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?