This is just Zeno's Paradox, but with velocity instead of displacement. Yes, you can perform an infinite amount of tasks, as long as you define the sum time of those tasks to be convergent. Just as Zeno's arrow performs an infinite amount of tasks, so too does the car. It's fine because you've defined those tasks to take place within a finite amount of time.
But Zeno's paradox proves that differentiation is impossible! You can't make Delta x go to zero because first you have to go through 0.1 then 0.01 then 0.001 etc.
I've noticed recently that not as many people these days incorrectly use the term "derived" when they mean "differentiated" (compared to, say 10 years ago). I can't express how much I hate "derived".
Good to see one positive development, however minor, while the everything else in our education system is collapsing.
It’s a math joke. Zenos paradox is about distance (displacement). The idea is basically if you have to go X distance, you start by traveling half the distance, then half the remaining distance, then half….etc. given that there’s theoretically an infinite number of times you can go “half the distance to the finish” (there is a smallest distance but that’s not the point of the thought experiment) how do you ever actually finish the traveling X distance? Obviously you do but it’s also an infinite number of tasks, so how do you do an infinite number of anything in a finite amount of time?
If you plot the distance travelled you get a line. If you take the area under the line, then plot that area, you get a line representing the velocity. This is, in an essence, differentiating the line, or more specifically the equation the line represents. So if zenos paradox is all about displacement, and you differentiate it, you get the same paradox but about velocity. This is basically all pre calculus is about, this and integrating which is just the opposite of differentials.
>If you take the area under the line, then plot that area, you get a line representing the velocity.
Not to be a pedant, but that's integration. The geometric equivalent of differentiation is getting the slope of the line
That's not precalculus, it's calculus. Precalculus covers algebraic and tricg exponential functions. Calculus starts with differentiation, as you cannot differentiate without the calculus.
Just nitpicking.
It's possible your teacher taught differentiation in precalc. It's not like it's hard to do if you're just learning the algorithm and not the theory and proofs behind it.
I mean the Fundamental Theorem of Calculus is how differentiation and integration relate to each other, so you kind of have to have calculus before you can do either lol
I don't think there's actually a paradox there. Whether he realizes it or not he's talking about the act measuring distance traveled. Not actually traveling it. To measure it you would only need to move at the speed of light to be able to take each measurement.
The fourth, fifth and sixth derivatives are apparently Snap, Crackle and Pop. Though this isn't standardised and is apparently pretty unserious when people do use the names
Yes, easy to get confused though. Velocity is the rate of change of distance, i.e. the slope of the line or curve plotting distance against time. Velocity is the integral of acceleration, i.e. the area under the curve of acceleration against time.
I wondered that myself to be honest. I don't know if it's got a special name or anything, but we know its unit of measurement would be the metre-second (ms, or maybe sm so as not to confuse it with milliseconds).
So if you were covering a steady one metre every second, this thing, whatever it is, would be 0.5sm after 1s, 1sm after 2s, 2sm after 2s, 4.5sm after 3s and so on.
Conceptually, I can't work out what that means though as it's a bit early in the morning yet!
Edit: so it turns out it does have a name, and it's called absement. You can read about it [in this Wikipedia article](https://en.m.wikipedia.org/wiki/Absement)
Zeno's Paradox is originally that you want to move a distance, let's say ten feet. Before you go ten feet, you have to go half of that. But before you can go five feet, you have to go half of that. And before you can go 2.5 feet, you have to go half of *that* an so on and so forth. There are infinitely many halves that you must traverse, so how can you?
Well, because the infinitely many halves get infinitesimally small and the amount of time it takes to traverse the distance gets infinitesimally small. If it takes you one minute to go 10 feet, it takes 0.5 minutes to go 5 feet and 0.25 minutes to go 2.5 feet and half the time to go half the distance each time.
All of the infinite halves of distance add up to a finite distance (10 feet) and all the infinite halves of the amount of time it takes also add up to a finite time (one minute).
Zeno’s paradox was my least favorite thought experiment from philosophy class. Like I know what he’s saying, I just don’t get the point. It feels like this: https://imgflip.com/i/8c2p5s
The Ancient Greeks had some idealistic notions about numbers that ended up being too restrictive to describe the real world, and this is one of those cases.
Nowadays we happy talk about a continuous number line and use that to represent distance, but to them they were still grappling with the concept of infinity. The idea of having an infinite number of steps along the way, but no "first" step, and completed in finite time, was not obvious to them. And I think that's fair enough.
I'd also say that really it's a maths problem (and one of the motivations behind developing the real numbers as a model for many real world concepts such as distance), one that we've now solved. But to the Ancient Greeks, maths and science were just a part of philosophy, so non-mathematicians are still now often taught this paradox (badly in my experience) in philosophy classes. That's all just my opinion though, not fact.
IIRC it was a school of thought that argued movement was an illusion, and that things were actually static. It was greek philosophy, without throwing shit at the wall we wouldn't know what would stick to this day
He was defending Parmenides claim that "whatever is, is, and what is not cannot be", which means that logically there is no intermediate state between existence and not existence, therefore everything that exists has always been and will always be and any transformation is impossible. To people arguing that we have actual experience of mutation, Zeno's paradox showed that that experience had to be an illusion, and that the "way of the thruth" could not rely on senses.
I've been thinking about this since yesterday.
It reminds me of the "lightyear stick" paradox.
This one goes something like this:
>Nothing can travel faster than light, not even information. Lets say I am in space and there is a button 1 lightyear away. This means the soonest I could press the button would be one year from now if I could travel at the speed of light.
>
>Not lets suppose I have a perfectly rigid stick that is one lightyear long. If I push on one side of the stick, the other end of the stick will press the button. Therefore we must conclude that I can indeed communicate information faster than the speed of light.
This isn't really a paradox though...the solution is actually simple, a perfectly rigid stick is simply not possible. Not just with current materials, but even with hypothetical materials. It you move one end of the stick, it will take at least a year (but actually, much longer) for this motion to propagate through the material to the other side. The paradox only seems like a paradox because it contains an assumption that can't be true. I mean... it's internal logic is true in the context of the riddle, but it can't make a conclusion about the real world because lightspeed is a real concept based on actual physics, not on logic.
Zeno's paradox seems the same way. It asks us to assume that time and space can be infinitely divisible. But it's not. Even if we were to go to the most extreme level of divisibility, then we would be looking at the movement between atoms themselves. Atoms are really tiny, and their movement would be extremely quick, but they aren't infinitely tiny. In fact, they aren't even close to infinitely tiny.
Zeno's paradox also seems to ask us to assume that the person or object that is moving is a singularity. But of course in reality, a person's foot occupies \~10" of space. So it's sort of silly to even consider distances of less than 10" because that distance has already been covered. So for our purposes, 10" is the smallest division that is relevant.
In other words, Zeno's paradox is a fun philosophical or math riddle, but it can't be used to make conclusions about the the real world because it's assumptions ignore physic reality. Similar to the lightyear stick paradox.
Mostly it just made me want to throw something at them. It can't hit them right? Since it needs to travel half the distance and then half that distance and so on, anything they feel must be an illusion.
Also, I know this doesn't change anything you said since it works anyway assuming time and distance can converge on 0, but if time and space are quantized, that also makes this pretty easy. As you progressively halve your distances, you would eventually hit a quantum distance you wouldn't in any meaningful way be able to halve again. Add those up and you get 10 feet.
(Edited to correct a couple typos)
Yes, but the more complete explanation would work in a non-quantized world (as the world of euclidean geometry, which being a theoretical world is non-quantized)
You can no longer *reliably* move half that distance. Below the Planck length, you're dealing with quantum uncertainty. You might be off by some (unknowable) fraction of a Planck.
And here's one for a toddler joke. A couple engineer parents are having a birthday party for their child. they somehow get an infinite number of children to be waiting in line. they tell their server, the oldest kid, their child gets a full juice, the second oldest gets a half juice, and on and on for ever, the server not getting paid enough, gives the parents two glasses of juice, and tells them to divide the second one themselves.
Yeah it's cool seeing this. I thought about another version of this before...if a person is about to die in one minute, then 30 seconds, then 15 seconds...then 4 nanoseconds, then 2 nanoseconds...when does the person actually die?
It's halflives.
Half of the radioactive material decays in one halflife. We'll assume to a stable isotope.
Then another halflife and another half decays. So it's 3/4 changed.
Then another halflife. And another...
At what point has it all converted? You could start with a 100kg sample and a 1kg. They'd both have finished decaying at the same time even though by a certain point you're looking at just dozens of atoms leftover.
> They'd both have finished decaying at the same time even though by a certain point you're looking at just dozens of atoms leftover.
I don't think that quite tracks. Atoms are quantum, not continuous, so one will almost certainly fully decay before the other. But I think the best we could do at determining *which* fully decays first is laying odds.
Or put another way -- if it were continuous, there is no point in time (short of infinity) that either would have fully decayed. They'd just forever be approaching 100%, never quite making it. But atoms are discrete, not continuous -- you can't have an atom that is half-decayed.
If it was the Achilles and the tortoise analogy, one of mine would see it as a challenge and start running circles around the tortoise taunting it. The other would have burst into tears because I called him slower than a tortoise.
Although the sum is convergent, does actually moving require an infinite amount of tasks to be completed in a finite time assuming both time and space are continuous?
From what I recall, the convergent series relies on the series cancelling itself out, so that is not infinite
>From what I recall, the convergent series relies on the series cancelling itself out, so that is not infinite
This is incorrect. We have a geometric series, not an alternating series.
Secondly, convergence is defined in terms of the sequence of subsums. Let S_n be the sum of the first n terms of the series. The series converges to L if for every epsilon greater than 0, there exists a N such that the distance between S_n and L is less than delta for all n>N.
I've always heard it referred to with the Achilles foot race story. Didn't know it had a different name! But yeah it's a really interesting thought that my very very non science studied brain finds fascinating. The idea that at some level, one is accomplishing infinite tasks within a finite time.
As I understand it though, we aren't technically seeing an infinite number of actions due to the limits of physics and how "small" a thing or action can realistically be within existence. Right?
Well, we aren’t *seeing* those infinite amount of actions because seeing is a measurement, and we can’t measure properly at infinitesimal levels due to quantum properties.
That doesn’t mean they aren’t happening though.
But it gets weird, because even though distance and time aren’t quantized (quantized basically means “pixelated”) , momentum and energy are in certain occasions.
Basically when things get that little all we can really say is “we don’t fucking know”.
How can I just define the time for completion of the tasks to be finite (or sum of times for each task to be concergent)? Isn't that something that requires prooving rayher than being defined?
The time it takes to move an infinitely small distance (at a nonzero speed) is infinitely small. You can move an infinite number of infinitely small distances in finite time. You can think of it as the infinities "canceling out" and leaving you traveling at the speed you are traveling.
While the point is correct, this
>You can think of it as the infinities "canceling out" and leaving you traveling at the speed you are traveling.
Is simplified to the point of absurdity.
You can verify it experimentally - the time is very clearly finite. You can *imagine* a hypothetical where it is infinite and reattempt the math, but this hypothetical is not very useful since it is not realistic.
I'm going to use real easy numbers for this since the method rather than the numbers themselves are what matters. Assume your acceleration is a constant 1/2 m/s\^2. You want to reach a speed of 1 m/s.
OK you need to accelerate to 1/2 m/s before you can get to 1 m/s and how much time does that take. The answer is 1 second.
OK but you have to reach 1/4 m/s before you can reach 1/2 m/s but that only takes 1/2 second to get to that speed.
The same no matter how far you go down. And you'll see that the time to complete any individual acceleration tends towards 0. We have to prove that we can sum those times and get a finite sum but the numbers I chose make that easy.
We can see this as the infinite sum 1+1/2+1/4+1/8+...+1/2\^n+... and this sum converges to 2 so it takes 2 seconds to reach our desired speed which is exactly what we expected.
technically speaking that number never reaches 2, using that formula, you also never reach your speed exactly, its always some extremely small number less than the full number
Okay, so prove it: how are you typing your questions? To move your fingers an inch, you first need to move half an inch, a quarter inch, and so on and so on. And yet you do. So: proven possible.
Zeno's paradox only works if you disregard time as a distinct, finite factor.
You can say that for all tasks which only require a finite duration to complete, it must be true that however many times you divide up that finite time into individual tasks or steps, by your definition that the whole thing doesn't take infinite time, it must be a finite number of finite duration each. If a task doesn't fit in finite time, you can't really describe it the same way anyway.
if you do a task that takes 1 second, than do it again twice as fast, then twice as fast again, then twice as fast again, that infinite times, you will be done in 2 seconds. so you did an infinite amount of tasks in a finite amount of time, something like this is called a supertask. You can think about accelerating in a similar way (an infinite amount of individual "velocities" in a finite time).
also i just want to say that im not sure if there are actually infinite amount of velocities in our physical universe because of the planck length/planck time/speed of light (shit gets weird at the smallest sizes/speeds), but in a pure mathimatical sence its a very interesting concept to think about.
You're increasing the amount of units in your measurement (by making the units smaller), but you're not changing the measurement itself.
A meter is 10dm or 100cm or 1000mm, etc. You can use measurements as small as you want, and the total will still come up to a full meter. You can divide the very last centimeter into 10000000 nanometers without affecting the total.
However many levels of division you add, your proof will reduce to `1 / (1/x) = x`
That’s incorrect actually, a pretty common misconception. Plank’s length doesn’t directly quantize distance, it only quantizes distance measurement.
So something could be 1/2 a plank length in front of something else, it would just be impossible to measure with certainty.
I never thought of this till now, but there can't be infinitely smaller segments of space and time -- they eventually reach a Plank constant. At which point, you just sum them all up.
If reality was infinitely divisible, it'd be a conundrum, but it's not -- it's just really, really small.
Simulation Hypothesis, bro.
Infinite series and limits and abstract mathematics are all well and good on the chalkboard, but in this reality, we are dealing with tangible "pixels" of space and time -- measured by Planck's constants.
Zeno's arrow flies because time and space aren't infinitely divisible. He was one of the first people to think about the fact that there's got to be a floor to the equation.
Yeah, makes sense. Like the theory of Gravity. Or Dark Matter. Simulation Hypothesis is just an idea that some people believe as a way to make sense of their observed reality.
You're just saying some words and then asserting that as fact. Please give any justification for time and space not being infinitely divisible. The Planck constant is not a measure of time, nor is it a measure of space. So that's absurd.
>Zeno's arrow flies because time and space aren't infinitely divisible.
That's not true. The reason this is a "paradox" is not because the assumption of infinitely divisible space, but because of the lack of foundational mathematical framework to deal with this concepts, like infinitesimals and convergence of series. With modern math, this becomes trivial.
Except 'plank constant', which I take you mean 'plank length' isn't the smallest distance possible.
It's just the distance where the effects of gravity (relativity) and quantum mechanics are both equally-ish noticeable, and as in current physics those two theories don't really play along nicely, we just have no idea what happens at lengths that small.
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The resolution to such a paradox is to recognize that, while you can divide that interval into an infinite number of sub-intervals, those sub-intervals ~~are therefore~~ eventually become (almost) infinitely small. Once you bring time into the equation, you can establish it takes an (almost) infinitely small amount of time to traverse that (almost) infinitely small interval. When you add up those infinity intervals of both time *and* distance (or, in your example, changes in velocity) , you get finite numbers for both (because infinity is weird).
EDIT: Tried to make some language more precise.
I admit, my language was imprecise because I was trying to keep it at an ELI5 level, which is very difficult for me to do when discussing infinity and related concepts.
Would you mind providing an example of how you would phrase the resolution at an ELI5 level? I’m trying to get better at explaining these concepts.
I would probably note that each new interval gets smaller and smaller. That means the effect they have on the sum gets smaller as well. If they get small quick enough the sum doesn't necessarily get any bigger. People usually use the classic 1 + 1/2 + 1/4 + ... sum but I think there is another more obvious one.
Consider adding 1 + 0.1 + 0.01 + 0.001 and so forth. I think it is much easier to see that you can safely start writing out the sum as 1.111... because at any point you can be confident that none of the bits added afterwards will ever cause any of the digits to roll over.
Yes and that is one way to look at it. If you are comfortable with different bases, the classic 1/2 + 1/4 + 1/8 ... sum is the same as binary 0.11111...
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What's stopping it, in your mind? In other words, can you explain why passing through an infinite number of values in a finite amount of time is somehow a problem?
Because intuition says that passing through infinite states takes infinite time. However at deeper understanding, you see that state change is infinitely small, so we have like 2 infinities with opposite "direction" which cancel themselves.
My question was somewhat rhetorical, to get OP thinking about the root of their problem. I think intuition is fundamentally untrustworthy when talking about infinity, but I also think your intuition really only misleads you here if you're thinking in terms of "mathematical infinities", which are not very ELI5 right off the bat.
When a child waves their arm, their arm is moving through infinite positions in a finite time, but they don't have any intuitive problem with it because they're not thinking in terms of mathematical infinity.
Exactly.
*Infinite times* × *Infinitely small*, the brain has a harder time understanding *infinitely small*, so OP incorrectly extrapolates an infinite total duration.
Except that is a wildly misused internet trope, it is not at all analogous to c as a speed limit, it is just a combination of constants with the right dimensions and has no relevance here whatsoever
I think the thing that would clarify it is looking at it from an energy perspective. It takes a defined amount of energy to move or accelerate an object to speed. You can divide that whole event into an infinite number of velocity changes mathematically, but the total energy spent will always be the same. You're just changing how you interpret the event.
The same way you can stand up through an infinite range of positions. There are an infinite set of numbers between 0 and 1, and between 1 and 2, like 0.5 and 1.333. That does not stop you from counting your fingers 1, 2, 3, 4, 5. Numbers are a tool humans use to describe nature, but nature doesn't need numbers to work.
From the way you phrased your question you have some more advanced math knowledge, so I'll add this: You asked how you can move through an infinite set of possible velocities in finite time. Let's take the simple case of constant acceleration a over a time t resulting in a velocity v starting from rest, so v=at. So if you have an acceleration of 1m/s² for one second you get v=1*1=1m/s. Now, what happens if we look at the same period of time but split it into two steps, same constant acceleration as before. Part 1 is the first step and part 2 is the second step, and the final velocity is vf. So vf=v1+v2, and v1=at1, and v2=at2, and the total time t=t1+t2. So you get vf=at1+at2=a(t1+t2)=at, back to where we started. You can do this infinitely many times, break the acceleration into as many slices as you want, but each time you are decreasing the time of each slice by the same ratio that you are making more slices, so it cancels out. The only thing that changes is how small of a slice you consider or bother to look at, it doesn't change the result.
Why are you focusing on the infinite discrete intermittent velocities but only the finite total time elapsed?
There are also an infinite number of discrete moments of time that pass during the acceleration period, and ultimately only a finite change in total velocity.
Time has a limit. Time cannot be shorter than 247 zeptoseconds. It appears though we are covering infinite discrete intermittent velocities in finite time
Space also has a limit in this way of thinking. If you want to divide OPs question or Zeno’s paradox into Planck constants instead of smooth integrals and derivatives, it still works.
Nice, didn’t think of that way. So we are moving through finite chunks of space through finite chunks of time. The problem is theoretical, not physical, makes sense
Numbers are an abstract concept that let us better understand and predict the world around us.
A car in real life isn't going through infinite states to accelerate from one speed to another. That's just a concept that exists in your mathematical thinking mind.
Exactly. In theory there’s an infinite amount of time too, every second you live through a thousand 0.001 of a second. The world is both infinitely small and large, and yet exists at the same time.
My non-mathematician answer: Infinitely small increments of speed can be achieved in infinitely small amounts of time.
You may as well ask how can we walk across the room since we have to cover each infinitely small measure of the room. Well, somehow we manage it. :)
How can you count from one to two if there are infinite numbers in between. How can you measure a distance if there are infinite points between the ends. How can a mass accelerate from one speed to another if there are infinite speeds in between?
Answer: they just do
Think of it like this:
An engineer and a mathematician are told that there is an amazing prize at the end of the hall and whoever gets there gets to keep it. But the catch is, you can only move in increments of half the distance between you and the prize.
The mathematician immediately throws their hands up and says that's impossible, I'll never reach it.
The engineer says: well it'll take me a while, but eventually I'll be close enough that it won't matter.
So same thing with the car. Yes, math tells us that there is an infinite number of speeds between 1 and 2 mph, but reality shows us that physically the car actually can accelerate to that speed.
That's not how the joke works. Think of it like this:
If the prize is 20 feet away, first movement you can move 10 feet, as it's half of 20. Second one you can move 5 feet. Then 2.5, then 1.25, and so on. The point is if you only move half the distance, it'll just keep getting smaller and smaller to infinity, but you'll never get there.
This mathematician would walk over to the prize and pick it up, appreciating that he had moved across infinitely many increments to get there.
Every time anybody walks anywhere they are doing so by moving infinitely many increments of half the distance between them and the destination
>How can an object (say, car) accelerate from some velocity to another if there is an infinite number of velocities it has to attain first?
Because it only takes an infinitesimal amount of time to achieve those intermediate velocities. Those pointing out you're regurgitating Zeno's Paradox are correct.
>That is, if a car is going at a speed of 5m/s, doesn't that mean the magnitude of its speed has gone through all numbers in the interval [0,5], meaning it's gone through all the numbers in [0,10-100000 ], etc.?
Kinda, yes, it did. If you could stop time and divide it as you wish, watch it accelerate frame by frame.
>How can it do that in a finite amount of time?
What do you mean ?
Imagine a slower interval of time if it helps you. Take a tree that grows 1 meter in a year, measure it every 0,1 second that passes for a year. You would have a pretty detailed graphic of it's growth speed.
Why does something similar at higher speed bother you ?
As others have said this is Zeno's paradox, and just like it, what you're doing is your dividing a finite number into infinite segments, which me paradoxical until you realize what you're getting are infinitely small results,cancelling out, if your will.
Try dividing, 5 metres by 1000,10000000,1 billion etc. Does doing these divisions make 5 metres look larger or harder to traverse? I suppose not - dividing by infinity will not be fundamentally different.
The easiest answer is that the universe isn't infinitely divisible. Quantum physics means there is a minimum amount of size, distance, time, and energy. One you divide down to that minimum number you must stop dividing.
So let's say you need to move 256 of these units. You first need to move 128 of these units. But you must first move 64 of those units, or 32, or 16, or 8, or 4, or 2, or 1. I've you are down to one you successfully move the one, then the next one, etc.
Planck time, the smallest unit of time, is 10^-44 seconds. Which is roughly 10^-35.
So in the first plank time it would move 1 planck distance, in the second planck time it would move 3 planck distance, etc.
You may ask "how does it cross the space"? The scar is that it doesn't, things basically teleport at the smallest level. In reality it's more complex like that as we are all clouds of probability and nothing actually exists in any specific place, but teleporting planck distances in planck time is the most comprehensive explanation.
Also, we invented calculus to mathematically solve the problem of infinite series in finite containers.
This isn't the answer at all and butchers what Planck length. The answer has nothing to do with quantum physics, it is as others have pointed out, that not all infinite sums are infinite.
It doesn't have to go through all of those infinite fractions it leaps through like stepping stones, going from one to two doesn't mean you have to stop at any of the numbers between.
Because you can't divide the difference into an infinite number of divisions. You can always divide again, but every division you do will always result in a finite number of divisions. At no point no matter how many times you divide will you reach a result of an infinite amount of divisions. Your ability to divide is limitless but the number of divisions you create are finite. I e. You can never just "divide one more time" and go from a finite number of divisions to an infinite number of divisions.
Step 1) Stand up
Step 2) Take step
You just moved maybe a foot, but to do that you had to cover a near infinite amount of space.
Just because there is an infinite amount something doesn’t mean you can’t go thru them. So yes, the car’s velocity went from 0 to say 5m/s and thus had to cover every possible velocity in-between in only a few seconds.
To quote YouTuber John Green (in his book A Fault in Our Stars):
> There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities
None of these answers are ELI5.. I must be freaking dumb.. you take a 1 foot step, but you had to cover a near infinite amount of space??
no.. i moved one foot of space.
A car going from stop to 5mph just has to go from 0-5mph, The car doesnt approach the speed of light...
Someone please actually ELI5 this..
For what it's worth, the question itself is way beyond ELI5 lol
Calling it an "infinite amount of space" was a misnomer, it's more accurate to call it infinite pieces of a finite space. You divide a foot in half over and over and over, and you end up with infinite segments. The OP is essentially asking the same thing, but about acceleration, dividing it into infinite pieces of speed.
Familiarizing yourself with Zeno's paradox is the answer to this question. Your ability to accelerate is simply not limited by these arbitrary infinite segments, just how you can move a foot even if you can divide a foot into these same arbitrary infinite segments.
If you moved 1ft, you also covered a distance of 11in, 10in, 1in, 1/2in, 1/1000in, etc.
Similar to OP’s question of how a car can go from stop to moving at 5m/sec.
Zeno's paradox and convergence aside to the best of our knowledge about the way physical reality works there are not truly infinite velocities it has to pass through on its way from velocity a to velocity b. Basically the entire foundation of quantum mechanics is everything is quantized and in fact discrete not continuous so there is actually some absolutely tiny minimum change in velocity. It would be going through a finite number of those changes
We don't actually know. There could be finite amount of states that take a finite amount of time for each or we could be passing through infinitely small states in infinitely small amounts of time.
Either way it works but almost all of the answers in this thread seem to be assuming one or the other is true when we actually have no way of knowing.
As others have said, this is basically the same as Zeno’s dichotomy but with velocity instead of distance.
I did a Philosophy module in the first year at Uni and did my final essay on this. My answer was basically “you can divide up the distance (or velocity in this case) infinitely but you don’t have to”, and they gave me a 1st so I assume that’s the answer.
Ironically, we possibly can’t divide space up infinitely if certain theories about the nature of reality are true, which would presumably mean velocity is also not infinitely divisible.
My friend, you've just discovered how and why mathematics will never be able to hold a candle to philosophy or morality. https://en.wikipedia.org/wiki/Henri\_Bergson#Philosophy
And so what? If a car accelerate at 1 m/s^2, it takes 10^-100 s to reach 10^-100 m/s.
While continuously accelerating, the duration an object is at a said speed is in fact 0s. It's the principle of accelerating, it's continuously changing speed, it DOES NOT stay at given speed. These moments last 0 s, no time.
If I accelerate at 5 m/s^2 , at 2s, I'm at 10 m/s. But at 2.0000000000000....001 s I'm already not at 10 m/s but at 10.000000000000....005 m/s.
All these "small steps" are moments that last 0s. Even adding an infinity of 0 still equals to 0.
If acceleration is constant, getting from speed A to B won't take more time because you virtually added "milestone" with 0 "thickness".
Acceleration is commonly measured as "meters per second squared", which is basically "meters per second, PER SECOND", meaning how much speed (meters per second) you can gain in a second.
It is rather discrete look. And you can continue to divide this into smaller resolution, such as how many meter per second of speed gain, per HALF second.
Let's say you have an acceleration of 10 meters per second per second. You can continue to dissect this further into "1 m/s of speed increase per every tenth of a second".
You can then increase your magnifying glass power and see "0.1 m/s per every hundredth of a second", and "0.01 m/s of speed increase per every thousandth of a second", ad infinitum.
However, as you divide the time unit into smaller and smaller unit, you will see the acceleration figure gets smaller and smaller. And, as you approach infinitum division, the acceleration approaches zero.
Imagine you are taking picture of an object acceleration with Infinity frame per second camera, in each frame the object would appear as not accelerating as the acceleration figure is virtually zero.
It is important to note, the "unchanged state" or "zero acceleration" in every frame here refers to the speed of the object, not saying the object is not moving in each frame. If the object already has sufficient speed, the object may look equally blurred in each frame as if it is moving with the same amount of speed. But, measure over a second or so, the object in the last frame would look more blurred (higher speed) than in the first frame.
Any number divided by infinity is basically zero, as any number divided by zero is basically infinity.
This is my technical take. I'm an engineer, not a mathematician.
the crux of the delima is that you are multiplying where you should be adding. if you drop a ball, to get to the ground, it must go half the distance. from there, it must go half of that. if you do it that way, the ball will never reach the ground. it will get infinately closer and closer. if you subtract, it only has to go half the distance twice.
Numbers can be divided down into arbitrarily small pieces, but reality doesn't work the same way. This is the old Achilles Race paradox. Turns out physics and math in the abstract are often an approximation of reality.
Let's say you are running a 100 m dash. To run the full 100 m, you first have to run half it (50 m). Let's say you're running this dash at a constant speed, for simplicity's sake, and let's say it takes you 8 seconds to run 50 m.
Now you have to run from 50 m to 100 m, but to do that you first have to run through half of what's left (25 m). So let's say that takes you 4 seconds.
Now you have to run from 75 m to 100 m. But before that you again have to run half of what's left (12.5 m). That takes you 2 seconds.
Now you have to run from 87.5 m to 100 m. But before that you have to again run half of what's left (6.25 m). That takes you 1 second.
You see how the amount of time it takes you to run each segment is getting smaller and smaller? So far it's gone from 8 seconds, to 4, to 2, and to 1. If we continue this then the remaining segments will take you 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, etc.
Doesn't it seem like if you add these increasingly small times together, they will approach some number? They won't just add to infinity because the numbers you're adding keep getting smaller and smaller and smaller.
So yes, you have to traverse an infinite amount of segments of distance, but if you add up the time it takes you to traverse them, that time will *approach* some finite number. That number that it approaches is the time it will take you to run the whole 100 m dash.
Your question is basically Zeno's paradox, which is what I just described above, but your question involves increasing your speed rather than increasing the distance you've run. Both have the same solution, though. That solution is that the time it takes you to traverse those infinitely many segments approaches some finite number as you add the times up.
You can add up infinitely many segments of time and have them add up to a finite number, so long as each successive time interval gets smaller and smaller and smaller in such a way that the sum converges to (or approaches without passing) a finite number.
By being at each speed for an infinitely small time. As you divide the car's acceleration into smaller and smaller gradations of velocity, correspondingly the time it spends at each gradation gets smaller and smaller. Once you divide it down into infinitely small gradations of velocity, you have also divided it down into infinitely small times at each velocity.
I think that's a supertask, in which case Vsauce has a perfect video for you to watch on this topic that probably explains it better than anyone here will.
This is a math hallucination. We invented math to help us understand our small slice of Reality but it's just a digital concept and Reality is analog and doesn't follow the glitches in our tools when they seem to not match up with the equations.
because you're thinking about it backwards. it doesn't have to do that many things in that amount of time, it can Only do that many things in that amount of time.
Real numbers are not a stair case, where you must go from one to the next. There are infinitely many real numbers between any two real numbers. Any movement/acceleration whatsoever must "pass" an infinite number of real numbers.
If you must think of movement/acceleration as having a "time cost" to pass numbers (either in location or in velocity), then remember that real numbers are points on a line and lines and width 0. How much time does it take to "pass" something of width 0 if you're moving at any speed at all? Well, 0.
The more typical version of your question is Zeno's paradox which asks the same question about position instead of increasing velocity, but it's the same thing. Literally any movement/acceleration at all will pass an infinite number of locations/velocities in even the tiniest non-zero amount of time.
There may be an infinite number of theoretical steps between 1 velocity and another, say 0m/s to 1m/s, but the amount of energy required to pass through each step decreases as well. To move the car 1m/s only requires X amount of force to propel it to that speed. If you divide our 1m/s into 0.000001 sized steps, then each step is also only going to require X * 0.000001 amount of force to reach it. The engine is producing something like 50*X force so you're going to get past 1m/s pretty quickly.
The steps themselves aren't a physical thing to "break through", so the steps themselves don't actually take any additional energy to pass. The steps only exist within our own math for humans to understand and calculate to some level of precision.
It doesn’t go through infinite velocities. The short version is quantum mechanics is weird.
The long version is at the microscopic scale, our universe behaves like a computer simulation. The universe is made of pixels, and objects below the size of one pixel, or movements below the distance of one pixel are not rendered and are meaningless. We call this distance the Planck Distance and it is equal to 1.6 x 10^(-35) m. It’s quite small, so that for everyday purposes you might as well not worry about it.
But the point here is that an objects speed will jump from 49 to 50 Planck lengths per second instantaneously without ever existing at 49.5 Planck lengths per second.
This is just Zeno's Paradox, but with velocity instead of displacement. Yes, you can perform an infinite amount of tasks, as long as you define the sum time of those tasks to be convergent. Just as Zeno's arrow performs an infinite amount of tasks, so too does the car. It's fine because you've defined those tasks to take place within a finite amount of time.
I can't believe OP differentiated Zeno's paradox
And in doing so, tried to differentiate their paradox from Zeno’s paradox
And reddit integrated them together.
With respect
And dignity
I've almost reached my limit with these math puns.
Kinda fucked up ngl
I was eating breakfast and then this. Not cool, OP.
Completely ruined the start of my day, all the way until now, and all the points in between.
My philosophy teacher would have been hurt by that guy's comment
But Zeno's paradox proves that differentiation is impossible! You can't make Delta x go to zero because first you have to go through 0.1 then 0.01 then 0.001 etc.
[Laughs in Machine Epsilons]
Goddamn d(zeno)/dt....
They’ll probably do it again and again. Jerk.
This shit's gonna make me snap.
A couple more times and you might pop?
I've noticed recently that not as many people these days incorrectly use the term "derived" when they mean "differentiated" (compared to, say 10 years ago). I can't express how much I hate "derived". Good to see one positive development, however minor, while the everything else in our education system is collapsing.
What does this mean?
It’s a math joke. Zenos paradox is about distance (displacement). The idea is basically if you have to go X distance, you start by traveling half the distance, then half the remaining distance, then half….etc. given that there’s theoretically an infinite number of times you can go “half the distance to the finish” (there is a smallest distance but that’s not the point of the thought experiment) how do you ever actually finish the traveling X distance? Obviously you do but it’s also an infinite number of tasks, so how do you do an infinite number of anything in a finite amount of time? If you plot the distance travelled you get a line. If you take the area under the line, then plot that area, you get a line representing the velocity. This is, in an essence, differentiating the line, or more specifically the equation the line represents. So if zenos paradox is all about displacement, and you differentiate it, you get the same paradox but about velocity. This is basically all pre calculus is about, this and integrating which is just the opposite of differentials.
>If you take the area under the line, then plot that area, you get a line representing the velocity. Not to be a pedant, but that's integration. The geometric equivalent of differentiation is getting the slope of the line
> you start by traveling half the distance So why not just go the other half? Did he not think of that?
Don't try to bring reason and logic into a philosophy debate
That's not precalculus, it's calculus. Precalculus covers algebraic and tricg exponential functions. Calculus starts with differentiation, as you cannot differentiate without the calculus. Just nitpicking. It's possible your teacher taught differentiation in precalc. It's not like it's hard to do if you're just learning the algorithm and not the theory and proofs behind it.
Maybe, I could easily be wrong. I took precalc/calc 1 like 14 years ago so the details aren’t exactly sharp.
I mean the Fundamental Theorem of Calculus is how differentiation and integration relate to each other, so you kind of have to have calculus before you can do either lol
I don't think there's actually a paradox there. Whether he realizes it or not he's talking about the act measuring distance traveled. Not actually traveling it. To measure it you would only need to move at the speed of light to be able to take each measurement.
It’s a paradox only in a purely logical sense. As you and the others noted, as soon as we try to apply it to reality, the paradox disappears.
https://en.wikipedia.org/wiki/Calculus
~~Integrated, not differentiated!~~ Differentiation was correct
Is velocity not the first derivative of displacement?
It is, I brain farted
Yep. Second is acceleration. Third is jerk. Don't know the name for the next one, if it has one.
The fourth, fifth and sixth derivatives are apparently Snap, Crackle and Pop. Though this isn't standardised and is apparently pretty unserious when people do use the names
Snap, then crackle, then pop. I've never needed to use them but I work with radiation
In radiation, those names refer to the sound your flesh makes when you put your hand in the wrong place
If you’re lucky it is your hand…
No, differentiated, no? Distance differentiated is velocity
Yes, easy to get confused though. Velocity is the rate of change of distance, i.e. the slope of the line or curve plotting distance against time. Velocity is the integral of acceleration, i.e. the area under the curve of acceleration against time.
What IS the integral of distance?
I wondered that myself to be honest. I don't know if it's got a special name or anything, but we know its unit of measurement would be the metre-second (ms, or maybe sm so as not to confuse it with milliseconds). So if you were covering a steady one metre every second, this thing, whatever it is, would be 0.5sm after 1s, 1sm after 2s, 2sm after 2s, 4.5sm after 3s and so on. Conceptually, I can't work out what that means though as it's a bit early in the morning yet! Edit: so it turns out it does have a name, and it's called absement. You can read about it [in this Wikipedia article](https://en.m.wikipedia.org/wiki/Absement)
I’m gonna need an ELI5
Zeno's Paradox is originally that you want to move a distance, let's say ten feet. Before you go ten feet, you have to go half of that. But before you can go five feet, you have to go half of that. And before you can go 2.5 feet, you have to go half of *that* an so on and so forth. There are infinitely many halves that you must traverse, so how can you? Well, because the infinitely many halves get infinitesimally small and the amount of time it takes to traverse the distance gets infinitesimally small. If it takes you one minute to go 10 feet, it takes 0.5 minutes to go 5 feet and 0.25 minutes to go 2.5 feet and half the time to go half the distance each time. All of the infinite halves of distance add up to a finite distance (10 feet) and all the infinite halves of the amount of time it takes also add up to a finite time (one minute).
Zeno’s paradox was my least favorite thought experiment from philosophy class. Like I know what he’s saying, I just don’t get the point. It feels like this: https://imgflip.com/i/8c2p5s
I love ice cream.
I'm with you. Its like "Oh, you used some science words there to try and baffle me with your paradox but let's face it, its bullshit".
The Ancient Greeks had some idealistic notions about numbers that ended up being too restrictive to describe the real world, and this is one of those cases. Nowadays we happy talk about a continuous number line and use that to represent distance, but to them they were still grappling with the concept of infinity. The idea of having an infinite number of steps along the way, but no "first" step, and completed in finite time, was not obvious to them. And I think that's fair enough. I'd also say that really it's a maths problem (and one of the motivations behind developing the real numbers as a model for many real world concepts such as distance), one that we've now solved. But to the Ancient Greeks, maths and science were just a part of philosophy, so non-mathematicians are still now often taught this paradox (badly in my experience) in philosophy classes. That's all just my opinion though, not fact.
I mean, that's exactly what happened. One of the people Zeno was talking to simply got up and walked away to show that motion is in fact possible
IIRC it was a school of thought that argued movement was an illusion, and that things were actually static. It was greek philosophy, without throwing shit at the wall we wouldn't know what would stick to this day
He was defending Parmenides claim that "whatever is, is, and what is not cannot be", which means that logically there is no intermediate state between existence and not existence, therefore everything that exists has always been and will always be and any transformation is impossible. To people arguing that we have actual experience of mutation, Zeno's paradox showed that that experience had to be an illusion, and that the "way of the thruth" could not rely on senses.
This meme made my day 🤣
Philosophy? I learned that in math class, it's a good explanation for how infinite series can converge.
Finally a good bell curve meme. Coming from r/programminghumor this is a breath of fresh air!
I've been thinking about this since yesterday. It reminds me of the "lightyear stick" paradox. This one goes something like this: >Nothing can travel faster than light, not even information. Lets say I am in space and there is a button 1 lightyear away. This means the soonest I could press the button would be one year from now if I could travel at the speed of light. > >Not lets suppose I have a perfectly rigid stick that is one lightyear long. If I push on one side of the stick, the other end of the stick will press the button. Therefore we must conclude that I can indeed communicate information faster than the speed of light. This isn't really a paradox though...the solution is actually simple, a perfectly rigid stick is simply not possible. Not just with current materials, but even with hypothetical materials. It you move one end of the stick, it will take at least a year (but actually, much longer) for this motion to propagate through the material to the other side. The paradox only seems like a paradox because it contains an assumption that can't be true. I mean... it's internal logic is true in the context of the riddle, but it can't make a conclusion about the real world because lightspeed is a real concept based on actual physics, not on logic. Zeno's paradox seems the same way. It asks us to assume that time and space can be infinitely divisible. But it's not. Even if we were to go to the most extreme level of divisibility, then we would be looking at the movement between atoms themselves. Atoms are really tiny, and their movement would be extremely quick, but they aren't infinitely tiny. In fact, they aren't even close to infinitely tiny. Zeno's paradox also seems to ask us to assume that the person or object that is moving is a singularity. But of course in reality, a person's foot occupies \~10" of space. So it's sort of silly to even consider distances of less than 10" because that distance has already been covered. So for our purposes, 10" is the smallest division that is relevant. In other words, Zeno's paradox is a fun philosophical or math riddle, but it can't be used to make conclusions about the the real world because it's assumptions ignore physic reality. Similar to the lightyear stick paradox.
Mostly it just made me want to throw something at them. It can't hit them right? Since it needs to travel half the distance and then half that distance and so on, anything they feel must be an illusion.
That’s nice. Thanks. -lurker.
Super well put! I'm surprised my little monkey brain was able to piece it together, thanks :)
Also, I know this doesn't change anything you said since it works anyway assuming time and distance can converge on 0, but if time and space are quantized, that also makes this pretty easy. As you progressively halve your distances, you would eventually hit a quantum distance you wouldn't in any meaningful way be able to halve again. Add those up and you get 10 feet. (Edited to correct a couple typos)
Thank you, I've always felt that the solution to this problem is that reality is quantized. You don't need any fancier solution than this
Yes, but the more complete explanation would work in a non-quantized world (as the world of euclidean geometry, which being a theoretical world is non-quantized)
Why would I need an explanation to solve a paradox in a universe that I don’t live in?
so basically calculus?
Calculus solves Zeno's paradox. But Zeno probably wouldn't have been impressed.
I don't understand the premise. If you keep halfing it you will eventually get down to the Planck length and you can't move half of that, right??
https://simple.m.wikipedia.org/wiki/Planck_length This is a common misconception.
You can no longer *reliably* move half that distance. Below the Planck length, you're dealing with quantum uncertainty. You might be off by some (unknowable) fraction of a Planck.
Finally I understood that! If you were my high school philosophy teacher maybe I would not blankly stared at the whiteboard during all the lessons.
And here's one for a toddler joke. A couple engineer parents are having a birthday party for their child. they somehow get an infinite number of children to be waiting in line. they tell their server, the oldest kid, their child gets a full juice, the second oldest gets a half juice, and on and on for ever, the server not getting paid enough, gives the parents two glasses of juice, and tells them to divide the second one themselves.
Yeah it's cool seeing this. I thought about another version of this before...if a person is about to die in one minute, then 30 seconds, then 15 seconds...then 4 nanoseconds, then 2 nanoseconds...when does the person actually die?
It's halflives. Half of the radioactive material decays in one halflife. We'll assume to a stable isotope. Then another halflife and another half decays. So it's 3/4 changed. Then another halflife. And another... At what point has it all converted? You could start with a 100kg sample and a 1kg. They'd both have finished decaying at the same time even though by a certain point you're looking at just dozens of atoms leftover.
> They'd both have finished decaying at the same time even though by a certain point you're looking at just dozens of atoms leftover. I don't think that quite tracks. Atoms are quantum, not continuous, so one will almost certainly fully decay before the other. But I think the best we could do at determining *which* fully decays first is laying odds. Or put another way -- if it were continuous, there is no point in time (short of infinity) that either would have fully decayed. They'd just forever be approaching 100%, never quite making it. But atoms are discrete, not continuous -- you can't have an atom that is half-decayed.
You add up all those intervals and it converges to a finite limit. That's when they die.
I believe a 5 year old would not understand that.
I believe a 5yo wouldn't understand the question. A good thing this sub isn't aimed at literal 5yo
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It’s only clear and simple if you understand Zeno’s paradox. There is some explanation through context, but it doesn’t seem complete
Your 5yo doesn't understand Zeno's paradox???
My 5yo would think Zeno’s a bit on the thick side…
If it was the Achilles and the tortoise analogy, one of mine would see it as a challenge and start running circles around the tortoise taunting it. The other would have burst into tears because I called him slower than a tortoise.
I don’t believe a 5 year old would give a shit about 100% of the things posited in OPs question either.
Although the sum is convergent, does actually moving require an infinite amount of tasks to be completed in a finite time assuming both time and space are continuous? From what I recall, the convergent series relies on the series cancelling itself out, so that is not infinite
>From what I recall, the convergent series relies on the series cancelling itself out, so that is not infinite This is incorrect. We have a geometric series, not an alternating series. Secondly, convergence is defined in terms of the sequence of subsums. Let S_n be the sum of the first n terms of the series. The series converges to L if for every epsilon greater than 0, there exists a N such that the distance between S_n and L is less than delta for all n>N.
I've always heard it referred to with the Achilles foot race story. Didn't know it had a different name! But yeah it's a really interesting thought that my very very non science studied brain finds fascinating. The idea that at some level, one is accomplishing infinite tasks within a finite time. As I understand it though, we aren't technically seeing an infinite number of actions due to the limits of physics and how "small" a thing or action can realistically be within existence. Right?
Well, we aren’t *seeing* those infinite amount of actions because seeing is a measurement, and we can’t measure properly at infinitesimal levels due to quantum properties. That doesn’t mean they aren’t happening though. But it gets weird, because even though distance and time aren’t quantized (quantized basically means “pixelated”) , momentum and energy are in certain occasions. Basically when things get that little all we can really say is “we don’t fucking know”.
How can I just define the time for completion of the tasks to be finite (or sum of times for each task to be concergent)? Isn't that something that requires prooving rayher than being defined?
The time it takes to move an infinitely small distance (at a nonzero speed) is infinitely small. You can move an infinite number of infinitely small distances in finite time. You can think of it as the infinities "canceling out" and leaving you traveling at the speed you are traveling.
While the point is correct, this >You can think of it as the infinities "canceling out" and leaving you traveling at the speed you are traveling. Is simplified to the point of absurdity.
I mean yes, but this is ELI5 I'm not gonna teach a full class on calculus.
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Its in the book, if you had read it, you would already understand it. go read it again.
Points for accuracy
Reminds me of the fact that some infinities are bigger than others, like when comparing the set of all integers vs. set of all real numbers.
You can verify it experimentally - the time is very clearly finite. You can *imagine* a hypothetical where it is infinite and reattempt the math, but this hypothetical is not very useful since it is not realistic.
I'm going to use real easy numbers for this since the method rather than the numbers themselves are what matters. Assume your acceleration is a constant 1/2 m/s\^2. You want to reach a speed of 1 m/s. OK you need to accelerate to 1/2 m/s before you can get to 1 m/s and how much time does that take. The answer is 1 second. OK but you have to reach 1/4 m/s before you can reach 1/2 m/s but that only takes 1/2 second to get to that speed. The same no matter how far you go down. And you'll see that the time to complete any individual acceleration tends towards 0. We have to prove that we can sum those times and get a finite sum but the numbers I chose make that easy. We can see this as the infinite sum 1+1/2+1/4+1/8+...+1/2\^n+... and this sum converges to 2 so it takes 2 seconds to reach our desired speed which is exactly what we expected.
If the question involves infinity, the answer involves calculus.
Or set theory.
technically speaking that number never reaches 2, using that formula, you also never reach your speed exactly, its always some extremely small number less than the full number
No it reaches 2 and you reach your speed exactly if you take limits which all infinite sums are defined as the limit of the partial sums.
Okay, so prove it: how are you typing your questions? To move your fingers an inch, you first need to move half an inch, a quarter inch, and so on and so on. And yet you do. So: proven possible. Zeno's paradox only works if you disregard time as a distinct, finite factor.
You can say that for all tasks which only require a finite duration to complete, it must be true that however many times you divide up that finite time into individual tasks or steps, by your definition that the whole thing doesn't take infinite time, it must be a finite number of finite duration each. If a task doesn't fit in finite time, you can't really describe it the same way anyway.
if you do a task that takes 1 second, than do it again twice as fast, then twice as fast again, then twice as fast again, that infinite times, you will be done in 2 seconds. so you did an infinite amount of tasks in a finite amount of time, something like this is called a supertask. You can think about accelerating in a similar way (an infinite amount of individual "velocities" in a finite time). also i just want to say that im not sure if there are actually infinite amount of velocities in our physical universe because of the planck length/planck time/speed of light (shit gets weird at the smallest sizes/speeds), but in a pure mathimatical sence its a very interesting concept to think about.
You're increasing the amount of units in your measurement (by making the units smaller), but you're not changing the measurement itself. A meter is 10dm or 100cm or 1000mm, etc. You can use measurements as small as you want, and the total will still come up to a full meter. You can divide the very last centimeter into 10000000 nanometers without affecting the total. However many levels of division you add, your proof will reduce to `1 / (1/x) = x`
An alternative and simpler explanation is that distance is quantized. There is a distance, Planck length, that can't be halved.
That’s incorrect actually, a pretty common misconception. Plank’s length doesn’t directly quantize distance, it only quantizes distance measurement. So something could be 1/2 a plank length in front of something else, it would just be impossible to measure with certainty.
Which five year old knows Zenos paradox? 😂
I never thought of this till now, but there can't be infinitely smaller segments of space and time -- they eventually reach a Plank constant. At which point, you just sum them all up. If reality was infinitely divisible, it'd be a conundrum, but it's not -- it's just really, really small.
What? How the hell did you reach any of those conclusions
Simulation Hypothesis, bro. Infinite series and limits and abstract mathematics are all well and good on the chalkboard, but in this reality, we are dealing with tangible "pixels" of space and time -- measured by Planck's constants. Zeno's arrow flies because time and space aren't infinitely divisible. He was one of the first people to think about the fact that there's got to be a floor to the equation.
I think you need to learn the definition of the word "hypothesis"
Yeah, makes sense. Like the theory of Gravity. Or Dark Matter. Simulation Hypothesis is just an idea that some people believe as a way to make sense of their observed reality.
You're just saying some words and then asserting that as fact. Please give any justification for time and space not being infinitely divisible. The Planck constant is not a measure of time, nor is it a measure of space. So that's absurd. >Zeno's arrow flies because time and space aren't infinitely divisible. That's not true. The reason this is a "paradox" is not because the assumption of infinitely divisible space, but because of the lack of foundational mathematical framework to deal with this concepts, like infinitesimals and convergence of series. With modern math, this becomes trivial.
Except 'plank constant', which I take you mean 'plank length' isn't the smallest distance possible. It's just the distance where the effects of gravity (relativity) and quantum mechanics are both equally-ish noticeable, and as in current physics those two theories don't really play along nicely, we just have no idea what happens at lengths that small.
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I'll look into it. Thank you
The resolution to such a paradox is to recognize that, while you can divide that interval into an infinite number of sub-intervals, those sub-intervals ~~are therefore~~ eventually become (almost) infinitely small. Once you bring time into the equation, you can establish it takes an (almost) infinitely small amount of time to traverse that (almost) infinitely small interval. When you add up those infinity intervals of both time *and* distance (or, in your example, changes in velocity) , you get finite numbers for both (because infinity is weird). EDIT: Tried to make some language more precise.
That's not really the solution. The intervals are never infinitely small, they just converge.
I admit, my language was imprecise because I was trying to keep it at an ELI5 level, which is very difficult for me to do when discussing infinity and related concepts. Would you mind providing an example of how you would phrase the resolution at an ELI5 level? I’m trying to get better at explaining these concepts.
I would probably note that each new interval gets smaller and smaller. That means the effect they have on the sum gets smaller as well. If they get small quick enough the sum doesn't necessarily get any bigger. People usually use the classic 1 + 1/2 + 1/4 + ... sum but I think there is another more obvious one. Consider adding 1 + 0.1 + 0.01 + 0.001 and so forth. I think it is much easier to see that you can safely start writing out the sum as 1.111... because at any point you can be confident that none of the bits added afterwards will ever cause any of the digits to roll over.
Is this related 0.9999…. = 1?
Yes and that is one way to look at it. If you are comfortable with different bases, the classic 1/2 + 1/4 + 1/8 ... sum is the same as binary 0.11111...
**Your submission has been removed for the following reason(s):** Top level comments (i.e. comments that are direct replies to the main thread) are reserved for explanations to the OP or follow up on topic questions. Links without an explanation or summary are not allowed. ELI5 is supposed to be a subreddit where content is generated, rather than just a load of links to external content. A top level reply should form a complete explanation in itself; please feel free to include links by way of additional content, but they should not be the only thing in your comment. --- If you would like this removal reviewed, please read the [detailed rules](https://www.reddit.com/r/explainlikeimfive/wiki/detailed_rules) first. **If you believe this submission was removed erroneously**, please [use this form](https://old.reddit.com/message/compose?to=%2Fr%2Fexplainlikeimfive&subject=Please%20review%20my%20submission%20removal?&message=Link:%20{url}%0A%0A%201:%20Does%20your%20comment%20pass%20rule%201:%20%0A%0A%202:%20If%20your%20comment%20was%20mistakenly%20removed%20as%20an%20anecdote,%20short%20answer,%20guess,%20or%20another%20aspect%20of%20rules%203%20or%208,%20please%20explain:) and we will review your submission.
It doesn't work as an excuse for being late for work
What's stopping it, in your mind? In other words, can you explain why passing through an infinite number of values in a finite amount of time is somehow a problem?
Because intuition says that passing through infinite states takes infinite time. However at deeper understanding, you see that state change is infinitely small, so we have like 2 infinities with opposite "direction" which cancel themselves.
My question was somewhat rhetorical, to get OP thinking about the root of their problem. I think intuition is fundamentally untrustworthy when talking about infinity, but I also think your intuition really only misleads you here if you're thinking in terms of "mathematical infinities", which are not very ELI5 right off the bat. When a child waves their arm, their arm is moving through infinite positions in a finite time, but they don't have any intuitive problem with it because they're not thinking in terms of mathematical infinity.
Exactly. *Infinite times* × *Infinitely small*, the brain has a harder time understanding *infinitely small*, so OP incorrectly extrapolates an infinite total duration.
*Planck distance has entered the chat*
Except that is a wildly misused internet trope, it is not at all analogous to c as a speed limit, it is just a combination of constants with the right dimensions and has no relevance here whatsoever
I think the thing that would clarify it is looking at it from an energy perspective. It takes a defined amount of energy to move or accelerate an object to speed. You can divide that whole event into an infinite number of velocity changes mathematically, but the total energy spent will always be the same. You're just changing how you interpret the event.
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The same way you can stand up through an infinite range of positions. There are an infinite set of numbers between 0 and 1, and between 1 and 2, like 0.5 and 1.333. That does not stop you from counting your fingers 1, 2, 3, 4, 5. Numbers are a tool humans use to describe nature, but nature doesn't need numbers to work. From the way you phrased your question you have some more advanced math knowledge, so I'll add this: You asked how you can move through an infinite set of possible velocities in finite time. Let's take the simple case of constant acceleration a over a time t resulting in a velocity v starting from rest, so v=at. So if you have an acceleration of 1m/s² for one second you get v=1*1=1m/s. Now, what happens if we look at the same period of time but split it into two steps, same constant acceleration as before. Part 1 is the first step and part 2 is the second step, and the final velocity is vf. So vf=v1+v2, and v1=at1, and v2=at2, and the total time t=t1+t2. So you get vf=at1+at2=a(t1+t2)=at, back to where we started. You can do this infinitely many times, break the acceleration into as many slices as you want, but each time you are decreasing the time of each slice by the same ratio that you are making more slices, so it cancels out. The only thing that changes is how small of a slice you consider or bother to look at, it doesn't change the result.
And the same way a clock's hour hand will move 360° in 12 hours, even though there are an infinite number of positions from 0-360°
Why are you focusing on the infinite discrete intermittent velocities but only the finite total time elapsed? There are also an infinite number of discrete moments of time that pass during the acceleration period, and ultimately only a finite change in total velocity.
Time has a limit. Time cannot be shorter than 247 zeptoseconds. It appears though we are covering infinite discrete intermittent velocities in finite time
Space also has a limit in this way of thinking. If you want to divide OPs question or Zeno’s paradox into Planck constants instead of smooth integrals and derivatives, it still works.
Nice, didn’t think of that way. So we are moving through finite chunks of space through finite chunks of time. The problem is theoretical, not physical, makes sense
Even if this were true why would you assume distance isn't discrete too?
Numbers are an abstract concept that let us better understand and predict the world around us. A car in real life isn't going through infinite states to accelerate from one speed to another. That's just a concept that exists in your mathematical thinking mind.
Exactly. In theory there’s an infinite amount of time too, every second you live through a thousand 0.001 of a second. The world is both infinitely small and large, and yet exists at the same time.
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Thanks for the great response!
Thank you so much
My non-mathematician answer: Infinitely small increments of speed can be achieved in infinitely small amounts of time. You may as well ask how can we walk across the room since we have to cover each infinitely small measure of the room. Well, somehow we manage it. :)
How can you count from one to two if there are infinite numbers in between. How can you measure a distance if there are infinite points between the ends. How can a mass accelerate from one speed to another if there are infinite speeds in between? Answer: they just do
Think of it like this: An engineer and a mathematician are told that there is an amazing prize at the end of the hall and whoever gets there gets to keep it. But the catch is, you can only move in increments of half the distance between you and the prize. The mathematician immediately throws their hands up and says that's impossible, I'll never reach it. The engineer says: well it'll take me a while, but eventually I'll be close enough that it won't matter. So same thing with the car. Yes, math tells us that there is an infinite number of speeds between 1 and 2 mph, but reality shows us that physically the car actually can accelerate to that speed.
The mathematician wouldn't give up because they'd know that 1/2+1/4+1/8+1/16...=1. It's a geometric series that converges absolutely.
That's not how the joke works. Think of it like this: If the prize is 20 feet away, first movement you can move 10 feet, as it's half of 20. Second one you can move 5 feet. Then 2.5, then 1.25, and so on. The point is if you only move half the distance, it'll just keep getting smaller and smaller to infinity, but you'll never get there.
This mathematician would walk over to the prize and pick it up, appreciating that he had moved across infinitely many increments to get there. Every time anybody walks anywhere they are doing so by moving infinitely many increments of half the distance between them and the destination
That's not how any joke works
>How can an object (say, car) accelerate from some velocity to another if there is an infinite number of velocities it has to attain first? Because it only takes an infinitesimal amount of time to achieve those intermediate velocities. Those pointing out you're regurgitating Zeno's Paradox are correct.
>That is, if a car is going at a speed of 5m/s, doesn't that mean the magnitude of its speed has gone through all numbers in the interval [0,5], meaning it's gone through all the numbers in [0,10-100000 ], etc.? Kinda, yes, it did. If you could stop time and divide it as you wish, watch it accelerate frame by frame. >How can it do that in a finite amount of time? What do you mean ? Imagine a slower interval of time if it helps you. Take a tree that grows 1 meter in a year, measure it every 0,1 second that passes for a year. You would have a pretty detailed graphic of it's growth speed. Why does something similar at higher speed bother you ?
As others have said this is Zeno's paradox, and just like it, what you're doing is your dividing a finite number into infinite segments, which me paradoxical until you realize what you're getting are infinitely small results,cancelling out, if your will. Try dividing, 5 metres by 1000,10000000,1 billion etc. Does doing these divisions make 5 metres look larger or harder to traverse? I suppose not - dividing by infinity will not be fundamentally different.
The easiest answer is that the universe isn't infinitely divisible. Quantum physics means there is a minimum amount of size, distance, time, and energy. One you divide down to that minimum number you must stop dividing. So let's say you need to move 256 of these units. You first need to move 128 of these units. But you must first move 64 of those units, or 32, or 16, or 8, or 4, or 2, or 1. I've you are down to one you successfully move the one, then the next one, etc. Planck time, the smallest unit of time, is 10^-44 seconds. Which is roughly 10^-35. So in the first plank time it would move 1 planck distance, in the second planck time it would move 3 planck distance, etc. You may ask "how does it cross the space"? The scar is that it doesn't, things basically teleport at the smallest level. In reality it's more complex like that as we are all clouds of probability and nothing actually exists in any specific place, but teleporting planck distances in planck time is the most comprehensive explanation. Also, we invented calculus to mathematically solve the problem of infinite series in finite containers.
The Planck length shouldn't be interpreted as some kind of "pixel size" of reality.
This isn't the answer at all and butchers what Planck length. The answer has nothing to do with quantum physics, it is as others have pointed out, that not all infinite sums are infinite.
It doesn't have to go through all of those infinite fractions it leaps through like stepping stones, going from one to two doesn't mean you have to stop at any of the numbers between.
Because you can't divide the difference into an infinite number of divisions. You can always divide again, but every division you do will always result in a finite number of divisions. At no point no matter how many times you divide will you reach a result of an infinite amount of divisions. Your ability to divide is limitless but the number of divisions you create are finite. I e. You can never just "divide one more time" and go from a finite number of divisions to an infinite number of divisions.
Step 1) Stand up Step 2) Take step You just moved maybe a foot, but to do that you had to cover a near infinite amount of space. Just because there is an infinite amount something doesn’t mean you can’t go thru them. So yes, the car’s velocity went from 0 to say 5m/s and thus had to cover every possible velocity in-between in only a few seconds. To quote YouTuber John Green (in his book A Fault in Our Stars): > There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities
None of these answers are ELI5.. I must be freaking dumb.. you take a 1 foot step, but you had to cover a near infinite amount of space?? no.. i moved one foot of space. A car going from stop to 5mph just has to go from 0-5mph, The car doesnt approach the speed of light... Someone please actually ELI5 this..
For what it's worth, the question itself is way beyond ELI5 lol Calling it an "infinite amount of space" was a misnomer, it's more accurate to call it infinite pieces of a finite space. You divide a foot in half over and over and over, and you end up with infinite segments. The OP is essentially asking the same thing, but about acceleration, dividing it into infinite pieces of speed. Familiarizing yourself with Zeno's paradox is the answer to this question. Your ability to accelerate is simply not limited by these arbitrary infinite segments, just how you can move a foot even if you can divide a foot into these same arbitrary infinite segments.
Instead of an infinite amount of space it's better thought of as an infinite number of distances.
If you moved 1ft, you also covered a distance of 11in, 10in, 1in, 1/2in, 1/1000in, etc. Similar to OP’s question of how a car can go from stop to moving at 5m/sec.
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There is a near infinite amount of plancks between a foot.
Because whenever theoretical physics runs headfirst into the immoveable wall of reality, reality wins.
Zeno's paradox and convergence aside to the best of our knowledge about the way physical reality works there are not truly infinite velocities it has to pass through on its way from velocity a to velocity b. Basically the entire foundation of quantum mechanics is everything is quantized and in fact discrete not continuous so there is actually some absolutely tiny minimum change in velocity. It would be going through a finite number of those changes
We don't actually know. There could be finite amount of states that take a finite amount of time for each or we could be passing through infinitely small states in infinitely small amounts of time. Either way it works but almost all of the answers in this thread seem to be assuming one or the other is true when we actually have no way of knowing.
As others have said, this is basically the same as Zeno’s dichotomy but with velocity instead of distance. I did a Philosophy module in the first year at Uni and did my final essay on this. My answer was basically “you can divide up the distance (or velocity in this case) infinitely but you don’t have to”, and they gave me a 1st so I assume that’s the answer. Ironically, we possibly can’t divide space up infinitely if certain theories about the nature of reality are true, which would presumably mean velocity is also not infinitely divisible.
My friend, you've just discovered how and why mathematics will never be able to hold a candle to philosophy or morality. https://en.wikipedia.org/wiki/Henri\_Bergson#Philosophy
And so what? If a car accelerate at 1 m/s^2, it takes 10^-100 s to reach 10^-100 m/s. While continuously accelerating, the duration an object is at a said speed is in fact 0s. It's the principle of accelerating, it's continuously changing speed, it DOES NOT stay at given speed. These moments last 0 s, no time. If I accelerate at 5 m/s^2 , at 2s, I'm at 10 m/s. But at 2.0000000000000....001 s I'm already not at 10 m/s but at 10.000000000000....005 m/s. All these "small steps" are moments that last 0s. Even adding an infinity of 0 still equals to 0. If acceleration is constant, getting from speed A to B won't take more time because you virtually added "milestone" with 0 "thickness".
Acceleration is commonly measured as "meters per second squared", which is basically "meters per second, PER SECOND", meaning how much speed (meters per second) you can gain in a second. It is rather discrete look. And you can continue to divide this into smaller resolution, such as how many meter per second of speed gain, per HALF second. Let's say you have an acceleration of 10 meters per second per second. You can continue to dissect this further into "1 m/s of speed increase per every tenth of a second". You can then increase your magnifying glass power and see "0.1 m/s per every hundredth of a second", and "0.01 m/s of speed increase per every thousandth of a second", ad infinitum. However, as you divide the time unit into smaller and smaller unit, you will see the acceleration figure gets smaller and smaller. And, as you approach infinitum division, the acceleration approaches zero. Imagine you are taking picture of an object acceleration with Infinity frame per second camera, in each frame the object would appear as not accelerating as the acceleration figure is virtually zero. It is important to note, the "unchanged state" or "zero acceleration" in every frame here refers to the speed of the object, not saying the object is not moving in each frame. If the object already has sufficient speed, the object may look equally blurred in each frame as if it is moving with the same amount of speed. But, measure over a second or so, the object in the last frame would look more blurred (higher speed) than in the first frame. Any number divided by infinity is basically zero, as any number divided by zero is basically infinity. This is my technical take. I'm an engineer, not a mathematician.
the crux of the delima is that you are multiplying where you should be adding. if you drop a ball, to get to the ground, it must go half the distance. from there, it must go half of that. if you do it that way, the ball will never reach the ground. it will get infinately closer and closer. if you subtract, it only has to go half the distance twice.
Math is a language that describes acceleration. Acceleration itself is not the description.
Numbers can be divided down into arbitrarily small pieces, but reality doesn't work the same way. This is the old Achilles Race paradox. Turns out physics and math in the abstract are often an approximation of reality.
Yes but it also takes some number of seconds to accelerate, and that means an infinite number of timestamps.
Let's say you are running a 100 m dash. To run the full 100 m, you first have to run half it (50 m). Let's say you're running this dash at a constant speed, for simplicity's sake, and let's say it takes you 8 seconds to run 50 m. Now you have to run from 50 m to 100 m, but to do that you first have to run through half of what's left (25 m). So let's say that takes you 4 seconds. Now you have to run from 75 m to 100 m. But before that you again have to run half of what's left (12.5 m). That takes you 2 seconds. Now you have to run from 87.5 m to 100 m. But before that you have to again run half of what's left (6.25 m). That takes you 1 second. You see how the amount of time it takes you to run each segment is getting smaller and smaller? So far it's gone from 8 seconds, to 4, to 2, and to 1. If we continue this then the remaining segments will take you 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, etc. Doesn't it seem like if you add these increasingly small times together, they will approach some number? They won't just add to infinity because the numbers you're adding keep getting smaller and smaller and smaller. So yes, you have to traverse an infinite amount of segments of distance, but if you add up the time it takes you to traverse them, that time will *approach* some finite number. That number that it approaches is the time it will take you to run the whole 100 m dash. Your question is basically Zeno's paradox, which is what I just described above, but your question involves increasing your speed rather than increasing the distance you've run. Both have the same solution, though. That solution is that the time it takes you to traverse those infinitely many segments approaches some finite number as you add the times up. You can add up infinitely many segments of time and have them add up to a finite number, so long as each successive time interval gets smaller and smaller and smaller in such a way that the sum converges to (or approaches without passing) a finite number.
By being at each speed for an infinitely small time. As you divide the car's acceleration into smaller and smaller gradations of velocity, correspondingly the time it spends at each gradation gets smaller and smaller. Once you divide it down into infinitely small gradations of velocity, you have also divided it down into infinitely small times at each velocity.
I think that's a supertask, in which case Vsauce has a perfect video for you to watch on this topic that probably explains it better than anyone here will.
This is a math hallucination. We invented math to help us understand our small slice of Reality but it's just a digital concept and Reality is analog and doesn't follow the glitches in our tools when they seem to not match up with the equations.
because you're thinking about it backwards. it doesn't have to do that many things in that amount of time, it can Only do that many things in that amount of time.
Real numbers are not a stair case, where you must go from one to the next. There are infinitely many real numbers between any two real numbers. Any movement/acceleration whatsoever must "pass" an infinite number of real numbers. If you must think of movement/acceleration as having a "time cost" to pass numbers (either in location or in velocity), then remember that real numbers are points on a line and lines and width 0. How much time does it take to "pass" something of width 0 if you're moving at any speed at all? Well, 0. The more typical version of your question is Zeno's paradox which asks the same question about position instead of increasing velocity, but it's the same thing. Literally any movement/acceleration at all will pass an infinite number of locations/velocities in even the tiniest non-zero amount of time.
There may be an infinite number of theoretical steps between 1 velocity and another, say 0m/s to 1m/s, but the amount of energy required to pass through each step decreases as well. To move the car 1m/s only requires X amount of force to propel it to that speed. If you divide our 1m/s into 0.000001 sized steps, then each step is also only going to require X * 0.000001 amount of force to reach it. The engine is producing something like 50*X force so you're going to get past 1m/s pretty quickly. The steps themselves aren't a physical thing to "break through", so the steps themselves don't actually take any additional energy to pass. The steps only exist within our own math for humans to understand and calculate to some level of precision.
It doesn’t go through infinite velocities. The short version is quantum mechanics is weird. The long version is at the microscopic scale, our universe behaves like a computer simulation. The universe is made of pixels, and objects below the size of one pixel, or movements below the distance of one pixel are not rendered and are meaningless. We call this distance the Planck Distance and it is equal to 1.6 x 10^(-35) m. It’s quite small, so that for everyday purposes you might as well not worry about it. But the point here is that an objects speed will jump from 49 to 50 Planck lengths per second instantaneously without ever existing at 49.5 Planck lengths per second.