Comment on Jeopardy wall calendar pretending that the coastline paradox doesn't exist

• clay_pidgin@sh.itjust.works ⁨3⁩ ⁨weeks⁩ ago

  If between 0 and 1 are an infinite number of real numbers, then between 0 and 2 are twice infinite real numbers, IIRC my college math. I probably don’t.

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• Capricorn_Geriatric@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

  It isn’t.

  When you look at the number of real numbers, you can always find new ones in both - you’ll never run out.

  That being said, imagine (or actually draw) two number lines in the same scale. One [0,1] the other [0,2]. Choose a natural number n, and divide both lines with that many lines. You’ll get n+1 segmets in both lines.

  When you let n run off into infinity, the number of segments will be the same in both lines. This is the cardinality of the set.

  But for practical purposes of measuring a coastline, this approach is flawed.

  Yes, you’ll always see n+1 segments, but we aren’t measuring thw number of distinct points on the coastline, but rather its length.

  If you go back to your two to-scale number lines and divide them into n segments, the segments on one are exactly two times larger than on the other.

  This is what we want to measure when we want to measure a coastline. The total length drawn when connecting these n points (and not ther number!) as their number runs off towards infinity.

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  • SmoothOperator@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

    But isn’t the issue that coastlines have a fractal nature? That depending on your resolution, you could have a finite or infinite length of a coastline? In which case measurement is hard to define.

    Talking about integrals, the fun part is that even with a coastline of indeterminate length, the area of a continent is easy to define to arbitrary precision - you can just define an integral that’s definitely inside the area and one that’s definitely outside the area, and the answer is between those two.

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    • Capricorn_Geriatric@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

      Talking about integrals, the fun part is that even with a coastline of indeterminate length, the area of a continent is easy to define to arbitrary precision - you can just define an integral that’s definitely inside the area and one that’s definitely outside the area, and the answer is between those two

      Is it?

      The main problem with a coastline’s shape isn’t the fractality of it, or the relative size we measure in (the “resolution”).

      It’s the fact that a coastline isn’t a static thing. The tides move the shoreline by up to a few meters.

      Then there are tectonic movements. These are much slower, but much more powerful: at one point Asia wasn’t even a thing.

      As you take the “resolution” up, yes - you’ll see various fractal-like behaviour.

      But, and thus is a big but: this will happen even if you take a straight ruler of, say, 1m in length (or, since we have to deal with every little edge case here, the part of it that actually measures out a meter). If you zoom in on it at the molecular and atomic levels, you’ll come across the same problem: a straight line isn’t a straight line! It just appears straight at the human “resolution” (scale).

      But does that mean a ruler measuring out 1m isn’t 1m long?

      Yes and no.

      Just like our coastline.

      T> alking about integrals, the fun part is that even with a coastline of indeterminate length, the area of a continent is easy to define to arbitrary precision - you can just define an integral that’s definitely inside the area and one that’s definitely outside the area, and the answer is between those two.

      Is it really?

      And what is the difference between length and area, other than the one dimension they are apart?

      What you’re taking as a common sense assumption for area is applicable equally to the length. Find two extremes, and the answer is somewhere in the middle. The less extreme those extremes become, the more accurate the approximation.

      Just as you can integrate the area, there must be an equivalent process to integrate the length.

      And again, any approximate curve of a coastline used to define the length of a coastline is a bigger assumption than a sufficiently sane “resolution” used to divide the curve into discrete intervals.

      Why does area get to be especially fun and definite while length , its one-dimension-away sibling doesn’t?

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      • SmoothOperator@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

        Why does area get to be especially fun and definite while length, its one-dimension-away sibling doesn’t?

        Excellent question, and as you yourself allude to, it’s a question of bounds. If you can establish and upper and lower bound on a quantity and make them approach eachother, you can measure it.

        On a finite 2d surface you can make absolute lower and upper bounds on any area - lower is zero, upper is the full surface. All areas are measurable. But on the same surface you can make a line infinitely squiggly and detailed, essentially drawing a fractal. So the upper bound on the length of a line is infinite. Which means not all lines have a measurably length.

        This extends naturally to higher dimensions - in a finite 3d space, volumes must be finite, but both lines and areas can be fractally complex and infinite. And so on.

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• AnarchoEngineer@lemmy.dbzer0.com ⁨3⁩ ⁨weeks⁩ ago

  The cardinality of the two intervals [0,1] and [0,2] are equivalent. E.g. for every number in the former you could map it to a unique number in the latter and vice versa. (Multiply or divide by two)

  However in statistics, if you have a continuous variable with a uniform distribution on the interval [0, 2] and you want to know what the chances are of that value being between [0,1] then you do what you normally would for a discrete set and divide 1 by 2 because there are twice as many elements in the total than there are in half the range.

  In other words, for weird theoretical math the amount of numbers in the reals is equivalent to the amount of any elements in a subset of the reals, but other than those weird cases, you should treat it as though they are different sizes.

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  • SmoothOperator@lemmy.world ⁨2⁩ ⁨weeks⁩ ago

    Sure, the length of the intervals is easily compared. But saying

    there are twice as many elements in the total than there are in half the range

    is false. This difference is the whole crux of the coastline problem, isn’t it?

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