What is the difference between Zeno's "Dichotomy" and Richardson's "Coast of England" paradox?

Question

We assume, though I believe it can be debated, that Zeno's "Dichotomy" paradox is apparently "unreal." We can treat any given distance as the sum of an infinite regress of smaller and smaller distances, so the person walking that distance never arrives, or never even starts. Yet we know this isn't what we experience. Aristotle and many others have provided solutions.

Richardson's paradox seems more or less the same. If we measure the "Coast of England" as the sum of smaller and smaller fractal units or "distances" the length of the coast grows larger and larger towards infinity. Yet is is a "real" phenomenon, as Richardson noted in actual coastal measurements in cartography.

So, I am assuming the time factor in the "Dichotomy" makes the difference? Treated as motion we are dividing the distance units by time units, arriving at an intersecting defined "limit," as in calculus. But how would this work with the "Coastline?" I have a feeling this is pretty simple, but requires a mental toolkit I lack. Incidentally, I don't think Richardson's "fractals" are really the issue here.

(As a complete aside, I'm trying to think various measurement paradoxes through in relation to the measurement of the "value" of wage labor in timed units and the cumulative "value" of Capital in Marx, just a fanciful exercise. But I'm only a hobbyist and nowhere near a clearly framed question on that yet.)

So, is there some way of introducing time limits into the Coastline paradox? Or is there some other resolution? Or some clarifying way to reframe the comparison?

Answer 1

The issue is not time but whether the mathematical series which models the sequence is convergent or divergent.

In the case of Zeno's arrow paradox, in which he argues that the arrow never gets there, the distance travelled at each step is one half of the previous distance. Given that the total distance is x and the initial step is x/2, this gives rise to the series:

distance = x/2 + x/4 + x/8 + ....

This is a convergent series, whose endpoint after infinitely many iterations equals x.

Similarly, if total flight time of the arrow over distance x/2 is t/2, this gives rise to the convergent series:

flight time = t/2 + t/4 + t/8 + ....

a similar convergent series, whose endpoint after infinitely many iterations equals t.

Thus, the infinite series of steps sums to a flight time not of infinity but of t and Zeno better get out the way quick. His paradox is exposed as a failure of understanding.

However the coastal paradox involves a fractal sequence. Here, each step in the crinkling-up increases the initial "smooth" length l by a fixed ratio r. After n iterations, the relevant equation is:

Total length = l x rⁿ

For a fractal, r is always greater than one, so as the number of fractal steps increases without limit, rⁿ tends to infinity.

Perhaps fortunately, coastlines stop being fractal once they reach the scale of a rock, pebble or, at worst, a grain of sand.

So, the answer to your question is all in the mathematics of the given series.