Isaac Newton elegantly answers all of Zeno's questions with the invention of Calculus. It's a branch of math that revolves around rates of change, infinity, zero, and approaching a number. Learn some of the theory behind Calculus. It's really useful.
These animations really are amazing, the ideas come across so clearly and it's such a pleasure visually. I studied philosophy in college and I think these are great introductions. Keep up the good work!
To put it in math terms, What Zeno's paradoxes are using are pure mathematics, while the observable real world is using discrete mathematics, ie physics. Real world units are not infinitely divisible as pure mathematical units are, therein lies the disparity. The flaw, so to speak, is that there is no such unit as a "moment" as Zeno described it in his second paradox, the "moment" either has an object moved from rest, or it has not, in fact, progressed through time. Of course, this lies predicated on only one of the two following facts being true, that: A) the world is infinite, and thus can encompass time completely, in which point there is the possibility that there is a distance traversed at which time has not elapsed, or B) that TIME is infinite, with no describable beginning or end, at which there is a unit of time small enough that an object in motion has not, actually, moved at all. Either one of those two statements being false or true is enough to discount Zeno's paradox, scientifically speaking.
1.616199(97)×10−35 metres is one unit of planck length, theoretically the smallest measurable length given uncertainty in the locations of particles on very small scales. 5.39106(32) × 10−44 s is one unit of planck time. The smallest measurable amount of time possible, because if the smallest measurable distance is one planck and the fastest measurable thing is a photon moving then it stands to reason nothing can occur faster than a photon moving a distance of one planck unit. However newton disagreed and showed that with calculus all movement occurs with infinitely small fractions of time and position which was the original use of the derivative. Certainly without calculus these paradoxes could be extended to absurdity using planck scale, but it is not necessary with calculus to show that all changes happen as a sum of an infinitely large amount of infinitely small changes.
Maxwell Hill But was Newton aware of a "cap" so to speak on the infinite division that the planck unit represents? I may be misunderstanding your argument, but calculus is a branch of mathematics, an inherently theoretical field, until we step into the realm of "applied" mathematics do we get things like physics, which DO depend on, as you put it, reduction to the absurd levels of distance and time, which approach, but do not actually arrive at "zero", which is the unit of no elapsed time, or no displacement. The logical question then, I believe, is 'does the planck unit translate to Zeno's "moment"?' Can we physically reach a unit of time, provided that we divide the units enough, reach a point where the displacement of a moving object is equal to zero? If we take Einstein's time dilation equation: delta(t) = 2L/c We can solve for t, where the length, or 2L, is equal to zero, would give us an answer of delta(t)=0, ie a unit of time equivalent to zero as well, and the ramifications to the two statements I extrapolated in the previous comment.
Ali Al-Shaikh He already said that you don't need a "cap" on the infinite division when using calculus. This is because you're already using real numbers. Real numbers are continuous. Zeno's paradoxes aren't paradoxes at all. They're just calculus problems.
These paradoxes can easily be averted by realizing the functions of the universe; to do this, We can't confuse observations with conclusions. "When a the bike reaches where the man was, he will be further ahead than where he was" is logical, but to think that the bike will increment smaller and smaller amounts to infinitely small scales is absurd. to clarify a bit better, look at the arrow idea; "it crosses a mid point to reach the end" makes sense, but there is no defined midway, as physics simply doesn't confine itself to such a rule. "Observations=Conclusions" is a dangerous system to use, because you must place reason between them.
Here's why I call bullshit on these paradoxes: Objects move at a rate of distance/time. The paradox assumes that the faster object cannot ever pass the slower object with a head start, because it must always catch up by some incrementally smaller distance. This ignores the rate of distance/time. As the distance gets incrementally smaller, the time passed gets incrementally smaller, and eventually, you would have to say that time has to "stop" for the faster object not to pass the slower object. But obviously, time does not stop, and the objects must both move at their assigned rates of speed.
Why would time have to "stop"? Well, if we are saying Planck Time actually applies to the behaviour of time, not just it's measurement, this is the case. But if we are saying time can be infinitely small, there is no reason for it to "stop".
R. Ringshifter I'm saying that the paradox requires distances that approach zero, which given the distance/time relationship, give time passed as approaching zero. When a number gets so small, it is effectively zero. What I'm saying is: the time that is part of our reality doesn't stop for the sake of a thought experiment. If the faster object only needs 5 seconds to pass the slower object with a head start, that 5 seconds WILL pass, and there's nothing a philosopher can do about it.
Aristotle brings this idea right down to Earth. If motion were to be infinitely divisible, time must also be infinitely divisible. Thus the infinitely many increments of distance can be completed by the use of infinitely many increments of time.
Makes perfect sense to me. I recognize change as another symptom of time that is exclusive to my current existence of experiencing things on a smaller scale. On this level things seem to flux and go through cause and effect, but in more simple and higher frequency matrices the illusion of change is less noticeable. It is the only way for a multiverse to work that it has some basic goal of unifying, and if all things seek unifying than change is certainly hard to argue for.
Zenoe's first paradox makes the assumption that they aren't moving at the same time. The distances are changing at the same time, one more than the other.
I've been watching you guys since the beginning and absolutely love what you are doing and where the channel is headed. I can't wait to see the future of Thugnotes/Wisecracks!
holy crap I've thought about this since I was a little kid. i realize that you could continue dividing an increment of measurement, so I thought there was infinite space between any two objects. i impress myself
One interpretation of Zeno's Paradoxes that I prefer is that it's like Schrodinger's Cat - an attempt to point out the absurd implications of his colleagues' ideas. (Schrodinger's Cat isn't saying you have an alive-dead cat, it's saying "what you're saying means there would be an alive-dead cat in this situation, therefore what you're saying is patently absurd and therefore probably wrong." Given the amount of time that's passed since the days of Greek philosophers, it stats to reason that there are a number of Zeno's contemporaries who we don't know about. Hell, there's plenty of scientists involved between Max Planck and the guys working at the LHC...I can name a handful of them, sure, but no way I can name all of the ones that made major contributions, let alone the probably-thousands of individuals who solved the little problems necessary to make that next big step. A lot of libraries burned between Zeno's writings and the present day. Also, the fact that the only similarly absurd "scientific theory" (see above on Schrodinger's Cat) is so often misinterpreted despite the relatively short period of time since its inception) could be seen as somewhat suggestive.
Zeno demonstrated that a universe in which time and space increments are subject to infinitely smaller regress is philosophically impossible. The paradoxes predict the concept of planck length and planck time. There's no such thing as an "analog" universe and indeed there COULD not be such a thing, as Zeno proved. This video offers a sophomoric understanding of these paradoxes. Zeno isn't showing that motion is impossible, he is showing that motion is impossible in the analog universe imagined by his contemporaries.
+swordsman3003 correct me if I'm wrong. But I don't think Zeno "proved" the impossibility of an anolog universe since Georg Cantor showed that mathematics can describe continuity in space and timy as long as you differentiate between infinities.
+Stephen Cesler Analog is to digital as continuous is discrete. Something like height is continuous, (you could be 6ft tall, 6.1ft tall, 6.01 ft tall, 6.0000000001 ft tall, etc), while something like number of siblings is discrete (you could have 0, 1, 2 .... etc). Quantum mechanics shows that elements of our universe are best thought of as being 'quantised'. Photons can't have any energy (i.e, photon energy isn't continous, or analog) but rather, can only have specific quantities of energy (they are discrete).
+swordsman3003 Actually, Zeno was, according to Aristotle (our principle source on the matter) trying to show that the world was unchanging. I will say that this video actually misstates the fletcher's paradox, which states that at any quanta of time the arrow is traveling neither to where it is (as it is already there) nor to where it isn't (as no time is elapsing for it to get there). A thing that is not traveling is stationary, which means at every quanta of time the arrow is motionless. Quantization does not resolve this issue. Leibniz and Newton do answer this question, however, with the assertion that impetus (what we now call "inertia") is an unseen element contained within the moving body. Thus, the arrow isn't motionless, as it still carries the potentia of motion within it, even if the viewer cannot perceive it.
So change can be observed only in relation to a particular end in time or space. When we designate the runners position as that end point, we aren't concerned with what happens once the motorcycle overtakes the runner. Just as when we properly recognize the finish line, we stop caring about their relative positions in space after they cross it.
This is the only 8-bit I'll contest, still love this one and all other ones. But superposition explains the possibility for change via time/relativity. We perceive light which is made up mostly of photons traveling through mediums so we can perceive them as light, but they do not comply to normal laws they can be brought back from time, duplicated, be in two places at once, travel fast enough to not perceive time and I'm sure a number of things I missed, but ask a physicist and they'll explain it way better :P.
Wow. This is awesome. You both finally made Zeno's arrow paradox clear enough to me that I can see why it's wrong, and made it clear to me why .999... = 1.0
Philosophical theories are so much more fun to contemplate when you're in high school and uni.... but not at my age. At this point it just terrifies me.
Well glad to see the entire comments section has stepped forward to point out how Zeno's paradoxes can be dismissed as bollocks by either obvious common sense, or a more nuanced, holistic (and modern) understanding of mathematics and physics. His mentor said something stupid with no real reasoning behind it (that the episode touched on at least) and his student defended them with half-assed arguments... although to be fair, they probably made more sense at the time since our understanding of various sciences wasn't as great.
+leviadragon99 Yes, to propose that change doesn't exist within the closed boundaries of our given system is preposterous, but to identify that we exist within a closed system, where time, space, energy and everything else that might exist are locked and bound to, is of essence.
The problem is that there's a limit to how short a distance in space or time can be (the Planck length), before the uncertainty principle makes the distance irrelevant. Of course, this may support Zeno's paradoxes, but in a slightly different way.
Beyond that, there's also the physics question of (and I'm going to butcher this terribly) "what happens if we put in enough energy to the point that a wavelength is shorter than a Planck length?" So even that since of like an infinitesimal still hasn't been called into question yet.
Zeno's paradox is the study of an asymptote on the negative X axis without considering that is continues onto the positive after reaching a point of "infinite distance" according the the Y axis using allegory.
When the biker slows down we often forget that time is slows down too. It's gets exponentially smaller based on the the difference between the bikers and runners velocities. This means that what Zeno describes does actually happen and can be measured with a graph that plots the distance between the biker and the runner. We can then take the data from that graph using only up until the point where the biker passes the runner. Instead of conventionally plotting the data we want to make the x axis equal to intervals of the biker reaching the runner's last recorded position. This does not alter the data but simply creates a different perspective of the original graph. If you were to visualize the entire race using this method it would appear like the spike of a wave that then levels out as the biker drives off into the sunset.
I used to be terrified of change. When my family moved to a place I had never been before and we had no family roots like before I was terrified. It was only days before the move that I made peace with myself that change is good and bad, but it's the law of life, and life is about living and living is about change.
Great video. Side note, The Fault In Our Stars by John Green has a scene where a character references Zeno's First Paradox, pretty cool to learn about it more here.
Zeno's paradoxes confused us until we invented calculus. Once you have the concept of a limit, the problems disappear, even if you assume spacetime can be infinitely divided. Check out a book called Zero: The history of a dangerous idea, really good book.
This philosophy was before the invention of 0. The motorcycle would pass the runner, decreasing from the head start to 0 when he catches up with the runner, then the distance would then increase. The second paradox sections the amount of time with the amount of distance it takes to hit the arrow. But the arrow starts at 0. the paradox assumes that you have to divide time and distance infinity times, but 0 represents the lack of time and distance. Not moving to moving is change
I think Zeno's paradox was resolved by Newton and Leibnitz and the invention of calculus in the 17th century. They showed how you may add up infinite numbers of infinitely small quantities and the result is still a finite number.
So basically Zeno is saying that if you stopped time, then there would be no change. But time actually passes in the real world, so his paradoxes are whack.
Well that and he didn't understand acceleration and ssumed that motion is constant. In terms of EM waves, you can't slow them down at all, since they are bound by physics to travel at 3x10^8 m/s everywhere (in terms of within atmosphere, measuring the distance/time of an EM wave/photon interacting with two separate particles will still be that number even if a beam of light fired through water across 10m will appear 'slower'). In this case, the runner/biker paradox works because if a photon is shot .1 second before another photon, then it will never be able to catch up and appear that motion is impossible (before we discuss observer/object relativity).
I don't think I quite grasped the first paradox, unless the motorcyclist decides to stop at intervals, or has agreed to travel at a lower speed surely the motorcyclist will just accelerate to a speed beyond that of the man and overtake him, regardless of point of view, the change in velocity has increased beyond that of the man, in which case distance has changed, can someone explain this to me
To be frank although it is quite obvious that even without acceleration the motorbike is going to be ahead in short time, when you take it like Zeno puts it is tough to explain. Mathematicians couldn't do it until Taylor's expansion. You can study it and would figure out how..
***** of course they did, don't be silly :p And the paradox is more simply explained like this: I want to clap my hands. Let's assume I only move one hand, that hand will have to move the whole distance from one hand to the other. But to do that it surely has to also move half that distance, and before that a quarter of the distance, an eight of the distance and so on. If these small movements can be infinitely small, my hand will never reach the other hand because it has to move an infinite ammount of small distances.
It's funny because that even though nothing really can be changed/created in the whole of the world, it seems like memories are still expanding. Which is even funnier, because we can't see memories, only imagine them. So does that prove that the only change possible is in a world where we can't see, touch, hear, smell, and taste? Even further than that, that ever changing universe thereby seems to become the only world that matters, because that is the only world that can change. So it is the only world we CAN change.
Space time behaves as observed (obviously) when observed via a confluence of the senses, creating the standard impression of "reality", but when the mind isolates and withdraws from the senses individually (as in Buddha Dhamma), the relativistic impression gives way to a monadal experience of here/now/all/nothing, devoid of self or world, here or there, future or past, energy or matter, light or dark, etc. While this all-encompassing void is the fundamental substrate of all possible experience, it is mutable unto knowing itself (the field state that occasionally crests into consciousness), and it is the presence of knowing within emptiness that differentiates one thing from another, giving rise to a relative universe from within the bosom of the absolute.
Zeno's paradox is put safely to rest by Special Relativity. Light moves at a constant speed. This speed is constant for all un-accelerated observers. What this means is that every motion in the universe has a context. Everything must be compared to the speed of light which cannot be slowed down in constant velocity, so every velocity can be unambiguously compared to a constant speed which works in every direction. However, it is interesting to note that when applying acceleration to a system Zeno's paradox re-appears. For relativistic acceleration gives the observer the impression that light can slow down. We discover that there is a way to accelerate in such a way that light approaches the speed of zero. It's this concept that I based my hypothesis on black holes which may reveal how galaxies form and cluster up. Come to my channel to find out more and look for my videos on the nonsingular hypothesis.
We've solved the second paradox. The problem is summing up the infinite sum 1/2 + 1/4 + 1/8 + 1/16 ... and so on, for the arrow must travel half of the total distance, then a half of that, and a half of that, and so on forever. The paradox is that this sum seems to never become 1 unit of distance. But we can apply infinite summation rules to show that this summation does in fact equal 1, meaning the arrow covers the distance it should. Let 1/2 + 1/4 + 1/8 + 1/16... = s Then 1/4 + 1/8 + 1/16... = s/2 Subtracting gives 1/2 = s/2 Hence 1 = s, the summation in question. Thus, though the arrow makes an infinite number of half steps, it will cover the whole distance.
All this tells us is that *Zeno* didn't _knew_ *Calculus/Mathematics* and/or Philosophers don't consider Math. Relevant: xkcd.com/1153/ ( See the title text , which says "The prosecution calls Gottfried Leibniz" )
the reason the paradoxes aren't applicable, is the discovery of the maximum speed and minimal distance. ie. the speed of light and the plank length. meaning their is a minimal amount of possible time, the time it takes for a photon to travel a Plank length.
Silly Zeno, you just don't know your limits! Seriously though, I was under the impression that Leibniz proved Zeno's paradoxes false. Is there some subtlety I don't know about that still bothers mathematicians?
+Philip Stuckey Actually, Archimedes was the first one to settle down mathematical Zeno's paradoxes in modern sense(though his own idea were not widely/accepted at the time). And if you are talking about calculus or theory of convergent, then though those were first precisely treated by Leibniz(I must say, in a damn excellent way) but put on rigorous way by Cauchy, Weierstrass and others. And infinitesimals were first rigorously treated by Abraham Robinson by non standard analysis. Conceptually, Zeno effect is still of importance (not paradoxical though) like in quantum zeno effect an zeno executions. That said, actually original Zeno paradoxes were not mathematical, it was about physics, more precisely about motion. If we generalize the concept to something like this - decompose some physical quantity (distance in first paradox), compute each part individually (makes geometric sequence in this case), add them up (converges to a finite value). He was saying how can you add infinite things in finite time. For example traditional epsilon-delta limit method doesn't adds these numbers it just gives the existence of bounds of finite sub-series. And non standard analysis inherently ain't constructive. Lastly, there is something of similar nature but much more subtle and paradoxical, Casimir force. The calculation of Casimir force is of above nature, i.e. decomposition -> computation -> addition, but the series it creates is not convergent (precisely, it become some constant times (1+8+27+64+...+n^3+...) ), and amazingly enough it gives the correct answer if we put a finite value instead of infinity (precisely 1/120 for (1+8+27+64+...+n^3+...) ). For completeness, there are methods discovered earlier which actually gives this value for this divergent series, but the reason that why it works is not clear.
The only way to reconcile this is if units of distance and time are finite. If they were, for instance, the Planck length and Planck time, there would be no such thing as an indivisible moment.
zeno's paradoxes kind of shows a misunderstanding of infinity. infinity doesn't prevent an arrow from reaching the target it allows it. (sorry I'm not good at explaining this but.. vihart has a great video on how 0.99 = 1 which uses a similar problem)
Mathematically, Zeno's "paradoxes" fail due to a convergent series. The series 1/2 + 1/4 + 1/8 + 1/16 ... = 1. Not close to one, but exactly one. It converges upon 1. Basically, this is how mathematicians solve this apparent paradox.
The first paradox implicitly assumes that the runner has a higher velocity than the biker. If you look move along with the biker you'll see that the runner would have a negative relative velocity and the ground an even greater negative relative velocity. So the critical point is when the biker and runner at the same displacement on the track. Basically, it's a restating of the second paradox. The second paradox is pretty straightforward when you keep in mind that if the arrow has a constant velocity subdividing distances subdivides time as well. The only issue is how anything at all gets from one "moment" in time to the next. If you believe that the Universe is a fixed-point and our brains just walk along it without inputing anything from a time before the point it's at you have to wonder why that causal relationship still exists and reserve that belief on where quantum randomness comes from when collapsing from a quantum state to a classical one.
As always a good episode. But gosh darn it we aren't made of numbers and theories. We're made of flesh and blood and atoms. These take up space with mass, and move in time with energy. So okay the arrow takes up it's space in a moment of rest. But the archer or bowman or kid on christmas puts in energy into the arrow shoving it through time. Then physics happens and the arrow hits the target. That's the problem with Zeno, he's speaking of math and infinities when he and every other thing in creation is physical and finite. I'm the first guy to say "we're all the universe!" and "our perceptions are flawed!" don't get me wrong. But these are truths that can't be proven by math because math is only a language to describe both itself and dreams of a stranger universe. The unity of the cosmos isn't homogenous, it is diverse like a body. A foot is not a hand but they are part of the body. Gurus and Qabballists and other mystic folk always say that our perceptions are flawed. And like all things we've made, including languages, they come from our flawed perception.
actually you can prove his 2nd paradox wrong with math. Potential energy of the arrow within the moment can be transferred into kinetic energy. I just wanted to clear up a principle you stated correctly. the first paradox can be theoretically proven false with the unit of a plank distance of 1.616199×10^−35 metres. At this level there seems to be no shorter distance at which subatomic particles can move.
duh Rooker The Planck lenght is the smallest *measureable* lenght (that one can measure), not the smallest possible lenght. Where did you hear that it is the shortest distance subatomic particles can move?
an arrow cannot possibly experience 0 time, because experience requires time. You cannot assume that an arrow in "zero time" has a speed of 0, because speed = distance/time, or 0/0, which is undefined. All this manages to prove is that 0/0 is in fact, undefined.
Doesn't work still. Long story short is that length and time isn't movement. Planck Length and time isn't as "recent" as you having internet might lead you to believe. They are all 20th century theories. Planck himself died on '47. That leaves A LOT of time to react to this paradox. Comenters aren't really at fault for getting it wrong (they are at fault for not reading it, but regardless). It's basically the "fault" of the uploader's presentation The whole paradox is simpler to explain and you'll understand why Planck length and Planck time do nothing in this regard. An arrow travels from point A to point B, say that distance is 20 meters (time isn't even necessary). But to get there, it must have passed at some point point C = 10 meters. To get to C it must have passed point D = 5 meters etc. *ad infinitum* Since a segment holds an infinity of points, that means that the distance cannot ever be passed, (else it would be finite). Planck length, which is the smallest possible length in space is still made up of numbers you can divide, so even though physically it cannot be "composed" of smaller distances, a distance is an abstract entity, so you can divide it by as much as you wish. When thinking of Zeno's paradoxes, it helps to think in terms of hyperbolic functions, and, as consequence, through limits x -> ∞. If you see someone doing that, that means he's on to the "big" question of Zeno, if not, don't waste your time.
I don't see how you can divide a planck length, I mean you can say it's half a planck length, but that's as meaningful as saying a square is a triangle.
Doesn't the first paradox only occur because time is removed from the equation? Am I wrong? It works if you judge the speed of the motorcycle by the man's previous position, the motorcycle is always one step behind. It works because their positions are measured at different points of time.
I dunno. I don't think our perception is flawed, but our ability to comprehend what we perceive definitely is. We live in a medium-sized world, far removed from the humongousness of stars or the minuteness of subatomic particles and forces. With paradoxes like this, we're just grasping for understanding based on the tiny bits of sensory data we're able to acquire.
I love this series!!!! I remember having to explain some of Zeno's Paradoxes in a presentation in my one required college math course (I was an English major). I think I explained them really well, but I still didn't buy them. Interesting concepts, though!
zeno's paradox has already been solved.It takes the arrow 2 seconds to hit the mark. Zeno's Paradox - Numberphile Just because there are an infinite number of steps to complete a task does not mean the task will take an infinite amount of time to complete them.
I don't understand the second paradox.. No matter how many times you halve a number (be it representative of distance or time), the remaining number will have a value. At what point would the arrow have not moved, and time still elapsed, if all you're doing is halving the distance each time you halve the time?
This one's easy. Paradox one, Space is infinitely divisible so if you take a closed section of space you can divide it in such a way that you will get an infinite number of divisions (one such way is described in the paradox). Now if you stop to measure each one of those divisions you will indeed be measuring for eternity and never get to the end point of the closed section of space. This is because you will have an infinite amount of measurements to make not because traversing that space is impossible. Pust another way, traversing from point A to point B and counting how many times you can divide the space between point A and point B are two entirely different exercises. Infinitely divisible does not mean infinitely long. Paradox two, The time of the arrow's flight is measured from the start of the arrow's motion, therefore there is no point in the measurement of that time that the arrow is not in motion. In other words from the "moment" the timer is not 0 the arrow already has motion.
HOLY SHIT I SAW THIS VIDEO AND IMMEDIATELY THOUGHT OF PARMENIDES. I thought to myself, there's no way wisecrack is going pre-socratic, its gotta be someone more recent, but you went there. I love you guys.
I think Zeno's second paradox doesn't work because the arrow is traveling at however many meters per second, not however many "halfway points" per second.
The first paradox is easily solvable with calculus. Suppose that the distance between the man and the bike is 2 m. The speeds are 1 m/s (man) and 2 m/s (bike). In 0s: s(bike) = 0m s(man) = 2m In 1s: s(bike) = 2m s(man) = 3m In 1.5s: s(bike) = 3m s(man) = 3.5m In 1.75s: s(bike) = 3.5m s(man) = 3.75m So, although the distance is an infinite series, also is time. The last step is infinitely small in space and in time. So, the sum of the distance travelled is finite and the time it takes for the bike to catch the guy is finite too. This is true even for continuous variables. It isn't necessary to consider Planck units.
Someone like Thomas Hobbes would probably have called this paradox nothing more than a semantics game. Zeno heavily relies on a semi-fictionalized concept of Infinity to come to all of these conclusions. Hobbes says that as long as you continue quantifying the characters in an infinite number - such as the distance between the cyclist and the runner - you procedurally and continuously dissolve the infinite by the assignment of specific, finite numbers. Just because Zeno phrases it by saying there is an "infinite" distance between the two bodies does not by necessity suppose that the space between them is somehow eternal/irreducible.
Zeno's paradoxes all seem to assume a continuous universe. Perhaps in formulating these paradoxes it occurred to him that the universe behaves vastly different on the quantum scale and ideas like planck length and discrete motion have to be entertained to explain such easily demonstrable actions.
Zeno's 2nd paradox implies there is a zero in time. When if it is to be measured there can be no zero. I.E. Time can be infinitely divided into smaller and smaller measures in which an object traveling, has kinetic energy, there is no measure of time no matter how small that the kinetic energy does not exist, only the perception of the motion is so infinitesimal as to appear stationary, it is in fact still in motion.
While I can't say as much as I'd like to on an episode on this,such math related questions* (I admit reason and logic in philosophy are two areas I'm not good at and am only beginning to venture into),such math related questions were they to be presented in school (along with "decision" centered math like game theory's stuff like the prisoners dilemma) would be another good way to motivate little kids to like math more. Don't get me wrong,it's great that ppl try to show kids the analogies btw sciences and math,but to get a good grasp of math with based on a notion of continuity (like the calculus used in physics and such) is a tedious and long road unlike how a lot of the stuff in game theory with it's objective to optimize agents and paradoxes like this,give you a pretty little glimpse of a hardly spoken/brought part of arguably more intuitive kinds of math. As a result it might kids to eventually engage with axiomatic systems like in symbolic logic."Better" ^uV then getting kids motivated to wanna get a grasp of applied probability for gambling,eh ;)?. *wikpedia says that "infinite processes" like with Achilles and the tortoise/the motocyclist and the runner here weren't really less effortful to solve till the late 19th century
“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. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful.” ― John Green, The Fault in Our Stars
Doesn't the motorcyclist and man paradox only work if we assume 0 acceleration from both of them (and of course, more specifically, not a positive or greater acceleration by the motorcyclist?)
It seems clear to me that not everything in the universe can be explained by thinking in the narrow constraints of what we know, yet we are also somewhat hampered by being unable to think outside of what we can comprehend. We know now that things can and do change all the time - such as changing energy states in atoms, or a fire giving off energy as heat or light. It's hugely central to our understanding of physics, especially quantum physics. Thus, it's not CHANGE that is impossible, it's our logical understanding of what change IS that is impossible. Change is not a finite thing. It's infinite. Therefore, it's too big of a concept for us to fully grasp. We can define WHAT change is, but not WHY change is. It's not an illusion, we just can't comprehend the idea fully. Let's look at the example of a sine wave on a graph. Sure, you can graph it with infinitely small points (and graph it more smoothly with every increase in how many points on the curve there are), but each and every one of those points fractions smaller and smaller. The units are not changing, the size of the units is changing. Therefore, there is no such concept as infinity, not in reality - infinity is just a word for an eternally continuous thing. But we can't quantify an infinite number of points on a graph, so we see it as a continuous graph, and conclude it therefore is a sine wave. The question is, how far down does an approximation go before it ceases to be an approximation, and just "is"?
The problem with Zeno's defensive argument lies in trying to explain motorcycles to his cohorts.
Q BAN Exactly.
The original argument is apollo and the tortoise.... Obviously wisecrack was just doing their 8bit shtick.
Ben Hinman The original comment was a joke... Obviously some people were doing their whole take jokes seriously schtick
"The distinction between past, present, and future is only a stubbornly persistent illusion" -Albert Einstein
Zeno, father of troll physics.
A philosopher is by definition a troll.
Zenos in a car driving somewhere with his parents, and he's like "are we halfway there yet?"
This is one of those ideas you can refute by kicking a rock.
Isaac Newton elegantly answers all of Zeno's questions with the invention of Calculus. It's a branch of math that revolves around rates of change, infinity, zero, and approaching a number.
Learn some of the theory behind Calculus. It's really useful.
Think Zeno must have had a few when he came up with those particular paradoxes.
These animations really are amazing, the ideas come across so clearly and it's such a pleasure visually. I studied philosophy in college and I think these are great introductions. Keep up the good work!
To put it in math terms, What Zeno's paradoxes are using are pure mathematics, while the observable real world is using discrete mathematics, ie physics.
Real world units are not infinitely divisible as pure mathematical units are, therein lies the disparity.
The flaw, so to speak, is that there is no such unit as a "moment" as Zeno described it in his second paradox, the "moment" either has an object moved from rest, or it has not, in fact, progressed through time.
Of course, this lies predicated on only one of the two following facts being true, that:
A) the world is infinite, and thus can encompass time completely, in which point there is the possibility that there is a distance traversed at which time has not elapsed, or
B) that TIME is infinite, with no describable beginning or end, at which there is a unit of time small enough that an object in motion has not, actually, moved at all.
Either one of those two statements being false or true is enough to discount Zeno's paradox, scientifically speaking.
1.616199(97)×10−35 metres is one unit of planck length, theoretically the smallest measurable length given uncertainty in the locations of particles on very small scales.
5.39106(32) × 10−44 s is one unit of planck time. The smallest measurable amount of time possible, because if the smallest measurable distance is one planck and the fastest measurable thing is a photon moving then it stands to reason nothing can occur faster than a photon moving a distance of one planck unit.
However newton disagreed and showed that with calculus all movement occurs with infinitely small fractions of time and position which was the original use of the derivative. Certainly without calculus these paradoxes could be extended to absurdity using planck scale, but it is not necessary with calculus to show that all changes happen as a sum of an infinitely large amount of infinitely small changes.
"only one of the facts are both true"
David Sanders Thanks for catching that, I fixed it.
Maxwell Hill But was Newton aware of a "cap" so to speak on the infinite division that the planck unit represents?
I may be misunderstanding your argument, but calculus is a branch of mathematics, an inherently theoretical field, until we step into the realm of "applied" mathematics do we get things like physics, which DO depend on, as you put it, reduction to the absurd levels of distance and time, which approach, but do not actually arrive at "zero", which is the unit of no elapsed time, or no displacement.
The logical question then, I believe, is 'does the planck unit translate to Zeno's "moment"?'
Can we physically reach a unit of time, provided that we divide the units enough, reach a point where the displacement of a moving object is equal to zero?
If we take Einstein's time dilation equation:
delta(t) = 2L/c
We can solve for t, where the length, or 2L, is equal to zero, would give us an answer of delta(t)=0, ie a unit of time equivalent to zero as well, and the ramifications to the two statements I extrapolated in the previous comment.
Ali Al-Shaikh He already said that you don't need a "cap" on the infinite division when using calculus. This is because you're already using real numbers. Real numbers are continuous. Zeno's paradoxes aren't paradoxes at all. They're just calculus problems.
This is one of the greatest RUclips series ever. It is very clever and well-narrated.
These paradoxes can easily be averted by realizing the functions of the universe;
to do this, We can't confuse observations with conclusions. "When a the bike reaches where the man was, he will be further ahead than where he was" is logical, but to think that the bike will increment smaller and smaller amounts to infinitely small scales is absurd. to clarify a bit better, look at the arrow idea; "it crosses a mid point to reach the end" makes sense, but there is no defined midway, as physics simply doesn't confine itself to such a rule. "Observations=Conclusions" is a dangerous system to use, because you must place reason between them.
Here's why I call bullshit on these paradoxes:
Objects move at a rate of distance/time.
The paradox assumes that the faster object cannot ever pass the slower object with a head start, because it must always catch up by some incrementally smaller distance. This ignores the rate of distance/time.
As the distance gets incrementally smaller, the time passed gets incrementally smaller, and eventually, you would have to say that time has to "stop" for the faster object not to pass the slower object.
But obviously, time does not stop, and the objects must both move at their assigned rates of speed.
Why would time have to "stop"? Well, if we are saying Planck Time actually applies to the behaviour of time, not just it's measurement, this is the case. But if we are saying time can be infinitely small, there is no reason for it to "stop".
R. Ringshifter I'm saying that the paradox requires distances that approach zero, which given the distance/time relationship, give time passed as approaching zero. When a number gets so small, it is effectively zero. What I'm saying is: the time that is part of our reality doesn't stop for the sake of a thought experiment. If the faster object only needs 5 seconds to pass the slower object with a head start, that 5 seconds WILL pass, and there's nothing a philosopher can do about it.
Paradoxes are a philosopher's toolset not their undoing you ignoramus.
Aristotle brings this idea right down to Earth. If motion were to be infinitely divisible, time must also be infinitely divisible. Thus the infinitely many increments of distance can be completed by the use of infinitely many increments of time.
these eps are really the top 3 things on youtube.
Makes perfect sense to me. I recognize change as another symptom of time that is exclusive to my current existence of experiencing things on a smaller scale. On this level things seem to flux and go through cause and effect, but in more simple and higher frequency matrices the illusion of change is less noticeable. It is the only way for a multiverse to work that it has some basic goal of unifying, and if all things seek unifying than change is certainly hard to argue for.
Zenoe's first paradox makes the assumption that they aren't moving at the same time. The distances are changing at the same time, one more than the other.
Really nice spritework, you guys. You made link and the old sage sprite from Zelda 2 very proud
Really? Then Battle of Olympus stole Zelda 2's engine and only changed the sprites :P
Dude, this is trippy as fuck. The way you illustrated it is like... whoaaooaaooowww....
Angry Foreigner Look at your channel now. All grown up, 3 years later.
I've been watching you guys since the beginning and absolutely love what you are doing and where the channel is headed. I can't wait to see the future of Thugnotes/Wisecracks!
holy crap I've thought about this since I was a little kid. i realize that you could continue dividing an increment of measurement, so I thought there was infinite space between any two objects. i impress myself
8-Bit Philosophy, each time you visit me it is as though a faerie has filled up my hearts. Thank you.
One interpretation of Zeno's Paradoxes that I prefer is that it's like Schrodinger's Cat - an attempt to point out the absurd implications of his colleagues' ideas. (Schrodinger's Cat isn't saying you have an alive-dead cat, it's saying "what you're saying means there would be an alive-dead cat in this situation, therefore what you're saying is patently absurd and therefore probably wrong."
Given the amount of time that's passed since the days of Greek philosophers, it stats to reason that there are a number of Zeno's contemporaries who we don't know about. Hell, there's plenty of scientists involved between Max Planck and the guys working at the LHC...I can name a handful of them, sure, but no way I can name all of the ones that made major contributions, let alone the probably-thousands of individuals who solved the little problems necessary to make that next big step. A lot of libraries burned between Zeno's writings and the present day.
Also, the fact that the only similarly absurd "scientific theory" (see above on Schrodinger's Cat) is so often misinterpreted despite the relatively short period of time since its inception) could be seen as somewhat suggestive.
Zeno demonstrated that a universe in which time and space increments are subject to infinitely smaller regress is philosophically impossible. The paradoxes predict the concept of planck length and planck time. There's no such thing as an "analog" universe and indeed there COULD not be such a thing, as Zeno proved.
This video offers a sophomoric understanding of these paradoxes. Zeno isn't showing that motion is impossible, he is showing that motion is impossible in the analog universe imagined by his contemporaries.
+swordsman3003 correct me if I'm wrong. But I don't think Zeno "proved" the impossibility of an anolog universe since Georg Cantor showed that mathematics can describe continuity in space and timy as long as you differentiate between infinities.
+swordsman3003 What do you mean by an "analog" universe?
+Stephen Cesler Analog is to digital as continuous is discrete. Something like height is continuous, (you could be 6ft tall, 6.1ft tall, 6.01 ft tall, 6.0000000001 ft tall, etc), while something like number of siblings is discrete (you could have 0, 1, 2 .... etc).
Quantum mechanics shows that elements of our universe are best thought of as being 'quantised'. Photons can't have any energy (i.e, photon energy isn't continous, or analog) but rather, can only have specific quantities of energy (they are discrete).
+swordsman3003 Actually, Zeno was, according to Aristotle (our principle source on the matter) trying to show that the world was unchanging. I will say that this video actually misstates the fletcher's paradox, which states that at any quanta of time the arrow is traveling neither to where it is (as it is already there) nor to where it isn't (as no time is elapsing for it to get there). A thing that is not traveling is stationary, which means at every quanta of time the arrow is motionless. Quantization does not resolve this issue. Leibniz and Newton do answer this question, however, with the assertion that impetus (what we now call "inertia") is an unseen element contained within the moving body. Thus, the arrow isn't motionless, as it still carries the potentia of motion within it, even if the viewer cannot perceive it.
what is the rate of change or how does change change???
Zeno's paradoxes themselves are complete and utter nonsense.
You say they have baffled mathematicians, but anyone with beginners level calculus knowledge would be able to show why it does move.
So change can be observed only in relation to a particular end in time or space. When we designate the runners position as that end point, we aren't concerned with what happens once the motorcycle overtakes the runner. Just as when we properly recognize the finish line, we stop caring about their relative positions in space after they cross it.
Until we found Calculus and we discovered that a sum of an infinite set can have a finite value.
This is the only 8-bit I'll contest, still love this one and all other ones. But superposition explains the possibility for change via time/relativity. We perceive light which is made up mostly of photons traveling through mediums so we can perceive them as light, but they do not comply to normal laws they can be brought back from time, duplicated, be in two places at once, travel fast enough to not perceive time and I'm sure a number of things I missed, but ask a physicist and they'll explain it way better :P.
Wow. This is awesome. You both finally made Zeno's arrow paradox clear enough to me that I can see why it's wrong, and made it clear to me why .999... = 1.0
More of these please... i frigging love them
This vid by far of all other 8bit vids is hard to comprehend,still its a great video
Philosophical theories are so much more fun to contemplate when you're in high school and uni.... but not at my age.
At this point it just terrifies me.
Well glad to see the entire comments section has stepped forward to point out how Zeno's paradoxes can be dismissed as bollocks by either obvious common sense, or a more nuanced, holistic (and modern) understanding of mathematics and physics. His mentor said something stupid with no real reasoning behind it (that the episode touched on at least) and his student defended them with half-assed arguments... although to be fair, they probably made more sense at the time since our understanding of various sciences wasn't as great.
+leviadragon99 Yes, to propose that change doesn't exist within the closed boundaries of our given system is preposterous, but to identify that we exist within a closed system, where time, space, energy and everything else that might exist are locked and bound to, is of essence.
Nah, calculus made them obsolete.
Calculus shows us how the motorcycle catches up to the man, but does it show us how it show us how the motorcycle surpasses the man?
The problem is that there's a limit to how short a distance in space or time can be (the Planck length), before the uncertainty principle makes the distance irrelevant. Of course, this may support Zeno's paradoxes, but in a slightly different way.
But acceleration?
Beyond that, there's also the physics question of (and I'm going to butcher this terribly) "what happens if we put in enough energy to the point that a wavelength is shorter than a Planck length?" So even that since of like an infinitesimal still hasn't been called into question yet.
Zeno's paradox is the study of an asymptote on the negative X axis without considering that is continues onto the positive after reaching a point of "infinite distance" according the the Y axis using allegory.
Great series
When the biker slows down we often forget that time is slows down too. It's gets exponentially smaller based on the the difference between the bikers and runners velocities. This means that what Zeno describes does actually happen and can be measured with a graph that plots the distance between the biker and the runner. We can then take the data from that graph using only up until the point where the biker passes the runner. Instead of conventionally plotting the data we want to make the x axis equal to intervals of the biker reaching the runner's last recorded position. This does not alter the data but simply creates a different perspective of the original graph.
If you were to visualize the entire race using this method it would appear like the spike of a wave that then levels out as the biker drives off into the sunset.
I used to be terrified of change. When my family moved to a place I had never been before and we had no family roots like before I was terrified.
It was only days before the move that I made peace with myself that change is good and bad, but it's the law of life, and life is about living and living is about change.
Man i love this series
Thug Notes! My dude! Big ups to ya.
Oh, how Zeno would've loved to be a student of quantum physics.
This is the best thing I've ever seen
Great video. Side note, The Fault In Our Stars by John Green has a scene where a character references Zeno's First Paradox, pretty cool to learn about it more here.
A riff on The Battle of Olympus for the NES, outstanding!
Zeno's paradoxes confused us until we invented calculus. Once you have the concept of a limit, the problems disappear, even if you assume spacetime can be infinitely divided. Check out a book called Zero: The history of a dangerous idea, really good book.
Quadrance not distance is used to consider the separation between points. Argument over Zeno.
Fantastic job guys!
Relativity blows a big whole in all of this!
I love this 8-bit philosophy!
Ooooh! Oooh! 0:16 text backdrop from Legendary Wings.... I fucking love that game
This philosophy was before the invention of 0. The motorcycle would pass the runner, decreasing from the head start to 0 when he catches up with the runner, then the distance would then increase. The second paradox sections the amount of time with the amount of distance it takes to hit the arrow. But the arrow starts at 0. the paradox assumes that you have to divide time and distance infinity times, but 0 represents the lack of time and distance. Not moving to moving is change
Love the excitebike bit, so many happy memories.
I think Zeno's paradox was resolved by Newton and Leibnitz and the invention of calculus in the 17th century. They showed how you may add up infinite numbers of infinitely small quantities and the result is still a finite number.
So basically Zeno is saying that if you stopped time, then there would be no change. But time actually passes in the real world, so his paradoxes are whack.
Well that and he didn't understand acceleration and ssumed that motion is constant. In terms of EM waves, you can't slow them down at all, since they are bound by physics to travel at 3x10^8 m/s everywhere (in terms of within atmosphere, measuring the distance/time of an EM wave/photon interacting with two separate particles will still be that number even if a beam of light fired through water across 10m will appear 'slower'). In this case, the runner/biker paradox works because if a photon is shot .1 second before another photon, then it will never be able to catch up and appear that motion is impossible (before we discuss observer/object relativity).
I don't think I quite grasped the first paradox, unless the motorcyclist decides to stop at intervals, or has agreed to travel at a lower speed surely the motorcyclist will just accelerate to a speed beyond that of the man and overtake him, regardless of point of view, the change in velocity has increased beyond that of the man, in which case distance has changed, can someone explain this to me
yeah they probably didnt have acceleration back then which is why there was this paradox
***** It's due to a lack of understanding how infinite numbers work. Zeno believed the sum of an infinite number of finite steps could not be finite.
To be frank although it is quite obvious that even without acceleration the motorbike is going to be ahead in short time, when you take it like Zeno puts it is tough to explain. Mathematicians couldn't do it until Taylor's expansion. You can study it and would figure out how..
HemeHaci is that .999999999... = 1?
*****
of course they did, don't be silly :p
And the paradox is more simply explained like this:
I want to clap my hands. Let's assume I only move one hand, that hand will have to move the whole distance from one hand to the other. But to do that it surely has to also move half that distance, and before that a quarter of the distance, an eight of the distance and so on. If these small movements can be infinitely small, my hand will never reach the other hand because it has to move an infinite ammount of small distances.
It's funny because that even though nothing really can be changed/created in the whole of the world, it seems like memories are still expanding. Which is even funnier, because we can't see memories, only imagine them. So does that prove that the only change possible is in a world where we can't see, touch, hear, smell, and taste? Even further than that, that ever changing universe thereby seems to become the only world that matters, because that is the only world that can change. So it is the only world we CAN change.
Zeno doesn't seem to understand the concept of acceleration
***** it's still obsolete
Nothing truly collides, they merely get in range of the normal force, and therefore never need to reach they're destination.
Space time behaves as observed (obviously) when observed via a confluence of the senses, creating the standard impression of "reality", but when the mind isolates and withdraws from the senses individually (as in Buddha Dhamma), the relativistic impression gives way to a monadal experience of here/now/all/nothing, devoid of self or world, here or there, future or past, energy or matter, light or dark, etc. While this all-encompassing void is the fundamental substrate of all possible experience, it is mutable unto knowing itself (the field state that occasionally crests into consciousness), and it is the presence of knowing within emptiness that differentiates one thing from another, giving rise to a relative universe from within the bosom of the absolute.
Lol, we used this during our philosophy exams, certainly makes tests more bearable.
The motorcycle paradox failed to take acceleration and the mans stamina into account
Zeno's paradox is put safely to rest by Special Relativity. Light moves at a constant speed. This speed is constant for all un-accelerated observers. What this means is that every motion in the universe has a context. Everything must be compared to the speed of light which cannot be slowed down in constant velocity, so every velocity can be unambiguously compared to a constant speed which works in every direction.
However, it is interesting to note that when applying acceleration to a system Zeno's paradox re-appears. For relativistic acceleration gives the observer the impression that light can slow down. We discover that there is a way to accelerate in such a way that light approaches the speed of zero.
It's this concept that I based my hypothesis on black holes which may reveal how galaxies form and cluster up. Come to my channel to find out more and look for my videos on the nonsingular hypothesis.
We've solved the second paradox. The problem is summing up the infinite sum 1/2 + 1/4 + 1/8 + 1/16 ... and so on, for the arrow must travel half of the total distance, then a half of that, and a half of that, and so on forever. The paradox is that this sum seems to never become 1 unit of distance. But we can apply infinite summation rules to show that this summation does in fact equal 1, meaning the arrow covers the distance it should.
Let 1/2 + 1/4 + 1/8 + 1/16... = s
Then 1/4 + 1/8 + 1/16... = s/2
Subtracting gives 1/2 = s/2
Hence 1 = s, the summation in question.
Thus, though the arrow makes an infinite number of half steps, it will cover the whole distance.
rates of acceleration are variables, variables can be changed. Zeno is a syntax error.
All this tells us is that *Zeno* didn't _knew_ *Calculus/Mathematics* and/or Philosophers don't consider Math.
Relevant:
xkcd.com/1153/ ( See the title text , which says "The prosecution calls Gottfried Leibniz" )
Damn! Mind is totally blown! All movement is the sum of a bunch of stationary moments in time?
the reason the paradoxes aren't applicable, is the discovery of the maximum speed and minimal distance. ie. the speed of light and the plank length. meaning their is a minimal amount of possible time, the time it takes for a photon to travel a Plank length.
Silly Zeno, you just don't know your limits! Seriously though, I was under the impression that Leibniz proved Zeno's paradoxes false. Is there some subtlety I don't know about that still bothers mathematicians?
+Philip Stuckey Actually, Archimedes was the first one to settle down mathematical Zeno's paradoxes in modern sense(though his own idea were not widely/accepted at the time). And if you are talking about calculus or theory of convergent, then though those were first precisely treated by Leibniz(I must say, in a damn excellent way) but put on rigorous way by Cauchy, Weierstrass and others. And infinitesimals were first rigorously treated by Abraham Robinson by non standard analysis.
Conceptually, Zeno effect is still of importance (not paradoxical though) like in quantum zeno effect an zeno executions.
That said, actually original Zeno paradoxes were not mathematical, it was about physics, more precisely about motion. If we generalize the concept to something like this - decompose some physical quantity (distance in first paradox), compute each part individually (makes geometric sequence in this case), add them up (converges to a finite value). He was saying how can you add infinite things in finite time. For example traditional epsilon-delta limit method doesn't adds these numbers it just gives the existence of bounds of finite sub-series. And non standard analysis inherently ain't constructive.
Lastly, there is something of similar nature but much more subtle and paradoxical, Casimir force. The calculation of Casimir force is of above nature, i.e. decomposition -> computation -> addition, but the series it creates is not convergent (precisely, it become some constant times (1+8+27+64+...+n^3+...) ), and amazingly enough it gives the correct answer if we put a finite value instead of infinity (precisely 1/120 for (1+8+27+64+...+n^3+...) ). For completeness, there are methods discovered earlier which actually gives this value for this divergent series, but the reason that why it works is not clear.
The Royal Academy of Science, headed by Newton, said it was Newton, not Leibniz, checkmate atheists!
The only way to reconcile this is if units of distance and time are finite. If they were, for instance, the Planck length and Planck time, there would be no such thing as an indivisible moment.
zeno's paradoxes kind of shows a misunderstanding of infinity. infinity doesn't prevent an arrow from reaching the target it allows it. (sorry I'm not good at explaining this but.. vihart has a great video on how 0.99 = 1 which uses a similar problem)
Mathematically, Zeno's "paradoxes" fail due to a convergent series. The series 1/2 + 1/4 + 1/8 + 1/16 ... = 1. Not close to one, but exactly one. It converges upon 1. Basically, this is how mathematicians solve this apparent paradox.
The first paradox implicitly assumes that the runner has a higher velocity than the biker. If you look move along with the biker you'll see that the runner would have a negative relative velocity and the ground an even greater negative relative velocity. So the critical point is when the biker and runner at the same displacement on the track. Basically, it's a restating of the second paradox.
The second paradox is pretty straightforward when you keep in mind that if the arrow has a constant velocity subdividing distances subdivides time as well. The only issue is how anything at all gets from one "moment" in time to the next. If you believe that the Universe is a fixed-point and our brains just walk along it without inputing anything from a time before the point it's at you have to wonder why that causal relationship still exists and reserve that belief on where quantum randomness comes from when collapsing from a quantum state to a classical one.
I think I might need to rewatch this a few times to completely grasp its meaning...
the things not happen or change to one point to other, there are continuously changing
Does plank length time help answer some of these paradoxes and show the separation of reality and maths?
As always a good episode. But gosh darn it we aren't made of numbers and theories. We're made of flesh and blood and atoms. These take up space with mass, and move in time with energy. So okay the arrow takes up it's space in a moment of rest. But the archer or bowman or kid on christmas puts in energy into the arrow shoving it through time. Then physics happens and the arrow hits the target. That's the problem with Zeno, he's speaking of math and infinities when he and every other thing in creation is physical and finite. I'm the first guy to say "we're all the universe!" and "our perceptions are flawed!" don't get me wrong. But these are truths that can't be proven by math because math is only a language to describe both itself and dreams of a stranger universe. The unity of the cosmos isn't homogenous, it is diverse like a body. A foot is not a hand but they are part of the body. Gurus and Qabballists and other mystic folk always say that our perceptions are flawed. And like all things we've made, including languages, they come from our flawed perception.
actually you can prove his 2nd paradox wrong with math. Potential energy of the arrow within the moment can be transferred into kinetic energy. I just wanted to clear up a principle you stated correctly. the first paradox can be theoretically proven false with the unit of a plank distance of 1.616199×10^−35 metres. At this level there seems to be no shorter distance at which subatomic particles can move.
duh Rooker I suck at math and science so I'll take your word for it. Thanks for the contribution.
Iacob Goldstein No problem. Have a good day.
Same to you.
duh Rooker
The Planck lenght is the smallest *measureable* lenght (that one can measure), not the smallest possible lenght. Where did you hear that it is the shortest distance subatomic particles can move?
You used Battle of Olympus! That game is great!
Bertrand Russell In his history of Philosophy demolishes this argument.
an arrow cannot possibly experience 0 time, because experience requires time. You cannot assume that an arrow in "zero time" has a speed of 0, because speed = distance/time, or 0/0, which is undefined. All this manages to prove is that 0/0 is in fact, undefined.
Wait, what about the "Planck length" and "Planck time"?
+michael einhorn vsauce made a video on the plank about 3 years ago
Ohhh
I guess your right
Doesn't work still. Long story short is that length and time isn't movement.
Planck Length and time isn't as "recent" as you having internet might lead you to believe. They are all 20th century theories. Planck himself died on '47. That leaves A LOT of time to react to this paradox.
Comenters aren't really at fault for getting it wrong (they are at fault for not reading it, but regardless). It's basically the "fault" of the uploader's presentation
The whole paradox is simpler to explain and you'll understand why Planck length and Planck time do nothing in this regard.
An arrow travels from point A to point B, say that distance is 20 meters (time isn't even necessary). But to get there, it must have passed at some point point C = 10 meters. To get to C it must have passed point D = 5 meters etc. *ad infinitum*
Since a segment holds an infinity of points, that means that the distance cannot ever be passed, (else it would be finite).
Planck length, which is the smallest possible length in space is still made up of numbers you can divide, so even though physically it cannot be "composed" of smaller distances, a distance is an abstract entity, so you can divide it by as much as you wish.
When thinking of Zeno's paradoxes, it helps to think in terms of hyperbolic functions, and, as consequence, through limits x -> ∞.
If you see someone doing that, that means he's on to the "big" question of Zeno, if not, don't waste your time.
I don't see how you can divide a planck length, I mean you can say it's half a planck length, but that's as meaningful as saying a square is a triangle.
Doesn't the first paradox only occur because time is removed from the equation? Am I wrong? It works if you judge the speed of the motorcycle by the man's previous position, the motorcycle is always one step behind. It works because their positions are measured at different points of time.
I dunno. I don't think our perception is flawed, but our ability to comprehend what we perceive definitely is. We live in a medium-sized world, far removed from the humongousness of stars or the minuteness of subatomic particles and forces. With paradoxes like this, we're just grasping for understanding based on the tiny bits of sensory data we're able to acquire.
I love this series!!!! I remember having to explain some of Zeno's Paradoxes in a presentation in my one required college math course (I was an English major). I think I explained them really well, but I still didn't buy them. Interesting concepts, though!
zeno's paradox has already been solved.It takes the arrow 2 seconds to hit the mark.
Zeno's Paradox - Numberphile
Just because there are an infinite number of steps to complete a task does not mean the task will take an infinite amount of time to complete them.
I don't understand the second paradox.. No matter how many times you halve a number (be it representative of distance or time), the remaining number will have a value. At what point would the arrow have not moved, and time still elapsed, if all you're doing is halving the distance each time you halve the time?
This one's easy.
Paradox one,
Space is infinitely divisible so if you take a closed section of space you can divide it in such a way that you will get an infinite number of divisions (one such way is described in the paradox). Now if you stop to measure each one of those divisions you will indeed be measuring for eternity and never get to the end point of the closed section of space. This is because you will have an infinite amount of measurements to make not because traversing that space is impossible. Pust another way, traversing from point A to point B and counting how many times you can divide the space between point A and point B are two entirely different exercises. Infinitely divisible does not mean infinitely long.
Paradox two,
The time of the arrow's flight is measured from the start of the arrow's motion, therefore there is no point in the measurement of that time that the arrow is not in motion. In other words from the "moment" the timer is not 0 the arrow already has motion.
I always knew that Kirby was the essence of all things.
anyone know the name of the game thats music starts at 1:16 ~ 1:24?
HOLY SHIT I SAW THIS VIDEO AND IMMEDIATELY THOUGHT OF PARMENIDES. I thought to myself, there's no way wisecrack is going pre-socratic, its gotta be someone more recent, but you went there. I love you guys.
I think Zeno's second paradox doesn't work because the arrow is traveling at however many meters per second, not however many "halfway points" per second.
The first paradox is easily solvable with calculus.
Suppose that the distance between the man and the bike is 2 m. The speeds are 1 m/s (man) and 2 m/s (bike).
In 0s:
s(bike) = 0m
s(man) = 2m
In 1s:
s(bike) = 2m
s(man) = 3m
In 1.5s:
s(bike) = 3m
s(man) = 3.5m
In 1.75s:
s(bike) = 3.5m
s(man) = 3.75m
So, although the distance is an infinite series, also is time. The last step is infinitely small in space and in time. So, the sum of the distance travelled is finite and the time it takes for the bike to catch the guy is finite too. This is true even for continuous variables. It isn't necessary to consider Planck units.
Someone like Thomas Hobbes would probably have called this paradox nothing more than a semantics game. Zeno heavily relies on a semi-fictionalized concept of Infinity to come to all of these conclusions. Hobbes says that as long as you continue quantifying the characters in an infinite number - such as the distance between the cyclist and the runner - you procedurally and continuously dissolve the infinite by the assignment of specific, finite numbers. Just because Zeno phrases it by saying there is an "infinite" distance between the two bodies does not by necessity suppose that the space between them is somehow eternal/irreducible.
Max Planck slayed Zeno by shooting him with the smallest arrow in the universe.
Zeno's paradoxes all seem to assume a continuous universe. Perhaps in formulating these paradoxes it occurred to him that the universe behaves vastly different on the quantum scale and ideas like planck length and discrete motion have to be entertained to explain such easily demonstrable actions.
Zeno's 2nd paradox implies there is a zero in time. When if it is to be measured there can be no zero. I.E. Time can be infinitely divided into smaller and smaller measures in which an object traveling, has kinetic energy, there is no measure of time no matter how small that the kinetic energy does not exist, only the perception of the motion is so infinitesimal as to appear stationary, it is in fact still in motion.
While I can't say as much as I'd like to on an episode on this,such math related questions* (I admit reason and logic in philosophy are two areas I'm not good at and am only beginning to venture into),such math related questions were they to be presented in school (along with "decision" centered math like game theory's stuff like the prisoners dilemma) would be another good way to motivate little kids to like math more.
Don't get me wrong,it's great that ppl try to show kids the analogies btw sciences and math,but to get a good grasp of math with based on a notion of continuity (like the calculus used in physics and such) is a tedious and long road unlike how a lot of the stuff in game theory with it's objective to optimize agents and paradoxes like this,give you a pretty little glimpse of a hardly spoken/brought part of arguably more intuitive kinds of math.
As a result it might kids to eventually engage with axiomatic systems like in symbolic logic."Better" ^uV then getting kids motivated to wanna get a grasp of applied probability for gambling,eh ;)?.
*wikpedia says that "infinite processes" like with Achilles and the tortoise/the motocyclist and the runner here weren't really less effortful to solve till the late 19th century
“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. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful.”
― John Green, The Fault in Our Stars
> there is a bigger infinite set of numbers
Jesus Christ, what a load of shit.
Doesn't the motorcyclist and man paradox only work if we assume 0 acceleration from both of them (and of course, more specifically, not a positive or greater acceleration by the motorcyclist?)
It seems clear to me that not everything in the universe can be explained by thinking in the narrow constraints of what we know, yet we are also somewhat hampered by being unable to think outside of what we can comprehend. We know now that things can and do change all the time - such as changing energy states in atoms, or a fire giving off energy as heat or light. It's hugely central to our understanding of physics, especially quantum physics. Thus, it's not CHANGE that is impossible, it's our logical understanding of what change IS that is impossible. Change is not a finite thing. It's infinite. Therefore, it's too big of a concept for us to fully grasp. We can define WHAT change is, but not WHY change is. It's not an illusion, we just can't comprehend the idea fully.
Let's look at the example of a sine wave on a graph. Sure, you can graph it with infinitely small points (and graph it more smoothly with every increase in how many points on the curve there are), but each and every one of those points fractions smaller and smaller. The units are not changing, the size of the units is changing. Therefore, there is no such concept as infinity, not in reality - infinity is just a word for an eternally continuous thing. But we can't quantify an infinite number of points on a graph, so we see it as a continuous graph, and conclude it therefore is a sine wave. The question is, how far down does an approximation go before it ceases to be an approximation, and just "is"?
Nice video guys.