If I shoot an arrow at a target, at some point it will reach one half of the distance to the target. Then it will reach one half of that distance. It will continue to reach the half of the previous half forever. So, how does it ever reach the target?
There must be some distance so small that it can't be divided in half. That distance is zero.
The time it takes for the arrow to reach one half of the distance, is one half of the time.
So the total length traveled by the arrow is one half the distance, plus one half of one half, plus one half of one half of one half, etc. This sum is a convergent sequence: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1. In total the arrow traveled exactly the whole distance, even though we have decomposed the traveled distance into an infinity of parts in our minds.
How much time did it take for the arrow to travel all these parts?
The time spent by the arrow on a given length of the travel is proportional to the length of that travel. If the target is 100 meters away and the arrow travels at 1 meter per second, then it will take half a second for the first half, then a quarter of a second for the next half of a half, then an eighth of a second for the next half of a half of a half, etc. The smaller the distance, the smaller the time. The total time spent by the arrow is the sum of the convergent sequence: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1. In total the arrow spent 1 second traveling the 100 meters, as expected, even though we have decomposed the travel time into an infinity of parts in our minds.
So, no paradox. Both the travel distance and the travel time can be decomposed into an infinite number of parts, but the parts get smaller and smaller, and their sum is finite, which means that after a finite time the arrow will have traveled the finite distance as expected.
Is there a distance so small it can’t be further divided?
The modern solution to this problem is the use of infinitesimals, as used by Leibniz and Newton in their development of the calculus. Infinitesimals are itsy-bitsy (being technical for a moment), and still controversial. See The Opposite of Infinity, by Numberphile on YouTube.
There does not have to be some distance so small that it can't be divided in half, to solve the paradox.
The infinitesimals referred to by Mark Andrews become the differentials in calculus as first invented by Leibnitz and Newton. With differential calculus, the so-called paradox you cite is easily shown to be not a paradox at all, but a perfectly solvable math problem.
Zeno did not have differential calculus at hand and thought he had disproved the reality of motion. He was wrong.
I do not wish to leave this post in original format as comments have made clear that my assumption was not totally accurate, see below.
In physics the shortest possibly lenght is called the Planck length.
Quantuum mechanics is a very well researched field, but cannot today be combined with general relativity. One way, I believe not beeing a specialist, to attack that problem is to postulate that there is a minimum length, a quantized size. The Planck units include not only lenght but also time, energy and temperature as piezes in a currently unsolved puzzle trying to join GR and QM.
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There is no consensus on whether length is quantized, ie that there is a shortest lenght or not. It seems like most lean towards that lenght is not quantized, ie that there is no shortest lenght as such. What seems to be a consensus is that there is a shortest lenght that can be measured. In effect this shortest lenght probably could not "meaningfully" be subdived into parts that can be measured which sort of answers the original question.
Regardless, this has moved from beeing a philosophical question into a physics discussion regarding the very front of theory where the consolidation of Quantuum mechanics and General Relativity is currently (well, since quite a few years) attempted. Sorry for taking your time.
Epsilon is the smallest value that can be added to a number such that n+ε>n. In practice I have mostly seen it used in computer modeling to solve the case that when you have material that is 1 unit thick and you drill a hole into it that is one unit deep that the hole might not go all the way through my instead drilling a hole that is one plus epsilon units deep, it will go all the way through.
double-precision, over a limit range so they're fairly close together, at least in the smaller parts of that range, making them usable as a discrete approximation of the continuous space of real numbers. This question is asking about actual physical space, not limitations of fixed-width number representations in computing that we use to approximate reality.
Commented
Sep 24, 2023 at 1:34
2^exp * mant, so they have the same relative precision over the whole range; the decimal equivalent is scientific notation with a fixed number of significant figures but any exponent. For example if you had 5 decimal digits, adding 1.0000 * 10^-9 to 1.0000 * 10^0 would round back down to 1.0, so we'd say that 1e-9 is less than epsilon. Similarly, 1e13 + 1e6 would be lost to rounding error; floating point has the same relative precision at every exponent so the representable numbers get farther apart.)
Commented
Sep 24, 2023 at 1:41
As far as natural numbers go, when they are represented as distances, they all admit of being broken in half. Zero is the sort-of-exception, except that 0/2 is determinately evaluable (unlike division by zero). But other things being equal, adding an infinite number of zero-length distances will not yield a nonzero distance since 0 + 0 + 0 + ... + 0 = 0. So though zero is trivially infinitesimal, it is not nontrivially infinitesimal, and will not by itself solve the given problem, regardlessX of how many times it is self-composed.
Bringing in transfinite ordinals will not be enough, either: the surreal infinitesimal 1/ω is divisible by 2, as 1/2ω, and so on. Or 1/ω > 1/ω1, which is > than 1/ω2, and so on, all the way to 1/V (for V = an entire numerical universe), which arguably does flatten out if 1/V/V = 1/V2 = 1/V, etc. But using V as a bona fide number is a costly maneuver, and few are daring enough to claim that physical spacetime is absolutely infinitely divisible, but the tendency is to think that the Continuum is relatively infinite instead, and hence not the size of an especially robust numerical universe.
Algebraically, we could of course define a number, let's call it e, which does not equal 0, but which is such that e/2 = e, and this would be some flavor of infinitesimal, seemingly indivisible, except that we might go on to define further numbers, let's called them E's, such that e/E doesn't equal e "just like that," and so on and on.
XAlthough, to be sure, an adventurous or creative set theorist might define some large cardinal number k such that "if 0 is added to itself k-many times, the result is > 0," which might be made to work in some exotic, if perhaps silly, region of the set-theoretic multiverse.
If you want an answer from physics, rather than philosophy, you're talking about The Planck Length which is about1.616255×10^{−35m}.
This is the distance light (or strictly speaking causality) covers in one unit of Planck Time, which is the theoretically smallest possible measurement of time.
Now, these are the points where our current understanding of physics indicates that there's nothing meaningful that can be measured below this. It isn't, however, to our knowledge and inherent "pixel size" of the universe. Although, it is possible that a GUT could find in the future that spacetime itself is quantized into these units (we just don't know that yet).
If you want a mathematical answer to Zeno's Paradox specifically, your answer lies in calculus, specifically in limits.
I'm not going to reproduce the full proof here, but it's trivially possible to prove that the geometric series $\sum{1}{\inf}{\dfrac{1}{2^n}}$ is equal to one.
Just because this is unintuitive, doesn't mean it's a paradox. Zeno just didn't have calculus. Unfortunately, the universe (and maths) doesn't owe us any duty to be understandable.
"Is there a distance so small it can't be further divided?" I searched through the existing answers and discovered that neither "atomism" nor "Democritus" are mentioned. They should be, since Democritus developed the theory of atomism which was a rival theory to Aristotle's conceptions. According to Democritus' atomism, there is indeed such a smallest distance. The meaning of atomism has varied through the ages. Notably, in the 17th century there was much debate over atomism/indivisibles, which was at the time thought contrary to catholic doctrine and therefore prohibited both by the jesuits and the catholic establishment in Rome. This created no end of trouble for Cavalieri who would explain in vain that his version of indivisibles are unrelated to the structure of the physical continuum. Ultimately, Cavalieri's religious order (the jesuats, with an "a") was suppressed in 1668. Some scholars speculate that the suppression may have been at least in part due to the fact that the order "harbored" scholars like Cavalieri and his student Degli Angeli, who engaged in dubious intellectual pursuits such as the theory of indivisibles. Galileo, of course, also had a theory of indivisibles that was different from Cavalieri's.
Is there a distance so small it can't be further divided?
There are certainly distances that WE (humans) can't further divide. The thing is we conceptually measure things by looking at them and comparing them to something else.
The problem is that if we go to the ridiculously small, we a) can no longer consider "looking at them" as a neutral act, but it becomes an active interaction changing the thing that we look at and b) we run out of things to compare something to.
So idk for a simplified picture think of "looking at something" akin to how bats "look" at something, that is they send out a shockwave of air pressure towards a target and measure what comes back. Now the shockwave of a bat scream is hardly going to move walls or humans, but if you go towards ever smaller objects it becomes more and more likely that the force of that shockwave is able to blow away the object itself. So by "looking at a thing", you change the thing itself and so the result of the measurement no longer tells you what the object looks like, because by the time you get the results the object might be different due to your interaction with it.
So you need not just the location, but also the momentum (where it is going and how fast), though these are 2 different measurements and they impact each other. That is measuring the movement of something means that it changes location and measuring the location sets a thing in motion. So there are limits to how accurate you can measure either of these two.
So you end up with something like Planck length and time, which mark the theoretical boundaries of our ability to measure things. That doesn't mean that there aren't things smaller than that, it just means that we can't see them and don't know of any measurable physical effects on those scales that we could compare them to.
So that's a very simplified version of the physics. Conceptually, though there's nothing stopping you from ever further divide something. Like 1/2, 1/4, 1/8, 1/16 ... is essentially just 1/(2^n) where n is part of the natural numbers. And as the natural numbers are infinite, you'll never run out of a next number to further divide that.
So yeah if you where to always take steps cutting the remaining length in half and would take the same time for any such step, then you'd essentially NEVER reach your target. In reality though once you've hit the Planck length you wouldn't be able to tell the difference between stagnation and progress making this kind of a useless endeavor.
Also if you were to shoot a real life arrow, then you'd not take equal amounts of time for a step of cutting the distance in half but you could assume an almost equal speed during these space/time intervals.
Meaning that if you have a constant velocity and a decreasing length, the time it takes to travel that distance is also decreasing by a factor of 2. So the distance to target is approaching 0, but so is the time to arrival.
And even if you argue, "but the arrow looses velocity over time". Sure but the rate of loss of velocity also approaches 0 with the difference in space and time approaching 0 and likely much faster than the other 2 so that difference becomes pretty negligible.
And in the limit of n going to infinity the sum over 1/(2^n) is 1 so you've traveled the whole distance in the whole time.
imagine. I can imagine the ocean being filled with pudding. Anything I can think I can imagine. Obviously, you are using some strange new definition of imagine I hadn't previously been aware of.
Is there a distance so small it can't be further divided?
Yes. A common example is a chess board. Pieces exist in distinct squares. Motion is jumps between squares. These squares are minimum indivisible distances. The diagonals of the squares are the next permissible distance (bishops). The third permissible distance is that of the knight.
Another example is a bitmap. There is a magnification limit for bitmaps. At this limit the pixel is the smallest distance and cannot be divided further.
The Planck length may be the "physical" example but the Planck length is the shortest measurable distance. A smaller distance may exist beyond our ability to measure (for now).
I'd say the width (diameter?) of one of smallest / most elementary unit of matter confirmed to exist by physicists would be the smallest unit of spatial distance that cannot be further divided.
