with C, divide it into A and B, then rotate B back to become a copy of A, and then rotate each of those to become copies of C. So you start with C, do some "cutting" and rotations, and you get two copies of C.

Finally, when you take a few of these and put them together, you get a full sphere, so you can't say they have zero volume.

Does that make sense? Does that answer your question?

Does that help?

Re: Conterintuitive facts in mathematics, CS, and physics

#300

Post by dartharva »

TMWNN wrote: post #297
dartharva wrote: post #265

Earlier quoted context omitted.

But that's almost never the case.

How do you know, unless you do the calculation?

Don't know about you, but I have never seen any place that charges more for one large pizza than for two medium pizzas.
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