i hate the coastline ‘paradox’ and every other ‘paradox’ that’s just a missing variable. “if we measure with a big resolution it’s a smaller number of units and a small resolution is a bigger number!?” that’s not a paradox. that’s just how that variable works always. it’s not confusing or interesting at all.
Jeopardy wall calendar pretending that the coastline paradox doesn't exist
Submitted 2 weeks ago by GraniteM@lemmy.world to mildlyinfuriating@lemmy.world
https://lemmy.world/pictrs/image/ecfd3de2-68ce-49ee-a1bf-fc1b1979ef69.jpeg
Comments
nublug@piefed.blahaj.zone 2 weeks ago
scott@lem.free.as 2 weeks ago
But if you shrink the “yardstick” down to an infinitesimally small size, the length, effectively, becomes infinite… and it’s the same for all coastlines. They’re all infinitely long.
… but some are longer than others. ;)
MrPoopyButthole@lemmy.world 2 weeks ago
Max Planck says no…
Lemming6969@lemmy.world 2 weeks ago
Didn’t calculus solve this stuff?
calcopiritus@lemmy.world 2 weeks ago
You can’t shrink the yardstick down to an infinitesimal size.
Coastlines are not well defined. They change in time with tides and waves. And even if you take a picture and try to measure that, you still have to decide at what point exactly the sea ends and the land starts.
If the criteria for that is “the line is where it would make a fractal” then sure, by that arbitrary decision, it is infinite. However, a way better way to answer the question “where is the line” is to just decide on a fixed resolution (or variable if you want to get fancy), which makes the distinction between sea and land clearer.
It is like saying that an electron is everywhere in the universe, because of Heisenberg’s uncertainty principle. While it is very technically true, just pick a resolution of 1mm^3 and you know exactly where the electron is.
theunknownmuncher@lemmy.world 2 weeks ago
Not all infinities are equal, friend. Asia does have more infinite coastline than other continents.
BetterDev@programming.dev 2 weeks ago
Its true that not all intinities are equal, but the way we determine which infinities are larger is as follows
Say you have two infinite sets: A and B A is the set of integers B is the set of positive integers
Now, based on your argument, Asia has the largest infinite coastline in the same way A contains more numbers than B, right?
Well that’s not how infinity works. |B| = |A| surprisingly.
The test you can use to see if one infinity is bigger than another is thus:
Can you take each element of A, and assign a unique member of B to it? If so, they’re the same order of infinity.
As an example where you can’t do this, and therefore the infinite sets are truely of different sizes, is the real numbers vs the integers. Go ahead, try to label every real number with an integer, I’ll wait.
Atlas_@lemmy.world 2 weeks ago
I’ll label every real number with the integer 1.
theunknownmuncher@lemmy.world 2 weeks ago
Go ahead, try to label every real number with an integer, I’ll wait.
Why would I be trying to do this though? You’ve got the argument backwards.
solarvector@lemmy.dbzer0.com 2 weeks ago
If your unit of measurement is 1 Asia coastline, all others would be some changing fraction thereof. Mathematical equation paradox maybe but hardly over that disproves the answer.
RattlerSix@lemmy.world 2 weeks ago
DragonTypeWyvern@midwest.social 2 weeks ago
Diddlydee@feddit.uk 2 weeks ago
It’s correct, though. You’d apply the same scale of measurements to all coastlines, and using a standard of 1km or 0.5km plot points, Asia wins.
DragonTypeWyvern@midwest.social 2 weeks ago
Leaving aside Planck scale, infinities can be larger than other infinities.
yermaw@sh.itjust.works 2 weeks ago
So when that kid said “well I hate you infinity plus a million” he was on to something mathematically?
tias@discuss.tchncs.de 2 weeks ago
No, but if he said “well I hate you two to the power of infinity” he would be.
stupidcasey@lemmy.world 2 weeks ago
Hmm, just because the distance measured varies based on the increments it is measured in doesn’t mean that using the same stick it wouldn’t be bigger.
cattywampas@midwest.social 2 weeks ago
Unless they’re assuming a certain resolution of measurement.
m4xie@lemmy.ca 2 weeks ago
Surely the coast of a continent of a given area can only have a finite theoretically maximum length even if the whole coast is a Hilbert Curve filling that area, because the minimum feature size is determined by the surface tension of water m
bryndos@fedia.io 2 weeks ago
My new years resolution will be to solve this paradox.
BedbugCutlefish@lemmy.world 2 weeks ago
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.
TropicalDingdong@lemmy.world 2 weeks ago
Some infinites are larger than other infinites.
cynar@lemmy.world 2 weeks ago
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.
GraniteM@lemmy.world 2 weeks ago
That’s a fair point. I forgot that some infinites are larger than other infinites.
SmoothOperator@lemmy.world 2 weeks ago
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.
clay_pidgin@sh.itjust.works 2 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.
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.
AnarchoEngineer@lemmy.dbzer0.com 2 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.
bryndos@fedia.io 2 weeks ago
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 . . .