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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#141

Post by TMWNN »

geogra4 wrote: post #100

>Two 12 Inch Pizzas have less Pizza than one 18 inch pizza Love this one. Always go for the biggest pizza you can

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.

Re: Conterintuitive facts in mathematics, CS, and physics

#142

Post by GolDDranks »

d_burfoot wrote: post #117

Surprising fact about the sun: it actually produces less heat per unit volume than a compost pile - this makes sense when you consider that fusion events are very rare. The reason the sun is so hot is that it has an enormous ratio of volume to surface area. Heat emission due to radiative cooling is proportional to the surface area, but heat creation due to fusion is proportional to the volume.

I learned about this many years ago, and even now, I am occasionally amused by this thought.

Re: Conterintuitive facts in mathematics, CS, and physics

#143

Post by dwohnitmok »

Kranar wrote: post #104
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…

> 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.

Ah a Platonist in the flesh. Don't see many of you on HN. I don't think real numbers truly, objectively exist and think of them more as artifacts of human thought, but that's a deep deep rabbit hole.

I'm kind of curious then, what do you believe the cardinality of the "real" real numbers is?

Re: Conterintuitive facts in mathematics, CS, and physics

#144

Post by Kranar »

tshaddox wrote: post #136
wodenokoto wrote: post #32

Earlier quoted context omitted.

It’s also terribly phrased. > The vast majority of real numbers can't be described. But it is impossible to give a single example. If we accept the first sentence and assume the second refers to indescribable numbers, then isn’t it obvious that we have no examples of things we cannot describe? If the second sentence refers to real numbers in general I can give one example or two.

How could you give an example? The example you give would be a description of a number.

He just means the phrasing is kind of poor. A general description of an undefinable number could be something like: given a sequence of Turing Machines, T_1, T_2, ..., T_n, where T_1 is the smallest representation of Turing machine over a grammar G, and T_2 is the second smallest representation of a Turing machine over G, and T_3 is the third smallest etc... take the limit of the ratio of said Turing machines that halt to Turing machines that don't halt as n goes to infinite.

I mean that's a description of some number, you could even write it out mathematically or write an algorithm to express that number. Of course neither the algorithm or the formula will ever converge and yet it will also always be bounded between 0 and 1 (hence it doesn't diverge to infinity).

So is that a description of a number? Well sure in one sense I just described it, there is only one single real number that can satisfy that description, and as I said I could in principle write it out formally and rigorously... and yet in another sense it also doesn't describe anything since no matter how hard you try, there will always be at least two real numbers that could potentially satisfy the definition and no way to eliminate one of them.

Re: Conterintuitive facts in mathematics, CS, and physics

#145

Post by fgonzag »

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%?

just as going from 98% effectiveness to 99% effectiveness halves your risk (you go from 2% chance of falling ill to 1%)

its a common concept in games in which armor follows a linear formula (each point of armor is more effective than the last when calculating effective health)

Re: Conterintuitive facts in mathematics, CS, and physics

#146

Post by Kranar »

dwohnitmok wrote: post #143
Kranar wrote: post #104

Earlier quoted context omitted.

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…

> 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. Ah a Platonist in the flesh. Don't see many of you on HN. I don't think real numbers truly, objectively exist and think of them more as artifacts of human thought, but that's a deep deep rabbit hole. I'm kind of curious then, what do you believe the cardinality of…

I think I'm with you on that. Real numbers don't exist in an objective sense, I mean they exist in the same sense that an Escher painting of a hand drawing a hand exists, but they don't exist in the sense that a hand drawing a hand actually exists.

When I was in high school I remember thinking that computers use the discrete to approximate the continuous and that it is the continuum that is real and the discrete that is an imperfect representation of the continuous. Then a high school teacher blew my mind when he told me to consider the opposite, that in fact it's the continuous that is used to approximate the discrete. The discrete is what's real and we humans invented the continuous to approximate the discrete.

That simple twist in thinking had a profound effect on me that influences me to this day 30 years later.

If anything I may have some extreme opinions that frankly no one takes seriously and I'm okay with that. For example I think the finitists had it right and infinity does not exist. There really is such a thing as a largest finite number, a number so large that it's impossible even in principle to add 1 to it. I can't fathom how large that number is, but there's physical justification to believe in it based on something like the Bekenstein bound:

https://en.wikipedia.org/wiki/Bekenstein_bound

At any rate, I like thinking about this stuff, I do appreciate it, but I don't take it literally. It's poetic, it can inspire new ways of thinking, but I also remind myself to compartmentalize it to some degree and not take these ideas too literally.

Re: Conterintuitive facts in mathematics, CS, and physics

#147

Post by IIAOPSW »

nighthawk454 wrote: post #68
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…

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…

Or everyone's newest favourite, the virus randomly mutating is "rare".

Re: Conterintuitive facts in mathematics, CS, and physics

#148

Post by Gibbon1 »

sib wrote: post #137
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%.

It's probably more intuitive if you say that it's "half as ineffective"

I first saw this in a discussion of power supply efficiency. A 95% efficient supply generates ~half the heat that a 90% one does.

Re: Conterintuitive facts in mathematics, CS, and physics

#149

Post by galaxyLogic »

stevetodd wrote: post #80
hinkley wrote: post #52

Earlier quoted context omitted.

What really throws people for a loop is the wait time after queuing starts. People expect the wait time to drop once arrivals return to normal, but there just isn't capacity to catch up. In the real world, except at the DMV, people give up and shorten the queue (or don't join it to begin with), causing the arrival rate to go below nominal allowing the workers to catch up. In must-have or automated situations they see…

This also explains the supply chain issues we are experiencing now because of the move to just-in-time manufacturing across the board.

Cool. Ships waiting to dock on California cost. This explains it!

Re: Conterintuitive facts in mathematics, CS, and physics

#150

Post by ncmncm »

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.

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