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 the number of distinct points on the coastline, but rather its length, i.e. the distance between these points.
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 their number!) as the number of points runs off towards infinity.
The solution to this “paradox” is probably closer to the definition of the integral (used to measure areas “under” math functions) than to that of the cardinality of infinite sets (used to measure the number of distinct elements in a set).
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.
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! Just by taking an optical microscope, you’ll see the inherent jaggedness (fractality) of our supposedly-straight ruler. It turns out our ruler just appears straight at the human “resolution” (scale).
But does that mean a ruler measuring out 1m isn’t 1m long? While it may not have tectonic or tidal movements, the molecules building up the ruler aren’t straight.
Does this mean the ruler, “zoomed in enough”, will appear to be of infinite length?
Yes.
But does that mean its length is infinite?
No. Its length is clearly 1m, +/- a small rounding error.
The same idea applies to our coastline.
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.
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 equally applicable 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 besides, any curve used to model the length of a coastline is a bigger assumption than a sufficiently sane “resolution” used to divide the curve into discrete intervals for the purposes of geodesic measurements. As you vary the number of reference points,the length will indeed increase. But after each successive round of refinement, the difference will be less and less, even though it will consistently rise. At one point, it will become insignificant enough.
Why does area get to be especially fun and definite while length, its one-dimension-away sibling doesn’t?
What about volume? Is it an unsolveable enigma like length, or a long-solved problem like area?
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 the number of distinct points on the coastline, but rather its length, i.e. the distance between these points.
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 their number!) as the number of points runs off towards infinity.
The solution to this “paradox” is probably closer to the definition of the integral (used to measure areas “under” math functions) than to that of the cardinality of infinite sets (used to measure the number of distinct elements in a set).
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.
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! Just by taking an optical microscope, you’ll see the inherent jaggedness (fractality) of our supposedly-straight ruler. It turns out our ruler just appears straight at the human “resolution” (scale).
But does that mean a ruler measuring out 1m isn’t 1m long? While it may not have tectonic or tidal movements, the molecules building up the ruler aren’t straight.
Does this mean the ruler, “zoomed in enough”, will appear to be of infinite length?
Yes.
But does that mean its length is infinite?
No. Its length is clearly 1m, +/- a small rounding error.
The same idea applies to our coastline.
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 equally applicable 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 besides, any curve used to model the length of a coastline is a bigger assumption than a sufficiently sane “resolution” used to divide the curve into discrete intervals for the purposes of geodesic measurements. As you vary the number of reference points,the length will indeed increase. But after each successive round of refinement, the difference will be less and less, even though it will consistently rise. At one point, it will become insignificant enough.
Why does area get to be especially fun and definite while length, its one-dimension-away sibling doesn’t?
What about volume? Is it an unsolveable enigma like length, or a long-solved problem like area?