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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#281

Post by Cheezemansam »

CJefferson wrote: post #182
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.

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".

One of the big things though, is that you can't know exactly what this division looks like. Like you said, at a high, non-technical level that is kind of fundamentally what is going on, with the caveat that the actual "cut" you are making isn't clean like splitting numbers down the middle.

Re: Conterintuitive facts in mathematics, CS, and physics

#282

Post by jerf »

curiousgal wrote: post #197
nighthawk454 wrote: post #68

Earlier quoted context omitted.

Yes, it doesn't explicitly state the rate or distribution of events. But it is a good reminder of what happens when your whatevers/second are pretty high - see the famous "One in a million is next Tuesday" [1]. "Rare" is soon if you roll the dice fast enough. Any time your service has a high TPS, your API gets a lot of calls, a button in your app gets pressed a lot, ... this applies. Critically, "a lot" is defined re…

> And I’ve seen some absolute doozies in my time – race conditions on MP machines where a non interlocked increment occurred (one variant of Michael Grier’s “i = i + 1” bug) I could not find any info about that bug, anyone got a link or a source?

I assume that bug is referring to the fact that while i = i + 1 may look atomic to you as a human, in the computer it turns into

    Read i into register.
    Add one to that register.
    Write i back to the memory location.
And there's a window during that "add one to the register" where you can obviously have something jump in and write something else to that memory location.

What happens on your real processor is more complicated since this is going to relate to cache coherency between the processors, not directly writing RAM at that point, and that's a deep rabbit hole. I couldn't describe it all in detail anyhow. But I can observe it doesn't take much at all to turn that one cycle vulnerability into something with a larger target.

Re: Conterintuitive facts in mathematics, CS, and physics

#283

Post by jerf »

simonh wrote: post #241
hamilyon2 wrote: post #186

Earlier quoted context omitted.

Given vastness of sun, it's age and existence of extremophiles, I would be surprised if there is no life there.

Life requires a stable environment in which persistent structures can be maintained over long periods of time. High temperatures are inimical to that. This is why we live in a part of the universe where temperatures rarely rise much above a few hundred degrees Kelvin. Above that most complex chemical structures break down.

Even more fundamentally, a life form needs to be able to pump entropy out more quickly than it comes in, no matter what its substrate is. With all that heat and light and magnetic fields running around and pressing in on any conceivable life form living in the sun, there's no way it could possibly pump it out fast enough.

With that analysis you don't have to get into the weeds of what exactly plasma and magnetic fields might theoretically be able to cohere into and whether it may be able to be life someday... it doesn't matter. There's no way sun life can pump out the entropy fast enough no matter what.

(On the flip side, one can imagine some form of nebula, gas-cloud life, but they would have to be so slow that there's no chance any of it could evolve into anything terribly complicated in the life time of the universe. If we ever did find some it would double as proof that there must have been some other life form that created it.)

Re: Conterintuitive facts in mathematics, CS, and physics

#284

Post by jerf »

mjburgess wrote: post #220
Pyramus wrote: post #215

Earlier quoted context omitted.

That is correct and should be highlighted more. In a way the real numbers are a model (or maybe a 'language') to describe physical phenomena. They work exceptionally well at that, but they are not backed by evidence and do come with (theoretical) limitations. This bachelor's thesis is a good starting point [1], search for 'finite precision physics' or 'intuitionistic math/physics'. [1] https://www.math.ru.nl/~landsma…

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 physics knows that neither theory has the domain of "the entire universe". This is not a wild claim by an HN commenter, this is consensus in the physics world, just perhaps not phrased in the way you're used to.

It's possible the eventual Grand Unified Theory will still have continuous space at its bottom, but it's also entirely possible it won't. Loop quantum gravity doesn't. And personally I expect some sort of new hybrid between continuous and discrete based on physics history; whenever in the past we've had a similar situation where it couldn't be X for this reason, but it couldn't be the obvious Not-X for some other reason, it has turned out to be something that had a bit of both in them, but wasn't either of them.

Re: Conterintuitive facts in mathematics, CS, and physics

#286

Post by BeetleB »

caf wrote: post #130
BeetleB wrote: post #85

Earlier quoted context omitted.

Gabriel's Horn was cool till someone pointed out to me that you can have a line of infinite length within a square (trivially). When comparing something of a certain dimension with something of a higher dimension, it's not at all surprising that the lower one can be infinite and the higher one finite. Usually it's phrased as "a finite amount of paint can paint an infinite area." But why do I need the Horn to realize…

The way I'd heard the paint comment was along the lines that "Gabriel's Horn can hold only a finite quantity of paint, but requires an infinite quantity of paint to cover the surface" . So if you think of it as a bucket that can't hold enough paint to cover itself, that is at least a little surprising.

But that's exactly my paint. If you allow for infinitely thin paint, then a finite volume of paint can always cover an infinite surface - you don't need Gabriel's Horn to show that.

If you don't allow for infinitely thin paint, then no - Gabriel's Horn surface cannot be painted even with an infinite amount of paint.

Re: Conterintuitive facts in mathematics, CS, and physics

#287

Post by svachalek »

> 11. Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it: https://en.wikipedia.org/wiki/The_Market_for_Lemons

This is new and interesting to me, although I think the phrasing of 11 is untrue as it's more about a cumulative effect in a market than an individual sale. Still I think this explains a lot of things in a way I've never really thought about it before. For example, dating apps.

Re: Conterintuitive facts in mathematics, CS, and physics

#288

Post by curiousgal »

svachalek wrote: post #287

> 11. Knowing just slightly more about the value of your car than a potential buyer can make it impossible to sell it: https://en.wikipedia.org/wiki/The_Market_for_Lemons This is new and interesting to me, although I think the phrasing of 11 is untrue as it's more about a cumulative effect in a market than an individual sale. Still I think this explains a lot of things in a way I've never really thought about it befo…

I remember I struggled to wrap my head around this in my microeconomics class when we first explored information asymmetry.

Re: Conterintuitive facts in mathematics, CS, and physics

#289

Post by tzs »

cedilla wrote: post #217
unholiness wrote: post #106

Earlier quoted context omitted.

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.

For all we know it's significantly older than counting. Pebbles representing bijections to wares like sheep (called calculi like in calculus) occur earlier than counting marks and much earlier than anything resembling numbers. There are still today human tribes that don't count at all.

> Pebbles representing bijections to wares like sheep (called calculi like in calculus) occur earlier than counting marks and much earlier than anything resembling numbers.

How was this determined? I wouldn't expect that using pebbles this way would leave any distinctive marks or damage or residue on the pebbles that would allow an archaeologist several tens of thousands of years later to tell that was what the pebbles were used for.

Re: Conterintuitive facts in mathematics, CS, and physics

#290

Post by Kranar »

Viliam1234 wrote: post #244
H8crilA wrote: post #81

Earlier quoted context omitted.

Just to make this more concrete: 1. There is a countable model of real numbers. 2. There even is a countable model of the entire set theory.

> There is a countable model of real numbers. What exactly happens when you try to apply Cantor's diagonal argument to this model? I guess that at some step, you get an answer like "outside of the model, yes this exists, but inside the model the answer is no", but I would like to see it precisely, how exactly the in-model reasoning diverges from the outside-model reasoning.

Consider that there are "real numbers" as in some kind of construct that we as humans are interested in understanding, and then there's ZFC-real numbers, which is an attempt to formalize "real numbers" rigorously. What we know is that we can never rigorously take "real numbers" and uniquely formalize them and so any formal investigation of "real numbers" will be prone to multiple interpretations.

Given this, consider all models M that contain some set R(M) that satisfies ZFC's definition of real numbers and where R(M) is actually countable. Furthermore M also contains some set N(M) that satisfies ZFC's definition of natural numbers.

Within this model M, since N(M) satisfies ZFC's definition of the naturals, it is ZFC-countable (that is it satisfies ZFC's definition of a countable set). Furthermore applying Cantor's diagonal argument to M, one can show that M does not contain a set that represents a surjection from N(M) to R(M), hence R(M) is ZFC-uncountable (it satisfies ZFC's definition of being uncountable).

That said, all this means is that ZFC-countable, and ZFC-uncountable do not fully capture what it actually means to be countable or uncountable. ZFC-countable means a set has the same cardinality as whatever set satisfies ZFC's definition of natural numbers, which is not the same as what we as humans consider to be actual natural numbers.

Similarly being ZFC-uncountable just means a set has a greater cardinality than the set that satisfies ZFC's definition of natural numbers, but that does not mean that such a set is actually uncountable.

There is no way to extend ZFC so that what we consider to be actually countable or uncountable has one single unique interpretation. If there were then we could claim that said unique interpretation captured precisely our notion of countable and uncountable.

What we can do is jump up a level to second order logic, and in that logic it actually is possible to have one unique interpretation of countable and uncountable sets so that there is a unique and countable set of naturals and a unique and uncountable set of reals, but second order logic comes with its own set of ambiguities and issues that for the most part mathematicians reject outright.

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