### Tolstoy's Math

Sarang, in comments, derides Tolstoy's analogy between math and history. I am inclined to cut Tolstoy some slack, because I quite like the analogy. But it does seem to be true that you don't need much math to show that the Achilles-tortoise "paradox" is not paradoxical at all.

Imagine that Achilles runs at 1 meter per second, while the tortoise runs at 0.1m/s. Imagine that Achilles starts 1 meter behind the tortoise. Then in one second, Achilles will have reached the place the tortoise started, but the tortoise will have advanced 0.1 meter. In another 1/10 of a second, Achilles will have reached this point, but the tortoise will have moved on. Hence the "paradox."

So set t = to the amount of time that it will take Achilles to catch the tortoise. It will take him one second, plus 1/10 of a second, plus 1/100 of a second, and so on.

t = 1 + 0.1 + 0.01 + 0.001 + ....

Set m = 0.1 + 0.01 + 0.001 + ...

So t = 1 + m.

Now multiply both sides by 10.

10t = 10 + 10m = 10 + 1 + m

Now subtract t from 10t:

10t - t = (10 + 1 + m) - (1 + m)

9t = 10 + (1 + m) - (1 + m)

9t = 10

t = 10/9

So apparently Achilles will catch the tortoise in 10/9 of a second.

Sarang, have I fucked something up?

Imagine that Achilles runs at 1 meter per second, while the tortoise runs at 0.1m/s. Imagine that Achilles starts 1 meter behind the tortoise. Then in one second, Achilles will have reached the place the tortoise started, but the tortoise will have advanced 0.1 meter. In another 1/10 of a second, Achilles will have reached this point, but the tortoise will have moved on. Hence the "paradox."

So set t = to the amount of time that it will take Achilles to catch the tortoise. It will take him one second, plus 1/10 of a second, plus 1/100 of a second, and so on.

t = 1 + 0.1 + 0.01 + 0.001 + ....

Set m = 0.1 + 0.01 + 0.001 + ...

So t = 1 + m.

Now multiply both sides by 10.

10t = 10 + 10m = 10 + 1 + m

Now subtract t from 10t:

10t - t = (10 + 1 + m) - (1 + m)

9t = 10 + (1 + m) - (1 + m)

9t = 10

t = 10/9

So apparently Achilles will catch the tortoise in 10/9 of a second.

Sarang, have I fucked something up?

## 6 Comments:

Seems right. (Easier way -- start Achilles at the origin, tortoise 1m ahead, equate their distances, so that t = 1 + 0.1 t; therefore t = 1/0.9.) But yes, that is how you would do it using infinite series.

That Zeno's paradox was ever thought paradoxical is something I find deeply weird.

I think the infinite series approach is useful because it makes explicit the point that even the sum of an infinite number of periods of time can add up to a finite - in fact quite short - period of time. This, it seems to me, is what really explodes the paradox.

But yes, amazing that it was ever considered a paradox.

yeah, like, to think that anyone ever thought 1 + 1/2 + 1/4 + 1/8 ... added up to anything other than 2 is pretty stunningck.

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