On some rare days when the sun is shining and I'm enjoying a well made kouign-amann (my favourite comes from b.patisserie in San Francisco but Patisserie Poupon in Baltimore will do the trick), I find a brief respite from my usual depressed state and take delight, if only for a brief moment, in the fact that mathematics completely resolved Zeno's paradox. To me, it is the quintessential example of how mathematics can fully solve a philosophical problem and it is a shame that most people still don't seem to know or understand this monumental fact. Although there are probably thousands of articles on Zeno's paradox on the internet (I haven't bothered to actually check), I feel like visiting it again today even without a kouign-amann in hand.
I don't know what the original statement of the paradox is but they all involve motion from one location to another like walking towards a wall or throwing a javelin at a target. When you walk towards a wall, you must first cross half the distance, then half the remaining distance, and so on forever. The paradox is thus: How then can you ever reach the wall, or a javelin reach its target, if it must traverse an infinite number of intervals? This paradox is completely resolved by the concept of the mathematical limit, which Newton used to invent calculus in the seventeenth century. I think understanding the limit is the greatest leap a mathematics student must take in all of mathematics. It took mathematicians two centuries to fully formalize it although we don't need most of that machinery to resolve Zeno's paradox. In fact, you need no more than middle school math to solve one of history's most famous problems.
The solution to Zeno's paradox stems from the fact that if you move at constant velocity then it takes half the time to cross half the distance and the sum of an infinite number of intervals that are half as long as the previous interval adds up to a finite number. That's it! It doesn't take forever to get anywhere because you are adding an infinite number of things that get infinitesimally smaller. The sum of a bunch of terms is called a series and the sum of an infinite number of terms is called an infinite series. The beautiful thing is that we can compute this particular infinite series exactly, which is not true of all series.
Expressed mathematically, the total time
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-4EINne.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-cIQkBy.png)
which can be rewritten as
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-IP9v_1.png)
where
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-7ywzgl.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-M7MDW5.png)
For simplicity we can take
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-t0jWUN.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-WxRKBD.png)
To solve this sum, the first thing is to notice that we can factor out
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-04vhah.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-zIlvRC.png)
The quantity inside the bracket is just the original series plus 1, i.e.
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-fRlQHK.png)
and thus we can substitute this back into the original expression for
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-StVpNe.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-YSaxgQ.png)
Now, we simply solve for
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-H5eCdz.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-3Ntswo.png)
Now, subtract
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-AX6uvw.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-p7hO2y.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-cDbEoo.png)
I never get tired of this. In fact this generalizes to any geometric series
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-s0zjzy.png)
for any
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-2gynLi.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-Aa2uWE.png)
Now, notice that in this formula if you set
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-f8ZFHu.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-zM1IEa.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-iYGEx2.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-ciUoMH.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-TRGl53.png)
![Math solved Zeno’s paradox Math solved Zeno’s paradox](https://m5.paperblog.com/i/258/2585581/math-solved-zenos-paradox-L-XuvIxM.png)