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Coastline paradox

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Re: Coastline paradox

#91

Post by strogonoff »

OskarS wrote: post #15

I’ve always found this argument very cool (and obviously true in a mathematical sense) but not really ”practically” relevant. Like, the only reason you would be interested to know the length of a coastline is because you want to know how far a ship would need to travel, and that is pretty well-defined. Or, like, if you wanted to do a naval blockade, you would need to know how many ships you’d need to cover the entire…

The coastline paradox is useful as an illustration of how any measurement or observation is performed against an abbreviated version of reality that corresponds to a specific practical purpose, whereas the underlying complete reality is never phenomenologically accessible. (Measuring the coastline to arbitrarily high precision would require an arbitrarily high amount of energy and time, all the while the coastline is undergoing natural processes that change the result of the measurement.)

Just how there can never be a complete map, there can never be a complete and provably correct model of reality. A sufficiently detailed map becomes the territory, and if you with your measuring apparatus are part of the territory then you’d need to recurse infinitely. The fact that there can never be a complete map (complete and provably correct model of reality) is irrelevant to natural science itself, but it’s relevant to how we think about natural science and its limitations when it comes to explanatory capabilities.

Re: Coastline paradox

#93

Post by dotancohen »

TylerE wrote: post #70
margalabargala wrote: post #66

Earlier quoted context omitted.

The lower bound is generally very easy to measure precisely. So a change in that would be immediately noticed. For example, how long is the North Carolina coastline? No idea. But we can measure the precise distance between the point where that state's border with South Carolina meets the ocean, and the point where the border with Virginia meets the ocean. The coastline must be at least this long. And if you measure y…

Much more complcated problem than you're implying as NC has no unbroken coast. It's all shoals and barrier islands - and when you're inside that, swampland.

That does not affect the lower bound as described.

Re: Coastline paradox

#95

Post by nonrandomstring »

AdamH12113 wrote: post #73
irrational wrote: post #27

I’ve never taken calculus (the last math class I took was an algebra class in high school in the late 1980s). But I read a description of calculus once that used the example of trying to figure out the length of a coastline and described how one could use smaller and smaller sections to estimate the length and this is what calculus is all about. But, this article doesn’t mention calculus at all, so now I’m wondering…

Your impression is exactly right. The difference lies in something called smoothness . Loosely speaking, a smooth curve is one where if you zoom in far enough at any point, the curve starts to look like a straight line. You might have to zoom in a very long way, but as long as you eventually end up with something that looks straight, it counts as smooth, even if the overall shape is very wiggly. Loosely speaking, smo…

Great explanatory writing, thanks.

> In mathematics, we say that the length diverges. If you were measuring a smooth curve, the length would converge on a single value -- incorporating more detail would change the length, but only by tinier and tinier amounts.

Can you say something bout the exponential curve in this context? IIRC whatever level we zoom to we see the same curve, neither converging nor diverging. Is there something special we should take note of here?

Re: Coastline paradox

#96

Post by heavenlyblue »

ilaksh wrote: post #18

I think I've heard this in reference to the difficulty of software estimation. The project details and potential problems are unknowns until you get to that specific part of the project. Even though theoretically you can see the "shape" and rough size of the project when zoomed out.

It's delusional. The only reason this is a problem in mathematics is because lines in mathematics are infinitely narrow, while in reality complexity is bounded.

In real life, specific unknowns in part of the project may turn out to be actually _easier_ than you thought, while in perimeter calculations it's always a lower bound at a higher scale.

Re: Coastline paradox

#97

Post by luca3v »

nonrandomstring wrote: post #95
AdamH12113 wrote: post #73

Earlier quoted context omitted.

Your impression is exactly right. The difference lies in something called smoothness . Loosely speaking, a smooth curve is one where if you zoom in far enough at any point, the curve starts to look like a straight line. You might have to zoom in a very long way, but as long as you eventually end up with something that looks straight, it counts as smooth, even if the overall shape is very wiggly. Loosely speaking, smo…

Great explanatory writing, thanks. > In mathematics, we say that the length diverges. If you were measuring a smooth curve, the length would converge on a single value -- incorporating more detail would change the length, but only by tinier and tinier amounts. Can you say something bout the exponential curve in this context? IIRC whatever level we zoom to we see the same curve, neither converging nor diverging. Is th…

The exponential curve does look like a line when you zoom in very close to a point. For example, f(x) = e^x looks linear with slope e^x near x. You can see that, for small epsilon, e^(x+epsilon) - e^x is approximately epsilone^x, with an error term of the order of epsilon^2 e^x

Re: Coastline paradox

#98

Post by skrebbel »

kqr2 wrote: post #71
chrisweekly wrote: post #51

Earlier quoted context omitted.

Yes, there's a classic blig post (sorry, no bookmark handy) about an analogous trip from SF to LA.... Also, (maybe counterintuitively), in software estimation the more fine-grained the estimates, the more likely they are to _underestimate_ project scope. For areas that are better-understood, the estimate will be (properly) limited in scope (reducing the overall buffer) -- which leaves less margin for error in the are…

https://techcrunch.com/2016/04/30/estimate-thrice-develop-on...

Woa, from way back when a traffic sign with the word “Estimate” was considered helpful imagery

Re: Coastline paradox

#99

Post by chrisweekly »

kqr2 wrote: post #71
chrisweekly wrote: post #51

Earlier quoted context omitted.

Yes, there's a classic blig post (sorry, no bookmark handy) about an analogous trip from SF to LA.... Also, (maybe counterintuitively), in software estimation the more fine-grained the estimates, the more likely they are to _underestimate_ project scope. For areas that are better-understood, the estimate will be (properly) limited in scope (reducing the overall buffer) -- which leaves less margin for error in the are…

https://techcrunch.com/2016/04/30/estimate-thrice-develop-on...

Thanks, yes - that's the one.

(Too late for me to correct "blig" -> "blog", whoops.)

Re: Coastline paradox

#100

Post by remram »

justinl33 wrote: post #72
remram wrote: post #57

Earlier quoted context omitted.

Interestingly there is no paradox for convex shapes, like measuring tape forms.

But isn’t the problem that the ‘object’ formed by the measuring tape/rulers isn’t representative of the (most likely) concave object you’re trying to measure.

It's not a problem for me or my waist in practice, no ;)
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