Відео

Quantum Mechanics has become the new String Theory! What a fall from grace

z^2 + c^p Easy

Ended too quickly

So you mean to say that there's at least one subset in the deconstruction of the sphere, that does not belong to the sigma-algebra?

mu(emptySet) = 0 is so Euclidean....

if you have a finite list you can't apply that diagonal theory: 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 Ah, now I see why the number of real numbers is actually 2^[aleph null]... The list of natural numbers is n long and the list of real numbers is 2^n long...

Can you do a comparison video showing how long it takes for a double pendulum to come to a complete stop against the time it takes for a standard pendulum simulating the same mass to come to a complete stop, if both pendulums begin in the same initial position? I can't find any good examples of this.

Beautiful, deserves 100x the current number of views

Choice is not solely responsible for 0 volume sets.

Euclidean Geometry is a type of game (a thought experiment) used for the purposes of showing relationships between mathematical operations. Specifically, during the time of its conception, it was used to visualize how algebraic operations can be expressed as relationships. The experiment starts by presenting the operator (typically, that means us) with some information about the game and a set of rules that we must obey as we interact with the experiment. You may have heard of "tangents", "angles", "intersecting lines", and all that familiar stuff taught to us in school. But, what it amounts to telling us is that a "rationally defined structure" (a geometric object) "is scalable". And, by scaling the objects we create up or down, we can prove that the object is an expression of mathematical operations. Well, specifically that it is an expression of "algebraic" operations. (Remember this.) In other words, so long as we can define it as a shape, we can prove it with mathematic operations. However, what our ancestors had not yet realized, even though they, themselves, stipulated within the rules of the experiment, they were very close to being able to define modern mathematics, such as calculus and derivitives, waves, oscillations. There were SO CLOSE! Even thousands of years ago. There were ALMOST there! But, here's what ultimately was overlooked during all that time, and only recently rationalized just 100 years ago by a man named Maxwell Planck while trying to quantize General Relativity. In Geometry, any shape we complete is an object we could express in mathematics. Any shape composed purely of finite expressions were accessible to our ancestors in the form of algebraic equations (or, expressionism). These objects are purely described today as polygons. But, in geometry, we are also told that we can carry out continuous operations, as well. It took people thousands of years to come up with continuous operations that, in the span of time since the game of geometry was composed and the activity of expressing mathematics had finally begun to describe curvatures, we've forgotten the meaning of the "operator". While the ancients understood operations as putting two quantities together, or dividing quantities in halve, they also understood the prospect of square relationships. All of these things they could easily describe as operations. And, in terms of what is doing this operation, Maxwell hit upon the rationality of an "observer" as a component of his theory of quantization. The ancients only describe quantities, and how these quantities relate to one another. Planck realized, without being able to express it clearly, that there is another component of geometry not yet rationalized, which was the "operator". Now, whether you define the operator as a conscious, observant being, or as an intersecting plane of reality crossing through the experiment, it doesn't matter what you describe this operator as. What you need in order to carry out any operation is an objective reference. In 2D space, we may commonly call this a 3D environment looking into a 2D environment. Physicists will say that to intersect a 3D environment, you either need a deity, an axis of time, or you need to add more expressionism (create new dimensions tied to specific relationships). All of these aspects are one and the same. The operator, the "observer" is an objective overview of the experiment. For example, in order to draw a circle, though we can't accurately describe this with algebra, we can do this with a continuous operation making use of algebraic expressions. And, that's an allowable operation in geometry. Anything that is continuous is expressed as a repeating series of algebraic expressions. This includes trying to define a circle by creating an infinitely small triangle to measure it's circumference, or creating a line whose length continuous without end (an infinitely expanding line). In geometry, we are allowed to use TIME as an operator. Whether you describe time as an instantaneous step or as a step carried out smoothly from start to end, the final result is the completion of an operation. The operator, the observer, isn't you, or a deity. It's time. Time is the objective operator in geometry. And, without time, we cannot define objects with infinite series operations. Without time, we cannot describe a circle. Without time, we cannot define volume. And now we know that a point represents a place without time. (But, don't forget, we're only talking about geometric relationships. This was a game about thinking of relationships. This is intrinsically not a model of the universe. But, today, people are starting to realize that this sense of rationality is tied to understanding our universe. Without even realizing it, the ancients already gave us the Unified Theory of Everything. We've had it all along.)

Nice video! I really like Your explanation! I just did not understand one point on 8:20 when You explained Dirac Measure. How S can contain x? If M is a sigma algebra and = P(X), it is a set of subsets of X. And S being a subset of M is subset of subsets of X, so set of sets of elements of X. And x is an element of X. So how x can be an element of S, if it only can be an element of an element of S? Maybe I’ve missed something, I would appreciate an explanation, thank You! ❤

Normal algebra: cannot even measure Sigma algebra: hold my beer

wow im a statistics freshman in Chinese college,your video truly help me understand what *space* means !!love from China.

Can you continue the series to get to lebesgue integrals and proofs of dominated convergence theorem? Would be sick with manim

Incredible introduction to Measure Theory! Giving out examples does wonders for the understanding of the theory. Can't wait to have more quick explainers on this topic. Also, why is it called "sigma" and "algebra"? Is the name arbitrary or is there some historical background?

7:05 Better to say more accurately 'pairwise disjoint'.

luv u

Fantastic video! Thanks again for the positively-biased visuals (black-on-white) 🤗

Clicked on this video bc thumbnail stutters when you scroll on your phone.

These are good videos but they always seem like the end way too early. My advice is to wrap everything up by applying what you have said to what you have shown in the intro. I have no idea what a sigma algebra has to do with the volume of a cube

Edit: Sorry, I missed the word 'disjoint'. My bad. Good explanation! It does not seem to me that the Dirac measure is a measure on the power set of X. One of the properties that you give for a measure is that the measure of the union of a collection of subsets is the sum of the measures of the sets. Consider the set X = {1,2,3}, and the Dirac measure where the fixed x=1. Now the measure of {1,2} is 1 and the measure of {1,3} is 1. Their union is {1,2,3}, which should be measure 2 by the above property, but is actually 1 by the definition of the Dirac measure.

I see you are still learning. But keep it up. The quality is quite high in this video.

Is this still manim? Looks good

You should make a followup where you explain how this helps us solve the problem with the cube!

GREAAAT VIDEO KEEP IT UP

Great explanation

I don't see the connection of any of the algebra with the cube you show at the beginning

Infinite chocolate bonbon hack?

An absolute masterpiece

Part regarding how finite subsets would make it work feels omitted. Worth making a follow up? I would approach it from non-measurability point of view, probably by looking at Vitali set and extending

ill watch your other videos this was great

The thumbnail of this video dances in really cool way if you scroll it back and forth. Because the details are really minute and are messing with the rastorisation.

Nice visuals! I've always found Measure Theory to be counterintuitive.

This thumbnail is so trippy when i was scrolling it was moving

At the time 1:15 you say that the map (0,2pi) --> R^2 x |-> (cos(x), sin(x)) « Will give you the whole circle back », how does that include the point (1,0) since 0 and 2pi are not to be mapped?

This playlist has been so helpful... do you plan on making more? 🥹

It really is a nonsensical axiom and it should not have been accepted (even if the resulting theory turns out to be consistent).

Does this mean, you can reach any point from any other point?

So, which part of the sphere is non-measurable? And why? Like, I can totally cut up an orange and real life and I can definitely always measure every piece I cut out.

A surface is not an infinite collection of points. A line is not an infinite collection of points. You can see this because a surface has an area but the area of a point is zero. Zero times infinity is still just zero.

I'm having trouble understanding the second criterion. Could you clarify what you mean by "orbit"?

Did he really need a million dollars? He’s right next to Pythagoras in history. Let that sink in.

And yet people still accept ZFC. So sad that people are obsessed with an axiom set that induces paradoxical results.

Lottery

Great video and awesome animations! Subscribed!

Maybe pointed out already but there is a y missing in the formula of the thumbnail

Set theory is beautiful 🚪

Mandelbrot powers

The paradox is clearly the result of poor axioms. This is inevitable when mathematicians and scientists stay too far removed from the real world (see the almost complete stalling of physics in the last 50 years)