For those of you who aren’t familiar: Achilles and a Tortoise race, however the tortoise is given a leading start. Achilles is at Point A, whereas the tortoise is ahead at point B. The race begins, and by the time Achilles makes it to point B, where the Tortoise used to be, it has reached point C. Then Achilles arrives at point C with the Tortoise at point D. So on and so forth, with Achilles never catching up to the Tortoise as per the paradox.
But he definitely catches the Tortoise eventually, right? The tortoise has a lower velocity, hence the head start, so after a certain amount of time the distance between points is smaller than Achilles and the Tortoise’s difference in speed. What, if anything, is paradoxical about the world’s most famous paradox?
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We use the word paradox in a couple different ways. One is in logical contradictions, i. e. “This sentence is false.” It is impossible to have the statement be either true or false, so it is a paradox. “A married bachelor” is another paradoxical statement, as you are definitionally bound by the word “bachelor” to be unmarried.
Another way we use paradoxes is in seemingly contradictory statement that do actually have an answer. Zeno’s paradox falls in this category. Like you say, there is the definite answer that Achilles does catch the tortoise, the seemingly contradictory part is the description that you have to cover an infinite number of ever smaller but never 0 distances to go anywhere, so how can you move?
The answer being that the time it takes to move those smaller increments also decreases to 0 at exactly the same rate.
Let’s give some formulas for Achilles and the Tortoise, to see how the times when Achilles reaches a point where the Tortoise was previously at keeps shrinking and stays below a definite time (when their positions are tied) rather than getting big.
Say Achilles has position A(t) = t at time t and the Tortoise has position T(t) = 4 + t/100 at time t, so A(0) = 0 and T(0) = 4: at time 0, the tortoise is 4 units ahead (meters, feet, whatever).
Achilles reaches position 4 at time 4: A(4) = 4. And at that time T(4) = 4.04 > 4, so the tortoise is still ahead at time 4.
Achilles reaches position 4.04 at time 4.04: A(4.04) = 4.04. And at that time T(4.04) = 4.0404 > 4.04, so the tortoise is still ahead at time 4.04.
Achilles reaches position 4.0404 at time 4.0404: A(4.0404) = 4.0404. And at that time T(4.0404) = 4.040404 > 4.0404, so the tortoise is still ahead.
Achilles reaches position 4.040404 at time 4.040404: A(4.040404) = 4.040404. And at that time T(4.040404) = 4.04040404 > 4.040404, so the tortoise is still ahead.
Despite the tortoise always being ahead of Achilles at these times, notice the times we are working with are always remaining below 4.040404040404… < 4.05. So it’s simply *false* that Achilles “never” catches the tortoise because such reasoning is just ignoring the actual passage of time by focusing on ever vanishingly small units of time. Once the time reaches 400/99 = 4.040404040404…, Achilles and the tortoise are at the same position: their positions are *tied*. And after this time Achilles gets ahead of the tortoise and remains ahead. Notice the time when Achilles and the tortoise are tied is the value of an infinite decimal 4.0404040404…, which can be thought of as a convergent infinite series 4 + .04 + .0004 + .000004 + …, which is related to the answer by u/EquinoctialPie saying that the resolution of the paradox is the fact that infinite series can have finite values. One does not need other tools from calculus (like integrals), but perhaps the series can be viewed in different ways related to those other tools.
You can graph this: plot y = t and y = 4 + t/100. For very small positive t, we have t < 4 + t/100 (Achilles is behind the tortoise), but when t = 400/99 the two lines cross, and when t > 400/99 the first line is higher than the second line (Achilles is ahead of the tortoise).
In summary, there is no actual paradox if you pay attention to *all* times instead of getting fixated only on quite small times. The error here is analogous to people who mistakenly think an unending decimal like pi = 3.141592653589… is an “infinite number” because it has infinitely many digits: this confuses the number of digits with the numerical value of the decimal.
Another cause/fix to it is to do with quantisation, and quantum physics.
This lead to a similar paradox called the UV catastrophe.
The premise was you could infinitely divide things, time and distance, so half of whatever measurement there is, but it turns out you can only get so far, and then it blends together in the Planck length and Planck-time.
It’s not a case of just being the smallest thing currently measurable, it’s more the limit on the granularity of the universe.
The UV catastrophe is what started quantum physics, the maths said that a black body (in simple words, something that radiates heat perfectly) would radiate energy at all different frequencies in differing levels. The maths divided these frequencies infinitely, and they all had at least some energy, at that would mean it would radiate infinite energy too.
It was figured out that you couldn’t just divide everything infinitely, it got down to discrete quantised steps, eg mini packets or “quanta” in these levels.
The Planck length is the smallest possible division of space and distance, and you can’t halve it, and the Planck time is the time it takes light to cover the Planck distance, eg the fastest possible thing covering the smallest possible distance, and it can’t get smaller.
Let’s give some formulas for Achilles and the Tortoise, to see how the times when Achilles reaches a point where the Tortoise was previously at keeps shrinking and stays below a definite time (when their positions are tied) rather than getting big.
Say Achilles has position A(t) = t at time t and the Tortoise has position T(t) = 4 + t/100 at time t, so A(0) = 0 and T(0) = 4: at time 0, the tortoise is 4 units ahead (meters, feet, whatever).
Achilles reaches position 4 at time 4: A(4) = 4. And at that time T(4) = 4.04 > 4, so the tortoise is still ahead at time 4.
Achilles reaches position 4.04 at time 4.04: A(4.04) = 4.04. And at that time T(4.04) = 4.0404 > 4.04, so the tortoise is still ahead at time 4.04.
Achilles reaches position 4.0404 at time 4.0404: A(4.0404) = 4.0404. And at that time T(4.0404) = 4.040404 > 4.0404, so the tortoise is still ahead.
Achilles reaches position 4.040404 at time 4.040404: A(4.040404) = 4.040404. And at that time T(4.040404) = 4.04040404 > 4.040404, so the tortoise is still ahead.
Despite the tortoise always being ahead of Achilles at these times, notice the times we are working with are always remaining below 4.040404040404… < 4.05. So it’s simply *false* that Achilles “never” catches the tortoise because such reasoning is just ignoring the actual passage of time by focusing on ever vanishingly small units of time. Once the time reaches 400/99 = 4.040404040404…, Achilles and the tortoise are at the same position: their positions are *tied*. And after this time Achilles gets ahead of the tortoise and remains ahead. Notice the time when Achilles and the tortoise are tied is the value of an infinite decimal 4.0404040404…, which can be thought of as a convergent infinite series 4 + .04 + .0004 + .000004 + …, which is related to the answer by u/EquinoctialPie saying that the resolution of the paradox is the fact that infinite series can have finite values. One does not need other tools from calculus (like integrals), but perhaps the series can be viewed in different ways related to those other tools.
You can graph this: plot y = t and y = 4 + t/100. For very small positive t, we have t < 4 + t/100 (Achilles is behind the tortoise), but when t = 400/99 the two lines cross, and when t > 400/99 the first line is higher than the second line (Achilles is ahead of the tortoise).
In summary, there is no actual paradox if you pay attention to *all* times instead of getting fixated only on quite small times. The error here is analogous to people who mistakenly think an unending decimal like pi = 3.141592653589… is an “infinite number” because it has infinitely many digits: this confuses the number of digits with the numerical value of the decimal.
Zeno argues that motion is impossible because in order to get from one point to another, an object must first travel half the distance, then half the remaining distance, and so on, ad infinitum. Since there are an infinite number of distances to be traveled, the object can never actually reach its destination.
In reality, Achilles will eventually catch up to the tortoise, despite the paradox’s argument that he cannot. Zeno’s paradox is based on the assumption that an infinite number of smaller and smaller distances must be covered in order to reach a destination, but in reality, there is a smallest possible distance that can be traveled, such as the Planck length in physics.
Furthermore, the paradox assumes that time is infinitely divisible, which is also not true according to modern physics. When the actual physical laws are taken into account, the paradox can be resolved, and it becomes clear that Achilles can indeed catch the tortoise, given enough time.
Zeno’s paradoxes continue to be interesting philosophical puzzles that raise questions about the nature of space, time, and motion, but their solutions lie in our understanding of modern science and mathematics, which provides a more accurate and realistic description of the world around us.
Let’s give some formulas for Achilles and the Tortoise, to see how the times when Achilles reaches a point where the Tortoise was previously at keeps shrinking and stays below a definite time (when their positions are tied) rather than getting big.
Say Achilles has position A(t) = t at time t and the Tortoise has position T(t) = 4 + t/100 at time t, so A(0) = 0 and T(0) = 4: at time 0, the tortoise is 4 units ahead (meters, feet, whatever).
Achilles reaches position 4 at time 4: A(4) = 4. And at that time T(4) = 4.04 > 4, so the tortoise is still ahead at time 4.
Achilles reaches position 4.04 at time 4.04: A(4.04) = 4.04. And at that time T(4.04) = 4.0404 > 4.04, so the tortoise is still ahead at time 4.04.
Achilles reaches position 4.0404 at time 4.0404: A(4.0404) = 4.0404. And at that time T(4.0404) = 4.040404 > 4.0404, so the tortoise is still ahead.
Achilles reaches position 4.040404 at time 4.040404: A(4.040404) = 4.040404. And at that time T(4.040404) = 4.04040404 > 4.040404, so the tortoise is still ahead.
Despite the tortoise always being ahead of Achilles at these times, notice the times we are working with are always remaining below 4.040404040404… < 4.05. So it’s simply *false* that Achilles “never” catches the tortoise because such reasoning is just ignoring the actual passage of time by focusing on ever vanishingly small units of time. Once the time reaches 400/99 = 4.040404040404…, Achilles and the tortoise are at the same position: their positions are *tied*. And after this time Achilles gets ahead of the tortoise and remains ahead. Notice the time when Achilles and the tortoise are tied is the value of an infinite decimal 4.0404040404…, which can be thought of as a convergent infinite series 4 + .04 + .0004 + .000004 + …, which is related to the answer by u/EquinoctialPie saying that the resolution of the paradox is the fact that infinite series can have finite values. One does not need other tools from calculus (like integrals), but perhaps the series can be viewed in different ways related to those other tools.
You can graph this: plot y = t and y = 4 + t/100. For very small positive t, we have t < 4 + t/100 (Achilles is behind the tortoise), but when t = 400/99 the two lines cross, and when t > 400/99 the first line is higher than the second line (Achilles is ahead of the tortoise).
In summary, there is no actual paradox if you pay attention to *all* times instead of getting fixated only on quite small times. The error here is analogous to people who mistakenly think an unending decimal like pi = 3.141592653589… is an “infinite number” because it has infinitely many digits: this confuses the number of digits with the numerical value of the decimal.
Achilles definitely catches the tortoise.
The way the story is told, they keep describing snapshots in time that are closer and closer to the time the tortoise is caught without telling you about that moment.
Like if he catches him at noon and you tell a story describing 11:59, then 11:59:30, then 11:59:45, then 11:59:52.5, etc.
You can certainly tell a story like that where you never get to noon. But we all know how time works. Eventually noon will come.
Another cause/fix to it is to do with quantisation, and quantum physics.
This lead to a similar paradox called the UV catastrophe.
The premise was you could infinitely divide things, time and distance, so half of whatever measurement there is, but it turns out you can only get so far, and then it blends together in the Planck length and Planck-time.
It’s not a case of just being the smallest thing currently measurable, it’s more the limit on the granularity of the universe.
The UV catastrophe is what started quantum physics, the maths said that a black body (in simple words, something that radiates heat perfectly) would radiate energy at all different frequencies in differing levels. The maths divided these frequencies infinitely, and they all had at least some energy, at that would mean it would radiate infinite energy too.
It was figured out that you couldn’t just divide everything infinitely, it got down to discrete quantised steps, eg mini packets or “quanta” in these levels.
The Planck length is the smallest possible division of space and distance, and you can’t halve it, and the Planck time is the time it takes light to cover the Planck distance, eg the fastest possible thing covering the smallest possible distance, and it can’t get smaller.
Another cause/fix to it is to do with quantisation, and quantum physics.
This lead to a similar paradox called the UV catastrophe.
The premise was you could infinitely divide things, time and distance, so half of whatever measurement there is, but it turns out you can only get so far, and then it blends together in the Planck length and Planck-time.
It’s not a case of just being the smallest thing currently measurable, it’s more the limit on the granularity of the universe.
The UV catastrophe is what started quantum physics, the maths said that a black body (in simple words, something that radiates heat perfectly) would radiate energy at all different frequencies in differing levels. The maths divided these frequencies infinitely, and they all had at least some energy, at that would mean it would radiate infinite energy too.
It was figured out that you couldn’t just divide everything infinitely, it got down to discrete quantised steps, eg mini packets or “quanta” in these levels.
The Planck length is the smallest possible division of space and distance, and you can’t halve it, and the Planck time is the time it takes light to cover the Planck distance, eg the fastest possible thing covering the smallest possible distance, and it can’t get smaller.
Outside of some tricks with language (“this statement is a lie”), you can’t have actual paradoxes. By definition, a real paradox is impossible. However, you can get things that seem paradoxical. Normally, these situations involve reasoning that seems good, based on common assumptions, that leads to a result that contradicts what we know to be true.
So, we know that a racer can overtake another racer that has a headstart on him. Zeno’s reasoning seems solid, but indicates that overtaking should be impossible. The common assumption that turns out to be wrong is the idea that space is infinitely divisible. We’re in a simulation, and you can’t have half-a-pixel, so you have to always move across the screen in discrete units.
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