How real are the real numbers, really?

[@12tone]

"This is crazy, but don't worry, there's way more" is my new favorite introduction.

[@adityakhanna113]

physics in a nutshell

[@LookingGlassUniverse]

Ahahaha! Thank you. And I agree, this is how physics and maths just seem to go.

[@oz_jones]

"When this baby hits 88 infinities per hour, you are going to see some serious shit."

[@trulyUnAssuming]

You know how there are so many rationals, that you can always find an infinite number of them inbetween each of them? Well this is crazy, but there is more says Meassure Theory: they are so few, that the length of all rationals is 0. Which means that there are so much more irrationals than rationals. :-p

[@rashrasos6553]

Ok

[@ScienceAsylum]

I just wanted to say this collaboration was super fun. So many good discussions in the making of these videos! ...and I think they both turned out better because of it.

[@LookingGlassUniverse]

I wholeheartedly agree that our videos came out much better because of those discussions! Thanks a lot Nick :)

[@usmcfutball]

Nick Lucid rules! And now LGU rules as well. The fact that I am subscribed to BOTH makes me one of the cool kids. Cheers!

[@clems6989]

The two if you could cause a breakdown in the spacetime continuum!

[@upandatom]

I like how you take seemingly simple concepts that we've all glided over and present them in a creative and unique way while outlining that they are not as simple as what we first thought :)

[@LookingGlassUniverse]

Thanks a lot! I really like how simple things are actually often not simple at all, so it's fun to make videos about those situations.

[@adityakhanna113]

The last line of this comment holds for just about everything.

[@adityakhanna113]

When there's a problem, I think quite a lot about it. My friends remark that I'm being *irrational.* I often reply "I just need to fill the gaps" It mostly takes a *real* effort, (hyperreal effort, I might add) to think up a solution. But once the *limits* of my capabilities are reached, the problem is over in a *fraction* of a second.

[@LookingGlassUniverse]

This is beautiful.

[@46pi26]

I have no idea who you are, but I want to be your friend.

[@crackedemerald4930]

Friends are an *integral* part of life.

[@LookingGlassUniverse]

A lot of people are commenting about the Planck length as 'the smallest scale'. This isn't true. In physics we still use continuous variables (so Achilles still runs along smoothly)- rather than a discrete grid. But I would love to discuss the possibility of having a grid at some point. It's a very interesting idea.

[@ScienceAsylum]

I'd like to add that, while Planck length has been /suggested/ as a smallest length in order to quantize spacetime for quantum gravity, we have yet to collect any evidence for it. Also, there's nothing all that special about the specific number. It could have just as easily been half of that or 100 times that.

[@kevinberg84]

The fact you would dismiss a smallest scale simply because "in physics, we still use continuous variables" demonstrates a complete lack of understanding of the distinction between abstract and applied math!

[@willys6403]

But...I thought Achilles preferred to glide.

[@nunyabisnass1141]

The Planck scale is irrelevant if you're not confined by real numbers like the Planck scale. For the purpose of explaining theory, this riddle does a fine job of illuminating certain numerical properties. However, in a context of practical logistics, Achilles has no set minimum distance of travel, only a set maximum relative to the total length of the race. D= -1/2b ÷ SUM...ish. Or (because my phone is retarded and I haven't done any real math in 20 years), the distance traveled in the real world can never be smaller than the whole units one is calibrated for. You're then left with a choice. Do you not move because the rules paradoxically require to move a distance that does not exist in that universe, or do you move only by the minimum whole unit? But because the purpose of the discussion isn't about the pragmatic real world semantics, as a property of propositions the original intent of the exercise remains true, despite how much I'd rather argue my cocktail party parody.

[@PikaChu-um2gt]

What happens when an observer is watcher a hypothetical object the size of the Planck length moving? Does it become shorter due to length contraction?

[@christopherblare6414]

Achilles doesn't run over the fractions. Not because he's smooth and fractions are jumpy like first proposed at a glance, but the opposite. The smallest physical scale that maintains any coherent meaning of distance is the Planck scale, he's slowly incriminating up his distance traveled by one Planck length and in doing so jumping over all the irrational, all the transcendentals, all but tiny fraction of the fractions.

[@jesscool1991]

Really brilliant video. As a Physics graduate myself, I've taken Advanced calculus and Fundamental concepts of Algebra this semester, two math courses, and it completely revolutionized everything I knew! My respect for math grew. Also, 3Blue1Brown is a great channel for understanding math well. Math is just brilliant! I wish I had 3 lives so that I can properly do Physics, Math and Philosophy in each of those, haha!

[@Kraflyn]

hyperreals next! And alephs! And continuum! Cheers! :D

[@LookingGlassUniverse]

Ahaha! That would be pretty fun actually...

[@tamptus3479]

@@randylejeune Do you like combinatorial game theory? I like thiese books: Winning Ways (Berlekamp, Conway Guy) On Numbers and Games (Conway) Mathematical Go Endgames: Nightmares for Professional Go Players (Berlekamp, Wolfe) Do like the Game GO?

[@kapick]

Awesome video! I'm no expert, but the Achilles you drew looks very much like Hermes :D

[@adarshkishore6666]

I really like how you researched into the origin of real numbers way back to the past to make this. Its like the numbers were saying all the time 'To infinity and beyond'.

[@nateg08]

I like how they both plugged for eachothers videos. I watched his first which peaked my interest in this video.

[@JoJoModding]

I think that Newton and Leibiz said: "Lets look what happens when that time gets closer and closer to zero". Thats a huge difference when compared to ".. that time is small, really small".

[@kumarpranshu2533]

But few say complex numbers are just as 'real' as real numbers. It's as strange as negative number is to positive

[@BangMaster96]

Some infinities are bigger than other infinities. A circle is not real, because it requires infinitely many points to be formed in the real world, but, with the concept of the circle, we get Pi, and with the Unit Circle, we get Sin, Cos, Tan, all of Trigonometric identities, which is amazing because it helps us model many real world phenomenons in Physics, something that can not physically exist in the real world helps us explain the real world, if that isn't crazy magic, then i don't know what is

[@theultimatereductionist7592]

I love our modern (over 100 years old) well-defined ideas of infinity & set theory & measure theory & the real & complex numbers & other number fields. All the stuff undergraduate & graduate math majors get. I have NEVER understood nor cared for nor loves this archaic Newtonian idea of "infinitesimals". What practical good can one do with it? What can one compute with it? I love how we now can define Lebesgue integration over subsets of the reals such as "the set of all numbers whose decimal expansion contains no 3s, 7s, or 9s, and 1s only at a prime-number position".

[@chiepah2]

The only other person who was able to explain concepts as well as you was my kindergarten teacher. She was able to teach a room full of 5 year olds the concept of multiplication (which for the longest time I thought was impressive) and the concept of zero as well as multiplying by it (which I have only recently realized how big a deal that was). Anyway all that to say, thanks for another great video.

[@LookingGlassUniverse]

Awww, that makes me really happy to hear. Have you gone back and told her you appreciated her teaching? I'm sure she would love it.

[@chiepah2]

It was in Germany and I don't even remember her name :(

[@LookingGlassUniverse]

That's fair enough! It's really nice you still remember her though :)

[@krzyszwojciech]

If there were infinite amount of places between two points in space and reaching any point in between would take a moment, then traversing that length would take infinite amount of moments. Infinite time. So it seems there must be a limit to the divisibility of space. The problem is - we don't know where is that limit, so we can't calculate in wholes, instead we must in parts. Real numbers, while I don't think they are in fact real [in the sense of representing how the world really is], are still very useful, because they allow for divisibility that's arbitrarily small. They allow us to accommodate the holes in our knowledge.

[@williamdavis2505]

krzyszwojciech an infinite sum of infinitesimals can converge to a finite number. So says Calculus and Analysis.

[@williamdavis2505]

For example sum (1/2)^n for n = 1 to infinity. There are an infinite number of terms in the sum but the sum converges to 1.

[@krzyszwojciech]

​ @William Davis When we talk about the nature of space itself as opposed to what you can do with concepts in maths, you don't suppose it consists of different chunks, some big, some half that size, and so on until you get something conceptually infinitesimaly small. We expect the nature to be uniform. That's a hypothesis that is most parsimonious here. And if space is uniform, then either everything consists of the infinite of infinitely small points, or from discrete chunks, finite in any volume, we just don't know how small they are. The problem with my initial post may be that possibly when the infinitely small space is traversed, if it happens in infinitely small time then it's somehow consistent. Still, belief in infinite divisibility of the actual reality seems rather strange to me. A belief that something physical is actually infinite or infinitely divisible carries with it a burden of proof. A useful mathematical tool based on real numbers (or other numbers with the same cardinality) is a piece of evidence in that direction, I would dispute however that it's sufficient, given that an alternative explanation is possible: again, the rational numbers being conceptualized as arbitrarily small due to our inability to know what the reality really consists of. Them being a tool, a mapping device that can conceptually have more resolution than the 'terrain' it's mapping. There needs to be an argument made why it can't be so.

[@srpilha]

The moment I heard "infinitesimals" I was shouting NON-STANDARD ANALYSIS YAASSSSSS You make this (former) mathematician very happy with your videos. :)

[@LookingGlassUniverse]

Yay! I so happy these nerdy things make you happy- me too :P

[@nosyarggrayson]

I just read Godel Escher Bach so I am quite familiar with this paradox. I really appreciate your illustrations and thoughtful videos. Thanks!

[@LookingGlassUniverse]

That's a fantastic book. Thanks a lot!

[@chrisg3030]

I get on a train and ask a fellow passenger if it goes to Cambridge. Yes comes the reply. Later on I look up from my paper to see us speeding smoothly through Cambridge without even slowing down. My protest is met with "Sure it goes to Cambridge, it just doesn't stop there". As soon as we mark any point along the path taken by a moving object,, whether with a name, a fraction or a furlong post, we're inevitably treating motion as if it were a series of stops or final destinations. The smooth is broken up into the rough. But it's a useful convention.

[@foobargorch]

The map is not the territory, the way we've managed to describe the universe so far involves this math, but that doesn't mean that this is inherent, and one could go further and say that real numbers do not actually exist, because any irrational number would require an infinite amount of calculations to get to the bottom of. I personally like to see it as if this mathematical construction is a model of possible universes, and it's fine enough that our actual universe can be described with it to arbitrary precision, but whether or not it's actually real is very much an open question. Max Tegmark has some interesting lectures about this but I can only vaguely follow along when he gets more technical.

[@LookingGlassUniverse]

Yes! Exactly. I am personally not a fan of the reals either, for many of the reasons you said. But I wanted to present this arguement for why mathematicians came up with the reals in the first place. Then I was thinking of a follow up video.

[@ScienceAsylum]

I'm probably going to have to do a follow-up video too. So many good questions and discussions in the comments.

[@3snoW_]

Why do you say that any irrational would require an infinite amount of calculations to get to the bottom of? If you look at 1/3, no matter how many digits you write in decimal notation, you'll never get to a number that multiplied by 3 gives you 1. You'd have to do an infinite amount of calculations to represent that number in base 10. Does that mean that 1/3 doesn't exist? √2 is an irrational, you'd need an infinite amount of calculations to write it in decimal, but if you draw a square with side 1, its diagonal has length of √2. So √2 is irrational and it definitely exists.

[@n4rzul]

+foobargorch "because any irrational number would require an infinite amount of calculations to get to the bottom of" That sounds an awefull lot like Achilles never finishes the race...

[@Nickelnine37]

But what if......... space and time was fundamentally discrete? That would get around all these problems right? If space and time really was continuous wouldn't you need an infinite amount of information to describe anything? This troubles me :/

[@zsoltnagy5654]

From the statement there to be other fractions between any two fractions doesn't follow, that there are no "gaps" between any fractions. At 3:33 it appears to be claimed, that given the fact of there to be other fractions between any two fractions, therefore there are no "gaps" between any fractions, which is of course wrong given the existence of counterexamples by any irrational/nonrational/nonfraction numbers.

[@mandisaplaylist]

When I was reading about this part of calculus for the first time, I came with this explanation: "dt" is a value that is going to become zero in some not-so-distant future. So you need to do all the dividing with it before it reaches zero and then you let it reach the zero and "eat most of the result".

[@orlandomoreno6168]

The universe if effectively finite. Deterministic. Computable. All you need are countable sets.

[@williamdavis2505]

I bet you are an applied Computer Scientist, and find the Simulatiom Hypothesis plausible.

[@alexhenderson-clark4741]

One question, are complex numbers only useful for describing waves? I only three years on college physics math knowledge. But I remember using them to replace sin and cos functions. Super interesting video however.

[@LookingGlassUniverse]

Yeah, great question. See, complex numbers are certainly useful for describing things (in particular, rotations), but does that mean that they're real? Or a convenient language? The same could be asked about the reals, and that's why I wanted to make a video about how reals correspond to 'real' things like length- not sure complex numbers do.

[@xnqmap]

May I ask: how do you come up with such a video? Like, how did you get the idea of making it, and how did you do your research to find out how real numbers were first introduced?

[@jenniemaes1967]

The real vs. rational debate can be side stepped altogether by another important set that Newton couldn't have known about: The Computable Numbers. Any number whose digits can be calculated to arbitrary precision by some algorithm is computable. This includes every integer and rational number, but it also includes pretty much every irrational number that we care about or can five a name to. e = Σ {k=0 to ∞} 1/k! tau = 8 Σ {k=1 to ∞} (-1)^(k+1) / (2k-1) etc. are all computable, and so are all the combinations you can come up with using √,×,÷,^,+, etc. You can still do calculus and get all your convergences, but here's the kicker -- the computable numbers are countable, because you can enumerate every computer program! tl;dr, if you want to ask which numbers "actually exist", I'd say the computable numbers are a good place to look.

[@LookingGlassUniverse]

Thanks for raising the computable numbers! But how can you get calculus to converge- they don't have the least upper bound property. Is there another way?

[@jenniemaes1967]

Looking Glass Universe: I need to admit, I didn't know about [Specker Sequences][1] before I read your reply! You're right, in general the least upper bound is lost, which makes both calculus and analysis trickier. You can get around the problem if you redefine "convergent sequence" slightly. In the definition of a limit, instead of saying "for every ε there exists a δ...," you have to say "there exists a computable function from ε to δ..." That change eliminates Specker's trick (which is to hide the halting problem in the rate of convergence). Now you are only considering limits that converge to computable numbers, and the computables can have calculus again! Thanks for replying, I learned a lot today because of your comment! [1]: en.m.wikipedia.org/wiki/Specker_sequence

[@BobWidlefish]

Infinities often reveal shortcomings in our knowledge of the correct unit and modeling framework. There's nothing particularly mysterious about it, the implications seem bizarre because infinities don't exist in reality, so it's like talking about a square circle -- no conclusions follow. The very concept of existence, correctly understood, is exclusive of infinities. If we confuse our concepts approximating reality and modeling with reality itself, the implications are likely to often be nonsensical.

[@sebastiangudino9377]

Infinity does exist in math, maybe not the way you think, but you can actually add an infinite amout of things and get a finite result, you should read about geometric series, or the infinitecimals in calculus and how they are not a problem in any way. There is even the numbers itself, you can always take a number and add 1 to it to get a larger number, so there is no 'largest number'. That means you can go on forever

[@KnowBuddiesLP]

Nick from the science asylum sent me :)

[@LookingGlassUniverse]

Welcome :D

[@arturzatorski595]

Has anyone else been sent here by Nick in 2019?

[@jaytaffer9641]

What was the book you showed as a screen-shot?

[@sterius5098]

Really, you will be really good at doing an introduction video to calculus.

[@adityakhanna113]

Well, I doubt that the resolution was necessary to do Physics. At that time, I don't know, but can't we call the planck length our finite tiny number? It's mostly a mathematical problem pertaining to function continuity. As far as I go. :/

[@LookingGlassUniverse]

It is a mathematical problem, but one that was necessary to solve for the sake of physics. But I agree with you that there may be other solutions that don't require the real numbers, by some how having a smallest length (maybe the plank length) etc. That would be cool!

[@CraftyF0X]

So, It is purely a mathematical problem, which requires a biological mind to manifest, which requires a physical world to exist in and operate... so is it a physical problem then ?

[@adityakhanna113]

+CraftyF0X Which requires complex chemical interactions, semiconductors and hardcore programming to be communicated to us. That doesn't change what it is.

[@CraftyF0X]

Sry I was drunk. I guess my point was that if you belive in physical determinism (which you should) then the appearance of such mathematical problem was an inevitable consequence of the laws of physics.

[@prodprod]

But physics has come to approach the problem of infinite divisibility in rather a different way than mathematics -- and that's that neither space nor time is actually infinitely divisible. You may be able, conceptually, to imagine a given segment of space, and divide it by half or by a billion or a trillion or by any amount -- and the same with any interval of time -- but you really can't do that with the planck distance or the planck time. Because the universe is quantized, there really are physical limits to how small things can get and how brief intervals can be in our universe, irrespective of how we imagine things might be on a conceptual number line. Imagining infinite divisibility leads us to things like the ultraviolet catastrophe -- and that's not a good place to be.

[@sumsriv]

this was a very fascinating video on the weirdness of math. thank you for your awesome work!

[@LookingGlassUniverse]

No problem at all!

[@AThagoras]

If my understanding of physics is correct (it is a big IF), then space is quantised, the real numbers are only a mathematical construct and there are only a countable number of points in space-time.

[@AThagoras]

@skubb Heave I was pointing out that the mathematics of real numbers may not correspond to anything that exists in the real world. I don't know of any way to conduct an experiment to prove that space is not quantized.

[@AThagoras]

​@skubb Heave In physics, we really only observe discrete particle interactions. Space, time, waves, fields etc are all mathematical models that describe the statistical properties of particle interactions at various scales. Current quantum field theory models are *extremely* accurate, of course, and there has to be a good explanation of why they are so accurate. One view of physics is that the only things that exist are particles and interactions between particles. In this model of physics, space, time etc are emergent properties of the network of particle interactions which is defined by rules governing possible particle interactions.

[@viralsheddingzombie5324]

There have to be "gaps" between the fractions. What we're really talking about is the structure of space itself.

[@GarryBurgess]

One might argue that the runner also has an infinite number of time fractions to cover all those infinite amount of distance fractions, unless of course time is quantized like a lot of things seem to be in physics these days.

[@ManojKumar-io2ok]

Calculating averages over infinitesimal seemed to be fun when starting with Calculus.

[@danielbigham]

The thing that has been creeping me out the most about real numbers recently is that a single irrational number can contain an infinite amount of digital information. And so, if the distance between two atoms is a Real number, then at some point as they move apart, their distance would require an infinite amount of digital information to encode. And boy does that sound weird / wrong. www.wolframcloud.com/obj/danielb/Published/amountOfDigitalInformationToSpecifyDistanceBetweenTwoAtoms.nb

[@DaveGamesVT]

If there exists the Planck length, doesn't that mean there would be a smallest amount of size/distance, and that you really couldn't just divide space into infinitely tiny fractions?

[@HDitzzDH]

2:30 It should say 793/1000 right?

[@RazorM97]

yes, but that's not that important

[@zokalyx]

Analysis at its finest !!! :)

[@jmiquelmb]

Next do a video of Aquiles circling around the complex plane

[@B._Smith]

I really like the art because it makes the maths familiar to the average person. (Not saying your art is average).

[@The6thMessenger]

I never thought numbers are real things, but just created concepts to convey ideas, for us to process patterns.

[@Eztoez]

I don't get the point you're trying to make. This is just a philosophical exercise in mathematical semantics. A person who - under constant motion - moves from point A to point B arrives at point B and will cover the distance between A and B. You can break down the amount of space between A and B down to the Planck length, but it is of no use in the macroscopic world.

[@katokianimation]

Matematically he goes trough every fraction and irrationals beetween exactly 0 and 1. That is the matematical expression of runing 1 unit smoothly. Phisically we can not messure infinite amount of fraction, so this is not a real problem. Math is not absolute. It is just a language. It can discribe reality and it can discribe things that are isn't real. It is more consistent and more unified then any other language but still it can cause paradoxons sometime but that dosen't mean the world dosen't make sense. It means there are limitations of any languge and we should be careful how we describe things.

[@0011peace]

there is a smallest distance you can travel its called Planck length. And, the smallest unit of time the time light takes to go one Planck length.

[@abhilashasinha5186]

Everything goes smooth because they move on infinitesimally small points; that's the thing about calculus.

[@andrewosegueda2283]

3:37 When the runner disappears because he passed the .793...., He does actually disappear and the point still exists (in some arbitrary point and time in space). I thought of a movie and how it moves frame by frame no one 2 frames are alike. In the real world and we move in time present time, its kinda like the movies and frame. We have to kinda have to accept the runner of the past, disappeared at some random point right?

[@tcaDNAp]

Leibniz looks so cute! EDIT: nuuuuu Leibniz don't cryyy

[@kingnabeel12]

Hmm it's been a couple of years since I took electricity and magnetism but dont complex numbers appear when solving some problem pertaining to circuits? So complex numbers are part of the real word in a sense?

[@LookingGlassUniverse]

Yeah, great question. See, complex numbers are certainly useful for describing things (in particular, rotations), but does that mean that they're real? Or a convenient language? The same could be asked about the reals, and that's why I wanted to make a video about how reals correspond to 'real' things like length.

[@ArgumentumAdHominem]

Maxwell equations in time domain are real, as are all associated observables. However, many problems from wave or circuit theory can be addressed much easier by mapping to the frequency domain using a Fourier transform. The transformed quantities are then complex. These quantities have certain physical meaning, and are very useful, but are not necessarily real themselves, as the observables are still real.

[@RickyForITZY]

I just watched this. I will start by saying it is almost 9 a.m. here and been reading stuff for hrs so if not so coherent you know why. But anyway, I think there are a few things I want to say concerning this that you might find interesting. First you aren't really talking about an actual infinite. In Philosophy of Mathematics there is a difference between what is called an Indefinite also called a potential infinite represented by a lemniscate, Cantor also called it a variable infinite and said it was an improper infinite and an actual infinite (represented by the Aleph Zero symbol, see "Contributions to the Founding of the Theory of Transfinite Numbers" for these distinctions). In the first infinity is an ideal limit (so a finite collection can indefinitely grow towards this infinity perpetually but never get there). In the second there is a collection in which the number of members is really infinite and complete, a determinate whole not growing towards infinity as an ideal limit. Taking this into account we can say a few things. First one cannot merely assume that the entire interval is composed of an infinite number of points, consider Zeno's opponents like Aristotle who would say that the interval (or simpler line) as a whole is conceptually prior to any divisions which we might make in it (for this critique see Adolf Grünbaum, Philosophic Problems of Space and Time). We can divide it indefinitely but not actually infinitely (Arguments that Aristotle are wrong here in that a potential infinite entails an actual infinity tend to commit a modal operator shift error, ofc the intervals also being unequal amount to a finite distance as you've shown and we all know. Holding to any Actual Infinite also begs the Question against certain Mathematicians who hold to a School of thought called Intuitionism in which only a potential infinite exists). Also by infinite numbers "really" existing. I wonder what you mean by that. It sounded like Platonism in which these numbers exist as an infinity of Abstract Objects or even maybe Concrete Objects as in some forms of Conceptualism but that begs the question against Nominalism and Nominalists like myself who reject such things. To be more precise one can be a Metaphysically Heavyweight Platonist in which these numbers are just as real as physical objects or be more Nominalist in taking them in a Metaphysically Lightweight Sense as merely semantic and there are different varieties of Nominalism on offer. A good book on Nominalism that might be up your alley is called "Science Without Numbers" by Hartry Field.

[@jerrymacdonald9252]

How do fractions mesh with Planck's constant? I've always wondered if movement was truly smooth across all infinitesimal points or does it jump it Planck units. maybe another way to ask is if you have two zero point particles next to each other at planck length, how many zero point particles can you fit between them? Can anything exist between them? Can anything exist in that space regardless? Is the foundation of spacetime analogous or digital?

[@LookingGlassUniverse]

This is exactly what I want to make a follow up video on. Because you're right, this seems like it's also a valid solution. But which is nicer? I don't know. I'm not a fan of the real numbers, so maybe I'd go for this one.

[@sciych]

Physically. Yes, there are always discontinuities. Quantisation of everything? So, as long as stuff remains quantized, smoothness is only a macroscopic phenomenon.

[@LookingGlassUniverse]

I don't think that's necessarily true. Even in quantum mechanics you have smooth parameters for time and space.

[@sciych]

+Looking Glass Universe That seems counter intuitive. Is that smoothness a mathematical construct(simplification or abstraction)or can we REALLY move smoothly? And doesn't wave nature destroy smoothness?

[@LookingGlassUniverse]

Yeah precisely, Plank length isn't the 'smallest length'- it's an important length scale at which we think quantum gravity comes into play.

[@jok2000]

oo x oo is still oo. Also Cantor not withstanding you can enumerate the reals thusly 0,0.0,...,9.9,00.00,...,99.99,... oo. Don't expect anyone in academia to accept it though. There are a few duplicates, but it works for me, I have an uncomplicated definition of Infinity.

[@haarmegiddo]

The name for real numbers has nothing to do with the real world. And this analogy really fails for the real world because it does not acknowledge the Planck's length. And that is why the tortoise would never beat Achilles in the real world in the first place. Math is just pure abstraction, there isn't something as math in nature. Physics is great because it uses that abstraction and puts boundaries on it so it can be used to approximately describe the real world.

[@UnforsakenXII]

Just finished a electrodynamics problem set, time to fall asleep to this.

[@carbrickscity]

There's always a gap between math vs physics in regards to infinity. i.e. physics would always appear to be sloppy compare to pure math, when physics always have to take the real world into considerations. While in pure math, there's no such constrain. We always want to be precise in math. Take a look at set theory. Pure math is miles ahead. I see physics and science are taking too much assumptions too often, and using the term infinity too often, even to things that are not infinite. And always just take things for granted and assuming things are infinite while there are no actual proof yet or cannot be confirmed. Always use the term infinity for the unknowns. i.e. infinite gravity, infinite density, infinite space, infinite time. infinite energy, infinite possibility, infinite this, infinite that, etc. Too many people saying the universe or this or that is infinite just as they are facts, while in fact there are just conjectures. Kinda similar to saying pi is 22/7 or some other fractions. Way too sloppy. Just smh all the time looking at that. While they can get a pass from regular or science people, it's a joke to pure mathematicians, especially to those who study set theory or infinities.

[@Falkdr]

an infinite number of steps doesn't make it "smooth", it just adds an infinite number of steps to a finite number of steps, so it doesn't explain why achilles reaches the end at all, imho.

[@TheBookDoctor]

3:33 - "Even though we showed that there are no gaps between the fractions." No, you never showed that. You just showed that they're arbitrarily close together, which is not the same as not having gaps between them.

[@LookingGlassUniverse]

That's what I meant by a gap :P a distance from one to the next. But of course there are some sort of 'gaps' between them- so the reals can fit in.

[@martind2520]

@TheBookDoctor I would define a gap as a non-zero distance without any of the item of interest (in this case fractions) with-in that distance. Can you find, on the number line, any such distance without any fractions in? Or do you have a different definition of "gap"?

[@davidwilkie9551]

Temporal Superposition-singularity is real connection, real->probability, e-Pi-i dynamic-identification of proportions/ratios of probability in transverse eternity-now, 1-0Duration interval of potential possibility. That's the infinite amplitude of 0-1 interval within which the divisible spectrum of frequencies is dominated dimensionally by 1, 2, 3, 4...to infinite connected phases locked in superposition. Ie AM-FM density-intensity of probabilities in potential possibilities. (Each of us needs to figure it out for themselves. Alice in wonderland asked lots of questions and looked for answers, no matter how extreme?) The "Looking Glass Universe" is a Feynman type guess about the apparent 3D-perspective of Polar-Cartesian Coordination in a brane..? A real reflection/image of Eternity Now/QM-Time, but based (biased) on another point (dominant singularity) of view. Function in-form-ation.

[@Rhannmah]

The outcome seems still paradoxical to me if you stay in the realm of pure mathematics. In the real world, I solve this, as Zeno's Paradox, with Planck Length. So there isn't an infinite number of fractions of a distance, it is finite, the basic unit of distance is Planck Length.

[@sciych]

Well. If we do move to the other school of thought as numbers as representation of an idea rather than entities. We can *use* complex numbers to represent AC circuit values like impedance and others. Also, in fourier analysis and QM too right? So, what does the real world represent? Aren't all those also a part of reality then? Are they also,IN THE REAL WORLD?

[@RickClark58]

SciYCH ! That is the question isn't it? To rephrase it, is math objectively real? Unfortunately there isn't a definitive answer. Personally, I am of the school of thought that math and science are simply man made constructs that we use to make sense of the world and are not real identities. The universe doesn't use general relativity when a photon moves from point a to b, it is rolling down the hill of curved space. A convenient fiction I think math has been called and I agree.

[@LookingGlassUniverse]

Very well put. This is kind of what this video was about. I didn't want to think of maths as a representation, I wanted to talk about 'real' things, like lengths. Lots and lots of things from maths are useful- complex numbers, quaternions, groups etc. But are they real? Or a "convenient fiction"?

[@kidamkolkoznam]

if our counting system is not based on 10 but on some other number maybe 12 or any other number, would some irational numbers becom rational? Lets say when base is 10, then 1/3 = 0.33333... (it goes on forever), but if base is 12, then 1/3=1.4

[@tfae]

No, 1/3 is rational in any base. A rational number has a decimal expansion that's either finite OR repeating. 0.333... is repeating, so it's rational.

[@kidamkolkoznam]

@@tfae I understand that and thanks. But you didn't answer my question. Would some numbers become rational with some base other than 10. Is it possible?

[@dchapero6929]

Seems the integers could be the basis of all numbers/proportions? Notice the singularities within the iterative sine function - they correspond with 1/2,1/3,1/4, etc... irrational numbers, like sqrt 2, show themselves too. The relationship between the cosine wave and sine wave net the geometry related to Spec Relativity. Quite cool. fqxi.org/data/essay-contest-files/Chapman_Integers_Prime_Numb.pdf

[@shuffle208]

But, physically thought he has to travel some distance, and the smallest distance he could possibly travel was defined by Max Planck called the "Planck distance"

[@Bestape]

I get this when space and time are different but how does it work with unified spacetime?

[@martensjd]

Of course, to "describe the real world," we need complex numbers, not just reals.

[@vojtech2304]

Ok, that is one fast Achilles running 126km/h :D

[@LookingGlassUniverse]

Achilles is "the fleetest of foot of all mortals" :P (from Godel Escher Bach)

[@danielsteel5251]

Even without specific expectations for seeing anything in particular in this video, the video was still somehow not what I expected. Going from Zeno to Robinson, like, real quick ... I can't explain the feeling. Probably some emoji captures it? I don't even know.

[@LookingGlassUniverse]

Lol! A good or bad feeling :/?

[@danielsteel5251]

Oh, I don't know. I mean that insofar as I feel anything, I'm confused. (This is not a complaint.)

[@whateverihateyouwtf]

but are we really talking about infinity? its a question about the real world and the real world has to deal with the planck length, right? no calculus necissary. just really big numbers right?

[@fernautilus9863]

Oh goshhhhh this almost made me cry

[@pavanpratapsinghchauhan6804]

loved it really............ waiting for 1m subs

[@LookingGlassUniverse]

Thank you!!

[@royendershade8044]

It is too much to assume Aquiles moves smoothly. It cannot be concluded that the universe requires real numbers, there are many other possible explanations, with less inconvenients and more adequate to the physiscs we observe that sefine a discrete universe, real numbers being a mere tool for us to manage the quantities involved. Very interesting video though.

[@NeoNick5435]

Ahh... NSAnal, how I've missed you so.

[@CristianGarcia]

What about planck space and planck time? These other concepts are more discrete in their nature.

[@morganjones7428]

woo hooo yess, I love this kinda stuff !

[@jamesstaggs4160]

If a frog jumps halfway to a wall every time he jumps, how many jumps will it take him to reach the wall?

[@MrCmon113]

Only even numbers exist. Odd numbers are human constructs. In any case, please tell youtube to stop translating the titles.

[@balazskecskemeti]

But if Achilles is moving at a constant speed, he can't be located at a well defined point on the line.

[@dlbattle100]

The real numbers when applied to the real world have always seemed like sophistry to me. When talking about the real, physical world, there is obviously a limit to the resolution. There's only so much information you can cram into a given region of space, yet the real numbers make no acknowledgement of this fact.

[@enterprisesoftwarearchitect]

Mathematical idealization - not a theory of everything. Numbers - even the integers are an invention. Whoever thought of numbers was brilliant - but we see they aren’t faithful representations of reality, it appears. Except for group theory ... that seems to be where integers are given “real” meaning.

[@enterprisesoftwarearchitect]

You should teach us about group theory!

[@Clouding-e4l]

Is infinite a real number or not😊

[@jqerty]

Thinking about infinity really confuses me. The same with real numbers. Glad I'm not a mathematician and I can remain agnostic about it :)

[@LookingGlassUniverse]

Hahaha! I think a lot of mathematicians are too.

[@felixwinchester9256]

but how do you finish something which doesn't have a last step???

[@ArgumentumAdHominem]

Surely you need like 2000 steps to run 1km distance. That number is quite finite. So if you don't want to have a headache, you can get quite far while dealing with finite numbers only But if you do want to have a headache, you can have it even without real numbers. For example, your speed is 1m/s that means that in 1s you cover 1m. There is infinite number of fractions that fit between 0 and 1. So you cover infinite number of numbers moving at that speed. And if you move at any finite speed, you will still be covering infinite number of numbers. I think this alone is sufficient to blow mind

[@felixwinchester9256]

but in the paradox there was a turtle he raced against ,which he gave 1 km headstart, shoudn't the turtle always be ahead by some distance? i mean he will never catch it right? but he does of course..

[@felixwinchester9256]

there were actually 9 of them ,but the joke in the end was good.

[@sebastiangudino9377]

You should watch the video on Supertask by Vsause

[@andywright8803]

Sorry, but in the REAL world, this doesn't happen. Spacetime is quantized. I know this means that real numbers don't exist, but this is the difference between pure maths and applied maths (ie what the world is really like)

[@williamdavis2505]

This vid may sound hopelessly abstract, and it is! Infinitesimals may only exist as a figment of our imagination. Worse, infinitesimals are required for smooth spacetime, as presumed by Einstein’s General Relativity, and result in singularities. Heisenberg uncertainty shows us arbitrary precision does not exist in the physical Universe. Bring us a theory of quantized gravity! Down with singularities!

[@mrfranksan]

Complex numbers are necessary (real) too. And they are part imaginary, unless “imaginary” is a totally misleading descriptor. Which perhaps it is.

[@jamesashons9227]

any volume can be divided into an infinite amount of ever smaller parts.

[@andywright8803]

james ashons but not in the real world you can't. Volumes smaller than the planck volume simply don't make sense.

[@jamesashons9227]

Andy Wright as far as we know

[@jamesashons9227]

real numbers are the best, because they are basic. you can't have a 2.7 collection of objects, but you can have a collection of objects that are that size.

[@andywright8803]

james ashons But that is a ratioal number as well as a real number. There 8s no problem with rationals, but real numbers include irrational numbers which cannot be expressed as a decimal in this universe, and which cann9t represent any actual thing in this universe as s0acetime is quantized.

[@mskiptr]

Do we really need uncomputable reals though?

[@jonascarrillo8699]

Ok ipm on a loop now.

[@intellectelite]

do a video about complex numbers in the real world.

[@LookingGlassUniverse]

I'm not sure what I'd say about it exactly. What would you say?

[@adityakhanna113]

+Looking Glass Universe I guess you have done one. Not fully. But it was a part in a video.

[@LookingGlassUniverse]

Yeah, I suppose in the Born's Rule video.

[@intellectelite]

Looking Glass Universe Complex numbers can be used to represent rotation and waves. Euler's identity can make the proof/derivation of trig identities much more smooth.

[@intellectelite]

Looking Glass Universe Conformal mappings!

[@ZardoDhieldor]

I don't like non-standard analysis (such a choice-infested area of mathematics *cough*) but luckily you can have infinitesimals without them thanks to differential forms! _Differential geometry to the rescue!_ :D

[@LookingGlassUniverse]

I have never hear of that! What is it? And what do you not like about nonstandard analysis? I'm really interested!

[@ZardoDhieldor]

Wait, you're a physicist and have never heard of differential forms?!? Please tell me you're joking! :D You have most likely already used them. It's a way to mathematically formalise infinitesimal line segments, areas and volumes and integrate over manifolds. They're written like _3x dx^dy_ for example, have you really not seen this before? :/ As about non-standard analysis: The existence of the hyperreals relies on the Axiom of Choice, a rather weird axiom of set theory which I try to avoid. It leads to some weird consequences like the Banach-Tarski paradox. Also there are a lot of results in mathematics where you use it to claim the existence of some object(s) you simply _cannot_ find explicitly. There was a huge debate about this in the mathematical community at the beginning of the twentieth century but nowadays it is widely accepted. I only found out about it by accident, really (because some proofs just felt weird), and now I'm bugging my fellow students about it like a loony! :D

[@rossjennings4755]

Hey, no math-shaming. A lot of what physicists do with differentials can be just as easily thought of as shorthand for limit arguments, something to do with non-standard analysis, or as differential forms. I agree that differential forms are pretty cool, though. There's some interesting conceptual discussions about what they are and how to put physics in terms of them on John Denker's website: www.av8n.com/physics/thermo-forms.htm, for example. I got a lot of my initial understanding of differential forms from those pages. Not sure exactly how I ended up there, though.

[@LookingGlassUniverse]

Haha! Sorry sorry, I have heard of them. I was expecting something completely different. But you're right, they do take the place of infinitesimals- I've only seen them in diff geo and GR, but I don't think I have a really intuitive understanding of them. But what don't you like about nonstandard analysis?

[@LookingGlassUniverse]

Oh cool! Thank you so much for sharing. I would love to understand them more. I really appreciate it!

[@markorfv252]

Numbers arent distances .. Without metric concept we should able to construct real numbers..

[@kevinbihari]

My head hurts

[@aceavian]

Awesome

[@benwincelberg9684]

But does every bounded set really have a supremum 🧐

[@reaganmaxwell9867]

Great Video; but, I do not know about this nick guy. He looks untrustworthy...

[@FlowerBoyWorld]

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