If we say that an instant of time has no duration, why does a sum of instants add up to something that has a duration? I have a hard time understanding this.
I think of one instant as being a 'moment' of time. Hence, the sum of many instants would make a finite time period (for example 10 minutes).
EDIT: Since I got so many great answers, I was wondering, if someone can also give a illustrative example, besides the pure math ? I am just being curious...
I believe you're asking about a paradox in the style of Zeno's paradoxes. Your paradox is most similar to 'Paradox of the Grain of Millet'. You want to know how an infinite sum of infinitesimal instants could equal a finite length of time, $$ \int~\mathrm dt = t. $$ Well, the above is nothing but $$ \int~\mathrm dt =\lim_{N\to\infty} \sum_{n~=~1}^N t/N = t, $$ This, and many of Zeno's paradoxes, are resolved by understanding calculus and infinite sums.
If we say that an instant of time has no duration,
It does indeed have no duration, by definition. The word "instant" has no special physical meaning here. It means the same as labeling regular X/Y coordinates on a sheet of paper with a number.
Such a coordinate (be it in space or in time) is something fundamentally different than an interval between two such coordinates: the coordinate does not have a length, while an interval does have a length. The coordinate does not even have length 0, it has no length defined.
why does a sum of instants add up to something that has a duration?
This is where you are mislead. "Instants" (i.e., coordinates) are not summed up. Intervals are, by summing up their lengths. But the operation of "summing up instants (coordinates)" is not a useful operation here because instants (coordinates) do not have a length, so there is nothing to sum up.
N.B.: if you want to find out about the "real" mathematics behind this, check out Rieman Integrals, measure theory, "countably infinite", "almost everywhere" and such terms. There is a fascinating world behind what you discovered.
N.B.: as mentioned in the comments, you can of course indeed sum up "instants" (i.e., intervals of length 0), but only countably many, and a sum of countably many 0 is still 0. In the lazy formulation in my last paragraph, this is taken into account.
Perhaps it's useful here to differentiate between a specific time (as in, a one dimensional representation of a specific instant or location in time) and a duration, which is the measure of difference between two specific times.
In this case hat you refer to as a summable 'instant' may actually refer to a delta of duration, for example the Planck Time - named after physicist Max Plank, which is the amount of time it takes a photon to travel the Planck Length, which according to physlink.com is
roughly equal to $1.6 × 10^{-35}m$ or about $10^{-20}$ times the size of a proton
Which makes the duration of Planck Time equal to
roughly $10^{-44}$ seconds
I offer this explanation merely as an interesting aside to the already accepted answer - which I think probably better addresses your question.
Phrases like "instant of time" are tricky. If you're not careful, you can dive down rabbit holes that lead to pedantic questions like "what does 'is' mean?" which do little but frustrate people. You've got a good question, just be prepared for the answers to be a little more dodgey than you might like.
One of the major keys to the question is in AnoE's answer. AnoE draws a distinction between an instant of time and an interval of time. This distinction is very useful because intervals of time tend to be pretty well behaved in our minds. We're really comfortable that 2 intervals of 1 second, when put end-to-end is an interval of 2 seconds. We can then add another interval of 1 second to our 2 second interval to get 3 seconds. 3 and 1 can make 4, and so on. Nothing suspicious here. We can even postulate that this process can go on "forever," even if we don't have a very strong definition of "forever" to work with. None of our concepts have ever been proven out as they stretch to "forever" because nobody has lived that long!
What about the other direction? If I have a 2 second interval, I can subdivide it into a pair of 1 second intervals. I can take one of those and break it up into a 0.5 second interval. That can be broken into 0.25 second intervals and so forth. It seems like you can repeat this process, just like we could repeat the addition of intervals, until we get to arbitrarily small intervals.
But what happens if we go further? What happens if we keep subdividing until the intervals are "infinitely small." As it turns out, I have to put "infinitely small" in scare quotes because that concept turns out to be really tricky to assign an exact mathematical meaning. Zeno's paradox was mentioned by innisfree. It's the idea that, if I were to try to walk from one point to another, I would have to travel half of that distance first, then I could try to walk to the endpoint. However, once I get to the endpoint, I also have to walk half way between that half-way point and the end point (a three-quarters point). You can repeat this process to show that it would take me an infinite number of steps to reach you. Zeno argued that that means you can never reach anywhere!
Of course, anyone over the age of 3 will point out that Zeno is wrong. We go places. It happens (or at least appears to happen, if you're a hyper skeptical individual). So there must be something wrong with the argument. The truth of the matter is that Zeno was not actually suggesting that we are incapable of moving, he was pointing out a fundamental issue with the way we conceptualize the world around us. He was pointing out that our models suggest the world acts in a way which it demonstrably does not.
You see lots of mathematical answers here because mathematicians were really the only one that grappled with infinity in a rigorous enough way to actually have to worry about Zeno's paradox. Most people could just say "oh, it works because... we see that it works every day." Mathematics and reality aren't divorced, but sometimes it sure looks like they're in a trial separation! They call things like Zeno's paradox "supertasks." These are tasks that take an infinite number of steps to complete.
And the mathematicians went forth and analyzed these supertasks. Over hundreds of years, they came up with incredibly powerful tools. For this particular topic, Calculus is the crowning achievement of their efforts. Calculus is a very rigorous way to handle these infinite numbers. More importantly, it is an example of where mathematics and reality agree -- the hypotheses put forth using calculus have an excellent track record of being provable in real life!"
And so we have tools to formally capture these concepts of an "instant of time." We can even aggregate them together into intervals using a calculus tool called "integrals." However, when we do so, we'll see an equation like $\int f(t)dt$, where f(t) is the value of f at "an instant in time." However, we can't ignore the "dt" part of that integral. It's the notation we keep around to remind ourselves of how calculus is actually handling intervals. You can get away with thinking of the "dt" as an "infinitely short interval," as long as you only do so formally. It has a very particular formal meaning that's slightly different. However, the key to all of this is that your intuition lines up with what the mathematics actually do. When you walk somewhere, you actually get there!
So it will be very hard to find a non-mathematical example of summing instants of time into something that has a duration, because it's very hard to capture the concept of "an instant in time" with enough formality to talk about summing them together outside of mathematics. I know of no other field which has a level of rigor sufficient to truly explore this concept. Philosophy might, though their approach may be very different.
I recommend two vsauce videos if you want to learn more. He's my personal favorite way to learn about mathematics, and especially tricky concepts like infinities and infitessimals.
Since I got so many great answers, I was wondering, if someone can also give a illustrative example, besides the pure math?
What you're looking for is an infinitesimal and why they're useful. Calculus and integration says that if you sum up an infinite number of infinitesimally small slices of a thing, you'll get the area of the thing. This turns out to be really useful to calculate things that change like curves or acceleration.
An infinitesimal is the idea that for any $a < b$ there is always a $c$ where $a < c < b$. You can always fit a number between two other numbers.
Similarly you can divide any number into smaller and smaller and smaller pieces.
You can go on doing this infinitely many times and you'll get an infinitely small number: an infinitesimal.
But it's not 0! It's the smallest possible number which is not 0. If you add them all up you get the original thing. That's integration. That's calculus.
But it's not a number! It exists nowhere on the number line. Like infinity, an infinitesimal is a concept. And because it's not a number, things get weird of you try to do normal math on it.
So an instant of time does have a duration. It has an infinitesimally small duration. It's the smallest slice of time which is not 0.
Here's Dr James Grime of Numberphile talking about infinitesimals and demonstrating their use.
I also highly recommend A Tour Of The Calculus by David Berlinski. It takes a literary approach to the history and purpose of Calculus. Even after having taken years of pre-calc and calculus, it gave me a deeper understanding and appreciation.
That's the math, but this is Physics.SE. How does this work out in reality? Can you have an "instant of time"? AFAIK there's no limit to how thinly time be sliced. Even the Planck Time puts no limit on how small an "instant of time" can be. We know of no quantum of time. And yet here we are.
For more on the physics of time, and why we even have time, you might be interested in MinutePhysics's ongoing series about Time & Entropy.
I believe your confusion comes as the result of not understanding what an "instant" means/implies. Lets start by clearing up the source of your misunderstanding. "...we say an instant of time has no duration...", this is not true! No mater how small you make your "instant," it is not zero. Therefore, since it is not zero, if you sum a number of them, you will have a larger (finite) interval (of time).
