This is a clip from a conversation with Sean Carroll from Nov 2019. Check out Sean's new book on quantum mechanics titled Something Deeply Hidden: amzn.to/2C6aCaf New full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days. If you enjoy it, subscribe, comment, and share. You can watch the full conversation here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-iNqqOLscOBY.html (more links below) Podcast full episodes playlist: ru-vid.com/group/PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 Podcasts clips playlist: ru-vid.com/group/PLrAXtmErZgOeciFP3CBCIEElOJeitOr41 Podcast website: lexfridman.com/ai Podcast on Apple Podcasts (iTunes): apple.co/2lwqZIr Podcast on Spotify: spoti.fi/2nEwCF8 Podcast RSS: lexfridman.com/category/ai/feed/
Having someone in this space representing engineering and computer science is the greatest gift you could have ever given humanity besides your actual work. The questions you ask, the standards you hold people to, the neutrality you maintain and the way that you respond are so different from other platforms that I have stopped listening to many others for conversations about random topics. Even three years from now you're still consistent. Thank you.
In pure math, infinites often reveal details about finite things which were otherwise. Like generating functions for integer partitions of n into distinct parts and odd parts, or the notion of a finite limit of an infinite sequence of real numbers, which is the fundamental idea in all of calculus.
Deeply absorbing discussion with Sean Carroll. It helps to have some knowledge of physics to understand his very cogent explanations and analogies, but it also helps to have Lex prompt him with real or feigned somewhat naive questions. I believe Sean and Richard Feynman to be very different people, but I also see parallels in their abilities to make the complex understandable.
I love discussions about the infinite as well as unique and interesting mathematical concepts. I am thoroughly math stupid, but this still is extremely thought provoking.
I usually don’t comment on videos, but as someone that genuinely respects these two scientists, I feel I have to say something. I don’t particularly feel the question of “What is Hilbert Space?” is a good one. 1st. As I’ve seen in some comments below, there are many types of Hilbert Spaces. The one that is discussed here is L2 over some domain (usually over all real numbers) which is used in Quantum Mechanics since they care about square integrable (lebesgue) functions (which means they can be turned into probability distribution functions). Again, one particular type of a Hilbert Space. 2nd. A Hilbert Space is a complete normed vector space whose norm is naturally derived from an inner product. There are so many inner product spaces that are Hilbert Spaces (and in turn Banach Spaces). As a mathematician, I understand the notion of a Hilbert Space to a physicist is in its utility, so I understand why they bypass the formalisms and go straight to the particular Hilbert Space which helps formalize their mathematics. Rant over. Enjoy your day!
Lex, our intuitions are limited, having developed only through evolution. Yet we've been able to re-purpose cognition to discover highly useful abstractions that enabled our modern world. We all need to be comfortable with the reality of this situation we find ourselves in and be less overwhelmed by knowledge which cannot be interpreted by archaic modules within our mental apparatus.
I love thinking about this stuff. Reality has been feeding my mind so much lately.. The amount of synchronicities I experience a day now is almost getting to be to much. I'm fricken addicted to the rabbit hole so bad but it's so hard to stop. Gotta cut myself off
Simple explanation. I needed that clarification, not being a scientist or mathematician myself. Musca gratitude and fist bumps to the host and to Sean Carroll!
I am nearly finished reading Max Tegmark’s Our Mathematical Universe and am seeking a bit more understanding of Hilbert Space. I found this and am seeking to understand the ideas. Thank you for what you are contributing to me and our fellow primates. I do have Sean’s book!
One thing that helped me grapple with the real ness of infinity isn’t that things can be infinitely large, but the question “how many discreet values are there between the number 1 and the number 2” and the answer there is also infinity. There’s no limit to how many times you can divide up that continuum into discreet values.
from me was the question in elementary school ''how many times does nothing fit into something?'' ,maybe it helped cause I assumed nothing as an entity , as something that exists
So, Hilbert space is a mathematical tool, like a stage used to represent a system’s information. And it may or may not correspond to actual space?? It has an upper limit of infinity but the systems represented in it are not necessarily infinite dimensional??!!
Understanding Hilbert Spaces is not easy to comprehensively understand without understanding of functional analysis, you must at least know linear algebra. They are vector spaces (with usually infinite basis, infinite dimensionality) that are applied with the inner product which has a complete Cauchy sequence (so you need some real analysis as well).
Shubham Dhingra lol. If you really want to understand what’s being spoken about here, I’d highly suggest watching “The Essence or Linear Algebra” on RU-vid. It’s very engaging and entertaining
Not complicating. In order to describe a physical space you only need 3 dimensions or 3 basis vectors . In order to describe other systems, such as economics, you need a lot more vectors or "factors". A dimension is just a vector that isn't affected by another vector. If I move vertically in the y direction, it doesn't affect affect how I move in the x direction. So it's not so hard to understand there exists a system in which several vectors are independent of each other, meaning they don't inherently affect each other but are necessary to describe the system. For those who are hung up on infinite, math is a tool that allows for an infinite amount of these dimensions or basis vectors. Carroll is trying to say that it's a possibility that a physical system has infinite amount of dimensions used to describe a system, although it's not something that is shown in science yet.
Life is the anti-entropic force carrier. Like in war, all the chaos and commotion converges to a finish, a treaty, a winner is assembled and the chaos ends.
The thing that Lex is uncomfortable about is that infinity does exist. Which means an infinite amount of Lexs exist, including one (for sure) that is not so bothered about infinity. We eventually all get to the end, but what happens when we get there destroys the ego.
If you did know the orientation and position of each water molecule in that bottle, what difference would it make? It's still a bottle of water regardless of how many times you think about it.
You would be able to tell the bottles past & future. Which would make a lot of difference. Because of quantum entanglement it would essential allow you to move forwards & backwards in time relative to the bottles light path since you would be an observer of the bottle relative to the stationary reference point of any outside observer.
I've never heard entropy explained as the amount we know or don't know about a system. I'm not sure if that is a way of characterizing whether there are many or fewer degrees of freedom?
Yeah, if I know what you’re going to do (microstate level) you have no freedom. If I have partial info (if it rains he’ll grab umbrella) you have less degrees of freedom than a totally random actor who may grab umbrella in sun or leave it in rain. That’s all it is. Check out Shannon information entropy for more.
TL;DR : infinity isn't a noun, its a verb. Infinity is not a number and its not real, just like numbers aren't real. It's a sequence/process/step using symbols (what we call numbers) to represent the quantity of things we can observe. It basically just tells you to do the same thing you did before (n+1 times). Or negative infinity is just to do it in the opposite direction (n-1) times. In programming, I can reach infinity by using a loop with its stopping point at one step ahead of my current step, while increasing the stopping point by one step each time I execute the code. Thus, at every step, I have "doing infinity." When people say you will never reach infinity, it implies there is a target value that infinity has, but infinity has no value because it is not a number, its an algorithm that can be used to get a particular value until some condition has been met, or keep going until the power goes out.
Why is there this mysticism around mathematical spaces on which physics takes place? Hilbert spaces are just complete normed vector spaces (you can move anywhere in space without fear of falling into an infinitely small hole) where the inner product (how you do addition) induces a (simply connected) metric space. A place where kinematics exist and is described mathematically
I think what Lex was getting at is this: Is Inifinity ABSOLUTELY necessary to COMPLETELY describe some phenomena about the PHYSICAL universe? Or is it just an extremely useful tool? It's important to note the PHYSICAL part. Sean Carroll is correct in saying that it's possible for some inputs that an algorithm will never stop. HOWEVER, this is an algorithm being run on a theoretical machine. A real machine will stop eventually due to entropy. Let's take space as an example. Maybe space is quantized at some level that is not possible to measure, and therefore useful to consider space continuous. This can limit the smallest possible wavelength of light and interactions between subatomic particles. I'm not a physicist, but its at least interesting to think of lol. Obviously, infinity is so useful that it would absurd to remove it. No one in their right mind would do that, but I still feel like the question of whether infinities really exist in the physical universe (not the space of ideas or numbers, etc because obviously in that space it does exist) is still a valid question. I am not fully convinced that infinities really do exist in nature.
Tyler Heers How many points in space exist between here and the moon? Hint: between any two points there are more points. That’s real, at least as real as it gets. There’s an example of actual infinity in reality. Actually there are infinite points between the 1 and 2 on your ruler, so you can see what one type of infinity looks like right in Front of you
@@deanodeboI think the point he's trying to make is that an infinity might not exist when your counting things in reality like planets or particles in the universe or worlds in the many worlds theory. I think lex makes this point also that he thinks people use the word infinite as a cop out instead of doing the math or saying I don't know
Is the interview high on something? He seems unsure, slurring his words, maybe confused about his reason for being there, and kind of all over the place. It's difficult to follow the path of questioning. I hope he's ok :(.
He is very intelligent. It is odd to see someone who can use this amount of there cognition. He closes his eyes to ponder and produce the correct wording. Please be respectful to computer programmers and mathematicians and physicists and professors of any degree. You are not on their level of intellect and it is easy to gang up and bully people who are better than you.
There is nothing natural about infinite. There can be no location in space or time in an infinite universe. In essence, if the universe is infinite in age, then it would have taken an infinite amount of time to reach this moment. Infinite cannot be reached by definition so by definition the universe and time itself had a beginning. You can apply the same logic to physical space
Physicalist-formalist notion of 'actual infinity' cancels possibility of motion - Zeno. Undecidablity of Halting problem is coherent with 'potential infinity'. Lex is giving Carroll's mumbojumbo infinity too much slack.
Well you should have studied & educated yourself while you were younger & in your pre-development stage. This would have allowed your brain to have a greater capacity for understanding & connecting ideas. But if you’re over 15-19 it’s too late. This is your own fault.
His definition of a Hilbert space is plain wrong from a mathematical point of view... A Hilbert space need not be infinite dimensional at all, and need not be restricted on quantum wave functions. EDIT: I get that he tries to explain these in layman terms, but he could've done so, without being factually inaccurate.
Infinite is a messure of time, since time won't last to the dead end of the universe when no structures, no gravity, no time, and no space could exist. When the universe loses track of time, infinite becomes finite. Lex is right.
I understand why there is a need from the public opinion to have a good analogue for mathematical objects, but at the end those are mathamtical objects. There is a definition for them and that's it. That is what should be learned in the school, that the mathematical truth is not a part of this material world and every analogue is faint and misleading.
Moving my hands towards the keyboard i traverse an infinite series of halves of the distance, all in the Now which is infinitely short in duration. My intuitive Euclidean space is conceived as infinite and perhaps cannot be conceived but as infinite. is not infinity all around us.
@DY_Physics sure, convergence is an Infinite process , hence infinity all around us (you cannot metaphysically comprehend motion otherwise. See Zeno et al.). and convergence has very little to do with time or the present which is Infinitely short. convergence has very little to do with the (perhaps necessary) Infinity of space. again, infinity is all around us.
@DY_Physicsoriginally i was posing this question to Lex who, if i understand him right, has problems with the notion of infinity being a part of the real physical world. my claim is that infinity as a concept and as a property of objects and changes (motions) in the world is not problematic. (meta)physically there is infinity everywhere. i claim that this hold for (strictly)physically as well.
Well the computer programme is not gonna run infinitely long, computer needs energy to run and computer, funny enough, will disintegrate due to entropy, so no, it is not gonna run for ever .
Theoretical physics seems very redundant, seems just like a bunch of mathematicians circle jrkin ideas and numbers. Like they're trying to get their peers to validate and agree that they're theories are viable. Then they can attempt to ask for more money , grants etc.
This has very little at all to do with life in reality. They are playing with math, as mathematicians do. But the interviewer seems to want it to be some kind of profound view of reality. Yes math is fascinating, and I studied it for years and that is my academic background. But you will not find a spiritual grounding in it. The far reaches of specific narrow fields of mathematics are filled with mathematicians that use mathematical analysis and synthesis to find abstract mathematical structures in their research. However, it is a waste of time for a layperson to try to have a discussion about these issues. Instead, it is for colleagues in the field to assist and review each other‘s research purely for the purpose of making progress. At the point where he is asking about infinity, this discussion might as well be one guy speaking German about bridge building with another guy speaking Swahili about sheep herding. Honestly, we know as much about the number one as we do about the concept of infinity. You could spend an entire life studying the number one, and never run out of genuine research direction
While everyone knows that God created the universe, we can see here that the atheists have created a fortification of "We don't know what created the universe".
@@OM-el6oy Lex is being way too nice and polite to the idiot SC - who in his last comment reveals that he utterly fails to comprehend undecidability of Halting problem and what that means to notion of infinity.
You can even say that a Hilbert space is a special type of Banach space where the norm is induced by the inner product structure 😂 an even more descriptive definition
I don’t think he did a great job making the distinction between mathematical Hilbert space and our actual world. He mentions it in the beginning, but near the end Lex keeps bringing up this skepticism about the infinity of dimensions, as if it were a cop out or something. I think Sean could have easily pointed out that Lex may be feeling uncomfortable because he is mistaking a function space for a physical space. Needing an infinite number of basis functions to describe a function in another base is totally common and well founded. Has nothing to do with how many dimensions we live in
Greetings. Max, I think you are right here about Lex's confusion. There is all the difference in the world between abstraction and physical reality, even though we may successfully use the former in describing the latter. (It's one of the ways in which reality is intelligible.) I suspect, though, that Lex was not asking quite the right question. It is not that infinity is 'unintuitive' or 'uncomfortable' as an abstraction. This would be mere unfamiliarity. (I suffer from it myself.) It is that it is hardly clear how infinity applies to physical reality. That's where the mess gets made and where the conceptual difficulties lurk. For all that, I think this is one of Carroll's finest RU-vid appearances. Instead of getting himself messed up with philosophies he only half understands (which he often does) he simply explains a very tricky abstract notion. Being no physicist, I am thoroughly glad to know what is meant by 'Hilbert Space' before I die. Now, in my lamentable ignorance (I am a philosopher, after all - and grieve over it during the long winter nights) all I need from Carroll is a similar clip but clear excursus on what exactly is meant by 'wave function' and the difference between the abstraction and the reality there. I must be asking for much!
Hi, it seems what the host is trying to say is that if a particle, or a a syatem or the whole universe has infinite parametres to describe it, that is pretty mind-boggling. Although, I'd like to think it's true)
I was thinking the same! Hilbert space is where the functions that describe a quantum system live; it's not the physical space where stuff really exist or happens. You can extract information about the system in physical space from its representative function in Hilbert space.
I think you could best describe it as there's always an infinite potential in projective space which is a general coherent sphere around its compressed focus. Now the finite is always presupposed, therefore to relate that construct to some higher potential mind you only need a general sphere to magnify its potential for its limited capacity for alternate transformations. Therefore to contain an infinity within reality would be completely redundant and therefore infinity is limited to the necessary space in which is needed to compute all possible states of a living system. Though the Infinity also expands through the resonance to each and every brother/sister star in a reflective capacity. Therefore even though it's spatially limited to its own general spear it's reflectively multiplicative within all spheres within space itself. You can think of this as the wave function of any reality has a nucleus focused in its infinite string of potentiality of which only has a developed interface that extends to a certain degree along that string, whereas the only infinite is the potentiality of the string itself. Though in reality you only need so much as the information can develop and that's why when you look at a wave function you only look at a constant frame of view in a relative frame of time. This is on top of the fact that we see everything through a lens so what we see in the world is already distorted to some limited potential and that lens of our eye represents the crystallized material form of the magnetic field of coherency of which is that higher electromagnetic field of the mind and the lens of which it views all things within ourself. Therefore you can think infinity is the negative image of the positive interface of which is developed into constructive form within that infinite potential of space from its finite reliable state. Therefore you need an infinite potential to define some limited potential within itself to begin with, otherwise you would be limited in what you could create and you would eventually fill up the whole system and it would collapse or crystallize and never grow or develop or allow for any new creation again.
This is awesome! Was hooked immediately by the math theory and thoroughly enjoyed Sean’s ability to meaningfully comment on complex ideas in plain and simple english
I argue that you CAN'T do anything mathematically with infinite, and that this is where brilliant people run into trouble. Infinite is not an actual numerical value, and hence cannot be plugged into an equation. "2 times infinite" would hence be as nonsensical as "2 times happiness". Infinite refers to something never ending, which is the real crazy part. We could swap the word with "eternal", and it becomes more clear why trying to make it a math term is silly.
Infinity... How long will say a software process take to halt...as long as a piece of string. I agree with lex, infinity, even if we divide it up like cantor, feels like a catch all. A way to claim we know something whereas in fact it just makes current math work. Maybe every spiralling incalculable is a unique fractal that deserves a name
I must say the interviewer has no intuivitive sense of how simple the concept of infinity is. Being uncomfortable with Infinity means you just dont understand it. There is no paradox or confusion regarding infinity.
My guy is a research scientist at MIT he has a PHD in computer engineering I absolutely guarantee he has a better understanding of infinity than you do buddy.
@NumbToons I think the more plausible answer is that you have a very limited understanding of what it means. I mean have you ever tried to write an AI computer program that uses the concept of infinity? No? Then have some respect. I'm sick of naive commenters thinking they know more than any professional. This is the kolch and Doppler effect in action. The stupid think that they are smart and the smart think that they are stupid
@@vicslav4030 I don't have anything against you or the interviewer, i understand what you are saying. Dont worry, im well aware of my lack of knowledge and intelligence. I might be able to convey my point across in a good real life conversation. RU-vid comments are just suitable for arguments, not discussions. Have a good day sir.
@NumbToons I'm sorry bro these youtube comments are virtually the only human interaction I have. I'm pissed off cause I have 5 years to live if I'm lucky this disease will rot my brain just like my mother which is terrifying I can't talk to anyone cause it's so sad.
Space.... a broader term for what makes up the greater part of the universe. The universe.... a term I, and others, use interchangeably with the name/idea/concept/tangible physicality of (to our corporeal forms) God. Just trying to make ya think ;)
You ask great /deep questions man and in a humble way, and the fact that you are educated in these fields allows you to press more deeply on these issues so some of these smart men can't "squirm out " as it were.
Most people with a basic college level background in a mathematically intensive field know what a Hilbert space is. It is not that deep and shouldn’t even require a very good mathematician/physicist to explain.
