The Biggest Ideas in the Universe | Q&A 2 - Change

[@filiphedvicak]

The quality of free education in 2020 is amazing! Very happy to be around

[@mittelwelle_531_khz]

But sadly enough (at least to me) it seems also scientific illiteracy is on the rise. And more ironically, the means by which most unscientific nonsense is spread nowadays are "social networks", which in their highly evolved form wouldn't even exist hadn't been their generations of scientifically minded people before.

[@SatishfiedGaming]

@@mittelwelle_531_khz p⁰

[@Psnym]

Very happy Dr. Sean Carrol is around :)

[@pierfrancescopeperoni]

Ikr, physicists who can freely speak to actually interested people without having to attract uninterested audience. We also have lectures of MIT and other universities for free. All this was impossible on TV.

[@nigelbrayshaw2709]

Easily my favourite astro/particle/quantum physicosmologist.

[@Ron4885]

Nigel, pretty much sums it up :-)

[@kjrunia]

Every time, I’m amazed by your clear and patient way of explaining fundamental and sometimes technical subjects. Love it!

[@freeair9460]

Sean I've loved science for 30 years but my teachers weren't all that bright or easy to work with. I'm greatful for you doing this. I learn so much and so much easier from you. Thank you

[@rtravkin]

16:00 You could also say that integration is harder (than differentiation) because it is defined as a limit of a sum whose *number of summands* gets *bigger and bigger* as the step goes to 0, so the sum's complexity is unbounded, whereas the derivative is the limit of just a simple ratio of differences-so the expression under the limit has *bounded complexity* . (A comment from a mathematician.)

[@michaeljburt]

Sean- you are a national treasure for the proliferation of physics! It's incredible how humble you are. Unfortunately, I have dealt with many physicists (I worked in photonics for a while) who do not have the patience to explain the fundamentals... awesome stuff. Hope to see more.

[@PaulinaFriedman1974]

International treasure :) Great explanations for those who are too mathematically-inclined, like yours truly. Thank you for those videos.

[@КонстантинНовицкий-е8о]

People forget, that Mathematic and Physiс are just reflecting the World, not vice versa... They think that they BUILD Nature laws when they do Math, or Physical experiments.... But real is different. They build nothing, just reflekts a little slise of the World...

[@suarezledo]

Sean: you are amazing. Thank you very much.

[@SerDesper]

Thank you a lot, Professor Carroll, for these wonderful series with a wonderful idea behind them! I don't know anyone who did this type of lectures before. By the way, probably one answer to the question "why integration is harder than differentiation" is that differentiation is essentially a local operator, a local property of a function, while integral depends on the whole function, thus it is global. And it perfectly makes sense that doing any "global" analysis is harder than doing a "local" one.

[@LearningWithSuj]

This is a fantastic series! Thank you, Dr.Carroll !

[@ThatNateGuy]

Dr. Carroll, you're one of the best communicators of physics and mathematics I've ever seen. Not only do you understand the material more than adequately, but you're also an excellent orator. I've been following your work for about a decade now and you only seem to improve. Thank you for all of your work!

[@matthewkrellwitz8973]

Sean Carroll is the Bob Ross of Physics.

[@McCaffreyPickleball]

Thanks for keeping this series going! :)

[@gelonzo71]

Love the new background. Looks like chalk on a a blackboard and instantly sets me in learning mode.

[@papsaebus8606]

Thanks for the dark background, I really appreciate it🔥🔥

[@ecojulie]

Thank you, Dr. Carroll, so much for this series. The hunger to learn is very much here. Grateful for the investment of your time to help us learn these ideas.

[@davdark433]

I absolutely love these videos and I have been taking notes which helps me to understand it a little bit better. I've even started trying to learn calculus.

[@Why_Alex_Beats_Bobbie]

Excellent video Sean! Let me try to add to the conversation a bit if I may, with respect to the question of why is Integration harder than Differentiation. One answer that trivializes the question is that we think of Integrals as anti-Derivatives (which is sound by the Fundamental Theorem of Calculus) so that approach makes derivatives more elementary than Integrals. This does not necessarily answer the question however, since it does not explain why we do not think of integrals as standalone quantities and instead we invoke a duality to calculate them. Digging a bit deeper we see that Derivatives are just limits of fractions while Integrals are limits of infinite sums. In principle, infinite sums are much harder to manipulate as evidence by the lack of closed forms for many such sums, including the famous Riemann zeta function (which relates to Riemann hypothesis). At yet a deeper level of abstraction, Differentiation is more "local" than Integration in some sense. Although both concepts require limits and thus an open region to be defined, we can talk about the derivative of a single point, while the integral below a single point is always zero. As a result, to get the structure of a function, we always need to talk about the integral of a non-trivial region. Hope this helps.

[@eradawnz]

Great comment 👍 Interesting ways to think about the relationship between differentiation and integration

[@StormyJoeseph]

Thank you so much for taking the time to make these videos. They are greatly appreciated!

[@3dlabs99]

16:00 Isnt it because getting the slope of a curve only takes a little bit of the local curve but getting the area depends on the entire curve up to the point. So you can get the slope by looking at a much smaller section of the curve and therefore its less complex.

[@user-qf3lq4zj8g]

Notability's *Dark* mode was *notably a bright* improvement! Thank you Sean!

[@ba0cbmft]

@9:00 I would argue that the resolution to Zeno's paradox is that the unitary measure is being changed at each interval which is reflected as the denominator in the ratio of distance and therefore once you pick a specific unitary measure, you no longer have infinite intervals through which to pass since the total distance can be measured in that unit.

[@pipertripp]

at 12:20, I think that that's called a Koch Curve. The interesting thing about it is that you can inscribe and infinite length inside a finite area.

[@pa28]

Sean, I am really enjoying this series. Very much looking forward to new videos. Not really a fan of light on dark written presentation, but I'm also pragmatic that it seems to be very popular. I think your compromise grey texture pattern is a good one. Thank you for doing this.

[@GandalfDoesScience]

"Is velocity really an instantaneous quantity?" Not going to lie, I had this same question in grade 11 physics. I hadn't taken calculus before and didn't end up taking it for another year, so I have to thank my highschool physics teacher for giving me a good lesson on the basics. :)

[@pipertripp]

3Blue1Brown talks about this in his Essence of Calculus series, near the beginning of it IIRC. Worth your time if you haven't seen it.

[@ToriKo_]

@@pipertripp thanks for the recommendation

[@pipertripp]

For sure!

[@sbares]

As for why integration is harder than differentiation, maybe it's more enlightening to ask the question "why is differentiation so easy?" We know the answer to that: it's because we have the chain rule. Together with linearity, and the product rule, this means the set of "easily differentiable functions" is closed under most everyday operations.

[@maratonec5]

amazing series, Sean, keep on, thanks

[@SonaliSenguptasengupso41]

Question 1- Well explained. Fractal question- great !

[@anvillal.7787]

Sean's taken "so be it" to another level.

[@akumar7366]

This is geat particular I have very limited knowledge on physics so Iam loving this, I have a keen interest in the origins of the universe and by logical progression to its end. I have to say I feel really attracted to Sir Roger Penrose CCC theory, Iam hoping this topic in general is covered ♥

[@NerdyRodent]

You can change the brightness, contrast and colours prior to applying the chroma keying for a cleaner key.

[@bobabernathy6108]

Some of the most valuable content regardless of quarantine, thanks and good job.

[@cututorials]

I use your videos as background noice when I study physics, simply because i find your voice soothing :D

[@brian-kt1rc]

The reason why the problem and its inverse are not equally easy is because the one way you are sort of unpacking it and increasing entropy which is always easier to do. The other way you are repacking it and so you are decreasing entropy when reordering it. This decreasing entropy is not the most natural direction for a state of nature to progress

[@r.murphy2311]

I think of integration being more difficult than differentiation as being similar to how it's usually easier to tie a knot than to untie that same not. Exceptions to this exist in both domains of course but I think that's a relatable way of thinking about it.

[@marianoamar3867]

As a very common math analogy, you can also take Powers and Roots, or Exponentials and Logarithms. They're inverses of each other, but one direction is VERY much harder than the other

[@CstriderNNS]

The reason Diff is harder then Int is because Dif deals through entropy, and Int works through the Shannon entropy . The amount of information in the shannon entropy , is way more ordered , hence the more difficulty in finding the Int .

[@deeptochatterjee532]

My answer to why differentiation is easier than integration is that we have functions that we know how to differentiate and integrate (polynomials, exponentials trigonometric, etc.). To make functions more complicated to differentiate or integrate, we generally multiply them or compose them together. We have rules for differentiating those sort of complications of functions that are very simple, but our rules for integration of those same complication don't always make the computation any easier. Notably, differentiation tends to separate out the complication, while the integration just integrates the same sort of complication but with different functions

[@ronmexico5908]

Sean, who's ideas interest you the most? Ed Witten, Hawking, Einstein or someone(or several) else? I find ideas that interest me the most kind of make time stop as my internal cpu goes close to 100%. One of your Q&A videos on your most recent book put my brain in overdrive.

[@gnarlsley]

These videos help so much because without school I just feel dumb. Watching your videos keeps my mind sharp (y)

[@bntagkas]

having a pretty hard time to follow most of it since my background on math is pretty non existant, but i must say i find this fabric dark background awesome

[@jainalabdin4923]

Time is discrete rather than continuous if you consider quantum mechanics, where nothing smaller than the Planck Length makes sense, and therefore nothing smaller than the Planck Time makes sense.

[@youtou252]

OMG during the previous video I was slightly annoyed by the light background and now it's fixed :D I love Sean so much

[@Cooldrums777]

I think that the answer to the last question discussing Higgs field and particle mass was the most interesting portion of the whole video. Would like to see more of that.

[@kelltiol]

Proof that NP>P: Let's look at a measurement of the slope of a curve in a complex system, let's say, the change in kinetic energy of a rock impacting dirt. The dispersion of kinetic energy occurring at the instant the rock's momentum begins changing is easily measurable; we've described a "Discreet" system that contains presumably known values (the rock's mass and relative velocity). This dispersion will occur OVER TIME, which removes our ability to use discreet measurement terms for the entire system while retaining precision. "How many newtons of energy is that rock about to deposit into the dirt over how much time?". Easy. versus: "How many individual crystals of silicon dirt is that rock going to displace by the time its kinetic energy finishes dispersing?". Hard. One of these questions, the force of the rock, is about something we can observe in the present. The other, the amount of dirt about to be displaced, has low knowability/precision. A given characteristic of a system at a point in time can be defined as the answer to a question about that system's previous or future characteristics. The P-value of a given question about a given characteristic of a system is dependant on whether the axioms used include more or less "chance", or apparent randomness, between the time of the question and the system's most known state (as a function of our observational ability). Now the same questions in reverse after the event: "How much dirt did that rock displace?" is now easy. Present. "How much force would that rock have to have in order to displace exactly the amount of dirt that it did?" is now hard. Past. The second question now requires non-discreet measurements to acheive precision. Asking high-precision questions with limited data relative to where (or when) you're currently observing from, is HARD! Imagine four parties present: 1) Observer (asker of question), 2) data available when question is asked (discreet&known values), 3} values that make the system non-discreet over time (MANY currently unknown values due to either lack of sufficient observation or high number of potential observations), 4) currently unknown value of the FINAL (or least-energectic-state) system. I think the 'question difficulty' we're looking at is the observers location in time relative to the data required to answer their own question, depending on the 'chrono-axioms' they used. **"Easily Solvable" questions require knowledge of a system (observations made relative to it). The number of potential observations that can be made is smaller than the number of potential 'system-states', because we can't see everything all the time. Therefore NP>P.** If we theoretically say that we can observe all systems, then you've accidentally skipped this whole question!!! HAHAHAHAHAHAHA ...I'm sure I could be quite wrong here, lacking much formal secondary education, but this feels pretty right. What does everyone think?

[@dave31415]

The second last question: Is there any evidence of the discrete nature of time? One is the Planck time. It is extremely tiny, on the order of 10^-43 seconds. It comes from an attempt to find natural or universal units. It might just be the smallest measurable time and not that time is actually discrete.

[@tomserb9752]

I think the reason you can't take a derivative of a fractal is simple: a fractal isn't a function - if you look at your sketch of a simple fractal you'll see that for some values of the horizontal axis you have more than one value on the vertical axis.

[@kisslaci36]

2 question for this Q & A: Time being discrete or not. Don't we have the Planck lengths, Planck time showing space and time are not divisible ad infinitum? And when you talked about Lorrentz invariance, I was never really sure on this point: do time dilation and space contraction only happen when accelerating and if so why? or how can it not matter what speed you're travelling with?

[@kenyo5087]

Absolutely love it

[@claritas6557]

The kitty at 40:00 really made me happy

[@xcq1]

What do you mean the videos get too long? I'm putting them on, hearing a few interesting thoughts, and *bam* it's over again. And I would have thought the question about 'Are Newton's laws true everywhere' wanted to hear some thoughts on dark matter and mond.

[@iruleandyoudont9]

you're a boss for doing this

[@therugburnz]

I like the chalkboard look for the notes. In uni I liked chalkboard in small classes and white background overhead projector displays in large lectures. In Maths I liked the chalkboard classes not only because it meant the class was small it also slowed down the instructors speech whose first and second language wasn't english.

[@astrorad2000]

Thank you for this wonderful series.

[@Earth4Mars]

Speed of light appears to have changed in different material because it takes longer distance as it proprgare through the material

[@whip8]

Thank you for the discreetness of time.

[@_John_Sean_Walker]

Much appreciated.

[@bryanroland8649]

The answers raise new questions. Isn't the Planck time the shortest possible time in which anything at all can happen and therefore a discrete unit of time?

[@andreac7a]

Great video. And the dark background is much much better!

[@davidwalden1441]

My half-hearted explanation of Zeno's Paradox is in what I call Zeno's Corollary. Which would say, in order to get half way he has to go half of that distance (1/4). But to get to the quarter point, he has to go to the 1/8th point and so forth. Therefore, Zeno never got anywhere because he never left. 🤣

[@MyWissam]

Looks nice, meaning great. Looks like an archetypal blackboard.

[@markdavis1338]

RE: Question1 about position...... I think I was confused because; I think you actually defined a way to find 'change of position' - not 'position' - something I equate as 'location'....... I'm thinking vectors will be introduced shortly....

[@judgeomega]

you got some really good questions.

[@astrojames]

Green screen technique is on point 👌

[@astrojames]

On iPad you can now use a trackpad so if you need a cursor you can use that.

[@whip8]

Great production improvements.

[@MrIlispy]

Sean congrats for the program. We love it. Pls just change the background behind you... This one makes you look like a local channel astrologist telling the romantic future to divorced housewives

[@DanDill]

Sean, a suggestion: When you are referring to a part of a drawing, use a different color so we can see what you are referring to.

[@wafikiri_]

A better suggestion: Sean, can you get a pointer showing on Notability?

[@MattOGormanSmith]

That background in Notability, I wondered if it was the herringbone weave of the Turin Shroud. I thought that would be a particularly advanced form of trolling :D

[@ssshurley]

Brilliant as usual! The end gets heavy, meta-heavy.

[@Chayonray]

Excellent video lecture Mr. Carroll! In listening to your response to whether or not time is continuous or discrete (and your explanation of Lorenz invariance), does Planck time make an argument for the discreteness of time in that time cannot be measured (in fact may not have a meaning) at quantities less than the Planck time?

[@flymypg]

I suspect the first question may have been about "actual" position: "I've been driving at 60 mph all day. How much closer to Chicago am I?" The underlying notion is that integrating velocity cannot tell you anything about the actual physical position itself, only how it changed. Additional information beyond just the velocity is required, namely the original or starting position. Many students have lost points for omitting a stand-in for that initial value: x-naught.

[@paulperkins1615]

The difference between telling a computer and telling a person is this. A computer wants all its inputs explicitly stated. A person makes assumptions, and if you state too many inputs that s/he was going to assume anyway, they will get bored and stop paying attention before you get to the point.

[@ytinformes2]

Cool! I got the answer in the Q&A7. The usual misinterpretation suspect, I suppose. What about this for the discreteness of time: The Planck time is the time it would take a photon travelling at the speed of light to across a distance equal to the Planck length. This is the 'quantum of time', the smallest measurement of time that has any meaning, and is equal to 10-43 seconds. No smaller division of time has any meaning. Just wondering.

[@trebledog]

In your example of velocity to x(t), i wish you could have computed the derivative using the factors, 1 hour, 60mph. Probably too straightforward, but same examplxe with the slope a smooth curve. I struggled with calculus I, II, and flunked III, 50 years ago. Not enough pot I think, but such a joy to actually solve limits and integrals. Differential equations is. Where I got off the bus. Thanks for this series foe my self esteem.

[@henrikwannheden7114]

The latency of Notability seems to be less than Notes Plus. Also a change for the good.

[@alexmartian3972]

19:14 "everyone knows the answer .. but no one has been able to prove". How do they know then? Why are everybody so sure? I think the answer to that paradox is that maybe nobody knows the answer, they just think it is most probable one.

[@exhibitexpressevidence9919]

Thanks!

[@bryandraughn9830]

I'm learning that "binding energy" contributes to the " rest mass". That mass is already energy and no "conversion" is necessary . Could you elaborate?

[@dakinmaher4522]

Sean Carroll you have rock star status!

[@physicsismyfiancee...1353]

Sir I think in the zeno paradox, does it not be 1/2,1/4,1/8...? Pls clarify this

[@itellyouforfree7238]

Here is a possible way of understanding why symbolic integration is harder than symbolic differentiation. The basic fact is that if you consider a certain specific set of functions, the elementary functions, and all possible ways you can compose them, then when you take the derivative you still get a function which is a composition of these elementary functions that we more or less arbitrarily selected. Why is that? Because there are two specific rules, the chain rule and the total/partial derivative, than enable you to differentiate a composition of functions whenever you are already capable of differentiating the single individual pieces. No analogous rules exist for integration. And in fact there are simple enough compositions of elementary functions whose integrals are not compositions of elementary functions. Have a look here en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra).

[@schelsullivan]

Your green screen lighting looks pretty spot-on now

[@dmi0]

So, if there's an elementary function x(t), and a function v(t) such that v = dx/dt, what's the complexity class of finding v given x, and what's the complexity class of finding x given v?

[@auxbonnieux]

"Get used to that." Lol

[@steadyeddy6526]

It always seems easier one way than it's opposite. Easier to go downhill than uphill. Easier to lower the weight, than lift it up.

[@derricksteed3466]

Zeno - nice explanation, but what about "The Stadium" paradox? You seem to be assuming the continuity of time.

[@DanielKarbach]

Indeed nice work on the green screen, it's almost unnoticeable now

[@Pjaypt]

Excellent background!

[@MyWissam]

Would you please elaborate on the interaction or collaboration of experimental and and theoretical physicists in coming up with tests of a theory. As you said, we don't test ideas, but theories...what kind of experiments are worth doing, or how is their worthiness determined in the community? If the only accepted experiments are ones coming from theoretical predictions, aren't we losing an element of "amateur" science ... by this I mean, for example, the way Faraday worked (I think), experimenting and observing, and later theory followed...

[@robinhodgkinson]

Until today I have never heard of Zenos’s paradox. But the exact quandary came to me as as a kid, and occasionally all my life it’s popped back into my head. And here I was thinking no one had every considered it. Lol

[@atmostud39]

For some reason I thought there was a discrete smallest length of time to coincide with the Planck length of space. (?)

[@Sednoob]

Thank you.

[@Snowypeak-e3n]

I wanted a teacher like you in School. ❤

[@apistogramma2296]

I'm still struggling with why kinetic energy = 1/2 * mass * velocity^2. Accelerating an object from (for example) 10 to 11 meters/second imparts more kinetic energy to that object than accelerating it from 3 to 4 meters/second? This confuses me. Does it have something to do with inertial frames of reference?

[@kevinb.3541]

If you remember that kinetic energy is the accumulation of momentum as the velocity changes, it's not so hard to see why. The accumulation of momentum as the velocity changes from v1 to v2, is just the integral of mv from v1 to v2 or equivalently the area under the curve y(v) = mv between v1 and v2. Thus it can easily be seen that the area under the momentum curve between v1=3 and v2=4 is less than the area between v1=10 and v2=11 (Since y(v) = mv is just a linear function). This is actually quite an important fact in spaceflight; If a spacecraft in orbit uses its engine when its speed is greatest (at its lowest point of the orbit), then it will gain more kinetic energy out of a given amount of fuel, than if it were to burn at the highest point of the orbit where its speed is the lowest (Look up: the Oberth effect).

[@apistogramma2296]

@@kevinb.3541 Thanks for trying to help me out. I get that the formula for Ek is the integral of m*v, I just don't understand why that should be. I've always heard 'momentum' as just being a colloquial term for kinetic energy.' So I apparently don't understand this other subtle definition of 'momentum' that is the 'rate of change' (i.e. the derivative) of kinetic energy.

[@JohnDlugosz]

watch "The Mechanical Universe", a PBS series from the 80's. The complete series is watchable for free on its own website. The first episode or two explains this as a foundation.

[@famistudio]

Green screen looks great now. No more green hair! Your definitely getting better at this. Keep it up!

[@silent_traveller7]

Great work Professor, enjoyed ur textbook on GR, but had to say your live lectures are more lively than ur texts.

[@emrazum]

yeah your green screen key is a lot better now (idk how it works look good tho) & longer videos are now all the rage

[@CliqueSpace]

Is the universe expanding, or is the speed of light decreasing?

[@КонстантинНовицкий-е8о]

I have set some questions for you in Facebook, that is correct place for it, or should I write them here?

[@SuperemeRed]

Wow, nice background

[@reisanibal1]

If p = np, then integration is as "easy" as differentiation.

[@davemagaldadze]

Sean drew a Coronavirus at 29:00 :(( Awesome videos, couldn't be more thankful!

[@ithruyou]

I don't think that calculus gives a resolution to Zeno's paradox. The argument with time will only end up with the conclusion that time doesn't flow. When the limit of a series exists, it exists because we defined so in certain cases. We cannot change the fact that if we admit that there exist infinitely many such mid points and that to get to 1 we need to pass through all of them one by one, then we cannot get to 1 because there are endless tasks to complete.

[@ithruyou]

I forgot to thank you for this series of yours. I am really enjoying it and learning a lot from it. Thank you.