*Correction:* Several people (who clearly know more about computers than me) have informed me there is more than one type of transistor: Field Effect Transistors (FETs) and Bipolar Junction Transistors (BJTs). The ones I showed in this video are BJTs, whereas computers use FETs. Whoops! I stand corrected.
Another thing, you can simulate huge registers (as big as memory (including HDD) allows) with byte arrays. Of course, it has an overhead of additional computations (to process an array as a number) and the slower type of memory you use, the slower the calculations will go.
The computers currently on the market mostly use FETs. But is is absolutely possible to build a computer using BJTs. After all, there are still some examples of computers that used solenoid relais as switches. So the example isn't necessarily wrong, it's just a bit uncommon ;) O, if you want to know why I think that I'm correct: I'm an electronics engineer. My specialization is technical computer science (building computers, processors, peripherals and writing the software needed to make it work).
There were a number of computer architecture problems in the video. Additionally, to what was mentioned already (as far as I read) the notion of letting "the point float around the 64-bit register" is nonsense in the context of floating point numbers (that were explained correctly afterwards). Instead, that's actually called fixed-point arithmetic (because the exponent is not saved dynamically within the number). None of the errors were important to the question at hand... but it hurts nevertheless to see them in this pedantic channel ;)
Also, 1 and 0 are flows in opposite directions (or flows of electrons and "holes"), "no flow" (or disconnected, "High-Z", etc.) state is not normally used to transfer data.
This is where Plato's perfect forms comes in. He recognized that drawing a circle will always give an imperfect representation of this perfect form. So, if you accept Plato's idealism, and theory of perfect forms, pi does exist as an idea, if not manifested in the material world.
@@TheRosyCodex Of course pi can never be measure accurately in the material world because it's not a rational number. That's why it only exists as an ideal.
@@MGSchmahl If circles exist, then radius and circumference exist. Pi is just a relationship between radius and circumference, so it exists regardless of whether you can actually measure it exactly. Then we have to ask ourselves, "does geometry and math exist?".
Pi clearly exists, as an abstraction, regardless of whether we can represent it perfectly in the physical world. I _think_ it was 3 Blue 1 Brown who pointed out, much to my surprise, that, unlike a circle, there is no simple analytical expression for the circumference of a general ellipse. However, that’s not really surprising at all in hindsight, because the only reason why a simple, analytical expression for the circumference of a circle exists, is that pi has been _defined_ largely just for that purpose!
@@mr88cet -1 rocks?, 1/2 rocks to the exact atom? what even is a rock? if we had 5 rocks they are anyways not going to be identical so if we cut 1 rock in half 5=6? to solve these problems we cant use rocks we have to use abstract rocks, might as well use abstract circles
@@DendrocnideMoroides, 0 rocks is more abstract a concept than 5, -1 rocks is more abstract still, and “i” rocks even more abstract still. However, I’d say that half a rock is still just rock, just smaller rock.
@@mr88cet for your last point that is why I said "we can't use rocks we have to use abstract rocks" if we had an abstract rock where all of them are identical then half of it would be different than the full.
Perfect timing. I love that this one vid clears up a million computational concepts and with complete nomenclature/vocabularly . You are up there for me, a modern day carl sagan / neil degrasse tyson for the community of us so inclined ones
Old joke. First kid in a mountain family to go to school is asked what he learned. Boy: Pi R square. Father: That's stupid. Pie are round. CORNBREAD are square.
This question works the same way: what's your definition of definition? Language is able to work with abstract truth and physical truth. When physics asks you about existence, ofc it asks if it physically exists or not and not the otherwise.
The answer is so simple that we often fail to automatically grasp it. If something exists, it must participate in causation with other things in the universe and vice versa.
@@foxpup It is not simple at all, It's quite the opposite. For any definition of existence, if you search hard enough, you can find counterexample. There is actually no objective line between existence and nonexistence, so either we draw a subjective line, or stop thinking "black and white".
Just to make sure everyone is on the same page of the "infinity=-1/2" thing, it's wrong. Well obviously it's intuitively wrong, but that doesn't mean much in math, I'd recommend looking up the debunks of it if you're crazy enough for some infinite sums.
I love it. Exactly my point when (in the first class of a course on Numerical Methods and Statistics) I give my students an Oreo cookie (one to each, of course, that they can eat at the end) and a measuring tape, and tell them to determine Pi, experimentally. But there’s more. Even if space and time were to be continuous, and assuming General Relativity is true (or at least a good approximation to Reality), if you tried to measure a circle, you (or the lab, or space capsule, etc.) would be changing the curvature of spacetime, rendering the measure of Pi meaningless. Of course, I don’t go that far in the classroom, but tell them of a circle drawn on the surface of the Earth (or on a horse’s saddle) :D PS: I forgot that I'd already commented this video. Nevertheless, then I addressed other aspects of the "realness" of mathematical objects.
If you go to the atomic scale then there is not something like line in maths or any "perfect" shape at all, so if ideal cicrcle doesnt exist than line,cube,etc.. doesnt exist. Interesting how its all depends on your side of view, your glases... Thank you Nick for spending your time to help us learn something new, again!
Pi may be irrational, but it is precisely defined by several different recursive functions. If it didn't exist it couldn't be precisely defined. i^2=j^2=k^2=i*j*k=-k*j*j=-1, space is the complex part of a quaternion, and time is the real part. time and space are fundamentally incompatible due to that (-1)^(1/2) issue, and both are continuous, thus both relativity and quantum mechanics are incomplete at best.
I think THE pi is the ratio of circumference and diameter of an ideal circle. A perfect imaginary circle. Nothing is perfect in this world. That's why we get something like pi too as imperfect.
@@djkm9558 See the Wallis formulas basically they are series expansions of various transcendental functions then you can choose how far along the expansion you calculate
Does π exist? Yes. My favourite definition of whether something exists this: it's real if it's necessary in order to describe observations. π is necessary in many fundamental calculations, so it exists. You cannot draw a line exactly π metres long or count out π dollars, but it exists.
Why is it necessary? Some alien civilization could easily use something like ellipse close to a circle with specific alternative for pi as approximation of round objects and if it is close enough it would be pretty much the same, just some additional problems with formulas, maybe it is a part of their culture to not think about circles. We don't have circles in reality anyway. So, pi is not necessary, it is just easier to use. So, your criteria for existence is strange, limited by your imagination and nothing like colloquial meaning of the word.
@@dmitriy4708 first of all, Pi isn't about circles. It pops up in areas of maths that have nothing to do with geometry and it's fundamental to maths itself. For example, Euler's Identity: (e to the iπ +1 +0). Second, it's not my criteria for existence. It' the criteria that physicists use.
@@mjmulenga3 1) No, Euler's identity is such because of a circle in complex coordinates plane. And it is not only that, it is a periodic function, so you provided only one example of infinite amount of answers to this. Pi is always about circles even if it does not seem so. And physical existence is having coordinates in space-time, nothing more.
@@mjmulenga3 So, Euler's identity would be e^(pi*i) = e^(-2pi^2*k)*(-1), where k is an integer. Not so pretty, right? If k is 0 we have simple solution of -1 that you presented. Such are periodic functions. Whatever, do not think that something is necessary just because we chose to use it or even if you do not have any idea how can we do it otherwise. Math is just a tool we use, it could be different potentially.
I liked the explanation of this on the numberphile, the Tree Gaps and Orchard Problems. Insanely simplified version is that if you point any one direct the actual chance of you hitting anything is basicaly zero, and yet all around you you have "trees"(all the real numbers). And of all the gaps in that plnatation we know so little of those directions that lead to 'numbers' like Pi, e, root 2 and so forth.
"2" is also an infinite sequence (unless it's a counting number). It's the ratio of the length of one side of a square to the length of two sides. Neither rational nor irrational numbers actually exist - they are limits. The "exactness" of the rational numbers is hypothetical (I.e. imagination) (but, of course, indispensibly useful).
Nah it’s an integer. I think a better way to get at your argument is to argue pi itself is a different number system entirely. One system is the integers, another the real numbers, and then the “irrationals”. The irrationals can just be seen as their own system with a different base system. Similarly these irrationals extend out perpendicular to the “xy” plane with its imaginary counterparts. Pi is an inherit part of imaginary numbers, so I’d argue it’s simply another base of another “number” system. I don’t think number is a great way to view it though. It’s a concept we use for math, which can be represented by a number. Other fundamental concepts we do real math on are zero and infinity. You can’t write the number of infinity. It just exists. Same for pi
Very good question. I think, first if all, we have to define what existing means. In the physical world there are many ideal concepts that really do not exist, not only pi, but I think the point is that they are useful for our purposes of abstraction and prediction.
Exactly! What is the difference between Pi and say number 1? Does number 1 exist in the physical world? Of course not! Four apples are of course real, but the number itself is an abstraction.
Pi exists as a concept, but like many concepts in pure mathematics, they cannot be realized physically in reality. Others include infinity, infinitessimals, transfinite numbers, ... Sometimes even mathematicians don't always agree. At one point in history negative numbers were considered controversial, but they are generally accepted now.
Nick, I have made a few comments on your vids, but have always forgot to mention how entertained and informed I am watching them. You are a Marvel. No not the comic book hero, a real hero of science education. Can’t get enough. My brain doesn’t turn to mush when watching, it gets strong and stronger. At my age, sixty-eight, that is a good thing. By the way, how many out there have ever used a slide rule to solve a problem. Yeah, I am that old. I missed the abacus though. Keep on Keeping on, man.
A couple more corrections. For a non-zero floating point number in binary the first bit is always 1, so double precision fp numbers have 53 bits of info in the mantissa, not 52 (as long as they're not "denormalized" - the small numbers near zero are handled differently). "no pattern to the digits" is also wrong. If there were no pattern, we couldn't compute the digits. Maybe you meant that we can't predict the digits in isolation without calculating all of the preceding digits? True enough, in decimal, but in binary we have a bit extraction algorithm that lets us calculate any binary digit of pi without calculating the others. It's called a "bit extraction" algorithm. en.wikipedia.org/wiki/Approximations_of_π#Digit_extraction_methods
Numbers do not exist: they are adjectives to nouns, only what is named ("noun-ed") exists. Numbers have no names (nouns) because they are adjectives, so they only exist the same as pink, tall or much do.
@@LuisAldamiz Yes. Exactly. *Numbers* are the names of the properties by which objects are distinguished. "Does '3' exist?" is as meaningless/ill-posed a question as "Does 'many' exist?" or "Does 'yellow' exist?"
The way I understand it, this video is asking: "are there two lengths or distances of physical objects whose ratio is exactly and precisely to the infinitesimal decimal equal to pi?" For instance while the question of a number's existence is a philosophical matter, we know that we can have "2" quantities of a certain object. We know that if Paul has 1 apple and Mark has 2 apples, the ratio between these two quantities is exactly 2. But can we have something that is exactly "pi", or is anything that we calculate as being "pi" or a product of "pi" just an approximation? This is an interesting question, but unfortunately it's a question that is impossible to answer. You could make an argument that if two objects are four meters apart and they move to each others until they touch, there has to be a point in time when their distance is exactly "pi" meters. But how can you prove it? It all comes down, as the video concludes, to whether space is continuous or discrete.
Everything you said applies to the length of a straight line or to a simpler fraction like 1/2. You can measure them more and more accurately but you'll never be exactly right because the atoms won't be perfectly positioned with respect to the two ends, except in very unusual circumstances. Ultimately we come to subatomic particles and then on to quantum mechanics and the probability of a particular position which won't stay still. With Pi the calculation is complicated by the deviation of the line resulting in us needing to approximate it in ever more accurate assessments but still the real issue is that we can't measure reality perfectly. If we discovered a way to measure the circumference of a circle as easily and accurately as the diameter, we'd no longer have an endlessly more and more accurate measure of Pi, we'd come to a conclusion, but we'd still not ever be 100% accurate. So I don't see the problem. Even 1/3 doesn't quite exist expressed in decimal, because you can't stop writing the decimal 3s. It's no solution to say 'repeating'... No one can measure that far. So all in all the fact is that there's noting special about Pi. What we have is a roughness in our ability to measure the universe.
Let's bring this conclusion to the end. Because if we take quantum fuzziness into consideration, the finle question is: Does ANY number exist (not limited to Pi or Root2). Because if we are unable to draw a perfect circle, we are also not able to draw a perfect line and also not a line with length 1 (or even if you define a given length as length 1 you will never be able to draw a second line with the same lenght and also never a perfect right angle to construct Root2). So in the real phyisical world, according to your method, NO numbers do exist. Great content, really enjoy your videos!
First question should be: "what do I mean by "exist". Unless you are proposing fringe geometric realism (or whatever that philosophy is called - I'm sure it has a specific name, sorry), Pi exists as a clearly definable concept, but concepts are not composed of matter or energy (they are constrained by c though, :)
they represent something, like a number on the number line,geometrical proportions and sizes, or logic itself.. objects, and they are just abstract it's also a language, but they are supposed to be this, and themselves whatever they might mean,positions from an undefined point namely 0 i guess which can be anywhere you go, so it's well defined,and there are the axioms and proofs(in real life too), this is what math is about and problem solving... however they are still abstract, and keep in mind that our brains do the math too so it would be a mental representation,but what isn't a mental representation actually?... anything is, and most of stuff around us is empty space and how our brain perceives it from it's huge scale. the ultimate question would be mind over matter.. is the circle actually valid even if it's just too perfect so that could be just an illusion of our brains?
Another insight about π (not my original idea to be clear, but something I'm finally starting to understand after multiple online discussions for a few different people): π isn't fundamentally about circles anymore than e is fundamentally about compound interest -- pi is actually more about _rotation_. Mathematicians define it as half of the imaginary part of the period of the exponential function, which is basically a fancy way of saying it's half the period of sin(x). It shows up in circles because sine is a cyclical function -- if you look at it's polar graph, it's just a circle of radius 1. But sine, cosine, exponentials, and complex numbers are all ways of representing rotations. This is summed up beautifully in Euler's formula, e^(ix)=cos(x)+isin(x). We can use this to represent an 2d rotation we like by changing the value of x. (We can even extend it to 3+ dimensions using homogeneous coordinates, which is really cool!) Since this is a complex function, it has 2 pieces, both of which have a period of 2π (which really makes me wish we used τ=2π instead, but we're stuck with π now). So if we plug in π for x we get the famous identity, e^(πi)=-1. This is really just another way of representing the same transformation given by i²=-1, and it's because there are 2π radians in a circle. The x value in e^(ix) tells you the angle (in radians) to rotate by, starting at (1,0) on the unit circle. π is just the number that encodes a 180° rotation from 1 to -1 on the unit circle. There's the circle definition again -- divide a circle's circumference by it's diameter and you've got the angle that rotates you around a semi-circle. But fundamentally, the rotation is what matters more. Anyone have any thoughts?
People often ask if pi exists which is both a valid and interesting idea to ponder, but I think that people tend to pick this number instead of a number like 5 because the idea of "fiveness" has a different feeling than the idea of "pi-ness". The number 5 is more commonplace to every day experiences. Every day we count things. We encounter collections of 5 things all the time. The number pi on the other hand while still useful is not as common to every day experiences unless you happen to be a mathematician or do measurements of circlular objects and their geometric properties on a regular basis. There is also the question of what is meant by a number existing. So my question is this: Does any number, be it 5, pi, -17, 2+5i, etc exist?
I’ve thought about that and have am come to the conclusion that only 1 is real because although there can be five apples, they all represent different places in spacetime. It gets a lil sticky if we say that 0 is also a number because… there is either nothing or something… but then that’s “two” things which in turn makes three things and so on… what was the question?
This video is great since it explores an idea that I wish got more philosophical scrutiny. Like pointed out near the end, the quantization implied by QM means that any ideal notion of the real number line (i.e., that which is encountered in a course on analysis) cannot have a physical implementation, since something continuous cannot be embedded into a quantized structure (I think, seems like a reasonable inference). There would be gaps. In fact, in such a world as implied by QM, can anything even be continuous? Motion and temperature fluctuations only seem to be continuous, as they are just a many of very little quanta lumps of certain things. But anyway, in such a world, things like e and pi might not be physically real (since they are irrational), and so the equations that we have describing reality that use those constants would, in fact, be describing a reality that is not *exactly* like ours (one is an idealized continuous reality, the other discrete). Would that make anything in a discrete reality just approximations, albeit very good ones, to an idealized, continuous reality? I mean, who really cares practically, since the approximations are very good anyway, but don't you want to know what IS?
Aren't all quantum things waves according to a recent Science Asylum video? Waves themselves are continuous even if we can measure them as discrete objects. Which means quantum is really about energy instead of objects. In that case maybe space is continuous but vacuum energy is discrete.
Unless you believe in a real Platonic 'world of forms', it is not possible to even imagine a perfect circle, since that would be a product of a quantised brain.
This philosophy you are describing dates to the 1890's, when Mathematics started to be tied down accurately enough to allow detailed scrutiny. The debate fell effectively into the constuctivist camp (you should only ever use mathematical concepts that can be constructed; non-constructivist proofs such as reducto ad absurdum should be avoided) and the idealist camp, who felt that mathematics would be too constrained by this requirement. At the time we understood it as an ontological (existance) argument (along the lines you present it here). It is really with Goedel that we understand mathematics as a branch of logic. The problem nowadays is understood as a problem of epistemology (knowledge) rather than ontology. So perhaps a better way to look at it is that both classical mechanics and quantum mechanics are encoded in quite tight logic, with logical rules that allow experimental setup and experimental outcome to be related to each other. There is a beautiful book by Errett Bishop that derives all of calculus, all the way into the Hilbert Spaces you need to construct what we understand of quantum mechanics in a constructivist fashion. The upshot of this is that you can always represent the quantum mechanics of the experimental setup to the accuracy you need, as long as you have a big enough Turing machine (viz a vie Computer). So it is not the mathematical representation that stymies us, but our cleverness in crunching the numbers.
Dear Nick, Its always a great pleasure to learn and watch your videos. You're really accurate within your explanations and you do not show off because of your knowledge. thanks for that !
Hoi Nick, About the question you answered at the end about the spinning of black holes. To me the real question is: why did there rise forces if something accelerates? Are those forces still there in an empty universe? Can you accelerate in an empty universe. Is acceleration always with respect to other bodies? And why does one feel a force while the other doesn't? How do you know who is actually accelerating? Questions Ernst Mach asked long before. Thanks for your very fun and educating video's! I'm a physics teacher from the Netherlands and this is by far my favourite TH-cam channel. Kees Stoop
As far as I know, binary digits 1 and 0 do not correspond to electricity present / absent. They are represented by different levels of voltage, called "high" and "low". In a digital circuit, when no electricity is present, the device, or at least part of the circuit, is unpowered which is kind of the same as "undefined".
Q: How precise is the number 1 since all the measurements of any standard unit vary across space and time? A: It depends on how "unit" is defined. Every definition is built of concepts and those concepts are based on human experience which is fallible on every level. The important thing to remember is that errors are often random and randomness comes from nature. Humans make mistakes though and nature even in it's randomness does not. In fact randomness is one of the surest ways to derive an absolute number since a truly random number will approximate over time to 1/2. So there you have a basis for the number 2 even before you've defined one or zero.
This is an excellent video! I would definitely say space is fundamentally discrete It follows on to another interesting point about mathematics and its relationship to the universe I think. We have many laws, fundamental constants etc, that would appear to be in place (by definition) before the big bang. This therefore gives the impression that they somehow 'exist' outside of the universe... which isn't a thing imo. The point I am trying to make (badly) is that if all the laws/constants/numbers are not emergent - then none exist! and if they are emergent, then they cannot be constant or unchanging because they rely on the state of the universe Nice puns btw!
@Tim Erskine; Rob Bryanton talks about the fine structure of the universe being essentially arbitrary and created at the moment of the big bang in some of his "Imagining the 10th Dimension" videos.
*technically*, you are wrong when you say that current either flows or doesn't flow in the transistors of a computer. There will always be leakage currents that depend on many factors (as a result of using p- and n-doped semiconductors), but we define arbitrary thresholds we think are good enough to be called a 1 or a 0. Most don't know that there is 3rd state known as the high impedance state or z state. This is even useful in some applications like data buffers for high speed busses in computing.
@@ScienceAsylum Haha I intended it more as a tongue in cheek joke about pedantry (I do love it though xP) but I'm glad you actually thought it to be constructive! I want to thank you by the way for your videos, they really are great! My fav scientist on youtube :3
@@manoo422 Definitely not always true, entirely depends on design, materials, manufacturing and even environmental conditions like temperature. If you have high frequency noise in your signal line to the gate of the transistor, you could induce this noise to your output that will begin to bounce and float between Vdd (or Vcc) (1) and ground (0). This isn't even the aforementioned high impedance state, where you isolate the output from both Vdd and ground leaving it floating. Design considerations need to be made when designing circuitry to avoid capacitive coupling and line inductances to not influence the function of the circuit. Hell, if you have sensitive enough stuff, a crappy, noisy switching mode power supply could induce severe high frequency noise through your Vdd to pretty much everything in your circuit and entirely muck up anything digital you're trying to do. Don't take the equipment you use for granted, a lot of careful compromises have been made to design it.
Oh my god, I just realized something. I was playing chess with an Australian friend of mine. He kept saying "Checkmate" even though I had an escape from check. I think he might have been saying "check, mate." not "checkmate."
Mother Nature continues to boggle our human curiosity... presenting us with a seemingly innocuous relation between the circumference and the diameter of a circle and then... revealing a surprisingly endless non-repeating pattern of decimal numbers for its numerical approximation. This is Mother Nature at its best - refusing to be pigeonholed by human understanding... Fundamental reality is much much more robust and magical than we can ever imagine... despite our blind faith in mathematical precision which at best is still just an approximation!!!
An excellent, as always, piece of teaching but I have some remarks. If pi, or indeed _any_ numbers exist, they could not be physical objects. They mould be abstract objects. Why? If the number 3 exists, it would be of no use asking _where_ it exists. It wouldn't exist anywhere. And there is only _one_ number three. The number three doesn't exist in each trio of objects. We simply use the one and only number three to count them as a trio in the first place. It, and its fellow numbers, would have their own special non-physical reality. These issues are what makes 'Do numbers exist?' a philosophical question. No observation or measurement can ever tell us, for measurement makes use of numbers, so if numbers are not real, then measurement uses something that doesn't exist. I suspect a better opening question might be: 'What sort of thing is a number?' 'Abstract objects' is one answer. There have been others.
I really enjoyed this video. I work in water treatment & pie is utilized daily. this video helped answer abstract questions my professors could not. I would love to see you tackle some water math
I think it depends on whether you are an idealist like Plato. In which case pi and other abstract forms are what are real, and our world is full of imperfect replications of them. For materialists, the physical world comes first and notions like numbers and pi are just human constructs to make sense of the physical world.
A very bright scientist once said he did not believe in irrational numbers because it would take an infinite amount of information to describe it, and the universe has finite information.
Pi is a ratio of two numbers, not a number itself, hence why in mathematics it is not expressed generally as a number but as the greek letter pi. This is the same with many other constants in mathematics, for example the speed of light, which itself is not a number, but can only be quantifiable based on two values, time and distance, both of which can be any number of units of measure.
I always accepted that pi was something abstract and can't really exist in a medium in any perfect form, and thus why I was content with pi being something purely theoretical to calculate theoretical concepts to help us understand the more real physical concepts of the world. *so why the hell does x^2 + y^2 = r^2 graph a circle* I mean I know logically _why_ it works, but... _it physically exists_, and I don't know how to cope with it, and now I'm here.
I would have liked to have seen a discussion of Plato's theory of Forms included since some thinkers, including Roger Penrose, think that numbers might exist in that realm. Apart from that, I enjoyed the presentation
After 5 shots of vodka and a cigarette, I realized that a circle cannot be quantized , because pie seems to be a never ending number that doesn’t seem to exist, so I went looking for TH-cam videos that explains this , this is the best one that I have found 👍🏻
It's not necessary to have infinitely small precision for pi to have physical meaning. Take a circle with a diameter of 10^1000000 meters. Its circumference uses 1000000 digits of pi if you measure up to a meter, so no need of quantum mechanics here
@@holomurphy22, get a copy of Maple (even a twenty years old version can do it), and you can ask for as many digits of Pi, as the computer memory (adjusted for internal representation of numbers in Maple) allows. This is the easy way. Alternatively, use a specialised program to compute as many as you want ...
@@holomurphy22, you're right, I missed the part about "physical meaning". My fault. I interpreted your answer as pertaining to the video's first theme: calculating Pi experimentally by drawing a circle. Nevertheless, giving "physical meaning" to the first 1000000 digits of Pi is not the issue here; for that matter, it does not differ from giving "physical meaning" to the first 1 or 2 first digits of Pi
@@MiguelGarcia-zx1qj I answered to the comment above of Charlie. I dont see how its the same to give physical meaning to 2 digits and 10000... digits. There are measure limits.
Yeah I'm starting to think pi isn't a number but maybe like a space of potentiality from which you could pull numbers to as much precision as you want. So 3.14 is a number, 3.1416 is a number, etc., just pick the one you need, call it "pi" for your purposes.
on that note about computers the 64bit float is just a standard, you can definitely use your own specialized number system on your computer to calculate pi. I mean you aren't going to get all of pi but you reveal more of ip the longer you let it run for.
For more random pedantry: Pi Day (as in the date of 3/14) also exists in much of Asia, as we use the YMD calendar. So "March 14th" would still be "3-14" for us, even if our full dates are written like 2019-03-14 versus 3/14/2019. So we're still with you MDY bros in celebrating Pi Day! It's those weirdos who write their dates like 14/3 you should be concerned about trying to ruin your fun with their lamer 22/7 Pi Day...
Pi exists but only as a tool. I come from an engineering background, and engineers (who have to deal with the real world) learn early that everything is an approximation. Wheels, bearings, gears, etc. aren't round... They are approximately round... round enough to work is "round".
Q: Are space & time continuous or discrete? A: Both. These 2 concepts are not mutually exclusive as you accept infinite resolution of a discrete system.
Nice explanations. The puns are painful, and what's with the seemingly random pictures? walnuts etc. I am now going in search of a few more pi related videos. I also just subscribed. :)
Logic and mathematics are just tools we invented to explain our reality. Their accuracy/validity of these things is directly tied to how well they explain our reality within the framework we have constructed. There is a great video on this sort of thing somewhere on TH-cam but for the life of me I cannot remember the title.
The longer this goes the more philosophical it feels versus how scientific it feels. PI goes on forever. The further it goes, the more it refines our circle. At some point in the real world the addition of more refinement would be meaningless in the real world. But a concept it real and in this case extends beyond the practicality of the physical and into the metaphysical. And that makes it beautiful. "Something there is in beauty which grows in the soul of the beholder, like a flower: fragile - - For many are the blights which may waste the beauty. For the beholder-- and imperishable - - for the beauty may die, or the world may die, but the soul in which the flower grows survives. " S. R. Donaldson
That ‘strangeness’ you’re picking up is Nick, I think. I just found this channel a couple of days ago and he is (imo) one of the best in the business. I like John and Hank Green too ... but this dude is extra-hilarious, making for, I think, a better entertainment and learning experience.
The 64-bit hardware architecture is NOT the floating point precision, it just happens to coincide with "Double" precision which is most common in scientific computing. "Float" (32-bit), "double" (64-bit) and "long double" (128-bit) can be stored/computed on any architecture. The architecture is only for memory addresses, which is why 32-bit hardware had a 4GB memory limitation.
The craziest thing about Pi to me is that it contains every possible configuration of any string of numbers of any length. Every string of digits of every length. Meaning Pi contains itself. And not just as a string from 3.14 to infinity. It contains that string plus 1. It contains that string plus that string on either side of itself forwards and backwards. Meaning that pi doesn’t just contain itself, it contains itself infinitely many times. But here’s the weird thing. If Pi contains itself, that means there’s a point at which Pi will appear to repeat, because if it contains itself, it has to have a string that starts 314159 and continues on for as many digits as we could possibly calculate. And we will never, ever find it. It is absolutely there within itself, but it cannot be teased out, because then it would be repeating a finite number back to you.
What do you by exist? Would the absence of its appearance in the strictest attempt to “find” it in real world applications necessarily mean it doesn’t exist? How do you know pi goes on forever? Can we even know whether pi does or does not exist, or can we merely give our best hypothesis?
I think it’s safe to say pi exists once you stop seeing it as a component of circles and instead one of distribution from a given thing. It pops up in tons of places without circles, and I personally just see it as the “fundamental unit” of any function. Any function can be represented by infinite Fourier series and pi itself is infinite. While QM argues infinitesimals don’t exist, that’s just some arbitrary meaning. You can zoom out to infinity and then your resolution is effectively infinity. Pi is not a “number” but rather a consequence of infinities and imaginary numbers that come from the representation of information we call math. Pi isn’t the only trancendental number, it’s just one of the most fundamental ones. I’d personally argue, however, that it’s just as important as Euler number since they cannot exist without each other when dealing with imaginary numbers.
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*Correction:* Several people (who clearly know more about computers than me) have informed me there is more than one type of transistor: Field Effect Transistors (FETs) and Bipolar Junction Transistors (BJTs). The ones I showed in this video are BJTs, whereas computers use FETs. Whoops! I stand corrected.
Another thing, you can simulate huge registers (as big as memory (including HDD) allows) with byte arrays.
Of course, it has an overhead of additional computations (to process an array as a number) and the slower type of memory you use, the slower the calculations will go.
But overall it doesn't make any difference since we are talking about infinities.
The computers currently on the market mostly use FETs. But is is absolutely possible to build a computer using BJTs. After all, there are still some examples of computers that used solenoid relais as switches. So the example isn't necessarily wrong, it's just a bit uncommon ;)
O, if you want to know why I think that I'm correct: I'm an electronics engineer. My specialization is technical computer science (building computers, processors, peripherals and writing the software needed to make it work).
There were a number of computer architecture problems in the video. Additionally, to what was mentioned already (as far as I read) the notion of letting "the point float around the 64-bit register" is nonsense in the context of floating point numbers (that were explained correctly afterwards). Instead, that's actually called fixed-point arithmetic (because the exponent is not saved dynamically within the number). None of the errors were important to the question at hand... but it hurts nevertheless to see them in this pedantic channel ;)
Also, 1 and 0 are flows in opposite directions (or flows of electrons and "holes"), "no flow" (or disconnected, "High-Z", etc.) state is not normally used to transfer data.
I came for the fun math, I stayed for the existential crisis.
Lol !
I came for the existential crisis, I stayed for the fun math.
I knew the video was gonna be good when this was the first comment I saw
Every-fucking-time
Swap the words "fun math" with Jack Daniel's, and you have an accurate description of my 40th birthday and the following day
This is where Plato's perfect forms comes in. He recognized that drawing a circle will always give an imperfect representation of this perfect form. So, if you accept Plato's idealism, and theory of perfect forms, pi does exist as an idea, if not manifested in the material world.
Actually no, even in a world of pure math, the ratio between a circle's area/circumference and it's radius is an incommensurable proportion
@@TheRosyCodex Of course pi can never be measure accurately in the material world because it's not a rational number. That's why it only exists as an ideal.
@@prschuster so you shifted the conversation from talking about pi to talking about what it means to exist itself... perfect!!!
It all depends on what it means to "exist".
@@MGSchmahl If circles exist, then radius and circumference exist. Pi is just a relationship between radius and circumference, so it exists regardless of whether you can actually measure it exactly. Then we have to ask ourselves, "does geometry and math exist?".
Pi clearly exists, as an abstraction, regardless of whether we can represent it perfectly in the physical world.
I _think_ it was 3 Blue 1 Brown who pointed out, much to my surprise, that, unlike a circle, there is no simple analytical expression for the circumference of a general ellipse.
However, that’s not really surprising at all in hindsight, because the only reason why a simple, analytical expression for the circumference of a circle exists, is that pi has been _defined_ largely just for that purpose!
I mean, ultimately, if we're talking about physical existence, then no number exists because all numbers are abstract concepts.
@@BerryTheBnnuy, absolutely true. However, the “5” in “5 rocks” is a little easier to visualize than pi.
@@mr88cet -1 rocks?, 1/2 rocks to the exact atom? what even is a rock? if we had 5 rocks they are anyways not going to be identical so if we cut 1 rock in half 5=6?
to solve these problems we cant use rocks we have to use abstract rocks, might as well use abstract circles
@@DendrocnideMoroides, 0 rocks is more abstract a concept than 5, -1 rocks is more abstract still, and “i” rocks even more abstract still. However, I’d say that half a rock is still just rock, just smaller rock.
@@mr88cet for your last point that is why I said "we can't use rocks we have to use abstract rocks" if we had an abstract rock where all of them are identical then half of it would be different than the full.
Perfect timing. I love that this one vid clears up a million computational concepts and with complete nomenclature/vocabularly . You are up there for me, a modern day carl sagan / neil degrasse tyson for the community of us so inclined ones
Neil deGrasse Tyson exists in the modern day.
Pi used to exist, but then I got hungry
@@kellyjackson7889 that's a typical american pie
Nah it's still in your poop but I know it's a joke
Old joke. First kid in a mountain family to go to school is asked what he learned.
Boy: Pi R square.
Father: That's stupid. Pie are round. CORNBREAD are square.
Fuck 😂
He really milked those puns.
What is your definition of "exist" ?
Something that is
@@arvidsalle2979 everything is when we say it is
This question works the same way: what's your definition of definition?
Language is able to work with abstract truth and physical truth. When physics asks you about existence, ofc it asks if it physically exists or not and not the otherwise.
The answer is so simple that we often fail to automatically grasp it. If something exists, it must participate in causation with other things in the universe and vice versa.
@@foxpup It is not simple at all, It's quite the opposite. For any definition of existence, if you search hard enough, you can find counterexample.
There is actually no objective line between existence and nonexistence, so either we draw a subjective line, or stop thinking "black and white".
-1/12 ...I see what you did there.
Hmm!! Yup I didn't notice but now I realize.. smart
What did he do exactly? Im stupid
@@saatviksingh -1/12 comes in Ramanujan summation of natural numbers.
To be exact, the Riemann function of -1 is -1/12
Okay let me make it clearer, 1+2+3+4+5+6+7...∞= -1/12. You can check for sum of all natural numbers, on the net
Just to make sure everyone is on the same page of the "infinity=-1/2" thing, it's wrong. Well obviously it's intuitively wrong, but that doesn't mean much in math, I'd recommend looking up the debunks of it if you're crazy enough for some infinite sums.
I love it. Exactly my point when (in the first class of a course on Numerical Methods and Statistics) I give my students an Oreo cookie (one to each, of course, that they can eat at the end) and a measuring tape, and tell them to determine Pi, experimentally.
But there’s more. Even if space and time were to be continuous, and assuming General Relativity is true (or at least a good approximation to Reality), if you tried to measure a circle, you (or the lab, or space capsule, etc.) would be changing the curvature of spacetime, rendering the measure of Pi meaningless. Of course, I don’t go that far in the classroom, but tell them of a circle drawn on the surface of the Earth (or on a horse’s saddle) :D
PS: I forgot that I'd already commented this video. Nevertheless, then I addressed other aspects of the "realness" of mathematical objects.
There's nothing wrong with commenting more than once, as long as you're not spamming my comment section 👍
I wouldn't mind betting space & time are in fact quantized, we just haven't figured it out yet.
The concept of Pi transcending algebra sounds totally cool!
If you go to the atomic scale then there is not something like line in maths or any "perfect" shape at all, so if ideal cicrcle doesnt exist than line,cube,etc.. doesnt exist. Interesting how its all depends on your side of view, your glases... Thank you Nick for spending your time to help us learn something new, again!
what about a circle carved from a sphere using inverse square law
Until you remember that even then this mathematics can describe reality so well
If you go down far enough then nothing exists, so to ask if something "exists" is already redundant in itself
Do any of us exist as anything other than legends in our own minds?
Inorganic chemistry is pretty reliant on geometric shapes, so I feel quite certain tetrahedrons and hexagons exist.
I'm a simple guy, when Nick uploads, I click, I like, I share
I'm a simple guy, and I do same think I like you
The puns were crazy. I'm surprised he didn't put a picture of a pie on the screen every thyme he said pi. :D
🥧🥧🥧🥧🥧
I sea what you did their!
(And don't forget the parsley, sage, & rosemary!)
Fred
Yeah, but then ... we all know, that pie exists. Until we eat it.
exist should be egg-sist 🥚🥚🥚
This guy is amazing. He dont tell you what to think or how to think but but makes you think.
Pi may be irrational, but it is precisely defined by several different recursive functions. If it didn't exist it couldn't be precisely defined.
i^2=j^2=k^2=i*j*k=-k*j*j=-1, space is the complex part of a quaternion, and time is the real part. time and space are fundamentally incompatible due to that (-1)^(1/2) issue, and both are continuous, thus both relativity and quantum mechanics are incomplete at best.
What’s really amazing is 314 spells PIE in a mirror.
illuminati confirmed
Ok .. that is Zen AF .. write it on paper and flip it around in the light .. LOL
Whoah, trippendicular.
No it doesn't?
@@tobyinsley9010 " 4" looks like " P" .. "1" looks like "I" and "3" backwards looks like " E" . you need to use your imagination , a tiny bit
I think THE pi is the ratio of circumference and diameter of an ideal circle. A perfect imaginary circle. Nothing is perfect in this world. That's why we get something like pi too as imperfect.
If its the ratio of the diameter and the circumference of an imaginary circle, how did they compute a trillion digits of pi?
@@djkm9558 See the Wallis formulas
basically they are series expansions of various transcendental functions
then you can choose how far along the expansion you calculate
Wellllll.......Not So Faaast!!!😂😂😂
Yes. In summary PI is not a number. it is a ratio. Problem solved. The question was wrong
Andrew Hart // Ratios aren’t numbers? What about 2/3?
Does π exist? Yes.
My favourite definition of whether something exists this: it's real if it's necessary in order to describe observations.
π is necessary in many fundamental calculations, so it exists. You cannot draw a line exactly π metres long or count out π dollars, but it exists.
Why is it necessary? Some alien civilization could easily use something like ellipse close to a circle with specific alternative for pi as approximation of round objects and if it is close enough it would be pretty much the same, just some additional problems with formulas, maybe it is a part of their culture to not think about circles. We don't have circles in reality anyway. So, pi is not necessary, it is just easier to use. So, your criteria for existence is strange, limited by your imagination and nothing like colloquial meaning of the word.
@@dmitriy4708 first of all, Pi isn't about circles. It pops up in areas of maths that have nothing to do with geometry and it's fundamental to maths itself. For example, Euler's Identity: (e to the iπ +1 +0).
Second, it's not my criteria for existence. It' the criteria that physicists use.
@@mjmulenga3 1) No, Euler's identity is such because of a circle in complex coordinates plane. And it is not only that, it is a periodic function, so you provided only one example of infinite amount of answers to this. Pi is always about circles even if it does not seem so. And physical existence is having coordinates in space-time, nothing more.
@@mjmulenga3 So, Euler's identity would be e^(pi*i) = e^(-2pi^2*k)*(-1), where k is an integer. Not so pretty, right? If k is 0 we have simple solution of -1 that you presented. Such are periodic functions. Whatever, do not think that something is necessary just because we chose to use it or even if you do not have any idea how can we do it otherwise. Math is just a tool we use, it could be different potentially.
Resorting to practical necessity is a very bad way to justify the existence of a pure mathematical object.
I liked the explanation of this on the numberphile, the Tree Gaps and Orchard Problems. Insanely simplified version is that if you point any one direct the actual chance of you hitting anything is basicaly zero, and yet all around you you have "trees"(all the real numbers). And of all the gaps in that plnatation we know so little of those directions that lead to 'numbers' like Pi, e, root 2 and so forth.
"2" is also an infinite sequence (unless it's a counting number). It's the ratio of the length of one side of a square to the length of two sides. Neither rational nor irrational numbers actually exist - they are limits. The "exactness" of the rational numbers is hypothetical (I.e. imagination) (but, of course, indispensibly useful).
mann ! Now that is a pov!
Nah it’s an integer. I think a better way to get at your argument is to argue pi itself is a different number system entirely. One system is the integers, another the real numbers, and then the “irrationals”. The irrationals can just be seen as their own system with a different base system. Similarly these irrationals extend out perpendicular to the “xy” plane with its imaginary counterparts. Pi is an inherit part of imaginary numbers, so I’d argue it’s simply another base of another “number” system. I don’t think number is a great way to view it though. It’s a concept we use for math, which can be represented by a number. Other fundamental concepts we do real math on are zero and infinity. You can’t write the number of infinity. It just exists. Same for pi
@@ptrkmr In plane geometry there are almost no numbers, and we assume that some exact shapes and relationships exist. Do they actually exist?
Very good question. I think, first if all, we have to define what existing means. In the physical world there are many ideal concepts that really do not exist, not only pi, but I think the point is that they are useful for our purposes of abstraction and prediction.
Exactly! What is the difference between Pi and say number 1? Does number 1 exist in the physical world? Of course not! Four apples are of course real, but the number itself is an abstraction.
Pi exists as a concept, but like many concepts in pure mathematics, they cannot be realized physically in reality. Others include infinity, infinitessimals, transfinite numbers, ... Sometimes even mathematicians don't always agree. At one point in history negative numbers were considered controversial, but they are generally accepted now.
This is true. We're using numbers or math to describe things. Working fine to a certain degree.
Congrats on your 2^7 * 10^3 subs
2^10*5^3
50³ + 3·10³
Fred
Out!! Just...get out, all of you!!!
AKA 128,000 subs
@@SoI- so ~ (1/2)e*10^5
Nick, I have made a few comments on your vids, but have always forgot to mention how entertained and informed I am watching them. You are a Marvel. No not the comic book hero, a real hero of science education. Can’t get enough. My brain doesn’t turn to mush when watching, it gets strong and stronger. At my age, sixty-eight, that is a good thing. By the way, how many out there have ever used a slide rule to solve a problem. Yeah, I am that old. I missed the abacus though. Keep on Keeping on, man.
A couple more corrections. For a non-zero floating point number in binary the first bit is always 1, so double precision fp numbers have 53 bits of info in the mantissa, not 52 (as long as they're not "denormalized" - the small numbers near zero are handled differently).
"no pattern to the digits" is also wrong. If there were no pattern, we couldn't compute the digits. Maybe you meant that we can't predict the digits in isolation without calculating all of the preceding digits? True enough, in decimal, but in binary we have a bit extraction algorithm that lets us calculate any binary digit of pi without calculating the others. It's called a "bit extraction" algorithm. en.wikipedia.org/wiki/Approximations_of_π#Digit_extraction_methods
I agree that we need to start with definitions. But you missed the most important definition of all: "What does it mean to say a number 'exists'?"
Numbers do not exist: they are adjectives to nouns, only what is named ("noun-ed") exists. Numbers have no names (nouns) because they are adjectives, so they only exist the same as pink, tall or much do.
@@LuisAldamiz
Yes. Exactly. *Numbers* are the names of the properties by which objects are distinguished.
"Does '3' exist?" is as meaningless/ill-posed a question as "Does 'many' exist?" or "Does 'yellow' exist?"
Physically exists*
The way I understand it, this video is asking: "are there two lengths or distances of physical objects whose ratio is exactly and precisely to the infinitesimal decimal equal to pi?"
For instance while the question of a number's existence is a philosophical matter, we know that we can have "2" quantities of a certain object. We know that if Paul has 1 apple and Mark has 2 apples, the ratio between these two quantities is exactly 2. But can we have something that is exactly "pi", or is anything that we calculate as being "pi" or a product of "pi" just an approximation?
This is an interesting question, but unfortunately it's a question that is impossible to answer. You could make an argument that if two objects are four meters apart and they move to each others until they touch, there has to be a point in time when their distance is exactly "pi" meters. But how can you prove it? It all comes down, as the video concludes, to whether space is continuous or discrete.
Good point!
Everything you said applies to the length of a straight line or to a simpler fraction like 1/2. You can measure them more and more accurately but you'll never be exactly right because the atoms won't be perfectly positioned with respect to the two ends, except in very unusual circumstances. Ultimately we come to subatomic particles and then on to quantum mechanics and the probability of a particular position which won't stay still.
With Pi the calculation is complicated by the deviation of the line resulting in us needing to approximate it in ever more accurate assessments but still the real issue is that we can't measure reality perfectly. If we discovered a way to measure the circumference of a circle as easily and accurately as the diameter, we'd no longer have an endlessly more and more accurate measure of Pi, we'd come to a conclusion, but we'd still not ever be 100% accurate.
So I don't see the problem. Even 1/3 doesn't quite exist expressed in decimal, because you can't stop writing the decimal 3s. It's no solution to say 'repeating'... No one can measure that far.
So all in all the fact is that there's noting special about Pi.
What we have is a roughness in our ability to measure the universe.
At 2:23 I almost had a myocardial infarction by laughing so hard that I was unable to inspire for several minutes.
Same here!
Let's bring this conclusion to the end. Because if we take quantum fuzziness into consideration, the finle question is: Does ANY number exist (not limited to Pi or Root2). Because if we are unable to draw a perfect circle, we are also not able to draw a perfect line and also not a line with length 1 (or even if you define a given length as length 1 you will never be able to draw a second line with the same lenght and also never a perfect right angle to construct Root2). So in the real phyisical world, according to your method, NO numbers do exist. Great content, really enjoy your videos!
First question should be: "what do I mean by "exist".
Unless you are proposing fringe geometric realism (or whatever that philosophy is called - I'm sure it has a specific name, sorry), Pi exists as a clearly definable concept, but concepts are not composed of matter or energy (they are constrained by c though, :)
my five one of the most favourite channels,
3Blue1Brown, VSauce , Science Asylum, Veratasium, The Physics Girl
I find mention of them here 🙂🙂🙂
Looks like I'm in good company 🤓
Awesome video connecting a lot of things you don't see together very often
The first lemma should obviously be: What does it mean for a number to exist?
cheers!
they represent something, like a number on the number line,geometrical proportions and sizes, or logic itself.. objects, and they are just abstract it's also a language, but they are supposed to be this, and themselves whatever they might mean,positions from an undefined point namely 0 i guess which can be anywhere you go, so it's well defined,and there are the axioms and proofs(in real life too), this is what math is about and problem solving... however they are still abstract, and keep in mind that our brains do the math too so it would be a mental representation,but what isn't a mental representation actually?... anything is, and most of stuff around us is empty space and how our brain perceives it from it's huge scale. the ultimate question would be mind over matter.. is the circle actually valid even if it's just too perfect so that could be just an illusion of our brains?
A=1, B=256, C=1.24, D=8.643586, Pi =3.14?????? Get it?
@@mishagjata7374 C*D/2 would give you something close to pi
@@RazorM97 Funny numbers :)
"does pi actually exist?"
Mr.crazy: quantum mechanics.
Another insight about π (not my original idea to be clear, but something I'm finally starting to understand after multiple online discussions for a few different people): π isn't fundamentally about circles anymore than e is fundamentally about compound interest -- pi is actually more about _rotation_. Mathematicians define it as half of the imaginary part of the period of the exponential function, which is basically a fancy way of saying it's half the period of sin(x). It shows up in circles because sine is a cyclical function -- if you look at it's polar graph, it's just a circle of radius 1. But sine, cosine, exponentials, and complex numbers are all ways of representing rotations. This is summed up beautifully in Euler's formula, e^(ix)=cos(x)+isin(x). We can use this to represent an 2d rotation we like by changing the value of x. (We can even extend it to 3+ dimensions using homogeneous coordinates, which is really cool!) Since this is a complex function, it has 2 pieces, both of which have a period of 2π (which really makes me wish we used τ=2π instead, but we're stuck with π now). So if we plug in π for x we get the famous identity, e^(πi)=-1. This is really just another way of representing the same transformation given by i²=-1, and it's because there are 2π radians in a circle. The x value in e^(ix) tells you the angle (in radians) to rotate by, starting at (1,0) on the unit circle. π is just the number that encodes a 180° rotation from 1 to -1 on the unit circle. There's the circle definition again -- divide a circle's circumference by it's diameter and you've got the angle that rotates you around a semi-circle. But fundamentally, the rotation is what matters more.
Anyone have any thoughts?
People often ask if pi exists which is both a valid and interesting idea to ponder, but I think that people tend to pick this number instead of a number like 5 because the idea of "fiveness" has a different feeling than the idea of "pi-ness". The number 5 is more commonplace to every day experiences. Every day we count things. We encounter collections of 5 things all the time. The number pi on the other hand while still useful is not as common to every day experiences unless you happen to be a mathematician or do measurements of circlular objects and their geometric properties on a regular basis. There is also the question of what is meant by a number existing. So my question is this: Does any number, be it 5, pi, -17, 2+5i, etc exist?
I’ve thought about that and have am come to the conclusion that only 1 is real because although there can be five apples, they all represent different places in spacetime.
It gets a lil sticky if we say that 0 is also a number because… there is either nothing or something… but then that’s “two” things which in turn makes three things and so on… what was the question?
This video is great since it explores an idea that I wish got more philosophical scrutiny.
Like pointed out near the end, the quantization implied by QM means that any ideal notion of the real number line (i.e., that which is encountered in a course on analysis) cannot have a physical implementation, since something continuous cannot be embedded into a quantized structure (I think, seems like a reasonable inference). There would be gaps.
In fact, in such a world as implied by QM, can anything even be continuous? Motion and temperature fluctuations only seem to be continuous, as they are just a many of very little quanta lumps of certain things. But anyway, in such a world, things like e and pi might not be physically real (since they are irrational), and so the equations that we have describing reality that use those constants would, in fact, be describing a reality that is not *exactly* like ours (one is an idealized continuous reality, the other discrete). Would that make anything in a discrete reality just approximations, albeit very good ones, to an idealized, continuous reality? I mean, who really cares practically, since the approximations are very good anyway, but don't you want to know what IS?
Aren't all quantum things waves according to a recent Science Asylum video? Waves themselves are continuous even if we can measure them as discrete objects. Which means quantum is really about energy instead of objects. In that case maybe space is continuous but vacuum energy is discrete.
Unless you believe in a real Platonic 'world of forms', it is not possible to even imagine a perfect circle, since that would be a product of a quantised brain.
Even if everything is continuous, formulas of Physics ARE still approximations with very real errors
This philosophy you are describing dates to the 1890's, when Mathematics started to be tied down accurately enough to allow detailed scrutiny. The debate fell effectively into the constuctivist camp (you should only ever use mathematical concepts that can be constructed; non-constructivist proofs such as reducto ad absurdum should be avoided) and the idealist camp, who felt that mathematics would be too constrained by this requirement. At the time we understood it as an ontological (existance) argument (along the lines you present it here). It is really with Goedel that we understand mathematics as a branch of logic. The problem nowadays is understood as a problem of epistemology (knowledge) rather than ontology.
So perhaps a better way to look at it is that both classical mechanics and quantum mechanics are encoded in quite tight logic, with logical rules that allow experimental setup and experimental outcome to be related to each other.
There is a beautiful book by Errett Bishop that derives all of calculus, all the way into the Hilbert Spaces you need to construct what we understand of quantum mechanics in a constructivist fashion.
The upshot of this is that you can always represent the quantum mechanics of the experimental setup to the accuracy you need, as long as you have a big enough Turing machine (viz a vie Computer). So it is not the mathematical representation that stymies us, but our cleverness in crunching the numbers.
Dear Nick, Its always a great pleasure to learn and watch your videos. You're really accurate within your explanations and you do not show off because of your knowledge. thanks for that !
Hoi Nick,
About the question you answered at the end about the spinning of black holes. To me the real question is: why did there rise forces if something accelerates? Are those forces still there in an empty universe? Can you accelerate in an empty universe. Is acceleration always with respect to other bodies? And why does one feel a force while the other doesn't? How do you know who is actually accelerating? Questions Ernst Mach asked long before.
Thanks for your very fun and educating video's! I'm a physics teacher from the Netherlands and this is by far my favourite TH-cam channel.
Kees Stoop
SNL:
"Is it pizza?"
"It's *almost* pizza"
Instant like on the video because "Pan out" joke
But then you'd have to press like again for laughing at his own cheesy pun xD
Same
Your channel is really good, cheers from France
Okay. I mean, physics is one thing but when you combine that, maths, and my beloved computer technology I'm on geek overload. Too good!
As far as I know, binary digits 1 and 0 do not correspond to electricity present / absent. They are represented by different levels of voltage, called "high" and "low". In a digital circuit, when no electricity is present, the device, or at least part of the circuit, is unpowered which is kind of the same as "undefined".
Q: How precise is the number 1 since all the measurements of any standard unit vary across space and time?
A: It depends on how "unit" is defined. Every definition is built of concepts and those concepts are based on human experience which is fallible on every level. The important thing to remember is that errors are often random and randomness comes from nature. Humans make mistakes though and nature even in it's randomness does not.
In fact randomness is one of the surest ways to derive an absolute number since a truly random number will approximate over time to 1/2. So there you have a basis for the number 2 even before you've defined one or zero.
This is an excellent video! I would definitely say space is fundamentally discrete
It follows on to another interesting point about mathematics and its relationship to the universe I think. We have many laws, fundamental constants etc, that would appear to be in place (by definition) before the big bang. This therefore gives the impression that they somehow 'exist' outside of the universe... which isn't a thing imo.
The point I am trying to make (badly) is that if all the laws/constants/numbers are not emergent - then none exist! and if they are emergent, then they cannot be constant or unchanging because they rely on the state of the universe
Nice puns btw!
@Tim Erskine; Rob Bryanton talks about the fine structure of the universe being essentially arbitrary and created at the moment of the big bang in some of his "Imagining the 10th Dimension" videos.
*technically*, you are wrong when you say that current either flows or doesn't flow in the transistors of a computer. There will always be leakage currents that depend on many factors (as a result of using p- and n-doped semiconductors), but we define arbitrary thresholds we think are good enough to be called a 1 or a 0. Most don't know that there is 3rd state known as the high impedance state or z state. This is even useful in some applications like data buffers for high speed busses in computing.
Thanks for the clarification. I didn't think about leakage, but that makes sense.
@@ScienceAsylum Haha I intended it more as a tongue in cheek joke about pedantry (I do love it though xP) but I'm glad you actually thought it to be constructive! I want to thank you by the way for your videos, they really are great! My fav scientist on youtube :3
@@sebastianjovancic9814 good one in pedantry, but also true :)
Leakage is irrelevant because threshold always results in 1 or 0 which is all that matters.
@@manoo422 Definitely not always true, entirely depends on design, materials, manufacturing and even environmental conditions like temperature. If you have high frequency noise in your signal line to the gate of the transistor, you could induce this noise to your output that will begin to bounce and float between Vdd (or Vcc) (1) and ground (0). This isn't even the aforementioned high impedance state, where you isolate the output from both Vdd and ground leaving it floating. Design considerations need to be made when designing circuitry to avoid capacitive coupling and line inductances to not influence the function of the circuit. Hell, if you have sensitive enough stuff, a crappy, noisy switching mode power supply could induce severe high frequency noise through your Vdd to pretty much everything in your circuit and entirely muck up anything digital you're trying to do.
Don't take the equipment you use for granted, a lot of careful compromises have been made to design it.
Oh my god, I just realized something. I was playing chess with an Australian friend of mine. He kept saying "Checkmate" even though I had an escape from check. I think he might have been saying "check, mate." not "checkmate."
Mother Nature continues to boggle our human curiosity... presenting us with a seemingly innocuous relation between the circumference and the diameter of a circle and then... revealing a surprisingly endless non-repeating pattern of decimal numbers for its numerical approximation.
This is Mother Nature at its best - refusing to be pigeonholed by human understanding... Fundamental reality is much much more robust and magical than we can ever imagine... despite our blind faith in mathematical precision which at best is still just an approximation!!!
I love how much you say "A Perfect Circle". One of my favorite bands.
An excellent, as always, piece of teaching but I have some remarks. If pi, or indeed _any_ numbers exist, they could not be physical objects. They mould be abstract objects. Why? If the number 3 exists, it would be of no use asking _where_ it exists. It wouldn't exist anywhere. And there is only _one_ number three. The number three doesn't exist in each trio of objects. We simply use the one and only number three to count them as a trio in the first place. It, and its fellow numbers, would have their own special non-physical reality. These issues are what makes 'Do numbers exist?' a philosophical question. No observation or measurement can ever tell us, for measurement makes use of numbers, so if numbers are not real, then measurement uses something that doesn't exist. I suspect a better opening question might be: 'What sort of thing is a number?' 'Abstract objects' is one answer. There have been others.
I really enjoyed this video. I work in water treatment & pie is utilized daily. this video helped answer abstract questions my professors could not. I would love to see you tackle some water math
It just has the problem of "do numbers exist"
Yeah, like I'd argue 1 doesn't exist or not exist anymore than pi does. It's just a quantity we chose to define.
The puns were crazy lol.
Insta like and share coz of "pan out" joke.
I think it depends on whether you are an idealist like Plato. In which case pi and other abstract forms are what are real, and our world is full of imperfect replications of them. For materialists, the physical world comes first and notions like numbers and pi are just human constructs to make sense of the physical world.
A very bright scientist once said he did not believe in irrational numbers because it would take an infinite amount of information to describe it, and the universe has finite information.
Him : OR DO I ??!!???
Michael from Vsauce : Now finally a worthy opponent
Clicked on the video to learn about Pi.
Now I know everything about a CPU.
You're welcome 😉
And the hunger drove him crazy & we got our video...
Many thanks for linking the continuous-discrete problem with quantum mechanics-relativity.
Pi is a ratio of two numbers, not a number itself, hence why in mathematics it is not expressed generally as a number but as the greek letter pi. This is the same with many other constants in mathematics, for example the speed of light, which itself is not a number, but can only be quantifiable based on two values, time and distance, both of which can be any number of units of measure.
No, pi is a real number. The reason we use a symbol is that it is irrational.
HAPPY 3.14 DAY!!!!!!!!!!!!!!!
Still waiting for the third of Quattuordecember to celebrate the actual Pi day.
Or celebrate it on July 22. 22/7 is a closer approximation to pi than 3.14.
Or for the thirty first of April, which obviously is closer (/s)
@@stevethecatcouch6532 Just barely!
|π - 22/7| = .001264489...
|π - 3.14| = .001592653...
@HaleyHalcyon - Gaming Channel no, what we do is say today is 2019/3/25, that's the way we do!
I always accepted that pi was something abstract and can't really exist in a medium in any perfect form, and thus why I was content with pi being something purely theoretical to calculate theoretical concepts to help us understand the more real physical concepts of the world.
*so why the hell does x^2 + y^2 = r^2 graph a circle*
I mean I know logically _why_ it works, but... _it physically exists_, and I don't know how to cope with it, and now I'm here.
Wow
I think space time is continuous but can only be observed discretely.
This is one of your best videos. You hit a lot of really crazy stuff here.
Really? I was just messin' around with this one.
2:18 I love how red + green = orange to my brain
I would have liked to have seen a discussion of Plato's theory of Forms included since some thinkers, including Roger Penrose, think that numbers might exist in that realm. Apart from that, I enjoyed the presentation
After 5 shots of vodka and a cigarette, I realized that a circle cannot be quantized , because pie seems to be a never ending number that doesn’t seem to exist, so I went looking for TH-cam videos that explains this , this is the best one that I have found 👍🏻
It's not necessary to have infinitely small precision for pi to have physical meaning. Take a circle with a diameter of 10^1000000 meters. Its circumference uses 1000000 digits of pi if you measure up to a meter, so no need of quantum mechanics here
@@holomurphy22, get a copy of Maple (even a twenty years old version can do it), and you can ask for as many digits of Pi, as the computer memory (adjusted for internal representation of numbers in Maple) allows. This is the easy way. Alternatively, use a specialised program to compute as many as you want ...
@@MiguelGarcia-zx1qj Alright but how does this relate to my comment?
@@holomurphy22, you're right, I missed the part about "physical meaning". My fault. I interpreted your answer as pertaining to the video's first theme: calculating Pi experimentally by drawing a circle.
Nevertheless, giving "physical meaning" to the first 1000000 digits of Pi is not the issue here; for that matter, it does not differ from giving "physical meaning" to the first 1 or 2 first digits of Pi
@@MiguelGarcia-zx1qj I answered to the comment above of Charlie. I dont see how its the same to give physical meaning to 2 digits and 10000... digits. There are measure limits.
By Mistake Discovered this channel.. Watched all videos for 2 hours straight..
I asked myself this question, but ended up going around in circles.
😂
I feel like i finally found someone who actually knows what they're talking about.
In England they say, you are a bloody brilliant man
I ate the pi, it's gone now...
Yeah I'm starting to think pi isn't a number but maybe like a space of potentiality from which you could pull numbers to as much precision as you want. So 3.14 is a number, 3.1416 is a number, etc., just pick the one you need, call it "pi" for your purposes.
on that note about computers the 64bit float is just a standard, you can definitely use your own specialized number system on your computer to calculate pi.
I mean you aren't going to get all of pi but you reveal more of ip the longer you let it run for.
Pi: exists
Science asylum:
dat Spock shirt tho
It would be better, grammatically, to call it “The Pedantry Game.”
For more random pedantry: Pi Day (as in the date of 3/14) also exists in much of Asia, as we use the YMD calendar. So "March 14th" would still be "3-14" for us, even if our full dates are written like 2019-03-14 versus 3/14/2019.
So we're still with you MDY bros in celebrating Pi Day! It's those weirdos who write their dates like 14/3 you should be concerned about trying to ruin your fun with their lamer 22/7 Pi Day...
Good vid. Love you♥️
Terrific video. Hats off
I'm a Quantum Guy so I think the idea of infinitely dividing space is inherently ridiculous, I've always thought that circles don't exist.
Does π exist?
Do we exist?
Am I writing this comment or am I?
I had the same thought, thank you for making this video 🙏
Wow . Thank you so much!!
Pi exists but only as a tool. I come from an engineering background, and engineers (who have to deal with the real world) learn early that everything is an approximation. Wheels, bearings, gears, etc. aren't round... They are approximately round... round enough to work is "round".
To be completely pedantic, floating-point numbers also have a hidden bit. So the significand at [5:00] is 53 bits long.
the by information you give is in all your videos more intresting than the actual main quuestions themself...this is a compliment
Q: Are space & time continuous or discrete?
A: Both. These 2 concepts are not mutually exclusive as you accept infinite resolution of a discrete system.
Nice explanations. The puns are painful, and what's with the seemingly random pictures? walnuts etc. I am now going in search of a few more pi related videos. I also just subscribed. :)
The clipart is just in case anyone was missing that particular pun. There's either clipart or a prop for every single pun (I think).
Logic and mathematics are just tools we invented to explain our reality. Their accuracy/validity of these things is directly tied to how well they explain our reality within the framework we have constructed. There is a great video on this sort of thing somewhere on TH-cam but for the life of me I cannot remember the title.
The longer this goes the more philosophical it feels versus how scientific it feels. PI goes on forever. The further it goes, the more it refines our circle. At some point in the real world the addition of more refinement would be meaningless in the real world. But a concept it real and in this case extends beyond the practicality of the physical and into the metaphysical. And that makes it beautiful.
"Something there is in beauty which grows in the soul of the beholder, like a flower: fragile - -
For many are the blights which may waste the beauty.
For the beholder--
and imperishable - -
for the beauty may die, or the world may die, but the soul in which the flower grows survives. "
S. R. Donaldson
Such a great video. So many great points!
Glad i discovered your channel. Entertaining and informative in a strange way. Thanks mate. Keep it up.
That ‘strangeness’ you’re picking up is Nick, I think.
I just found this channel a couple of days ago and he is (imo) one of the best in the business.
I like John and Hank Green too ... but this dude is extra-hilarious, making for, I think, a better entertainment and learning experience.
The 64-bit hardware architecture is NOT the floating point precision, it just happens to coincide with "Double" precision which is most common in scientific computing. "Float" (32-bit), "double" (64-bit) and "long double" (128-bit) can be stored/computed on any architecture. The architecture is only for memory addresses, which is why 32-bit hardware had a 4GB memory limitation.
sir, your so funny and education and openminded in any subject in science sir
Long time watcher, first time commenter - you broke my brain bro.
Keep it up :)
Great video!
The craziest thing about Pi to me is that it contains every possible configuration of any string of numbers of any length. Every string of digits of every length. Meaning Pi contains itself. And not just as a string from 3.14 to infinity. It contains that string plus 1. It contains that string plus that string on either side of itself forwards and backwards. Meaning that pi doesn’t just contain itself, it contains itself infinitely many times.
But here’s the weird thing. If Pi contains itself, that means there’s a point at which Pi will appear to repeat, because if it contains itself, it has to have a string that starts 314159 and continues on for as many digits as we could possibly calculate. And we will never, ever find it. It is absolutely there within itself, but it cannot be teased out, because then it would be repeating a finite number back to you.
But PI doesnt repeat
What do you by exist? Would the absence of its appearance in the strictest attempt to “find” it in real world applications necessarily mean it doesn’t exist? How do you know pi goes on forever? Can we even know whether pi does or does not exist, or can we merely give our best hypothesis?
I think it’s safe to say pi exists once you stop seeing it as a component of circles and instead one of distribution from a given thing. It pops up in tons of places without circles, and I personally just see it as the “fundamental unit” of any function. Any function can be represented by infinite Fourier series and pi itself is infinite. While QM argues infinitesimals don’t exist, that’s just some arbitrary meaning. You can zoom out to infinity and then your resolution is effectively infinity. Pi is not a “number” but rather a consequence of infinities and imaginary numbers that come from the representation of information we call math. Pi isn’t the only trancendental number, it’s just one of the most fundamental ones. I’d personally argue, however, that it’s just as important as Euler number since they cannot exist without each other when dealing with imaginary numbers.