Achilles and the tortoise (Zeno’s paradox revisited)

I have a question.

Zeno’s paradox of Achilles and the tortoise goes a bit like this. Achilles races a tortoise and, in deference to good sportsmanship, Achilles decides to give the tortoise a head start. Say the tortoise is given a 100m head start and that Achilles travels 10x faster than the tortoise. When Achilles travels 100m to arrive at where the tortoise started the race the tortoise has moved ahead another 10m. When Achilles then travels another 10m the tortoise has again moved on another 1m. When Achilles travels another 1m the tortoise is ahead by another 100cm and so on and so forth. If length is infinitely divisible then logic suggests that Achilles can never catch up with the tortoise. This is, of course, absurd.

Or is it?

To make the calculation easier let’s make Achilles 2x faster than the tortoise. Say that Achilles travels at 2m/s and the tortoise at 1m/s. Achilles gives the tortoise a 1m head start. After 1 second Achilles would have travelled 2m and the tortoise 1m. This means that after 1 second Achilles has caught up with the tortoise (assuming, of course, that they are travelling in the same direction ie we are only concerned with the scalar value of distance travelled). Although intuitively obvious you can also set up a simple calculation to come to the same conclusion.

A mathematician may look at Zeno’s paradox and resolve it in what I intrinsically feel (ie what I feel deep down in my guts) is a much more satisfactory method. The Achilles and the tortoise paradox is a convergent infinite series and can be solved as such. The above example has come straight out of Numberphile’s YouTube video which can be seen here: https://www.youtube.com/watch?v=u7Z9UnWOJNY. For ease of reference the relevant snapshots from the video are arranged below:

Set up an equation for the distance that Archilles has to travel to catch up with the tortoise. That distance (which is an infinite series) is denoted “S”.

Solving the equation starts by calculating what 1/2 of “S” would look like.

Subtract “S” from “1/2 S”

Voila. S = 2m

The ability to solve the asymptote of this convergent series is as rewarding as it is simple (especially when it’s done by someone else). Note that the resolution of the paradox lies in the solution of an infinite series.

Ask a physicist to resolve the same paradox gives what I believe is a fundamentally different answer. The glib response by a physicist to a child with only high school physics knowledge would be that the scalar value of distance (ie length) is not infinitely divisible. You can step down length until you get to subatomic levels - say, somewhere in the order of Plank’s length - after which this scalar value becomes meaningless (both literally and metaphorically). This “phase-shift” is physics measurement problem taken from different angle.

The problem I have is that both these solutions can’t both be correct. The mathematician’s solution is the solution of an infinite series. The physicist proposes that such an infinite series cannot be mapped in the real world. In other words, the mathematician would say that you don’t need discretisation to solve the paradox but, with current tools, the physicist cannot deny the possibility of such a solution.

Occam’s razor favours the poignancy of the mathematician’s solution. Then again, the mathematician does not necessarily trouble himself with the frustrations of real world data.

The other problem I have is that I am neither a mathematician nor a physicist.