TITLE: How was Zeno's paradox solved using the limits of infinite series?
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: 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.