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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#271

Post by hikarudo »

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.

Feynman talks about exactly this in his book "Surely you're joking, Mr. Feynman":

"Then I got an idea. I challenged them: "I bet there isn't a single theorem that you can tell me - what the assumptions are and what the theorem is in terms I can understand - where I can't tell you right away whether it's true or false."

It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?"

"No holes."

"Impossible!

"Ha! Everybody gather around! It's So-and-so's theorem of immeasurable measure!"

Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms."

"But we have the condition of continuity: We can keep on cutting!"

"No, you said an orange, so I assumed that you meant a real orange."

So I always won. If I guessed it right, great. If I guessed it wrong, there was always something I could find in their simplification that they left out.

Re: Conterintuitive facts in mathematics, CS, and physics

#273

Post by 123anonanonanon »

yodelshady wrote: post #253
Aerroon wrote: post #200

Earlier quoted context omitted.

#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…

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.

Re: Conterintuitive facts in mathematics, CS, and physics

#274

Post by [deleted] »

reidjs wrote: post #39
whiterock wrote: post #2

very cool collection - truly something to pick from at a party. should have much more upvotes imho.

I have actually told this one at a party (well, not exactly a rager) "Two 12 Inch Pizzas have less Pizza than one 18 inch pizza." and we did the math to prove it to ourselves. Blew our minds... an actual valid use for middle school math.

[deleted]

Re: Conterintuitive facts in mathematics, CS, and physics

#275

Post by ColinWright »

CJefferson wrote: post #270
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.

True, that is the subtle bit -- but I think most people misunderstand (I'm not saying you are!), and don't realise you can always split an infinity into two -- this is just about splitting a sphere into some 'point clouds' such that you can cleverly stitch them back together into the same original space. In particular, the 'cutting' really makes no real-world sense (at least, as far as I can understand).

I agree that the "cutting" makes no real world sense. In a way, that's one of the points of the exercise.

And I agree that most people don't (initially) understand that an infinite set can be divided into two infinite sets that kinda "look the same", such as dividing Z (or N) into the evens and odds.

But BT is more than that. What follows isn't really for you, but is for anyone following the conversation.

Let's take a set A. It's a subset of the unit sphere, and it's a carefully chosen, special set, not just any random set. It's complicated to define, and requires the Axiom of Choice to do so, but that's what the BT theorem does ... it shows us how to define the set A.

One of the properties of A is that we can rotate it into a new position, r(A), where none of the points of r(A) are in the same position as any points of the original position, A. So the sets r(A) and A have a zero intersection. For the set A there are lots of possible choices of r ... we pick a specific one that has some special properties. Again, the BT theorem is all about showing us how to do this.

Now we take the union: B = A u r(A)

The bizarre thing is this. If we've chosen A and r (and therefore by implication, B) carefully enough, it ends up that there's another rotation, call it s, such that s(B)=A, the set we started with.

So whatever the volume of A, the volume of A u r(A) must be twice that, but that's B, and B can be rotated to give A back to us. So B must have the same volume as A. So 2 times V(A) must equal V(A), so A must have zero volume.

Well, we can kinda cope with that.

But if we've chosen A carefully enough, we find that a small, finite number of them, carefully chosen and rotated appropriately, together make up effectively the entire sphere (we miss out countably many points, but they have zero total volume, and we can fix that up later). So if finitely many copies of A make up a solid sphere, they can't have zero volume.

And that's the "paradox".

The conclusion is that we can't assign a concept of "volume" to the set A, and this is explained a little more in a blog post I've submitted here before:

https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheo...

There's a lot more going on than just the "I can split infinite sets into multiple pieces that kinda look the same as the original", although that is certainly part of it, and lots of people already find that hard to take.

To any who has got this far, I hope that's useful.

Re: Conterintuitive facts in mathematics, CS, and physics

#276

Post by akomtu »

d_burfoot wrote: post #114

Here is one of my favorites. Take a light source and shine it at a filter that is polarized up/down that is in front of a filter that is polarized left/right. None of the light will get through: the first filter removes all of the L/R components of the light, and the second filter removes all the remaining light. Now add a third filter, between the two, which is polarized at 45 degrees. Now some of the light goes thr…

My hunch is that light does get thru the first two filters, but it gets "squeezed" so it's hard to see it. The third filter "unpacks" the light.

Re: Conterintuitive facts in mathematics, CS, and physics

#277

Post by kcolford »

BalinKing wrote: post #176
0xmarcin wrote: post #154

Earlier quoted context omitted.

Also this only works when the highly contested Axiom of Choice is used.

But isn't Banach-Tarski one of the main reasons why AoC is highly contested in the first place? (I don't know much about higher math, so this only my amateur impression.)

It's also contested because while intuitive to a countable set of countable sets, generalizing to higher cardinals is not intuitive. Although the same could be said of the Power Law as well...

Re: Conterintuitive facts in mathematics, CS, and physics

#278

Post by syncsynchalt »

kergonath wrote: post #164
syncsynchalt wrote: post #31

Earlier quoted context omitted.

> 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) .

I'm a civil engineer. In statute terms pound is the unit of force and slug is the unit of mass. That might have colored my thinking.

Re: Conterintuitive facts in mathematics, CS, and physics

#279

Post by mint2 »

tharkun__ wrote: post #78
mint2 wrote: post #73

Earlier quoted context omitted.

But that 99% water simplification is needed for the purpose of the exercise. The left out overnight is basically an absurd statement that is intended to confuse and not really related to the actual question. The statement could be “left in a dry environment until” or simply “left to dry a few weeks”

You definitely could write that but it wouldn't change anything and you could make the same "argument" you are making now. "The left to dry a few weeks is basically an absurd statement that is intended to confuse" and it still wouldn't be true. It's not intended to confuse at all. It's intended to get the point across that you let this imaginary thing dry from 99% to 98%. They could've said "sponge" and let it out to…

Wow by the downvotes I learned people react extremely negatively to any critique of math word problems. Spherical cows and 99% water potatoes are fine, those over simplifications are required for the analysis.

Saying Leaving the potatoes out overnight to imply they halved in mass, sounds as reasonable as “the potatoes were in the ground for 12 hours and then doubled in amount, what percent of water are they now?” It’s so gratuitous and requires ignoring all other laws of physics, while the goal of the spherical cow type simplification is to only ignore a few key challenging ones.

Re: Conterintuitive facts in mathematics, CS, and physics

#280

Post by BigJono »

throwaway8451 wrote: post #203
Aerroon wrote: post #200

Earlier quoted context omitted.

#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…

OK, let's assume you have 75% elemental resistance. You also take 15% reduced damage and 10% less damage. You have 5% chance to avoid elemental damage and 10% to dodge it and are under the effect of Elemental Equilibrium and Gluttony of Elements. How much better is it to just kill everything before it can kill you?

You joke, but PoE's use of stuff like 'increased/decreased' and 'more/less' to distinguish between additive and multiplicative calculations is one of the smartest game design decisions I've seen.

The game has a lot of seemingly arbitrary distinctions and concepts that you just have to learn over time, but once you actually learn them, the consistency of it all makes it very easy to handle the large amounts of complexity in the game.

It's completely unbearable going back to other games that just say crap like "+30% to x" without actually distinguishing between the different ways calculations can be done, forcing you to either experiment endlessly, look up every tiny thing on a community wiki, or just wing it.

It's a nice contrast to something like WoW where every time you get a new item you just chuck the item code into some ten million LOC simulator and fuck around with limiting permutations until it doesn't take 15 minutes to run, just to find out through some totally opaque process that you have 189 more dps. And then 2 months later you find out there was a bug in the simulation and the item you deleted 1.9 months ago was actually better.

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