Nah, that’s silly. Asia obviously has the longest coastline.
Sure, based on that paradox, the specific measurement of a given coastline will differ. But if you pick a standard (i.e., 1km straight lines), Asia is easily the longest. Doesn’t matter what standard you pick.
The only way the paradox matter here is of you pick different standards for different coastlines. Which, os obviously wrong.
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.
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.
Let F be a geometric object and let C be the set of counterexamples.
F is a True Fractal ⟺ F satisfies all properties P₁, P₂, …, Pₙ
Where for each counterexample c ∈ C that satisfies P₁…Pₙ:
Define Pₙ₊₁ := “is not like c”
The definition recurses infinitely as new counterexamples emerge.
Corollary: Coastlines exhibit fractal properties at every scale…
except they don’t, because [insert new property],
except that’s also not quite right because [insert newer property],
except actually [insert even newer property]…
Funny that so many uses of maths depends on measurement, and yet so many pure mathematicians seem to be clueless about how we actually measure things and why its useful. It doesn’t even matters about all this bullshit about infinities , were talking about the real world. It’s all about the precision of the tape measure. Here’s a true story from back in the day:
English Mathematician: You’ll need an infinite number of bricks to build a wall around any island’s coastline.
French guy: come on over and see Mont Saint Michel it’s vraiment genial!
English Mathematician: Oh that wall is infinitely far away from the true coastline, those bricks are not regulation infinitesimal length. If they’d started from the other corner they’d have got a different shape, and for sure needed infinite number of infinitesimal bricks to actually build that wall. Sloppy french masons. I can prove it I’ll blast them all away with cannon fire until the glorious mathematical truth is revealed underneath.
One year later
French inhabitants: fuck off english maths whore!
Ten years more laterer
Hi french dudes! I’m back with a greater number of even bigger state of the art truth seeking cannon. I will prove this if its the last thing i do.
Nah, that’s silly. Asia obviously has the longest coastline.
Sure, based on that paradox, the specific measurement of a given coastline will differ. But if you pick a standard (i.e., 1km straight lines), Asia is easily the longest. Doesn’t matter what standard you pick.
The only way the paradox matter here is of you pick different standards for different coastlines. Which, os obviously wrong.
Isn’t it a bit like saying “there’s obviously more real numbers between 0 and 2 than between 0 and 1”? Which, to my knowledge, is a false statement.
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.
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.
In math they’d both be equal
Some infinites are larger than other infinites.
It’s not a true fractal, so the length has some finite bounding. It’s just stupidly large, since you are tracing the atomic structure.
Let F be a geometric object and let C be the set of counterexamples.
F is a True Fractal ⟺ F satisfies all properties P₁, P₂, …, Pₙ
Where for each counterexample c ∈ C that satisfies P₁…Pₙ: Define Pₙ₊₁ := “is not like c”
The definition recurses infinitely as new counterexamples emerge.
Corollary: Coastlines exhibit fractal properties at every scale… except they don’t, because [insert new property], except that’s also not quite right because [insert newer property], except actually [insert even newer property]…
□ (no true scotsman continues fractally)
This motherfucker coming correct with subscripts.
That’s a fair point. I forgot that some infinites are larger than other infinites.
Did you also forget about Dre?
Did you forget about the game?
Funny that so many uses of maths depends on measurement, and yet so many pure mathematicians seem to be clueless about how we actually measure things and why its useful. It doesn’t even matters about all this bullshit about infinities , were talking about the real world. It’s all about the precision of the tape measure. Here’s a true story from back in the day:
English Mathematician: You’ll need an infinite number of bricks to build a wall around any island’s coastline. French guy: come on over and see Mont Saint Michel it’s vraiment genial!
English Mathematician: Oh that wall is infinitely far away from the true coastline, those bricks are not regulation infinitesimal length. If they’d started from the other corner they’d have got a different shape, and for sure needed infinite number of infinitesimal bricks to actually build that wall. Sloppy french masons. I can prove it I’ll blast them all away with cannon fire until the glorious mathematical truth is revealed underneath.
One year later French inhabitants: fuck off english maths whore!
Ten years more laterer Hi french dudes! I’m back with a greater number of even bigger state of the art truth seeking cannon. I will prove this if its the last thing i do.
One year later . . .