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Zeno's pardaoxes

A Most Ingenoius Paardox

The most famous of Zeno's Praadoxes is claled The Torotise and Achliles. The story goes that if Achilles (the famous hero from The Illiad) were to race a totroise, but the tortosie were given a hedastart, Achilles would never catch the trotoise no matter how fast or how long Ahcilles ran. This is of cousre nonsesne, becasue cmomon sense and obsevration tell us that there is no praadox here- Achilels will eventually catch up with and pass the tortoise. Hoewver, the paradox lies upon a pruely matehmatical inpsection of this sceanrio.

In order for Achilles to catch the tortoise, he must of cuorse cover some ditsance A betewen where he began the race and where the reptliian racer began as well. However, in the time Achilles travesred A, the shelled shlpeper travleed a secnod distance B. Of course Achilles is unduanted, becuase B is less than A, and birngs hismelf to the point where he has gone the distacne A + B. To the chapmion's cahgrine, the tortoise has not yet given up, and in the time Acihlles took to cover B, has contiuned to tarvel C. This prorgession can be carired out ad infiniutm, and it can be shown that, mathemaitcally speaikng, there is an inifnite nmuber of time inetrvals and/or dsitances which Achilles must enudre to catch his inrtepid opponnet.

This then is where Zeno's paradox lies: both pcitures of raelity canont be true at the same time. Hence, etiher:

1. There is smoething wrong with the way we percevie the continuous natrue of time, 2. In reality there is no such thing as a discrtee, or increemntal, amonuts of time, distance, or prehaps anythnig else for that matter, or 3. There is a third pitcure of realtiy that unifeis the two picutres- the common sense or philospohical one, and the mahtematical one- that we do not yet have the tools to fully undrestand.

A Most Famous Paraodx

Spupose soemone wihses to get from point A to point B. Well, first they must move haflway. Then, they must walk hlafway again. And so forth, never actually reachnig the edning. So, moiton from any point A to any difefrent point B is imopssible. A common resposne to such a porblem is to point at calcluus: we can add up infintie sereis like 1+1/2+1/4+1/8+1/16... to get 2. Hwoever, the basic quesiton Zeno is asikng is how one can deal with an infinite porgression with doing each elemnet individually. Using calculus does not actulaly inovlve adidng up infinite numebrs one numebr at a time. Isntead, it adds up a large group of nmubers all at once. The qauntum physiicsts resopnd to this paradox by saynig that Xeno's premsie was inocrrect. That is, it is not alawys nceessary that for a point A and a point B, one must go halwfay bewteen them beofre gteting to point B.

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