Zeno's Paradox and the Planck Length | Answers With Joe
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- čas přidán 8. 09. 2024
- Today I discuss how some infinities are bigger than other infinities with Zeno's Paradox.
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TRANSCRIPT:
Zeno was in a cult.
Back in ancient Greece, hero cults were a thing, and kind-of a big thing. A Hero in Greek culture was a man who had died and was now revered as not quite godlike, but close.
They were somewhere between man and God.
And followers would build temples in their names and establish churches around them, and devout followers would pledge their lives to their ideals.
But Zeno was a part of the first known hero cult based around a philosopher. A philosopher named Ameinias of Elea, who created the Eleatic School of Philosophy.
The Eleatics, led by the Philosopher Parmenides, believed that the senses cannot be trusted to reach truth, that only by thinking and logic can you arrive at truth.
They also believed that change and even motion was nothing but an illusion brought about by our senses.
To prove this idea, Zeno posited a series of paradoxes, the most famous being the Arrow paradox and the Achilles and the Tortoise Paradox.
Much like my spinning example earlier, the arrow paradox says that for an arrow to get from the bow to a target, it must pass through an infinite number of halfway points to get there, and the Achilles and the Tortoise paradox takes the use of movement a bit further by saying that if Achilles were racing a tortoise and gave the tortoise a head start that every time he got halfway to the tortoise, the tortoise would have moved forward a small amount, therefore even though he’s going much faster, could he ever actually catch the tortoise?
Now logic says that of course the arrow reaches the target, there’s not some invisible field keeping it from touching, and of course Achilles beats the tortoise.
These are some of the first examples of a rhetorical device called reductio ad absurdum, where you disprove a statement by showing that it inevitably ends with an absurd result.
Other versions of this that came later include Gabriel’s cake, which says if you slice a piece of cake in half, then slice one of those halves in half, and again and so on into infinity, and then stack each of those layers on top of each other, the cake would stretch to infinity.
Now these are all space paradoxes, but there’s also one that divides time called Thompson’s Lamp.
It says you take one minute and turn a lamp on and off at every halfway division, so turn it on at 30 seconds, turn it off at 15 seconds, turn it back on at 7.5 seconds, on at 3.75, and on and on until it’s blinking so fast we wouldn’t be able to see the difference. When the minute passes, is the lamp on or off?
Now, there have been countless resolutions of Zeno’s paradox through the years, some philosophers going so far as to reasoning that space and time don’t exist… which is kind-of what Zeno was going for…
But mathematical constructs worked to prove that the sum of infinitely decreasing quantities could result in a finite number.
And the idea for a limiting value to an infinite process is central to calculus, which relies on infinitesimals in order to ascribe a finite number to an infinite number of pieces.
Thanks, Sir Isaac Newton!
But maybe the easiest answer is simply that you can’t divide time and space forever. There might be a real, physical limit to smallness. Enter Max Planck.
Max Planck was one of the most substantial physicists of the early 20th century who proposed an elemental size to the fabric of spacetime, which he called a Planck Length.
The Planck Length is insanely small. It’s 1.6x10-35 meters.
To give you an idea of how small that is, This (graphic - 15 zeroes) is the length of a proton. To get to Planck’s length, you have to add not 5 (graphics change for each), not 10, not 15, but 20 zeroes. It’s one hundred quintillionth the length of a proton.
• Planck Length - Sixty ...
He came to this by combining three fundamental constants, Gravity, the speed of light, and his own Planck’s constant.
And from there, he created Planck Time, the time it takes for light to travel one planck length.
So if there truly is a smallest indivisible length of time and space, Zeno’s paradox is solved. But many still aren’t so sure.
An intuitive way to resolve Zeno's paradox: Just as you cut the distance in half with each step, you likewise cut the time it takes to make each step in half. And while it does take an infinite number of steps to reach your destination, the time it takes to make those steps becomes infinitessimally short.
As for the Planck length, it is NOT "the smallest element of space", it is the shortest length (or time interval) that can be MEASURED. There is no evidence whatsoever that there is in fact "a smallest indivisible unit of space and time", and if there were, what Planck actually demonstrated was that we would not be able to measure it, no matter how precise our technology.
The reason why it cannot be done is because the shorter the distance you try to measure (e.g. using a beam of light), the larger that light beam's momentum will become (due to Heisenberg's Uncertainty Principle). To reach down to the Planck scale would require so much energy to be concentrated in such a small space that it would create a black hole (with an event horizon the size of the Planck length). Concentrating the energy even higher would just create a larger black hole.
Thanks. This makes thing a lot clearer.
That's really interesting
If you can't measure something smaller, then it might as well be the smallest unit. Functionally a unit is only useful if it could be differentiated. So if we could measure planck length but not smaller, then for all intent and purposes it is the smallest unit that MATTER for practical purposes.
it's like how a painting is judged by what it looks like in the human visible spectrum. It doesn't matter that other wavelengths exist, the visible colours is what matters. In the same way we don't need to worry about levels of precision that can't be detected; by definition if the difference is undetectable then it doesn't exist and has no effect.
On the other hand, if something that is different by less than a planck length mattered and has an effect, we would have detected the undetectable and broke the rule. QED unit differences only matter if they are detectable.
Its not a paradox when you consider that time flows(without pausing) , by you chopping each time and distance into smaller and smaller segments( time and distance have a one to one correspondence ) means nothing but showing that infinity exists in a finite value, which Cantor already did more than a 100 years ago. As an object moves through space over a distance at a certain speed , since flow of time is constant , as you get closer to the end of distance the frequency at which you will be getting a new value will approach infinity.
A infinite number of mathematicians walk into a bar.
The first one asks for a pint.
The second asks for half a pint.
The third asks for a quarter pint.
The bartender stops them and pours 2 pints and says
"Know your limits"
Master Wayne"
"Batman has no limits"
"Well you certainty do sir"
Commenting just to say that this made me chuckle
4 logicians walk into a bar. The bartender aks, "So, 4 pints of beer?"
The first one, "I don't know."
The second one, "I don't know."
The third one, "I don't know."
The forth one, "Yes."
The 5th one is shot before he can ask, then time stopped.
@@gkess7106 Entropy was levelled. There was a spark and then a bang
"If there is anything that makes us human, it's our ability to make up problems that don't really exist" amen!
See politics.
Side note: see capitalism for solutions to problems you didn’t know you had but found the solution to them on TV / YT
@@sinistersparky9657 This is a very interesting math problem. It had been a riddle for ages.
So Xeno studied a school of philosophy that said our senses cannot be trusted, only logic.
Then Galileo comes along and states that all logic must be checked against reality.
And it's Galileo's conclusion, formalized by Roger Bacon the "first scientist" which works.
Retired librarian
@@veralenora4033 This issue is about Matematics, not Philosophy
I've had one of the worst 3 days of my life , and your video helped keep my mind busy. You are the best Joe!
Sorry to hear that! Hope the rest of you week goes better. The Joe wills it.
OK... but it must have seemed infinitely long.
Actually, calculus _used to_ use infinitesimals, but they were superseded by the idea of _limits_
But infinitesimals are still there, in their own number line, called the _hyperreals_ (there are also the _surreal numbers_ )
Also, _I love_ *markdown*
In the old days we were [b]awesome[/b] enough to use bbcode... :(
I failed Pre-Cal in college and I'm pretty sure that shines through.
Joe Scott nah. this is a pretty common misconception. And I don't know how education was in America, but I can say with certainty that over here, education truly sucks. So not really your fault you failed.
lol, respect what he is doing, dont come with propaganda. please keep the channel politics free
Just saw this. I don't get it why political topics are like this. How about a civilised discussion on it? Wouldn't hurt anyone. It'll just be fun :)
Hm, if you split a cake in half, and keep doing that eventually you create a nuclear explosion XD
Cut the cake in half, eat half. cut remaining in half, eat that, repeat... Infinite cake!
@@stephenblessed92 Let's spit a cake. I only get the first three pieces. You get the rest which is infinite.
No, your partner just punches you in the face for ruining the cake they just made...
Is the cake yellow?
A very simple answer to Zeno's paradox exists in basic math: When you halve the distance you also half the time it takes to travel that distance, keeping your time to travel any given slice of the movement at a constant rate. By the time you near (thought experiment) infinite points or slices, the time is reduced to an infinith (not a word...yet) amount of time. Voila! The turn is completed unhindered by the halving affect.
Also without understanding this you wouldn't be able to explain how you halved the distance in the first place. To go 20 feet you first need to go ten feet. But how did you go ten feet if you first would need to go 5 feet.
A simple mathematical solution to Zeno's paradox is that although arrow will have to travel infinitesimal number of fractions to reach its target which seems to be never ending, but the time taken to travel each successive fraction also decreases by half and hence the time taken to traverse the final infinite fraction will be infinitesimally small i.e. zero. Hence the arrow completes its flight as it covers the final infinitesimal fraction in zero time.
I like it, a finite distance in a finite time frame, discrepancy is nil,
Zeno didn't understand that infinite sums can converge to finite values. If you sum the times the arrow takes to get to the next point half the distance to the target, you get a finite time equal to the distance divided by the speed.
Zeno's "paradox" illustrates a really important point. Paradoxes arise because of our misconceptions (or assumptions) about nature. Nature always proceeds. You can't break nature, you can only break your understanding of it.
Jess Stuart
That's not the problem. We know that an infinite number of things can equal to a finite amount of something. The problem is how do you reach infinity in the first place?
Each increasingly small step towards the target takes a (proportionally) increasing small time. And just like the sum of the infinite series of distance slices sums to a finite number, the infinite series of time slices does too.
1/2+1/4+1/8...=1 this is true for distance and time (and apples and ...)
Zeno paradox is not a paradox (mathematical or real), is the cleaver question from a deep thinker who did not know about infinite sums. His question was just a couple of 1000 years to early for the answer.
The referenced Veritasium video explains this nicely.
@@richardyoung3074 like h1a8 said how do you reach infinity in the first place if you're going only step by step without skipping a single step in the first place what is so hard to understand?!?
@@trevorallen3212 You don't. Infinity itself is an illusion.
Actually by switching the lamp on and off at an increasingly more rapid rate, at the end of the time period it is in a superposition of the on and off states; at least until the waveform collapses. Or until the switch wears out and breaks, in which case the lamp is off.
Or the switch lever breaks in ON position, so the lamp is on. Result of the supertask is not the superposition though.
It's both. Everett was right.
I love this video, but you have got two of Zeno's paradoxes confused. The arrow paradox is the idea that at any one moment - assuming time is finite - the arrow cannot move as it is occupying a space its own size and cannot move in between the instants. The dichotomy paradox which you mention in this video works with the example of an arrow, but is a different paradox. Zeno uses the idea of an athlete running from one end of a stadium to another, and its often called the stadium paradox.
exactly, well said
Zeno : creates a bunch of paradoxes
Calculus : *AM I A JOKE TO YOU*
This is a channel. Calculus is completely hypothetical and only approximates the real world. Every mathematician recognizies that the “continuous change” in calculus does not perfectly represent the real world.
@@sovietsandvich8443 But calculus WORKS. It allows us to create ideas which can be converted to engineering. That passes the test from science theory to science law. (I know Joe uses the word theory as a final conclusion, but he's wrong. Einstein's work is no longer the theory of relativity, its' accepted as the Law of Relativity.)
Vera Lenora but the measurements and calculations of calculus can only be approximated. The concept of planck length and Planck time should be enough to show that “continuous change” cannot really exist. It’s similar to how 2*pi*r is the circumference of a perfect circle, but no real perfect circle can exist.
Vera Lenora obviously speedometers work based on calculus, but the measurements will never truly be exact. The whole concept of a limit is that you continuously approach a given value. Derivatives and integrals are based off of limits, so they are approximations as well.
@@sovietsandvich8443 Thank you.
I think even in a continous universe, you solve the problem of zenos paradox by thinking that in each smaller path aquiles do in a small amount of time, due his constant speed, so in infinitesimals distances he does in infinitesimals amounts of time, and this sum of time converges to a finite time when he passes the turtle
I think Xeno's paradox demonstrates the absurdity of "infinity" in the real world. On that note, do a video on G64 and TREE(3) and explain how we know TREE(3) is larger than G64 when we can't calculate either. :)
"There's nothing more human than the ability to make up problems that don't exist" , amen bro.
the "is the lamp at the end on or off" thing is kinda like asking whether infinite is an even or an odd number
Basically, yes. And the answer is simple.
Modulate the lamp by a function that is equal to 1, whenever the lamp is on, and equal to -1, whenever the lamp is off. Clearly, that function does not converge.
@@lonestarr1490 well that just gives you "there is no answer" which Is hardly an answer, and if you treat the sequence of alternating 1 and -1 as if it's a convergent sum even though it's not, and you set it equal to a value, that value winds up being 0, which would indicate that the light wouldn't be in any state, but if you had to say it was, then it would be half on
Faaaar from "resolved". Here is the real "arrow" paradox from Zeno: "If everything is either at rest or moving when it occupies a space equal to itself, while the object moved is always in the instant, a moving arrow is unmoved."
So how does something move some planck lengths between planck instants, if it is standing still in every instant?
i spent my whole day reading about how time doesn't actually exist and i was like 100% convinced that it has to be true but now this video is giving me an existential crisis
Zeno Paradox is a rhetorical question. Similar to the what came first the chicken, or the egg scenario, where despite intelligent people knowing the chicken came first, many other people have, and will continue to argue that the egg came first, and those people are wrong to argue the egg side of the debate.
That may not be the best example, but Joe has my brain fried from being mind-blown binge watching his video's, and it was the best example i could think of at the moment to describe the inaccuracies of Zeno Paradox lolz ❤️👍
I once read that the smallest amount of time was the time between when the traffic light turns green and when the guy behind you honks his horn.
Is that not a "New York second"?
That needs to be a shirt... "Give them a Planck Length and They'll Take A Proton."
That is a really good shirt idea. Well played old chap, well played.
Joe, I never hardly ever comment, but I had to thank you for not mentioning Xenu,
backwards and forwards, not,
that was heart-warming-ly great.
Great that you gave a brief overview of Zeno, but there is way more to this than you mentioned! Zeno's paradox of space and time can only be explained by quantum physics and Heisenberg's uncertainty principle which unfortunately you do not mention at all. There is an excellent book about this and I highly recommend it. Cheers
How do I finish watching this video, it's infinitely long
Abhi Jamwal Just watch the first half, then half of the second half, then the next eighth, then the next sixteenth, etc
Abhi Jamwal I just finished it. I must have unlimited bandwidth and infinite charge on my phone.
It took an infinite number of times for me to get some of those lines out.
Au Contraire, the video is 7.086462259951e+45 Plank time units long!
Pay Attention Dude!
Zenos paradox I did think of it once without ever knowing of zeno, what happens if you divde time and space, movement is possible how exactly time is possible how exactly, I can sympathize with zeno because that idea was very personal to me sorry, an insult to zenos idea is an insult to me because i take that personal because i like that way of thinking, divide time infinitely divide space infinetly, thats the kind of guy i am. anyone who is against philosphy is against me by defaul and knowledge and science and mathemeatics.
I love how you move around the screen with every edit cut. It breaks the monotony usually created by edit cuts.
Zeno's pardox is misrepresented often...and so is here. The function is an ever more accurate description of the moment before the defined ending moment. The hare passing the turtle. The arrow reaching the target or whatever you wish to use. The correct question is How do I describe the position/time just BEFORE completion with infinite accuracy. As the number of moments (or distances) necessary to complete any action increase to infinity, the time taken for each moment approaches zero with equal vengeance.
I admire you for having the courage to do a downward camera shot on the top of your head. If I were to do such a thing it would highlight my thinning hair. Not balding, just thinning.
The reason you can reach your end destination on the chair is because you do not halve two travel in halves;You may travel 1 whole, 3/4 ,16/27 18/19/ or 1/2 at a time.
"if there is anything we humans are good at...." great ending.
I've got my cynical side. :)
The answer to this question by the way is being stupid
I have heard another version of Zeno's Paradox, does a Black Hole have a bottom. We call the point the singularity but that just means we have no idea what's past it. Though Planck again steps in, with some physicist saying that a Black Hole has bottom and its end point is not 0 or infinity. But that all the matter and energy inside is squeezed down to the Planck width. That a Black Hole is really a balance of forces, where it's the matter and energy falling in that keeps it going. But once that stops, like after the heat death of the universe, then the other fundamental forces start pushing back against the gravity and evaporation via Hawking Radiation is accelerated.
Max Planck's first dating attempt. " Hey there doll. Want to see the length of my plank"?
Doll: 🤭Oh my. It's so tiny.
Max: 😕....🤔.....😃... "To the chalk board"!!!
My apologies. It's late and l was bored.
This is a catastrophe that wouldnt occur if you were measuring a black body instead
...
Eh? Eh?
...ill see myself out
@@kendomyers noooooo🤣🤣🤣 that's a phallic fallacy!!
I have seen many mathematicians struggle with this paradox, because they are so focused on mathematical equations that they cannot see the obvious answer: the space between two defined points is not actually infinite.
Please talk more about that mythical creature who brain washed souls! That sounds fascinating and I can’t find anything on the inter web about it
Zeno was a monist. He put forward his paradoxes to demonstrate that if there is more than one thing, then any change in the relative position of those things (i.e. movement), leads inevitably to logical absurdity. Zeno's position is really a very subtle one. It isn't open to refutation by mathematical or physical argument. In modern day language he is saying that, ontologically, all we can say is that only one thing exists for certain, and that is the Absolute. Division or partition of the Absolute into discrete things is a mere social construct, a fiction to which we are habituated.
Reductio ad absurdum doesn't always work. Schroedinger's Cat was an attempt to show that a quantum notion was absurd, and therefore incorrect...
The idea of a limit is by itself completely sufficient to explain Zeno's paradox, full stop. An infinite series of steps CAN add up to a finite number. 1/2 + 1/4 + 1/8... does IN FACT equal exactly one, not infinity. Though interesting, the Planck length discussion really doesn't have anything to do with the topic.
You do a great job with these, even the difficult ones.
I disagree, conceptually the paradox is airtight in practical physics it is irrelavant, infinity is not made of the sum total of numbers approaching infinity, the minor discrepancy amounts to the same as a large discrepancy, however you represent it.
In fact, the zeno paradox is a matter of observation, indeed, someone who can observe the event in infinitely small time periods will observe that it is not possible to get from point A to point B.
Zeno's Paradox has always seemed to me something that would be considered very profound by people who are very high.
Lol
zenos arrow paradox is actually not another version of his dichotomy paradox but is an independent paradox. It states that in one instant an arrow that is flying cannot be moving but since time is made up of instants then motion must never happen. (Not saying this troll just wish to imform :) )
The first problem I see with Zeno's paradox is in the hypothesis itself. This exercise requires a known end point and therefore debunks itself. Without an end point there can be no halfway point. Even adding "time" to the math doesn't ruin logic. If you divide the distance by half of the previous distance then, assuming same rate of speed, the next duration is also halved (both distance and time). Since a halfway point requires an end point we already know the total distance and time. Zeno is just dividing by half and never reaching the end point, still not exceeding the total time.
I think Zeno's Paradox demonstrates the fact that things do not actually travel continuously. They travel discreetly. So on at least some sort of subatomic level, the atoms that make up our being have to disappear and reappear. So in a sense, we, on a very small scale, are teleporting across small distances. So we don't travel linearly... so there is no Paradox. But the Paradox does at least demonstrate that linear movement is not possible. Pretty insane when you think about this was thought up thousands of years ago. I think this is where quantum mechanics probably comes in to save the day.
The reason the paradox 'works' is because there is one crucial ingredient that was left out of it, which is that fact that you'll need an infinite amount of time in order to mark a point between the arrow and the target. So the paradox is actually not a paradox if one of the variables in the equation is an infinite number. So for the paradox to be current it should actually start with the sentence: Let's assume you have infinite amount of time. Then if won't be such a surprise that if you have infinite amount of time you can keep dividing distance without getting anywhere.
"if theres anything that makes us human, it's our ability to make up problems that dont exist" -Joe Scott
Zeno's paradox ignores the relationship between time and space; and ignores quantization. The fact that quantization (a fundamental limit to the divisibility of time\space\matter\energy) is a fundamental feature of reality is well demonstrated by the Black Body Problem in physics (insert tangent here). Also, in the Achilles and the Tortoise version, Zeno's ignores rate of travel (velocity) in favor of dividing the distance into infinity. What he didn't do is to recognize that the "infinite" number of half-way points are being crossed in time intervals that are decreasing in magnitude at a commensurate rate. Eventually, Achilles crosses all of the "infinite" number of mid-points in near-zero time; and because of quantization, he is able to cross the finish line. It's a false paradox that requires people to ignore demonstrated aspects of reality; and maybe that was Zeno's point: that pure logic can fail when it becomes unduly myopic.
It's possible physical space has no reality in and of itself, inter- subjective appearances, in which case space is a moot point,
Or, the zeno paradox which is completely irrelavent to physics, applying abstractions to the empirical world obviously fails, it is not a paradox, you could say the universe's primary state is a dynamic flow, slices of time and space are superfluous in our reading of physics, it seems to be the problem of motion which is expressed differently in a concept articulated-
A train and a fly collide, the fly and the train hit a moment of 0 speed, in order for both to change velocity, that's the paradox
It's not a paradox, by analogy a water mill utilises the water to push the gears to turn the mill, all of the causal chain has side effects, there is no static state at any given time,
I'm wrong probably can anyone fill me in?
There's a limit of how much you can divide something. But what's interesting is that there seems to be no limit of how much you can multiply something. There's no theoretical limit of how big a volume can be as far as we currently know.
Excellent video. The premise of an infinite number of decreasing values also agrees with my heretical assertion that x/x = 1 (where x is identically equal to x for all intermediate values) as x approaches (and reaches) 0, thus proving that 0/0 = 1
I sooooo appreciate these videos as the way explained in a way that can be conceived.
The Planck length does have very profound implications, it shows that the universe is finite in the small as in the large, meaning if you were to cut the smallest length in half (planck length) that item loses locality, meaning it is everywhere at once!!! I have an idea for a video to: ZPE zero point energy and the fact that their is more of it now than was in the past an how it effects time, electrons and also the speed of light, its truly mind blowing to consider the implications on how much it effects our perception of time and how it has shaped the very world and universe we live in:) If interested I can point you in the right direction for content.
Yeah, please share. :)
I never heard about this paradox. Really cool stuff.
Zeno's paradox is an infinite sum that converges, therefore we can have a finite result.
This is actually hugely consequential! Consider the that the quantized nature of everything is a necessity for our experience of reality at all (meaning we can move from one place to another). Someone designing our reality would have to produce this basic concept as a substrate to construct reality. I think it provides evidence that we are living in a simulation (though I'm having a little trouble building a formal argument).
Are you assuming that time actually exists and flows? Can we really be sure causality is aligned with our arrow of time if we include quantum events?
hm... So possibly the the concept of non-euclidian spacetime preempts zeno's paradox. Meaning it's only paradoxical because of the way we perceive spacetime? Trying to grok this video for inspiration czcams.com/video/YycAzdtUIko/video.html
The number of Divisions keep going higher to supposedly infinity, but as the the number of Divisions go higher to infinity, the size of those divisions go down to zero. And when zero(nothing) and infinity (everything) collide. You get something. The limiting value. They are all around us. Everytime you go from a 0 to 1 or 1 to 2.
Tufrah of Nizam is an other solution to this problem. That is a body jumps/leaps from one point of space to another point of space without being on any one if the intermediate points of space b/w the two . This is in harmony and In conformance with Plank’s Solution. There must be a minimum dustance for Tufrah طفرة
Thank you Joe for sharing your version of thinking about Zeno's paradox.
there are 4 zeno's parodoxes. Either time and space can be divided to infinity; or one of them can, or the other, or they both can't. Each paradox excludes one solution until there is non. All of those can be solved by just saying that 0,(9)=1.
This is why I have always asserted that the fifth physical dimension is scale (relative to space time) We cannot travel in it (size or speed) beyond out very narrow limits and we cannot observe what is beyond our limits - so we assume that’s all there is. Just like a two dimensional being would imagine a three dimensional world!
Plot twist: Zeno makes a reappearence in both Black Hole Physics and Quantum Mechanics. So called "tortoise coordinates" may be used to get around the "mathematical singularity" that pops up at the event horizon of a BH.
And then there is Zeno's paradox in QM that if you measure something in quick enough succession, the state seems frozen in time... so the particle doesn't, like, age.
The lamp is off bc the starting point is that of an even iteration where beginning-and end are undefined. Thus if one continues infinitely, one leaves the lamp off to the conclusion the iteration, like wise if the lamp starts the iteration in the on position the end of each iteration would be off.
You're really good at this. It's a nice summary of information
Thanks!
Joe Scott vjop
Thanks for another fun video! Don't know about everyone else though, but I'm struggling to see how these are paradoxical. Taking the arrow hitting the target example. If the arrow takes 1 second to go from the bow to the target, if you observed this for 1 second, you'd see the arrow hit the target.
If you watch it for 0.5s. Then another 2.5s. Then another 1.25s etc you'd see the arrow get closer to the target in smaller and smaller frames. As such, if you continue to watch for smaller and smaller lengths of time, you'd see the arrow get to the target making smaller progress each time. Take this to infinity, you'd never see the arrow touch the target unless you had the full 1s of time watched. 0.99999... seconds wouldn't be long enough to see it hit.
All I am seeing here is a Greek guy tell us that you can break things down into an infinite number of tiny pieces to describe the infinite lead up to an event happening.
So to answer the question about whether the bulb would be on or off... It would never reach a point of fixed state. So it would be described as alternating between on and off at an increasing frequency.
To oversimplify, if I told you it took 2 seconds to do something. At any point in time before 2 seconds that thing would not be done.
Do a video on the Mandela Effect please. I have been doing my research and still haven't concluded between Psychology and Parallel Universes. It'll be awesome to hear your side of it.
Browse my channel, it's one of my most popular videos.
Another great example of how we misunderstand infinity is the idea that in an infinite universe infinite things must happen.
The number of rational numbers between 0 and 1 is infinite. The number 15 does not occur in there.
The solution to Thompson's lamp paradox is just that the function O(t) which is 1 if the lamp is on and 0 if the lamp is off isn't defined for t > 1. The paradox only defines that function for 0
If you divide space into an infinite # of segments between two points then there is no space between the steps. If there is no space between the steps then it would take no time to cross a step. If it takes no time to cross a single step, then it would take no time perhaps to cross an infinite number of infinitesimal steps. But perhaps the combination of infinite and infinitesimal gives rise to what we know as finite.
with regards to zenos paradox, its depends on how you define “you” when calculating position. does “you” have a length or is “you” an infinitely small point somewhere in the center of the front of you and the back of you
Xeno's Paradox as stated considers a constant speed. Achilles is accelerating, speed considered over decreasing time. Time is the factor which solves the apparent paradox.
The reason why a 'Plank-like' measure is important is that it will be the 'line where the physical can no longer be measured because it is no longer an object. See: Cartography and the measure of coastlines.
Hey Joe, I watched half your video last night, and half of what was left tonight. I watch half of what remains tomorrow night, and so on.
Can't wait to get to the end 🙂
the reason the faster passes the slower is because the universe ticks ... on, off, on, off.... and the universe is a grid of plank size boxes where matter occupies at least one of the boxes. Eventually at a tick off the faster is behind the slower tick on, beside each other tick off, tick on the faster is past the slower.
The heck? How did Planch get that number from? "Well, you see young man, it's a little thing called: I made it up. I didn't want to sound too ridicolus when I invented it, so I gave it a super small (not infinite small) value, so people would think it was legit".
And this is the point when someone tells me where he really got that astronomical small value from.
you don't have to go down to the planck length, only down the a quantum step. Quantum steps are always whole numbers and not divisible... once you get to that point, the turtle can only move one quantum step while Achilles moves 10.
In Zeno's paradox we deal with speed (distance over time) and negate the time and only focus on distance. As the distance between Achilles and the tortuous approaches zero, so does the time frame. The reason he never get to the tortuous is the time has stopped.
Assuming Planck was right about Planck length and Planck time, does that mean wavelengths of light can only take on discrete values that are multiples of the Planck length? How would redshift and relativistic effects affect this?
i am not very well educated on the subject but from my limited understanding, yes, the universe is discrete. energy can only be transferred in packets of energy (QUANTities) of a certain smallest amount. from this follows that measurement is dependent on those limitations, which makes the planck length the smallest measurable distance.
But being 'measurable' and 'existent' are two different things; even though the uncertainty principle limits accuracy, that doesn't mean automatically that spacetime itself is discrete. Sure, assuming this may be practical for calculations (e.g., in lattice QCD), but it brings in difficulties of its own: for one, we're not allowed to use the differential machinery we're used to, as it is--a major shortcoming, IMO.
Good question
A photon would collapse into a black hole well before its' wavelength approached Planck length. We're going to need a much more sensitive method of measurement if we want to do actually useful work at that sort of scale
Great explanation. Thanks.
You don't need something like the Planck Length or the idea that space doesn't exist in order to solve Zeno's paradox - you can just stay inside the logic of the thought experiments and think them through to the end: E.g. when you say by turning in your chair you pass an infinite amount of points, then that means the distance between these points becomes infinitely small. Therefore the time needed to get from one point to the next goes towards zero. That way it is possible to actually reach the destination point within a finite amount of time. Most of the other thought experiments mentioned work similarly. Infinite sums that produce a finite result are a very common thing in calculus.
The problem with Thompson's lamp is that the thought experiment only defines the state of the lamp _before_ the end of the given time period. Therefore you can't derive any information about the lamp's state _at_ the end of the period. It could be anything.
Superposition?
Both on and off?
@@larryscott3982 No, it's more like saying: Every Monday, Wednesday and Friday I wear red socks, every Tuesday, Thursday and Saturday I wear blue socks. Now deduce: Which socks do I wear on Sundays? It can't be deduced - I might be wearing any or no socks at all.
In pre-calculus we blew through this with a section called "growth rates of infinity", so basically infinity divided by infinity makes a finite number.
The answer to Thompson's lamp paradox is that you will never reach the minute because it doesn't matter how much times you cut time in half you'll never reach the end because you can only go half way
If there is a smallest unit of space, zenos paradox is no longer well formed. This is because the "halfway" point between to places does not have an exact value: it would have to be rounded. If it were rounded, depending on whether you use a floor or ceiling function, you could get finite or infinite points the arrow must reach before hitting the target.
I believe that when you indicate a measurement, you do just that, ID a length let's say.
Therefore logically "1/2 the way there" is a determined length. I think you are saying the opposite and I disagree with the opposite.
@@charlesdegruchy9927 im sorry I dont think i understand. ID a length? What do you mean?
What im saying is that if space is discrete IE there is a minimum distance that can be traveled, then thereare two possibilities: the number of minimum distances to be travelled is odd, or it is even. If it is odd, it is indivisible by two, thus rounding must occur. If it us even, then eventually you will reach and odd number or 1, both of which are undivisible by 2, so rounding must occur. You feel?
When you mentioned 'SuperTask' at The Beginning, i thought you nailed this, but then, you didn't ! A SuperTask is an Invention, A Definition, which means that you can make it anything you want, & in this case, you've created a rule that doesn't make any sense. It's a collection of words that seem to make sense, but it doesn't. It's a trick. A cheap trick. A much better Trick is Saint Anselm's Ontological Argument, that was 'Solved' by Simon Blackburn that essentially asked ; Is Sherlock Holmes Smart ? ( The Answer isn't Yes or No. )
Infinity is fiction. Useful in some cases, but both has never and could never exist in reality. I'm honestly a bit surprised that the idea of infinity has such staying power. It's never been observed, certainly, and is almost entirely a construction of grammar (like omnipotence, omniscience, etc), and it leads to paradoxes left and right. With any other idea at all, when a paradox is caused by its introduction, we throw that idea out. That's basically HOW we distinguish between truth and falsehood. But when it comes to infinity... I guess it's just really easy to imagine on a sort of intuitive level like "well what if I never stop ever?" (and then ignore the heat death of the universe making that impossible).
Oh yes, I didn't mean to downplay it's usefulness in math. That's what I was alluding to when I said "Useful in some cases." I've read a great deal about mathematical infinity, the Continuum Hypothesis (actually, I think it's been proven? I suppose it should be the Continuum Theorem now), and different degrees of infinity. It's a big part of the surreal numbers Conway came up with. I still expect surreal numbers and surreal mathematics to play a larger role in physics in the future (they've shown up as explaining some of the paths particles would need to travel for pilot wave theory to be correct) similar to how complex numbers were well developed mathematically before it was discovered that they can be used to model warped spacetime and are useful for dealing with relativity. So yes, it's very useful. But still, fiction.
@@DustinRodriguez1_0 I might question its usefulness in maths... but that's just me. Are you familiar with Steve Patterson? You should look for his CZcams page, I think you might enjoy and agree with much of his philosophy of mathematics
@@DustinRodriguez1_0 check out
czcams.com/video/HRPonozrpZA/video.html
Heat death is a theory.
The heat death of the universe does not disprove infinity. It just proves the universe is finite.
Joe, your sense of humor is awesome.
With Thomson's lamp , when t=1 min lamp will be on, since we started with lamp off. By the way I had to use an analogy to arrive at that answer , not guessing.
Super position? Change is the illusion created by the observer interacting with the system? space is a bunch of slices of instances like camera film, the observer is the projector. by observing the space slices the observer creates the illusion of change time and multiple observers. How many angels can dance on a pin head? The answer is all of them. interesting thought experiment.
(5:45) That statement *ABSOLUTELY* needs to be on a T-shirt.
Great, now I have to go and see if you have it on a shirt.
Consider this situation. If a girl is standing at one end of a long corridor and you are standing at the other end, if you move half the distance to her and then half that distance, and so forth, will you ever reach her? A scientist would say no, you will never reach her, because moving half whatever the remaining distance each time, there will always be a separation. But an engineer would say, "I can get close enough to kiss her".
How about covering "The sum of all positive integers equals -1/12" or 1+2+3+4......= -1/12
Eh, that's the point,
Well, you can divide length and time; it's called speed ;)
...and thank you Leibniz (co-discoverer of the calculus).
A related problem is motion from rest. To start from complete rest (zero speed) and to commence motion; no matter how slowly you 'accelerate' there must be a point in time when you instantaneously change state from 'at rest' to some small but finite speed. This requires infinite acceleration and therefore an infinite force.
Thanks Obamacare! I mean Zeno...
I like it
Planck's idea of minor slices of space as a resolution to Zeno's paradox is inaccurate, slices of infinitesmal size/time are irrelevant, there is merely an exchange of energy, you press the accelerator, petrol goes to the spark plugs the small explosions move the pistons that turn the wheels and off you go, Planck was looking at the concept from Zeno's idea of an abstract space, to begin with the premise of static states completely misconstrues the actual state of the physical universe, whether the universe is infinite or not is irrelavant every motion has a causal chain, a water mill utilizes rivers as an inefficient mode to turn the mill wheels and so on, I like the idea of the neccessity for an infinite amount of force to influence the motions of the universe, but from another point of view the energy need only be as great as the universe finite or infinite as the case may be, maxwell's demon is a thought experiment demonstrating attrition, regardless it is a change of state energy can't be created or destroyed so far as the scientific community is concerned I think..
But maybe I'm wrong?
This is by far, my favorite video of all you have made.
I've read that most of these are just different versions of the same thing. Anyway, as far as Achilles & the tortoise, I always thought that maybe the rate at which he crosses succeeding half-distances actually reaches or exceeds infinity. Besides, in the Achilles & the tortoise paradox, Zeno seems to concede that they are in fact moving, so you could make the same assumption about the arrow & go, "In your face, Zeno!"
I thought it was going to be some sort of Quantum foam solution to or undermining of Zeno's paradoxes. However, it refutes them in the same spirit that Dr. Johnson "refuted" Berkeley on matter.
That thought experiment with the light ball and the time it made me think of how the particles are in superposition, so the the light ball might be on and of at the same time when you go so small in time if that makes any sense.
Hey I have always wanted to know what makes living things cells stay together and not just fall apart? please do a video on this.
Semipermeable membranes.
Bill Morris the random field of quantum probabilities is adequate to maintain Newtonian physics, why? We'll have to wait for a theory of everything it can't be explained away by concepts of biology it is a reiteration of the question not an answer
This is already at least partially solved. Space/time is a hologram and holograms are fractal. So, yes movement is basically an illusion although relatively its real enough and works. Also they are confusing the map for the territory or something like that. The symbolic realm of math and symbols isnt practical in the physical. No perfect circle and no jumping half the distance forever. You would have to relatively shrink your size each time to keep doing that. The holographic theory does conjecture a limit to the resolution of size on the 2D surface of information. But, fractals do go on forever. The problem with many scientists is that they arent smart enough to know that something can be true from a point of view and false from a relatively different point of view. So, space/time doesn't really exist (ive seen it fold into 2D) and yet it does because imagination works well enough.
one possible solution can be that in extreme slow-Mo even uniform velocity has very short and quick cycles of higher and lower (negative) instantaneous velocities,or some "tiny stoppages" that just give the appearance of uniform velocity
Kind of like a bulb in an AC circuit that flickers but appears to humans as uniform intensity light
Its not a paradox when you consider that time flows(without pausing) , by you chopping each time and distance into smaller and smaller segments( time and distance have a one to one correspondence ) means nothing but showing that infinity exists in a finite value, which Cantor already did more than a 100 years ago. As an object moves through space over a distance at a certain speed , since flow of time is constant , as you get closer to the end of distance the frequency at which you will be getting a new value will approach infinity. When it actually gets to infinity the frequency stops, hence we have arrived at our distance across infinitely many points in a finite time.
@@singh2702 I meant it like an alternative solution also did some ganja
@@dnaann1867 We all here tryna find solutions to problems that don't exist.
@@singh2702 Achilles and the Tortoise is still disputed,hard to tell if every series will be converging
@@dnaann1867 The only thing you can actually say is finite but has infinite points is distance, hence when you keep dividing each subsequent segment by 2 you will never reach the end point. Distance is static but time flows that's how we complete the distance not by taking limits.
This is my favorite episode from this channel
Watch out for Audible. I subscribed for the free audiobook. I then cancelled the subscription. Yet they still billed me for a Romance novel. They slipped that one in very subtly so that I didn't notice. I then had to cancel that one too to prevent further billing. So for one "free" subscription I had to cancel twice and was billed once. Watch out they don't catch you too - I would definitely not go with them again.
The Planck Length isn't the smallest unit of length. It's the smallest that makes sense to us right now as we understand things. We already know that 1/2 Planck Length is a length. The Planck Length is basically how far a particle's location smears into the space around it. Trying to measure more accurately than that doesn't make sense because the smear of the particles is larger than your length of measurement so you can't be more exact. That even applies to particles that are smaller than the Planck length. They smear too much to be measured more accurately.
Would be awesome if Joe Scott, or anyone, can explain: 1) how Planck reached that conclusion of planck length (minimum length possible), and en even more, 2) why the other planck units are not minimum possible, like, planck mass, planck energy.... I really wondered this question for many years if any one can answer!
Thank you.
True story, I actually had a segment recorded that discusses that but it somehow didn't make it in the video. He came up with it by creating an equation that factored in 3 other constants, gravity, the speed of light, and Planck's constant. Not sure how he came up with the equation, though. en.wikipedia.org/wiki/Planck_length
A great video as always!
Thanks!
Answer to "Thomson's lamp": there is not enough information to determine lamp's state at the moment 60 sec.
That puzzle is based on the fooling of intuition. In the problem, lamp state is defined for all moments of time on the half-closed interval [0,60) (moment 60 sec not included). So, intuition suggests that state in the moment 60 sec must be determined but this suggestion is wrong.
You are true that we will never know, though even people who agree that Planck's number could be divided further are not wrong.