The Biggest Ideas in the Universe | 19. Probability and Randomness
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- เผยแพร่เมื่อ 5 มิ.ย. 2024
- The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody.
This is Idea #19, "Probability and Randomness." In which we accept that none of us is Laplace's Demon, and in the real world we act under conditions of incomplete information, necessitating a turn to probabilistic reasoning. We talk a bit about what that it, how it works, and how it applies to statistical mechanics.
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#science #physics #ideas #universe #learning #cosmology #philosophy #probability #bayes - วิทยาศาสตร์และเทคโนโลยี
Erratum of the week: Apologies to Andrey Kolmogorov, whose name I inexplicably spelled "Komolgorov." Caught by @veleronHL.
Keep these up they’re awesome! As a math student, these have helped me too appreciate the use of geometry, topology and algebra.
Sean what software are you using to film/create these videos? (I assume you're writing on an ipad with the apple pencil?)
@@obiwankenobi07 he discussed that in one of the q&a videos
David Taylor which one do you remember? Thank you
@@obiwankenobi07 th-cam.com/video/m2qrXl0g0OE/w-d-xo.html
That’s dedication to watch yourself for an hour in order to write down everything you wrote the first time!
At least Sean doesn’t have to write and talk at the same time for this lecture!
(How many times has he said he shouldn’t try to do that.)
Any animator will tell you, it's easier to draw the lips to match the dialogue than to sync the acting to match the animation.
"It's amazing that I got this far in lecture on probability without mentioning Bayes..."
Yea, what are the odds?!
I feel so privileged listening to you teach. Thank you for doing this.
Sean, thank you so much for your willingness to do this series.
19:32 "What does physical probability mean? That's much dicier." A brilliant pun, even if unintentional.
Hi sir I'm your biggest fan and you are a Muse for me in physics.
You're one of my favorite humans.
Thanks, Sean. Congrats on 100k subscribers!
Where does probability come from?
Sean: That's much dicier!
"Funny Fact": The "second solution" for the Nortons Dome makes use of Newton's second law to "prove" that Newton's second law is wrong. It looks strange!
In other words, it uses the fact that force causes acceleration to show that at the summit there is acceleration without force.
PS: Congratulations on the content Mr. Carrol. I'm a fan!
"My prior that my math is correct is about 50%"
give or take some credence
I love these sessions with Dr. Carroll. They are fascinating to this Canadian university professor of psychology with a passion for physics. Sadly, once we all return to the classroom from online teaching when the pandemic starts to recede, these sessions from Sean will likely stop or at least become more infrequent. Pity that.
I guess the smartest person is one who updates their credences when they obtain new information, but always has doubts.
Thanks again Professor Sean, for these wonderful, enlightening & inspirational lectures. I’m sorry I don’t have a specific comment or question, but I just feel obliged to express my gratitude for the time & effort you devote, in sharing your knowledge & expertise with us - for free. And congratulations on reaching 100k subscribers. (ONLY 100k. Is that all ? The rest just don’t know what they’re missing !!)
what an extraordinary culture. Thanks a lot.
I have to say I never understood Quantum ideas until I heard you, Brian Greene and Jana talk about it.
Thank you so much for taking the time to teach all of us!
Congrats on 100K you're awesome!
Congratulations on your 100K subs! And thanks SO much for this.
I love science and cant begin to explain what your explanations mean to me.
It’s my goal to “unify” the fundamental science stuff with fundamental Karate theories and deliberate practise.
Incredible presentation Sean Carroll! you are awesome, you seem to talk complex theories in the most simple language!
I’m predicting that the next video will be about entropy with a prior of .9
So awesome. Love all the videos!
I feel so lucky that i get to experience these.
Love this stuff so much. I'm more of an autodidact, but still, taking online classes. Love it!
Admirable effort, bear in mind that true mastery of a subject requires practice, though! So hope you have a place to make some exercises on subjects like these, to really get a feel for it.
Making Physics Fun Sean. Good stuff.
Congrats on 100k Sean
simply great!
I sure enjoy these. In some ways they augment the Leonard Suskin courses beautifully.
Appreciate the content.
Excelente Physics material on videos through internet. Congratulations
I f****ing love you
Corrupted files plague us all!
The 19'th century is indeed interesting, thanks for mentioning 👍 I see my thoughts as vectors, truth as a point and interaction as a searchlight.
I understand maybe 5% of each of these videos but boy are the fun to watch
Thanx Mr Carroll
THX for the great effort.
μ is the sound made by spherical cows.
Γ is probably related to the use as the Gamma function, by way of entropy.
The laws of probabilities verses randomness are very interesting topics. One question that I have is: In an isolated system, do entangled wave functions of particles effect the various states and overall trajectory of the system? This is an amazing topic!
I love Sean's voice.
How many times have I watched one of Sean’s lectures late at night and woken up to see the video has ended. His voice is soothing.
Now I watch the videos in the daytime so I don’t miss anything.
Impressive editing...........when is your first motion picture coming out? And also thank you for these videos because they help dumb people like me seem smart when we talk about manifolds and spinor networks..............well as long as there are NO follow-up questions. Those are always a nuisance.
Kudos for the spherical cow land reference.
Sean, “r” in Norton’s dome’s shape equation refers to the distance on the surface, not the radial distance from “z axis”
100 k 👏🏽👏🏽❤️
A great video once again, that TH-cam are determined to ruin with constant advertisements!
It would be good to see a practical application of probability calculation within QM. If we take a single photon for example, its treated as a wave when travelling and then a particle as it interacts with matter. But as a wave we don’t know where the photon will appear, so there must be an explanation for this which includes probability.
Credence and frequenism: I take it to mean that if an event has probability p, then from all events with probability p (any sort of event) the fraction that will be true is p.
Very nice overview of the subjective vs objective probability problem. I think the entanglement is in the core of this problem and it is still difficult to say, but my feeling is there is a chance the world is fully deterministic on the objective sub-atomic level. What do you think?
We feel what is happening in future is changable by what we do now, only because we are subjectively looking from withhin a deterministic bubble, but not knowing the elements of this deterministic buble that makes the future fully predictable and we will never be able to predict the future, but this doesn't mean it is not determined from the beginning :) Just a point of view.
Amazing stuff. (Anybody please let me know what pen is he using.)
One approach to probabilities is to consider that there are a growing number of ideas wrought by competent men and women in physics that should work, but do not pan out in experiments. What is the probability that something important and universal is being overlooked? One number in mathematics that, to me, does not get enough attention is the number, 2. Before jumping off into many worlds, I think, one must pass muster at two worlds. Time is the realm of dualism. Is there a counterpart of the Universe that is being omitted at our peril? Are two-verses implicit in reality, yet overlooked? What I think imaginary math expresses is despite the violation of logic, -1 x -1 = -1 ^2 is true math, but we do not know why.
Yet, applications of imaginary math enabled the finding of the positron, the development of quantum mechanics, developing and manipulating alternating current, and many more uses. So here are my questions: Since the legitimacy of using imaginary numbers is now well established, should we not explore more deeply why a minus times a minus is a minus? Isn't it true that the universe we recognize may have a counterpart? Should not it be likely that minus one in the realm of the counterpart universe be plus one, and a brand new, vast, realm then opens up for us to grow into? Quantum entities seem to me to be tailor-made to exist in a two-part universe. Gravity may not remain so mysterious, if we know what is its counterpart. Dark (invisible) matter may become visible, and maybe there is a counterpart to entanglement which, when known, will make a lot more sense than it does now.
What was the Probability that Ariel shows up Randomly?
I get the feeling Sean really doesn't like Ernst Mach
in Norton dome formula you quoted the unit at the left hand side is a length but at the right hand side we find also time,due to g; is there perhaps a constant with dimensions somewhere?
❤ Very good 👍🏼
Indeterminacy is one of the biggest ideas in the universe!
The idea that you cannot give exact quantitative values to any physical system
This idea makes the idea of laplace's demon to be shown to be physically impossible, just like a perpetual motion machine is physically impossible!
Anyway I think much of the spooky action ascribed to quantum physics is just a basic misunderstanding between our Common Sense intuition that things have exact physical values like someone is exactly 6 ft tall. When in reality nature is in a constant state of flux and there are no exact values.
There are physical systems like the motion of the planets which are very simple and can be predicted with a high degree of certainty
But most physical systems are more like the weather and any attempt to measure them actually influences their future Behavior
So most of the physical systems especially here on Earth are very complicated and any attempt to actually measure and interact with them is going to affect them in ways which cannot be known beforehand
This is why the behavior of quantum physics is considered weird because any attempt to measure the system has incredible effects, because at the scale of quantum physics photons of Light Produce big effects.
[paused at 26 mins]
One way to convert subjective "one off" event probabilities to an "objective" probability is to use simulations. If you were to make a model of the relevant parts of the universe surrounding some event (e.g. winner of ball game) and run a very large number of simulations (with the input parameters to each simulation slightly permuted to model observational error), you would (assuming your simulation was good) tend to get results that simulated reality fairly accurately... if you applied this to the flip of a coin, given the initial energy supplied to the coin could only be crudely estimated, about 1/2 of your simulations would come up heads, and 1/2 tails.
Basically, an ensemble model.
At the beginning of the video there is an example where Newton's laws aren't deterministic because a certain differential equation doesn't have a unique equation. This is where it is important to be clear about the formal mathematical assumptions. Newton's laws *are* deterministic as long as the forces obey certain regularity properties. If you assume that formulas for forces are continuously differentiable, i.e. the potential is twice continuously differentiable, then there is a unique deterministic solution. The problem here is that the dome has a shape that is not twice continuously differentiable: r^1.5 has an infinite second derivative at r=0.
Hi Sir, I have a simple question. Inside a factory at the end of the shift a supervisor and his co-worker are counting the produced objects, the objects are approximately the size of a tennis ball. It is their daily routine,the worker counts the objects as he takes it from the production lot and puts it inside a bag. The role of the supervisor is to keep watch so that there is no mistake while counting. One fine day, before starting the counting process, the supervisor looks at the lot and writes done some random three digit number as quantity of the produced items, in short he assumes that the actual quality will probably match with that number. Now the question is what are the chances of the actual quantity matching exactly with that random number?
In Many Worlds, should the sum of probabilities of all outcomes of an event be the amplitude square of your branch, and thus very very small?
Bayes theorem looks very frightening as an equation but on a Venn diagram it is trivially obvious as the ratio of two areas, one of which is "your world" and the other which is the size of the event in your world.
You are starting to rock a Corona mane worthy of a leader of a lion pride. Grrrraaaooow!
Question: I always thought the *Three Body Problem* in orbiting planets was the most obvious example of classical indeterminism. What is the difference between Norton’s Dome and rigorously defined mathematical chaos?
Principal Principle was Major Major's high school homeroom teacher
Sean, if probabilities “work” in quantum mechanics according to the Born rule, why do we have Bell’s inequalities?
Sean nice background
26:30 ish - Cryptography. You don't have to try it an infinite number of times, just a finite but large number of times. The probabilities are objective and discrete.
Schrodinger's wave function's unitary evolution takes care of any hidden variables, making the function completely deterministic. Maldacena conjectures that the whole universe is the unitary evolution of a single probability wave function, but we will never know the algorithm, responsible for Bohmian determinism.
P= f(x,Vi)
Everett's self-locating probabilities... seem possibly frequentist to me. Think of little stick man dropping into the top of a Galton board, bouncing down hitting each pole and recording what bin little guy ends up in. Each bounce is a many-worlds split. Well, experiments in quantum mechanics ultimately count proportions of outcomes to compare the Born rule to the prediction of a theory. Little guy can not self-locate at each bounce, only the final bin. The "infinite" or big-number bounce results are the equivalent of long-run frequency.
(Just enjoying being provocative, not claiming this view is correct.)
Thank you, Professor Sean. I have a question, which may be more philosophical, than scientific? It would appear, from our perspective everything generally spins to the left. Planets orbiting suns to the left, all the way up to the galaxy, turning to the left? Why do our wrist-watches, clocks, and round faced timekeepers in towers, and reality turn clockwise? This is counter to the natural counter-clockwise motion in Spacetime that can even be found in sprouting plants as they grow.
Northern Hemisphere bias. There are clockwise plants and counter-clockwise plants.
You might make the argument from amino acid handedness, but clocks pre-date our knowledge of that.
Thanks for calling it Bayes Law and not Bayes Rule. Pet peeve of mine.
(According to me, Bayes rule is a decision rule, a function: observation ->action)
In a video about probability and randomness I'm tempted to do a pun asking "What are the odds of your tablet writing file corrupting for this video" but I won't.
I was also tempted to do a pun about the element Sodium but I though "Na".
Sean, regarding Norton's dome; is it a costume? Are we draping on one thing the appearance of another? Consider that an object can be motionless (velocity zero) yet in the midst of motion, as in the case of a ball at the apex of its trajectory after having been thrown upwards vertically on earth. This is quite different from a motionless thing engaged in no motion, like, say, a rock on the ground of some uninhabited, windless and seismically inactive desert. In the case of the thrown ball, it leaves your hand with speed, slows down under the drag of gravity as it ascends, "pauses" at the top, a point at which it has neither speed nor direction, then begins its descent. Though both the rock in the desert and the thrown ball at its zenith share for a moment, if it can be called "a moment," the trait v=0, it would be wrong to equate their overall states. Isn't the ball on Norton's dome really in motion, but with things presented in such a way that we catch it with momentary (arbitrarily long moment) zero velocity?
Thanks
? It is fair to say that Bohmian mechanics has all the problems of non locality and what the wavefunctions *really* physically is, plus it also has extra variables?
Can you do one on information? I have trouble understanding this concept more than anything else, it often seems like its being used to mean different things but no distinction is ever made.
It's the properties of the wave function (or particle) in question, iirc
In terms of communication, this is Niels Bohr reincarnation level. Precise and yet never trivial.
good stuff. Can you believe this is free? We live an interesting time
So this is a bit of a half baked thought. But, you say that things that only happen once can’t have a frequentist probability, because multiple times need to be sampled for that to make sense. This kinda reminds me of the arguments when people are first introduced to derivatives of “well you can’t have an instantaneous velocity because the notion of “how far you travel over time” requires us to think about what happens to something between multiple times (and not just an instant). So, do you think there’s a way to get around having to use multiple trials for a frequentist perspective using limits or something, like was done in calculus with velocity?
You need to make an episode on dimensions and string theory
Ramon Atila - the word you are looking for is "please".
For the Q and A, can you explain the nondeterminism in classical mechanics a bit more. In particular, I found this article which argues against this idea:
blog.gruffdavies.com/2017/12/24/newtonian-physics-is-deterministic-sorry-norton/
Hello Sean
Layman here. A thought occurred to me while watching some videos of yours and the JR podcast you were on and I wanted to ask a question since I do not have the knowledge of how to do it.
If you were to view spacetime as a liquid and reality as a ever expanding table. Like when you spill water and it pours over the edge, it is slow at first but speeds up as more molecules go over and more are pulled. Could that be what dark energy is. Spacetime pouring over an edge and dragging more with it as it goes. However, because our "table" is expanding the edge keeps moving.
Not that it'd be a literal edge mind. Just an idea I had. Spacetime seems to have a surface tension like water does and gravity being a indent in that surface (at least how it's often pictured) I wondered if other fluid like dynamics would also occur.
@Jeffrey Simmons I didn't think we were losing it per say. Nor a actual edge but spacetime bleeding into to something leading to the acceleration. Some other that pulls and adds to the acceleration of our universes expansion. The edge would be everywhere and nowhere at the same time since there isn't a true edge to the universe. The pulling increases but since spacetime increases too it never is fully "pulled or depleted" but the pulling still adds some acceleration. I just used the water analogy since it is easier to picture the effect I was thinking of. The water never depletes and the table is always expanding from whatever point you pick so the "edge" would constantly be in a different spot relative ro you as the observer but the "pulling" effect is still there. Like I said though lol. Layman but very interested in these subjects and question I had as to what might explain Dark Energy. If I am wrong I am wrong. Just pondering. I guess what I am thinking is what if our universe bleeding into another dimension and that adds energy to our own.
Can someone explain to me how the universe can be determined if random events can happen (eg chaos theory, a particle pops into existence)? It seems to me that even though it is conceptually possible to be able to wind back the hand of time to the Big Bang, given today’s data, and therefore be able to know the history/story of every particle, that this necessarily implies that every particle will do the exact same thing if time started again. There just seems to be too many random quantum events, let alone macro and conscious events, for the history of every particle to be determined.
Thanks Mr. Carroll. Statistically statistics lie or my interpretations are off the chart down the sides of your laptop.
I keep grinning at the choice of background
hmmmm...was that file actually corrupted?
or is Mr Carroll using it as an example of probability and randomness....
(ahem) what. are. the. chances.
When there are virtually an infinite decoherences occuring, the slice of the universe in which we live should become thinner and thinner, so loosing energy. Where will this end? Why some slices are bigger than others?
36:40 "Who is this person walking into my room with this box of gas? And how do you know they aren't tricking you?"
If I don't know who the person is asking me to measure gas in a box, given only the number of atoms and the total energy, they were tricking me.
Subjectively, strangers handing you boxes of hydrogen do not appear to have your best interests at heart.
[edit] 36:55 "Like, they could easily make it so that all the molecules are moving to the left, right? And they hit the wall and they all move to the right!"
What?!? HOW! TEACH ME THIS EASY THING!
Wanna buy a box of ideal gas? It's Methane - from a spherical cow of course !
"Sometimes when presented with evidence that the theory is unlikely, people just become more attached to it." Ah, so we are talking about String Theory in this series. :)
The principle principle sounds very like Occam's razor. The most 'obvious' explanation is probably correct.
And thank you Sean. Such an amazing series. I'm loving this you are such a great teacher. Thank you. 😍
Is Bayes theorem mixing probablity types: subjective (prior / posterior); objective or at least not agent based subjective (likelihood)? Do we care?
Interesting point. In practice even un-conditional probabilities are estimated subjectively, so it’s all the same soup.
You know how your preview frame right now is just the first frame, so you get this slightly uncanny "the picture comes alive" effect at the start? If you ever decide to do a youtube clickbait face preview, it should be the same way - you make the face, then just go right into the into.
Las Vegas was built on this Equation... not in your favor. The only game who gives you a 50/50 % "chance" is colors on roulette. Well a lil bit less than 50 since you have a green too...
"Intrinsically cool"
Probability = risk = thrill seeking?
David Lewis is the Principal Principal of the Principal Principle
Love SC videos but have issues with some of his more philosophical claims. At 51 minutes - gas in box probability distribution can be interpreted as an objective fact about the gas atoms. Based on a central limit theorem we can quantify the expected number of atoms at each velocity with a fairly exact value, an arbitrary small standard deviation. No need for subjective probability.
I was really interested in the comments on determinism. I'm not sure how a 'clockwork universe' with determinism works in theory. Infinite precision is impossible in theory as well as in practice for many reasons. Irrational and real numbers destroy infinite precision and most functions don't have precise solutions (consider the simple case of the three body problem). Newton's equations do not have precise solutions except in trivial cases. Determinism as a philosophical concept seems illogical to me. A universe that obeys Newton's laws cannot be fully deterministic because the solutions to newton's equations mostly have infinite imprecision.
Thank you Simon for your interesting and considered reply. I’m not clear as to the distinction between theory and practice in this context. Either the universe is deterministic or it is not. I might agree that Newton’s equations are deterministic in theory, but the universe is not.
The concept of a unit circle is theoretical and there is no such thing in reality. Integer measurements are impossible as any scientist will confirm. The fact the cos pi is -1 is true if we define angles in terms of pi. Simplifying in this way is a common feature of physics. The speed of light is set to be 1 and so is Plank’s constant to make calculation easier. These are conveniences. Pi divided by pi is 1 and pi to the power of 0 is 1. The universe is not so convenient. Cos (pi/6.34567…..) is the kind of thing we find in nature.
The problem is not simply difficulty with irrational numbers but also with real numbers. Determinism is about predicting the paths, motions, evolution and interaction of particles. For each particle we need three real (infinitely imprecise) spatial coordinates and an infinitely imprecise momentum in the context of all other particles at the same infinitely imprecise time coordinate. Forget for a moment the issues with simultaneity over spatial separations, at a single time we cannot know both direction and velocity/momentum of particle.
It is not simply the difficulty of getting infinite numbers into my head. It is that I cannot specify infinite coordinates such that I avoid chaotic solutions to Newton’s deterministic equations. The greater the time interval between the prediction and the position, the greater will be the amplification of any imprecision in the initial conditions. I think QM also destroys the idea of determinism but that is another story.
How can He sleep with such a full Brain?
Hello smart people,
I wonder about something in the many-worlds interpretation. It's clear that if we consider the universe to be the whole tree that branches off at every quantum measurement, then the universe is purely deterministic. But concretely, the world as we experience it finds itself only inside of one branch of that tree and as far as I understand, the process of how the world "chooses" a branch to go with is left unexplained, or in other words, it is random, hence many-worlds has a big hole in it. The underlying assumption here is realism, the idea that there's an objective reality that is the same for everyone. But it seems that realism is completely at odds with the many-worlds interpretation. Am I wrong ? Or maybe there are good reasons why realism isn't a practical/reasonable assumption ? If my understanding is correct, then wouldn't it be better to have a quantum mechanics interpretation that supports both realism and determinism, which are fundamental to science ?