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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#301

Post by zarzavat »

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…

The real numbers are a man-made axiomatic system. They were developed to make analysis mathematically rigorous to the high standards of pure mathematicians.

The real numbers are popular outside of mathematical analysis because they provide a "kitchen sink" of every number you could possibly need.

The downside is that the reals include many numbers that you don't need. The number 0.12345678910111213... is a transcendental real number, but it is not very useful for anything. It is notoriously difficult to prove that a given number is transcendental, i.e. part of the uncountable part of the reals and not the countable algebraic subset. Which is ironic because the uncountable part is infinitely larger!

I'm not suggesting that physicists should drop their Hilbert spaces. Rather that a distinction should be drawn between mathematical model and physical reality.

-

As for whether spacetime is countably infinitely divisible:

Infinity is big. Infinitely small implies that if you used all the atoms in the universe to write in scientific notation to write 10^-999..., that space would be more divisible than that. In fact for whatever absurdly tiny number you could think of, perhaps 1/(TREE iterated TREE(3) times) spacetime would be finer than that.

I'll admit it's possible, but I have trouble believing it.

Re: Conterintuitive facts in mathematics, CS, and physics

#303

Post by trevortheblack »

> 21. In two dimensions, there are infinitely many regular polygons. In three dimensions, there are five Platonic solids. In four dimensions, there are six platonic polychora. In all higher dimensions than four, there are only ever three regular polytopes. (Maths 1001, Richard Elwes)

This implies that Platonic Solids are the 3D analogue of the 2D regular polygon. This is not the case.

The Platonic Solids are merely all of the convex solid regular polyhedra. When the caveats are removed there are (at least) 48 regular polyhedra: https://www.youtube.com/watch?v=_hjRvZYkAgA

Re: Conterintuitive facts in mathematics, CS, and physics

#304

Post by mjburgess »

zarzavat wrote: post #301
mjburgess wrote: post #291

Earlier quoted context omitted.

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…

The real numbers are a man-made axiomatic system. They were developed to make analysis mathematically rigorous to the high standards of pure mathematicians. The real numbers are popular outside of mathematical analysis because they provide a "kitchen sink" of every number you could possibly need. The downside is that the reals include many numbers that you don't need. The number 0.12345678910111213... is a transcende…

Well functions have properties in virtue of being defined over the reals, eg., sin(x) --

I don't see that these properties are incidental.

Yes they obtain in virtue of /any possible "dividing" discrete sequential process/ never terminating, eg., space being "infinitely divisible".

However I dont think this is as bizarre as it appears. The issue is congition is discrete, but the world continuous.

So we are always trying to project discrete sequential processes out onto the world in order to reason about it. Iterated zooming-in will, indeed, never terminate.

I dont see that as saying anything more than continuity produces infinities when approached discretely. So, don't approach it that way, if that bothers you.

Re: Conterintuitive facts in mathematics, CS, and physics

#305

Post by czzr »

jankovicsandras wrote: post #256
zarzavat wrote: post #192

Earlier quoted context omitted.

> Spacetime is real-valued There's really no evidence for this, as far as we know the real numbers are a pure mathematical invention and don't have any physicality. Even if you want to say that spacetime is dense (i.e. infinitely divisible), there's an infinite number of fields like that, the real numbers are just a convenient superset. There's no evidence that spacetime is dense either, and many practical ways in wh…

I have very little understanding of physics and math and might be wrong. But if we accept that the Planck length is the smallest possible length and the Planck time is the smallest possible time, then it seems logical that the universe is an integer lattice of these. ("Spacetime is not real-valued") https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_units#Planck_time

It’s not known if those are fundamental limits.

However the idea that space time is discrete is a reasonable hypothesis to test, we don’t currently have any ways to probe at those resolutions, though.

Re: Conterintuitive facts in mathematics, CS, and physics

#306

Post by Viliam1234 »

Ansil849 wrote: post #71

For some reason, the one I have the most trouble intuitively grasping is "Two 12 Inch Pizzas have less Pizza than one 18 inch pizza."

To make it simpler, let's assume that the shape of pizza is a square, i.e. we are talking about a 12×12 inch pizza and a 18×18 inch pizza. This increases the size of both pizzas by the same ratio, so it shouldn't change the answer to "are two small pizzas smaller than one large pizza?".

Now let's measure the size in a new unit which is 6 inch long. So the small pizza is 2×2 units, and the large pizza is 3×3 units.

twice 2×2 = twice 4 = 8

3×3 = 9

Re: Conterintuitive facts in mathematics, CS, and physics

#307

Post by UnFleshedOne »

lapetitejort wrote: post #25

My favorite paradox in physics: what happens when you spin a disk at relativistic speed? The circumference should contract, since it's parallel to the direction of motion, but the radius is perpendicular, and thus should not contract. https://en.wikipedia.org/wiki/Ehrenfest_paradox

I think if you rotate the disk at maximum speed ignoring all material material properties, all points on disk will approach speed of light and you will end up with "spin through" because layers won't be able to maintain same angular speed (needed to consider the thing a disk) with fixed linear speed...

Re: Conterintuitive facts in mathematics, CS, and physics

#308

Post by I_complete_me »

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…

Here's my attempt to understand what's going on:

100g strawberry total weight where the 100% is made up of 99% water and 1% solid matter. 100g strawberry total weight where the 100g is made up of 99g water and 1g solid matter. 99/100 = 0.99

50g strawberry total weight where the 100% is made up of 98% water and 2% solid matter. 50g strawberry total weight where the 50g is made up of 49g water and 1g solid matter. 49/50 = 0.98

Re: Conterintuitive facts in mathematics, CS, and physics

#310

Post by skrtskrt »

123anonanonanon wrote: post #273
yodelshady wrote: post #253

Earlier quoted context omitted.

A similar but not quite the same mechanic is fuel economy. Let's say you have two vehicles, both doing 10,000 miles per year. One gets 10 mpg and the other 50. Would you rather upgrade the 10 mpg vehicle to 13 mpg, or the 50 mpg to 100 mpg? Not only should you pick the former - you should pick the former even if you could upgrade the 50 mpg vehicle to run on nothing .

It is the assumption that the two cars always make the same number od miles indepedently od the cost that is unusual and unintuitive.

If you think of it as a family that keeps obstinately driving both cars the same amount despite massive cost differences its weird but you could think of it as a mixed fleet of delivery or work trucks that are all needed regardless, the question is how you manage or prioritize upgrading them.
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