And the more of that detail you try to include in your measurement, the more your measured length will increase. No matter how far you zoom in on a part of the coastline, you always find more detail - bays and peninsulas, cliffs and inlets, pits and protrusions, tiny bumps, and so on. The coastline paradox is a situation where that doesn't happen. Loosely speaking, smooth curves behave well when you use calculus on them. You might have to zoom in a very long way, but as long as you eventually end up with something that looks straight, it counts as smooth, even if the overall shape is very wiggly.

Loosely speaking, a smooth curve is one where if you zoom in far enough at any point, the curve starts to look like a straight line. The difference lies in something called smoothness. The fact that there can never be a complete map (complete and provably correct model of reality) is irrelevant to natural science itself, but it’s relevant to how we think about natural science and its limitations when it comes to explanatory capabilities.

A sufficiently detailed map becomes the territory, and if you with your measuring apparatus are part of the territory then you’d need to recurse infinitely. Just how there can never be a complete map, there can never be a complete and provably correct model of reality. (Measuring the coastline to arbitrarily high precision would require an arbitrarily high amount of energy and time, all the while the coastline is undergoing natural processes that change the result of the measurement.) The coastline paradox is useful as an illustration of how any measurement or observation is performed against an abbreviated version of reality that corresponds to a specific practical purpose, whereas the underlying complete reality is never phenomenologically accessible.

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