It says that theoretically Achilles can’t overtake the tortoise, but I don’t the thinking behind that.
In: Mathematics
The argument is that while the tortoise is slow moving and Achilles can run at any speed, if the tortoise is given a head start he can never catch up.
The logic behind the argument was that whenever Achilles reached where the tortoise was, the tortoise would have moved on. So given a head start, when Achilles reached the head start the tortoise would be further ahead. Achilles catching up to him again would have the tortoise again past that point.
Of course, the logic is a fallacy since it assumes that each catch-up/move-on action is a segment instead of a continuum. Achilles doesn’t actualize reaching the tortoise’s head start point by stopping, just as his speed would easily continue and overtake the tortoise.
Subdividing the race distance into infinite segments does not preclude the reality of traversing those segments. The race is real, the segments are imaginary structures.
They knew the tortoise would lose, but didn’t have the math to fully prove why. The question is a thought exercise to examine how the arrow/Achilles catches up to the tortoise, while cutting time into smaller and smaller pieces until the moments in time are infinitely small. By doing this is appears to stretch out the number of steps the arrow/Achilles needs to take to infinity in the closing moments of the chase.
The math to reconcile what they knew to be true but couldn’t formalize mathematically was discovered between the late 17th century and the early 19th century.
The ancient Greek philosophers actually got fairly close to discovering calculus and limits, but didn’t quite make the final steps, and then after their time that area of science was neglected for a long long time until the late Renaissance / early Enlightenment eras. At least in western history, I’m not sure of how other areas fared but it seems they also stalled out at similar points.
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