Zeno’s paradoxes of motion – particularly “Achilles and the Tortoise” and the “Dichotomy” (aka the “Racetrack”) – are maybe the most well known and historically popular paradoxes in western philosophy. The consensus among philosophers is that they were solved by the insights of calculus. Even those who see a problem beyond what calculus resolves tend to think that set theory takes care of the loose ends. I will argue that neither calculus nor set theory satisfactorily deal with all of the problems involved in Achilles and the Dichotomy, and that the paradoxes have yet to be fully solved. I will further argue that a lesser known of Zeno’s paradoxes – the “Plurality” – also remains unsolved.
I won’t assume prior knowledge of either the paradoxes, calculus, or set theory, and will take care to introduce relevant concepts as they arise. I must also acknowledge that I am myself neither a professional philosopher or mathematician. Despite my attempt to thoroughly research the topic, it is more than likely that I overlook important points (I’ll tackle the paradox of the Preface some other day). Anyone who thinks they see an error, or something that I may be missing, is encouraged to message me.
The paradoxes and the calculus solution
“Achilles and the Tortoise” – they are having a race. Achilles knows he is twice as fast as the Tortoise, and so, wanting to spare the Tortoise’s feelings, decides to give it a head start of 10 meters. It takes Achilles a certain amount of time – let’s say 10 seconds – to catch up to where the Tortoise started; in this time the Tortoise has advanced, albeit, his overall advantage has halved (he’s now at the 15-meter mark).…