Math solved Zeno’s paradox

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 t it takes for an object traveling at constant velocity to reach its target is

t = \frac{d}{v}\left( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots\right)

which can be rewritten as

t = \frac{d}{v}\sum_{n=1}^\infty \frac{1}{2^n}

where d is the distance and v is the velocity. This infinite series is technically called a geometric series because the ratio of two subsequent terms in the series is always the same. The terms are related geometrically like the volumes of n-dimensional cubes when you have halve the length of the sides (e.g. 1-cube (line and volume is length), 2-cube (square and volume is area), 3-cube (good old cube and volume), 4-cube ( hypercube and hypervolume), etc) .

For simplicity we can take d/v = 1. So to compute the time it takes to travel the distance, we must compute:

t = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}\cdots

To solve this sum, the first thing is to notice that we can factor out 1/2 and obtain

t = \frac{1}{2}\left(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}\cdots\right)

The quantity inside the bracket is just the original series plus 1, i.e.

1 + t = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}\cdots

and thus we can substitute this back into the original expression for t and obtain

t = \frac{1}{2}(1 + t)

Now, we simply solve for t and I’ll actually go over all the algebraic steps. First multiply both sides by 2 and get

2 t = 1 +t

Now, subtract t from both sides and you get the beautiful answer that t = 1. We then have the amazing fact that

t = \sum_{n=1}^\infty \frac{1}{2^n} = 1

I never get tired of this. In fact this generalizes to any geometric series

\sum_{n=1}^\infty \frac{1}{a^n} = \frac{1}{1-a} - 1

for any a that is less than 1. The more compact way to express this is

\sum_{n=0}^\infty \frac{1}{a^n} = \frac{1}{1-a}

Now, notice that in this formula if you set a = 1, you get 1/0, which is infinity. Since 1^n= 1 for any n, this tells you that if you try to add up an infinite number of ones, you’ll get infinity. Now if you set a > 1 you’ll get a negative number. Does this mean that the sum of an infinite number of positive numbers greater than 1 is a negative number? Well no because the series is only defined for a < 1, which is called the domain of convergence. If you go outside of the domain, you can still get an answer but it won’t be the answer to your question. You always need to be careful when you add and subtract infinite quantities. Depending on the circumstance it may or may not give you sensible answers. Getting that right is what math is all about.

4 thoughts on “Math solved Zeno’s paradox

  1. 1. This geometric series is one of my favorites (partly because its one of the few i can remember how to solve).
    I also like 1+2+3+4+.. , 1-2+3-4+… exp^x… sum (a_.i^n ) and permutations
    (I can’t even remember all the terms like arithmetic, geometric, harmonic…mean. Even ‘worse’ is when one has equations and theorems named after people.)

    (Many of these series are connected to formulas partly due to Ramanujan–I forget the site —maybe ‘mathstackexchange’ — they have whole lists of integer or other sequences.
    (Wikipedia also has many, as does the ‘handbook of math functions’ by Abramowitz and Stegun.
    The ones i mostly look at have to do with decompositions of a number into primes.

    Looking at these reminds me of looking at the mountains in the Himalayas (i went to Ladack once–part of Kashmir, where again there is another conflict–i think muslims vs hindus) and Alaska/Canada along the Yukon river in winter.
    You see all these snowy mountains that look like ice cream cones , and say ‘that looks nice but I dont think i can go there’. (Besides, in Alaska it was –45F).
    Professional climbers do but that takes equipment, training, and alot of money and help usually. I dont climb anything more difficult than what you find in West Virginia which are hard and cold enough (can be -15F or at times coder.) (In Ladack you can get to maybe 20,000 feet before you hit the ice and cliffs and have to turn back. In Ladack i tried once to go visit Tibet–just over the border—i followed the trail through the snow the Tibetans used but didn’t get far–military stopped me.I also wasnt weaing any shoes—i figured ‘ i’m american and we’re exceptional so if Tibetans can walk with no shoes i easily can’. Not exactly the wisest idea but typical of my hiking ideas–eg ‘travel light’—so if its not too cold 1st thing you do is hide your tent, sleeping bag, extra clothes and most food–one can probably find food on the way and pick up your stuff on the way back . I sometimes have decided maybe i should have carried my stuff–misery now or later.)

    2. I once had a series that came out of some combinatoric ‘puzzle’ or ‘paradox’ (actually involving the 2nd law of thermodynamics) which ended up with a series (involving prime numbers). I didnt have a calculator so each day i’d calculate a few more terms to see if my algorithm ‘worked’. I think i got out to maybe 20-30 terms and it always worked. I wanted a ‘closed form solution’ (not a recursion) and looked in Abramowitz and Stegun but didn’t find it.
    (I have a feeling its actualy an easy problem which has been solved but I didn’t find the solution–or it may not exist.)

    Some of these formulas actually lead to probability distributions that look like variants of the Boltzmann distribution. You sort of have the ‘temperature’ of the number sequence. Ramanujan has a famous one which turned out be false–failed after a few million terms–so someone found a correction maybe 10-20 years ago (on arxiv).Sort of like an extra term in the ‘boltzmann distribution’.
    This might be like setting your GPS to take you to some distant planet. You get out a few billion miles and find out your algorithm took you to the neigborhing planet which is uninhabitable and once you get in its gravitational field you can’t get to the nearby one you wanted to go to–unless you install the correction factor.

    . Zeno’s paradox i call the tortoise and the hare.
    hare is racing the tortoise and gives it a head start. I like variants or counterexamples when the hare can’t catch the tortoise . For example, suppose the tortoise reproduces along the way -so then you have a whole lot and increasing number of tortoises (the baby tortoises also reproduce) all racing with the hare–and they take off on different routes.
    Maybe hare can’t beat all of them. (This may be realted to Goodstein’s theorem and the busy beaver problem.)
    There are also possibly ‘ultrafinitist’ variants. this may be an example
    (i dont really understand the paper and just found it—the author seems to be ‘real’ but also have some ‘out there ‘ ideas (also involving quantum theory .

    I think ultrafinitists deny the existence of most (ie irrational) real numbers and infinity.

    I think there are versions of pjhysics that also assume space-time-energy-information is discrete.
    (The old native american idea that ‘its turtles all the way down’ –eg you reduce quarks and electrons into their component turtles, and decompose those into smaller turtles , ends with the smallest turltle known to Man.
    I used to catch snakes basically to look at and maybe keep a few days to show people and then let them go . Ii call these 3 day pets
    I was always somewhat interested in the biggest snakes–maybe up to 7 feet was biggest i caught–and smallest–like a 4 inch snake .I’ve also seen snapping turtels as big as maybe 3 feet including the tail and as small as maybe 2 inches. I call the small ones ‘wannabe snakes and turtle’–because they act as hostile as the big ones but can’t back it up. ) I still catch these at times(its sort of an instinctive habit like a cat and a bird or dog and squirrel) though i decided its basically cruel–most very much object to being picked up.
    I caught one big water snake a few weeks ago—-previously it didn’t seem to mind but this time it ripped me up. It was the last really warm day—and perhaps it knew it didnt have time for a visit. I’m still trying to figure out if i’m going to have to see a doctor –i sort of wish it would just fix itself.

    Edward Nelson of Princeton was also an ultrafinitist. (He had a ‘stochastic mechanics theory’ which was bascially a ‘hidden variables theory’ but I think he acknowledged it didn’tget around Bell’s theorem. . He also tried in his last days to prove Peano arithmatic was inconsistant. Terrance Tao i gather found a flaw in his proof so he retratced it. (apparently used an axiom of infinity to prove infinity didn’t exist.)

    My variant of the zeno paradox might be solvable by a ‘super turing ‘ or quantum computer. the hare can race all the reproducing tortoises at the same time and beat all of them.

    (another version has achilles racing the tortoise to the galapagos islands. the turtles is going to visit relatives and to warn them acchilles is coming. he’s going to sell half the turtles to pet stores and eat the other half. depending on the structure of the mathematical universe the the turtles will either survive or else go extinct.


  2. Hello;

    Zeno’s paradox is indeed a little marvel in its own right. It seems all too simple at first, but it’s got layers and layers of depth that can be brought up.

    One such layer is to consider the physical phenomenon zeno is discussing : the observed fact that walking from point A towards point B will get you there. It’s a scientific fact (omitting of course trivial scenarios with obstacles and such). From this perspective, the formal fact that the corresponding geometric infinite series converges is not an explanation of the latter. It is just an equivalent representation.

    Think of it this way : suppose we don’t actually live in the physical world, but in the abstract world where “walking” refers to the act of adding stuff that get infinitely smaller and smaller. Again, in such a world, the convergence is just an observed phenomenon. Now, if we were to come up with the analogy of a man from the physical world walking from point A to point B, would it really be an explanation of the observed phenomenon ?

    Here’s then what I believe : The two questions (1., why do we always get to B when we walk towards it from A, and 2., why do the sums of infinite series that get smaller converge) are actually equivalent. They are two representations of the same problem.

    It might seem like a bit of a stretch. Mathematics, however, is not “unique”. If we meddle around with its axioms, we might just find a new mathematics where the familiar convergent series diverge. Hence, as long as we can construct a mathematics in which those series diverge, our mathematics is not necessarily a “resolution/explanation” of Zeno’s paradox (until we can prove that no such mathematics can exist –but again, according to what criteria ?). In this regard, Zeno’s paradox is a sort of calibration test between two àpriori non-unique worlds, the physical and the abstract. There is indeed nothing that inherently forbids other worlds fundamentally different from ours to exist (even physics has made peace with this possibility). Likewise, it would be completely arbitrary to claim that mathematics is unique, even if we live in a multiverse. The interesting bit however is that, from an exterior perpective to both (indulge here, please) there isn’t even an àpriori property that could distinguish between this two types of worlds, in my opinion (although, I still lack arguments to support this belief). It means the multiverse isn’t necessarily “homogeneous”; it could be inhabited by physical as mch as abstract worlds –and they walk in pairs somehow. And we luckily found which mathematical world is parrallel to ours; Zeno’s paradox is one indication of it.

    There is one other layer still, which I believe Zeno were perhaps thinking about it but couldn’t quite formulate it (?). That is scale transformation.

    Let me reformulate Zeno’s paradox : I’m at A. I start walking towards B. I first cross half of the distance. At each distance crossed, I shrink proportionally. At half the distance, I’m half my initial size. You’d see that at each step, I go back to the same initial state : suppose I’m crossing 4 meters, at a speed of 2 meters/sec. So it should only take me 2 seconds to cross. However, at half the distance, I’ve shrunk to half my size and so my step : I now walk at a speed 1 meters/sec and have 2 meters to cross. See that this new situation is exactly the previous one : I have to cross a distance at a speed of half that distance per second. And so on and so fourth. (Note : you can think of it as my world getting bigger at the same rate of my walk).

    When we introduce scale transformation, we’ll see the true mystery of Zeno’s paradox. The observed fact then, and the convergence of the corresponding mathematical representation, is only possible in the case of scale invariance.
    So a new questions arises : why do we not recede to infinity as we move ? Or rather, does our world admit a spatio-temporal limit (in that, at some point you cannot shrink anymore as you walk) ?


  3. @Atsa Thanks for the post. The article is technically correct although a bit fussy. It just says you also have to assume that velocity is constant all the way to infinitesimal scales. I think that’s pretty obvious and doesn’t need a modern theory of physics. So if the time it takes to cross a divide scales with the length then the same result holds. I’m still sticking to math solved Zeno’s paradox or at least provided all the tools to make it not a paradox.


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