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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...

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    $\begingroup$ No worries, no headaches. Because there is no proof, even at high resolutions, of any hint of a discrete nature in time. $\endgroup$ – user108787 Oct 31 '16 at 10:27
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    $\begingroup$ Compare to Euclidian geometry in which line segments of finite length are composed of points with no dimension. $\endgroup$ – dmckee Oct 31 '16 at 14:59
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    $\begingroup$ I think it must be a mistake to say "an instant has no duration", when we mean, an instant's duration is infinitessimally small and cannot be percieved. $\endgroup$ – Jodrell Oct 31 '16 at 17:02
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    $\begingroup$ "Hence, the sum of many instants would make a finite time period." many is infinite in this context. That changes a lot. It's like asking how many points is there in a line segment. Or how big is a point. When two inifinities meet the "magic" happens. Those are mathematical concepts, and you need to understand how those concepts transfer into physical reality. The infinite amount of infinitely small chucks cannot be practically translated into physical world, but they are useful as intermediate concepts, that allow us making predictions about it. $\endgroup$ – luk32 Nov 2 '16 at 11:51
  • $\begingroup$ You are asking a good question, but it's a tricky one to answer. One of the major issues is that words like "moment" and "instant" are tricky little buggers that have given philosophers difficulty for thousands of years. Calculus was developed around tools whose sole job is to provide an answer to your question. You may benefit from adding to the question what sort of philosophical or mathematical grounds you find reasonable for answers to build from. $\endgroup$ – Cort Ammon Nov 2 '16 at 17:12
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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.

  • How to count past infinity - In this video, VSauce provides an excellent introduction to infinity, and more importantly, a great discussion of all of the quirky oddities that arise when you try to capture the concept of "infinity." I find it very helpful to see just how many contortions mathematicians must go through to capture these concepts of infinities and infinitesimals without generating really boggling results.
  • Supertasks - This episode is direct to your question. It explores the concept of tasks that have an infinite number of steps yet complete in finite time
  • The Banach-Tarski Paradox - If you feel like going down a rabbit hole, the Banach-Tarski paradox is an example of just how strange the world of mathematics can get as you try to apply "obvious" claims to infinite sets. It's not required reading, but if you want to see the sorts of fits people have trying to make sense of infinities, this is about as "special" as it gets.
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  • $\begingroup$ Wow ! Thank you so much for this great answer. You basically summarized all the previous answers and gave me a great insight into the logic and thinking behind math and reality. Thank you also for your links !! Very nice subjects ! $\endgroup$ – james Nov 4 '16 at 15:17
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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.

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    $\begingroup$ I am not sure that understandig of calculus and infinete sums is sufficient for this. I think measure theory is needed. The collection of points in a given plane that are a fixed distance from a given point is a circle. - This circle has a circumference, which is a positive length. However the points existence is independent of the circle, so they are not infinitesimal arc segments. $\endgroup$ – Taemyr Oct 31 '16 at 12:25
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    $\begingroup$ This is the one that says it is theoretically impossible to shoot a tortoise with an arrow because the arrow always has to cross half the remaining distance to the tortoise before it can reach it? Such a shame about all those dead experimental tortoises. $\endgroup$ – Michael Richardson Oct 31 '16 at 15:09
  • $\begingroup$ @MichaelRichardson, not quite - the one with the tortoise is the paradoxon that you can sum up an infinite amount of non-zero numbers 0.5, 0.25, 0.125 and so on and end up with a finite number (i.e., limit = 1). $\endgroup$ – AnoE Oct 31 '16 at 16:15
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    $\begingroup$ Isn't this a simple as saying, just because an instant of time is so short that you cannot percieve its duration, it doesn't mean it has no duration. If you have an infinite amout of instants thay can combine to any time period you like. $\endgroup$ – Jodrell Oct 31 '16 at 16:59
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    $\begingroup$ @MichaelRichardson, you are mixing two paradoxes: shooting an arrow on a target AND Achilles chasing a turtle. Please do not shoot arrows at turtles, unless you really, really need to. Disproving paradoxes not included :-) $\endgroup$ – Rolazaro Azeveires Nov 1 '16 at 9:04
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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.

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  • $\begingroup$ Thank you very much for your answer. Would you mind to elaborate a bit further your assumption that an instance of time is the same a labeling a regular coordinate ? $\endgroup$ – james Nov 4 '16 at 15:05
  • $\begingroup$ @totyped, Collins English Dictionary – Complete and Unabridged: "2. a particular moment or point in time". Point = 0 dimension = not an interval. Other dictionaries also have the definition "1. an infinitesimal or very short space of time; moment." which is fine with me. I chose to pick the first interpretation in my answer because that interpretation is where the OP's confusion ("paradoxon", which it really isn't) came from in the first place. thefreedictionary.com/instant for plenty of alternative definitions. $\endgroup$ – AnoE Nov 4 '16 at 16:27
  • $\begingroup$ @AnoE both coordinates and instants could be summed: coordinates, because they are elements of a vector space (or affine if you want), and instants as measurable sets, they have null measure, and the sum of a countable set of instants is well-defined (measure is defined upon a $\sigma$-algebra) and is still null. What can't be summed up is the measure of an uncountble family of subsets, such as an interval made of points, so you can't state that the measure of each subset is null, and indeed is false. $\endgroup$ – Annibale Apr 9 '18 at 7:06
  • $\begingroup$ @Annibale: I have trouble understanding the intention behind your comment; you seem to be saying the same thing that I wanted to express in my answer. Did you mean to agree with the answer, or do you think there is some part of it which I should improve? $\endgroup$ – AnoE Apr 9 '18 at 8:33
  • $\begingroup$ @AnoE you wrote: "But the operation of "summing up instants (coordinates)" is not a defined operation because instants (coordinates) do not have a length, so there is nothing to sum up." so I want to correct your statement in this way: instants have a lenght, and is 0, and it can be summed up, but only countable times. So what fails is not the length definition of instants, but the possibility of sum them up in an uncountable family, and intervals are uncountable families of instants $\endgroup$ – Annibale Apr 9 '18 at 8:40
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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.

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    $\begingroup$ Note that there is nothing intrinsically special about the Planck Time, i.e., it does not "quantize" time (at least we don't know whether it does; it does have a special role in some theories but nothing generally accepted). $\endgroup$ – AnoE Oct 31 '16 at 14:46
  • $\begingroup$ You'r right, I've updated my answer to reflect this better. $\endgroup$ – Tom Tregenna Oct 31 '16 at 14:49
  • $\begingroup$ Well, if we say space is discrete at Planck sizes, which most theoreticians say has to be, then time has to also, from relativity you can transform them. There is no reason to think its not true in the small but becomes true above Planck sizes $\endgroup$ – Bob Bee Oct 31 '16 at 18:31
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    $\begingroup$ @BobBee I'm not aware of a survey that shows that most theoreticians in the field say that space-time is quantized at the Planck length & time scale. If you have a relevant link it'd be great if you posted it. Certainly, many theoreticians believe that space-time is probably quantized, and if it is, the scale at which quantization occurs is most likely to be roughly in the vicinity of the Planck scale, but (as yet) there's no hard evidence of this. $\endgroup$ – PM 2Ring Nov 1 '16 at 5:16
  • $\begingroup$ (cont) OTOH, there are suggestive hints that this is the case, eg, the (informational) entropy of a black hole is given by $S = kA / 4$, where $A$ is the area of the event horizon in square Planck lengths and $k$ is Boltzmann's constant, but of course that formula would not look so neat if different units were chosen to express Boltzmann's constant. BTW, space-time quantization doesn't imply that space-time is composed of 4D space-time "pixels": such a model is hard to reconcile with Lorentz transforms and Heisenberg's uncertainty principle. $\endgroup$ – PM 2Ring Nov 1 '16 at 5:17
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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.

  • $1 < 2$ then $c$ can be 1.5.
  • $1.5 < 2$ then $c$ can be 1.75.
  • $1.75 < 2$ then $c$ can be 1.8.
  • and so on...

Similarly you can divide any number into smaller and smaller and smaller pieces.

  • 1, divide it in half, you get 0.5.
  • 0.5, divide it in half, you get 0.25.
  • 0.25, divide it in half, you get 0.125.
  • and so on...

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.

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    $\begingroup$ Word of warning for the OP: this answer is describing non-standard analysis. It's a perfectly good way to think about calculus, but it's, well, non-standard; mathematicians might look at you a bit funny if you talk about "an infinitely small number". $\endgroup$ – knzhou Nov 1 '16 at 2:23
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    $\begingroup$ -1 for "You can go on doing this infinitely many times and you'll get an infinitely small number: an infinitesimal". For that to make sense you would have to explain how to do it "infinitely many times". $\endgroup$ – Martin Argerami Nov 1 '16 at 11:21
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    $\begingroup$ @MartinArgerami To actually slice something infinitesimally you'd need infinite time but you'd run up against the Planck Length first. The wonderful thing about math is you don't have to explain how to do it in the real world, you do it conceptually. Just as infinity is a consequence of the integers, infinitesimals are a consequence of the real numbers. As for any "largest" integer $n$ you can get a larger one with $n+1$, for any "smallest" real number $n$ you can always get a smaller one with $n/2$. But I think you know all that so I'm puzzled. $\endgroup$ – Schwern Nov 1 '16 at 21:24
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    $\begingroup$ @Schwern: precisely. The same way you don't have a largest integer, there is no smallest positive real number. You can create the infinitesimals, which are not numbers; they have to be created carefully, and it is usually not immediately obvious which reasonings you can translate from $\mathbb R$ to your extension of $\mathbb R$. In the particular case of continuously dividing by 2, what kind of infinitesimal do you get? And why? $\endgroup$ – Martin Argerami Nov 1 '16 at 22:34
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    $\begingroup$ @MartinArgerami I agree, but I'm not sure why this is relevant. The answer isn't intended as a rigorous proof and discussion of infinitesimals. The OP wanted an illustrative example of how you can sum up infinitely many "instants" and get something finite. I chose infinitesimals as a way to define an "instant of time" that's a pretty good combination of basic math and intuitive understanding. Are your comments the "mathematicians might look at you a bit funny if you talk about 'an infinitely small number'" that knzhou warned about? :) $\endgroup$ – Schwern Nov 1 '16 at 23:00
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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).

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protected by ACuriousMind Oct 31 '16 at 15:08

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