How was Zeno's paradox solved using the limits of infinite series?
This is a not necessarily the exact paradox Zeno is thought to have come up with, but it's similar enough:
A man (In this photo, a dot 1) is to walk a distance of one unit from where he's standing to the wall. He, however, is to walk half the distance first, then half the remaining distance, then half of that, then half of that and so on. This means the man will never get to the wall, as there's always a half remaining.
This defies common sense. We know a man(or woman, of course) can just walk up to a wall and get to it.
My calculus book says this was solved when the idea of a limit of an infinite series was developed. The idea says that if the distances the man is passing are getting closer and closer to the total area from where he started to the wall, this means that the total distance is the limit of that.
What I don't understand is this: mathematics tells us that the sum of the infinite little distances is finite, but, in real life, doesn't walking an infinite number of distances require an infinite amount of time, which means we didn't really solve anything, since that's what troubled Zeno?
Thanks.
It is often claimed that Zeno's paradoxes of motion were "resolved by" the infinitesimal calculus, but I don't really think this claim stands up to a closer investigations. It depends on very specific notion of what it would mean to "resolve" the paradox, and it looks to me that this notion can only be arrived at if one deliberately sets out to reformulate the paradox into a problem that
- Is somewhat related to the classical description of the paradox, and
- feels non-trivial enough that it's a problem worth solving, yet
- is simple enough that contemporary mathematics has a solution to it.
In this way the original "therefore Achilles never overtakes the tortoise!" becomes tacitly transformed into "WHEN does Achilles overtake the tortiose?" and once we get a numerical answer to the latter, the popularizers will claim that the original contradiction has been resolved.
Actually, the first step in resolving the paradox must be to understand why it was supposed to be paradoxical in the first place. Zeno's paradoxes rely on an intuitive conviction that
It is impossible for infinitely many non-overlapping intervals of time to all take place within a finite interval of time.
Only if we accept this claim as true does a paradox arise. But why should we accept that as true? If we reject it, then the paradox disappears -- and we don't need any infinitesimal calculus, or any concept of limit, to deny this premise. On the contrary, it must be up to anyone who wants us to accept it to provide some kind of argument -- and then he has the trouble of explaining why his pet principle doesn't conflict with reality, as demonstrated by Zeno!
Part of the story here is that the historical record is too fragmentary for us to know what the historical Zeno actually intended with the paradoxes. None of his own writings survive, and all we have to go by is other classical authors, who discussed the paradoxes for their own purposes, and just gave credit to Zeno for coming up with the example.
On this sparse record, it is entirely possible that Zeno never intended to convince anyone that motion is impossible. Instead, it is easy to imagine that, for example, one of Zeno's pupils may have tried to use the indented principle above in an argument, and then the master immediately shot that down by using the "paradox" as a vivid practical example of how there can be infinitely many instants or intervals in a finite period of time.
"mathematics tells us that the sum of the infinite little distances is finite, but, in real life, doesn't walking an infinite number of distances require an infinite amount of time?"
no it does not. Why do you think that it would? That's the whole point. If the distances you walk are small enough, you can do that in a finite time (in real life!)
Mathematics is an abstraction, true, but most of the time it is developed to explain (some parts of) reality. You seem to think that what goes on in mathematics has no bearing with "real life".
If the subject walks at a constant speed, you don't need calculus to calculate exactly when he will arrive at his destination. Just use the simple speed-equals-distance-over-time formula. This formula was unknown to the ancient Greek philosophers. They had no notion of actually measuring speed. It wouldn't be for another thousand years after Zeno's time that Galileo would formulate:
$$s=\frac d{t}$$
Today, we know that in going from point $A$ to another point $B$, we will pass through infinitely many other points along the way. If we associate an event with our arrival at each of these points, then infinitely many events will have occurred in the interval. The modern mind has no trouble with this notion.
Zeno's paradox, today, is mostly irrelevant, other than marking an important stage in the history of philosophy. Let's formulate it in its essential form:
If we sum infinitely many parts, the result can't be finite.
It makes use of the concept of infinity in its premise but there is no such thing as infinity. Zeno does not know what infinity is, we don't know what infinity is. By using such a vague term in his argument's premise, Zeno sacrifices rigor even before starting. In short, it is a bogus argument from the get go.
Taking a look at one example of the argument, namely tortoise and the hare, Zeno says "tortoise needs to advance the half of the way first, then half of the remaining way...". This in turn is problematic. Zeno does not believe these infinite parts can make a finite whole but he has no problem of chopping down that same finite whole into infinitely many parts in the first place. How can he know this is possible? All sorts of bogus reasoning bound to follow once you start to tinker with infinity. In short:
There is no such thing as infinity. Even if there is, it is only a metaphysical concept and can not really be used in physics and geometry in a computational way.
In calculus, we don't talk about infinite sums per se. Take a look at the sum:
$$ 1 + 1 / 2 + 1/4 + ... = 2$$
Those three dots could make you think that we are summing infinitely many terms. In reality, what calculus tells you is the following:
$$\lim_{n \to \infty}{\sum (1/2)^n} = 2, n \in N$$
and this has nothing to do with infinity. It tells you that you can make this sum as close to $2$ as you need. More rigorously, for all $\epsilon > 0$, there exists an $N$ such that
$$\mid{\sum (1/2)^n} - 2 \mid < \epsilon$$
for all $n > N$. The use of $\infty$ in limit above is just a notation, not to suggest a metaphysical use of the concept. It just tells you that you can make this sum as close to $2$ as possible, if you so desired. This in turn suggests that you can use these symbolic-analytical tools of calculus in your calculations for similar observations. The sum above was already $2$, clear from the observation, irrespective of any mathematics. Calculus just gives you tools that you can employ when thinking about such observations (measurements). Imagine the following computer program:
sum = 0
n = 0
epsilon = 0.00000002 // your choice of epsilon
while absolute_value(sum - 2) > epsilon:
sum = sum + (1/2)^n
n = n + 1
print n
Calculus tells you that this program will terminate and print some number, no matter how small you chose epsilon to be.
On the other hand, completely unrelated to the phenomenon of infinity, there is a physical observation; that the hare actually catches the tortoise. We want to compute the amount of time required for hare to catch the tortoise. Calculus just gives you a way to do this computation, again it has nothing to do with infinity. Computation is already finite, the result is already observed to be finite.
Zeno believed he had grasp of some abstract concepts and used his bogus reasoning about these abstract concepts to deny physically observed phenomena. Calculus is doing the other way, it constructs abstract concepts to explain (to a desired degree of exactness) physically observed phenomena. Philosophically speaking, what Zeno did is a fundamental error in thinking: trying to base physics onto metaphysics. This is bound to fail. This is the reason why Metaphysics was the last book in the shelf of Aristotelian curriculum. Aristotle clearly knew that you need to base your metaphysics onto physics, not the other way around. That's why you had to study physics before metaphysics, if you wanted to be a student of Aristotle (or more like peripatetic school of philosophy). Metaphysics, the word itself, means literally "after the physics" in Greek. This is the reason. But of course, metaphysics was still the "first philosophy", philosophy of first principles. This meant its truths were foundational to all else, even though we arrive at those via physical phenomenon, sensation and observation (by abstracting and conceptualizing on those).
So in short, Zeno's paradoxes were not paradoxes but were just errors in his thinking. It was not evident at the time since humans had more vague notions of concepts like number, measurement, infinity, time, motion etc. Calculus is not resolving this so-called paradox, it does something entirely different. Along the way, calculus refines our understanding of above-mentioned concepts to a point that we realize there was not an actual paradox in the first place.
"but, in real life, doesn't walking an infinite number of distances require an infinite amount of time,"
Not if the amount of time it takes to walk each distance is proportional to the distance. The sum of the times, no matter how we split it, even infinitely, will be proportional to the sum of the distance. i.e. always the same and finit.
But I don't think the issue is time; it could just as easily be claimed my wall in an infinite distance away from my because it's an infinite sum of finite distances. I think the issue isn't how can an infinite sum not be infinite, but how can anything finite contain an infinity.
I like to think of a hopping flea hopping across a table. Each jump takes him half-way to the edge and he does each jump in half the time of the previous one. So there'll be no point in time where the flea reaches the end of the table and says "at last". At one point of time the flea will be struggling with dim prospect of an infinite number of jumps ahead of her, and the very next moment (ha! see what I did there?) the flea will be completely done. but at exactly 2. Well, we really don't know what precisely two looks like.
I'm not sure that infintismals and limits actually "resolve" these issues. In a way, they seem simply to be a way of saying "now, settle down, kids; we know that we can split a finite amount into an infinite sum of decreasing amounts, so it shouldn't shock us that and infinite sum of decreasing amounts will add to a finite amount-- so let's quit trying to be smart-asses and stoners about it and start trying to use this as a practical tool".