Recently a lot of videos have popped Up for me claiming that you can’t accurately measure the coastline of a landmass cause the smaller of a “ruler” you use, the longer of a measure you get due to the smaller nooks and crannies you have to measure but i don’t get how this is a mathematical problem and not an “of course i won’t measure every single pebble on the coastline down to atom size” problem”. I get that you can’t measure a fractal’s side length, but a coastline is not a fractal
In: Mathematics
The coastline paradox isn’t necessarily stating that you “can’t accurately measure a coastline” because making that statement would depend on a definition of “accurately.” Even if your definition of “accurately” was on the sub-atomic scale, then measuring a coastline would be difficult but, in principle, not impossible. (Though this is true about measuring anything.)
Instead, the coastline paradox says that as your definition of “accurately” changes, the resulting measure of the length of the coastline will change in unexpected ways. It’s not a paradox to say that greater accuracy will change the measurement in some way. We might expect that there is some “true” answer that inaccurate measures will only approximate. Sometimes they will be too high, and sometimes they will be too low. What’s unexpected about coastlines is that increasing accuracy will almost always *increase* the measurement. This has to be taken into account in a couple of ways:
* When comparing coastline measurements, it’s important to ensure they were both taken with the same “ruler”
* Unlike a case with symmetric noise, it’s harder to use statistical tricks to glean the “true” measure from several noisy measures. If the noisy measurements lay both above and below the most accurate measurement, you could take a bunch of noisy measurements and average them to get a good idea of the true measurement. This doesn’t work with coastlines, so that’s the sense in which you “can’t accurately measure a coastline.”
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