 Hey guys, here's a short video explaining why calculus does not solve Zeno's paradoxes I'm not the first to point out these ideas, but I hadn't seen a really good Resource anywhere that was concise and didn't involve calculus So I wanted to lay out the basic concepts for you so everybody can understand them. So we'll start with a Infinite series so take the series one half plus one fourth plus one eighth Plus one sixteenth and so on you may have seen this series before You may also see this part of the series Where it says equals one that the series equals one, okay? The problem with a lot of math education is people don't seem to ask two really critical questions here first of all What are these little three dots mean when we say and so on what does exactly that mean? The normal intuition is something like and the series continues off Infinitely into the distance the numbers just continue going on. There's no end to this series now in this video I'm not going to explain why that's a dubious concept. That's not the focus of this video But it's worth asking what do numbers work that way can can an infinite series Continue off even if the series hasn't been constructed by anybody Where does this series come from but second question is more important Which is the focus of this video which is say what does the equals sign mean? okay Critically important, this is not something unique to me, but everybody needs to understand this The equals sign in mathematics is a symbol that means different things in different contexts So it doesn't always mean equals as in one plus one equals two There's this perfect identity between the numbers on the left hand and the numbers on the right hand What is two? Well, it is the same thing as one plus one in This context when you're talking about summing an infinite series The equals sign does not mean Equality it actually means something different. That's intentional the concept that we're going to be talking about was come up with uniquely to say this is not Equals in the regular sense now why they chose the equal sign good heavens. I guess I understand because it's really practical I won't get one to the details of why it's really practical, but it causes all kinds of logical confusion as People seem to think that calculus solves the most paradoxes and it doesn't So another way you can describe this series you could say the series Converges to one that's another word that has a very precise meaning You could describe it as saying the series gets Increasingly close to one as you add up a half and then a half of the previous term and a half of the previous term So you go a half a quarter and eight to 16th a 30 second and so on the farther you go The closer the sum will be to one Increasingly close you could also say that some is getting Arbitrarily close to one as close as you like to one While never fully reaching Equality with one again, that's the point of the concept of convergence is to say this is a non identical Relationship it's different than normal addition Getting extremely close to a thing is different than actually reaching a thing Another way you can describe it one can be understood as the limit of the series This is going to this is a concept that's central in calculus. We're not going to dive into but you can understand it this way as well The one can be understood as the smallest value that the series never reaches That's kind of an interesting way to think about the limit in this context as You add up this infinite series and you get closer and closer and closer to one you can kind of think of one is this Endpoint that's actually kind of beyond the edge You can't ever reach that point. It's the smallest amount that that series is never going to add up to Another word that's used to describe this is an asymptotic Relationship which I'll illustrate in a way that people will be able to grasp in just a second The point is that asymptotic relationships this getting close to a value, but never reaching a value is explicitly logically distinct from Normal summation so Thinking that calculus solves Zeno's paradoxes is conflating the asymptotic relationship with an equality relationship that is a logical and profound error and in fact that right there is sufficient To say calculus doesn't solve Zeno's paradoxes for that very reason it's dealing with asymptotic relationships and not equalities Okay, so let's I'll give you a visual example so you can visually see what I'm talking about So this is a simple graph of a function. Don't be scared. Everybody can understand this Function is just a thing that turns inputs into outputs And this is a graph of how it's turning those inputs into outputs So the function is f of x equals 1 over x. Okay, what does that mean? Well, if you input 1 into the function you will get out of the function 1 over 1 in this case so we'll plot that there 1 over x. So whatever x is the function is outputting 1 over x So if it's 2 if you put 2 into the function we get 1 over 2 and you can see where the next point is If the input of the function is 4 then the output of the function is 1 over 4 and Similarly with 8 and so on So this continues whatever number you put into the function it outputs 1 over that number You can understand this the following way as x increases f of x shrinks or As x increases f of x tends to zero. That's another term that sounds like calculus But it's pretty straightforward to understand Okay, the bigger the x the smaller the 1 over x and you can see that relationship graph there. Okay, so here's a question Could there be a large enough x such that f of x Equals zero doesn't approximate zero. We're gonna use equals in the strict logical sense that there is an identity between f of x and Zero another way to ask the question at any point. Does the line? Ever touch the x-axis if you look at that graph it goes way out to the right, right? You can imagine we're inputting a hundred trillion trillion trillion into the function and what comes out as one one hundred One one trillion trillion trillionth, right just tight, which is gonna be so close to the x-axis as to be practically there for Useful purposes, but it's still logically not touched So the question is at any point as far along as you go on that axis does that line ever actually touch the x-axis The answer is no it kind of makes sense It doesn't matter how big a number you plug into that function You're always going to have one over whatever the number is and that's another way of saying that is you're always gonna have Some amount left over Regardless of how big that underline number is Okay, so let's port this back to Zeno's paradox. So with Zeno, let's you see the graph here Let's consider the point on the far left of the line zero and then the point on the far right of the line one So that's the totality of the distance that has to be covered So our runner before going from zero to one he has to cover half of that distance in Zeno's paradoxes Okay, so he has to go half and then from that halfway point to the number one He has to go half of that distance which would be a half of a half Which is a quarter and then from that point he still has half the distance remaining So he has to go an eighth of the distance and he has to go a sixteenth of the distance Then he has to go a thirty second of the distance and so on So the so according to Zeno if space is infinitely divisible Then you have a never-ending series of half points that the person has to cross He can never complete all of them and so he concluded motion is impossible Well, we don't have to conclude motion is impossible though the logic of his argument is actually good So this looks very similar to what we just saw a half plus a fourth plus an eighth plus a sixteenth dot-dot-dot and Then we're out. We want to know. Okay. How could this series ever? Equal one Zeno isn't asking. How could the runner get arbitrarily close to one? He's not saying how could the runner get ever so our Fraction of a fraction of a fraction so close to the finish line. He wants to know how could he complete The race how could he get across the finish line? Yes, you can describe it as getting increasingly close to one another way of putting this is at every point according to Zeno and this Way of thinking there is always an infinite number of points remaining So wherever he is in that race if he's at the halfway point the quarter point the sixteenth whatever it is There is always Not just additional points remaining but an infinite number of points remaining. How the hell does that work? So even if he's a 1 1 trillionth of the way to getting to that final destination If space is infinitely debisable then well He's still got a myriad of infinities to cross and in fact There's an even further logical problem was which is how does he even get to the halfway point in the first place? Because even to go from zero to the halfway point means you have to cross a halfway point in the first place So before going a halfway you got to go a quarter and you got to go an eighth and you got to go 16th so you have infinities in both directions that are all nested in front inside of each other Which is a problem if you follow that logic just like the line will never meet the x-axis and the series whenever Literally equal one the runner will never Reach the end of this infinite series is never going to reach an end of crossing all of the points So this is valid reasoning and if you follow along with the intuition of the first two parts You'll see yeah, that actually makes sense if there's an infinite amount of points between here and there at every point There's always going to be an infinite more. He couldn't make progress. Yes, of course These values never these values converge, but they never actually reach equality Well, what does that mean does do we conclude with Xeno that motion is impossible? No, we don't there's actually a really simple concrete logical resolution to Xeno's paradoxes spaces not infinitely divisible one of the premises of this whole idea is that Between any two points there isn't there is a third point. There's a middle point. There's a halfway point Well, if space isn't infinitely visible, that's not true There's at some point along that series you get to the point that's right adjacent right next to the endpoint There's no middle point between them. The best analogy is to pixels on your computer screen Between two pixels. There isn't the middle pixel. It's a pixel side by side to another pixel That's fine. That actually solves all of Xeno's paradoxes. All we need is space to be discreet and It's no more complex than that For a long period of time people thought space couldn't be discreet because of this reason or that reason and math would break And it's impossible. All of those turned out to be wrong. We've seen in the 20 in the 20th 1st century discreet mathematics is Superior in my opinion to Continuous mathematics. We have the math of discreet space. It's not a big deal This is clearly the simplest resolution We don't have to have nested infinities inside of infinities. Everything is clear logical and precise again I'm not the first person to point this out. There've been plenty of people beforehand But based on my conversations with various individuals, it seems very clear that the math education that they're receiving Blurs the line between asymptotic relationships and Equalities which is just a travesty and hopefully this video will help clear that up once and for all