Your last few videos in this channel is killing it. Need it. Much need it ❤️

I think the opposite. All i see here, is just mathematicians coming up with new approaches to existing problems (made by previous mathematicians) and publishing new approaches. These are not results, and i feel like these are practically useless. So sad to see, that the education system embraces pointless research in such overly sophisticated, yet never applied, fields of science! What a shame, as it happens on the background of the world in fires, looking for help...and what is give?...some over-engineered half solution for made up problems...

Please Riemann hypothesis progress updates...

Uh yeah no, I have to correct you. The continuum hypothesis is UNDECIDABLE in ZFC. Meaning there is no way to decide it. There is nothign to SOLVE there, there is nothing unanswered. It was resolved and understood many many decades ago. We KNOW it is independent and we cannot say c=Aleph_1. We can assume it axiomatically if we so want, or assume its negation and both are EQUALLY valid. What you're talking about here is adding an axiom to create a NEW axiomatic system where we CAN say it, but that does not mean it was "resolved" or anything because we already knew the answer.

Thanks for sparing me the trouble of watching. As anything significant could be buried in such an annal. The only real breakthrough in lamestream science is how to get them to shield for a plasma environment while still thinking almost exclusively in terms of 'heat'. The almost being the novelty. Electricity still being a dirty word in space. Hence its smell at first described from the suits after a spacewalk as that of electric soldering was then peppered with burnt chicken and BBQ insinuations to make for the usual clumsy narrative reminiscent of the sticky tape on the supposed lunar landing module. Ah, who knows what's in the peel of an onion? It's a slow boil to get to the truth and for the cluttered cosmogony of the believers it seems all too much useless toil...

Same! I think in no small part it’s because we have all these KZfaq channels focusing on them! I’d much rather watch Videos about science, exploration and learning than MSM garbage that divides us. Science brings us together!

BS. Gravity (as a separate force) is a hoax. It has been done for a reason.

cringe bruh, stop commenting, ratio + yb better

I love these videos too !!

Quantum gravity is far more simpler to calculate than what modern day Physicists have known to be true.

I was thinking the same thing

Since there are no physical infinite machines so there must be mathematical operators that eliminates these infinities?

Was about to post something similar. The recent famous paper in this area is Jacot et al with the NTK in 2018. It’s also not clear to what extent this explains practice. Eg see the work of chizat and Bach on lazy training.

@@lexusmaxus or is infinity an inversion in some way

Obviously

Congratulations!

What's the paper?

Too bad ieee is just an international conglomerate of science paper resellers. I, and everybody else in this planet want to know why are you writing these papers, and what is you contributed progress. Sorry for the negative tone, and congrats to the publishing 😉

@@zfyl Exactly. Why is everyone writing these papers? And if it is for advancement of science, why is it not accessible to the general public? Is science a business - it is but many intellectuals do not want to see it as such because they want to believe that they do it for "a bigger cause" while in reality they do it selfishly which accidentally sometimes might actually do good, without the original intent being so. Please do not point to Arxiv.

@@sampadmohanty8573 don't you just have to pay for access?

wtf

Some bot stole your comment and slightly reworded it lmao

That's a funny looking cat .

Hi I could take 2 fists of shrooms and it still would have same sense to me as it does right now. Im glad You are happy tho.

@@WsciekleMleko yeah but just a few years ago you could've ate 2 ounces in your example. That's some crazy advancement, only a matter of time

Yo I worked on QML too back in college! I used to devour papers by Anatole Lilienfeld and Matthias Rupp coz of how interesting they were. Gaussian and Laplacian Kernels were the bread and butter of my Kernel Ridge Regression models and I was pleasantly surprised to see kernel vector spaces here lol It's one of the dark horses of ML

The continuum of natural numbers?

@@alexantone5532 Continuum is just a nickname for cardinality of real numbers.

and the new result makes it seem it is less likely that continuum is aleph1, which was Cantor's guess that seemed the most plausible for many years

if you take an infinite number of line segments one centimeter each..then you have an infinite line..this set of line segments are No = aleph 0 infinity..the line is one dimension object.. but! * if you take a square.. one square centimeter in size..the parallel straight sections that make this square up are infinite.. but the set of them is aleph 1 in size..and square in 2 dimension object.. could that be the key of dimentions ? especialy when we have fractals objects to describe?

@@EM-qr4kz I'm not sure if you have some typos or I'm not exactly understanding what you're trying to say. But your analogy of a straight line being the natural numbers and then extending it to a square seems to me like Cantor's prove that the rational numbers are countable and hence of the same cardinality as the natural numbers.

this lol, give us more breakthroughs!

same. I now jack about math, but its so fascinating.

You believe in black holes? Seriously?

Geosquare the GOAT

Support Vector Machines is somehow equivalent to neural networks?? Who knew!?! I would love to see the proof. I might lack the fundamentals to understand everything but it might be interesting anyway...

This has actually been around for 3 decades now. Since the 1990s in fact

Thanks to RBF kernel

yeah me too

No. It means it still sucks half of the time. And in this case i bet it sucks much more than a half. And it means it's useless to watch it since you end up in the same spot you started but fooled & getting more arrogant having an opposite feeling

@@szymonbaranowski8184 when I was a teenager, I was reading Hawking, Brian Green etc and understand maybe 10% the first time. I would read and read again the pages and chapter to understand more each time. The world is a complex place. As a scientific researcher, I face everyday this complexity. Over simplifying is possible and useful. Kurtzgesagt is a pretty neat example. However, in some cases, in my opinion, if you still want to go far, you can't explain it in 10min simply. But well, you are perfectly free to disagree .

I was like. Damn... I know all of these ideas when I was watching it. I guess I can finally taste the fruits of my university education.

@@matthewtang1489 They told me the quadratic formula would be important, but no one said I'd ever need to know set theory. Oh such ripe fruits .-.

is this single handedly makes all this breakthrough just a simple revisiting of an existing conclusion?

the real MVP

@Gergely Kovács: no. Neal's work dealt with neural networks with a single hidden layer, this breakthrough studies the limit of width for deep neural networks, i.e, many hidden layers.

@@daviddodelson8870 Thanks for the clarification. Strange, though, that it took 25 years to take that next step.

We made machines but we don't know they perform task? sounds counter intuitive

Ever done a task that just works, you don’t know how, it just works? Yeah this is it. It’s actually pretty cool

@@ajaykumar-ve5oq more like we can measure how well they perform tasks, so we don't care about the whys :) transparent statistical methods exist and are widely used, just for AI black box methods perform better most of the time

this video is idiotic, since we do know how they work and we have been visualizing them for ages now

Saying we don't know how neural networks work is a stretch to the same caliber of saying we don't know how cars work.

This question was answered in 1964 by Paul Cohen and Kurt Gödel. The Continuum Hypothesis (CH) is _independent_ of Zermelo-Fraenkel axioms (plus the axiom of choice). In other words, standard mathematics cannot prove it nor it's negation. You can, however, extend standard mathematics to include the CH or some other axioms. David Asperó et al "breakthrough" doesn't use only standard math. They only proved the equivalent of two axioms that are known to imply one particular hypothesis that is incompatible with CH... The video is unfortunately very superficial and gives the false idea of an "answer" to a problem that, in my opinion, is already answered.

well... the keyword 'H' being hypothesis of course there's also the "incompleteness theorem", and extending the "axioms" might lead to inconsistency. Indeed "standard math" can't touch it, however including CH might be a little too much. Maybe I was just too "classically" educated, but still... interesting, as was the video, i think.

@@ferdinandkraft857 it independent from ZFC doesn't mean that it's neither true or false. Axiom of pairing, axiom of infinity, axiom of union, etc.. are all independent from each other and we all know they are true. If anything non standard is just a conventional there wouldn't be ZFC as we know it, only ZF. Gödel himself believed that the Continuum hypothesis was wrong, without prove nor disprove rigorously, we still can use logical deduction and reasoning to get a agreeable answer.

@@Noname-67 "Axiom of pairing, axiom of infinity, axiom of union, etc.. are all independent from each other and we all know they are true." define "true" none of them describe the physical universe, there's no reason someone can't say they're false and work with that

okay, this is actually a fair point totally agree

Also because it's (sometimes) beautiful and can engage deeply

That’s everything though not just math

what good is about these things? whom this will benefit?

@@zfyl If you want a really accurate answer, then It is "what" will this benefit which is mainly all of our technology. And only if they're used right, then they'd improve the quality of life overall; but no guarantee on human behavior.

We dont know the size of set of real numbers, we just know its bigger the aleph0. It can be aleph1, aleph2 .... even can be monstrously big as aleph_omega_1 etc. And in current state of most widelly accepted axiomatization of mathematics called ZFC is impossible to sovle continuum hypothesis. One watching this video get the impression that real numbers are aleph1 in size which is not true.

@@IvanGrozev yeah this video is all wrong.

Google is not altruistic ;)

@@martinschulze5399 as long as someone is doing something

i have great respect for the scientists working at google, but as a company it is inevitable that their goals are not always allied with humanity's interests

@@willlowtree yeah my comment wasnt about about goals or anything. just that theyre doing so much science and sharing a lot of it with the world

@@martinschulze5399 Water is wet. Nothing is altruistic.

Parameters are whatever the starting nodes pick up and layers are layers, right? Or are parameters the starting nodes themselves?

@@shambhav9534 In a simple feed-forward neural network like a multilayer perceptron, you can represent a neuron / node by the equation y=h(w*x + b). x is what goes into the layer that neuron belongs to (if it's the first hidden layer, x is just an unchanged input feature vector), y is what goes out. w are the weights (all the edges) connecting all the neurons in the previous layer with the one in the current layer we're currently looking at, b is a bias. '*' is a dot product. h is a nonlinear activation function. The union of all weights and biases of all neurons between all the layers are the parameters which are learned during training.

@@Irrazzo Okay I get it now.

Just one more thing about layers: instead of thinking of layers in terms of the nodes of which they consist, you can also think of them in terms of the data that flows through your network (the x's and y's). Then, layers are different, increasingly abstract representations of your data, connected via transformations, or functions. And the complexity, the 'billions', are due to the enormous size of the function space of the overall function (transformation) which the network approximates by a series (or rather, composition) of functions which only slightly differ from one to the next.

@@Irrazzo I understood nothing but I do think I understand layers. They're layers which modify the starting input and at the end that input becomes the output. I tried(just tried) to make a neural network back in the day, I think I know the basics.

Kernel machines are linear, whereas neural networks are, generally, non-linear. Showing that an infinite-width network can be reduced to linear essentially raises suspicion about whether finite neural networks can be simplified in some novel way as well. The consequences could be groundbreaking.

@@PedroContipelli2 arent all layers of neural network just linear functions of the previous layer, so technically isnt it possible at some conditions a multi layer neural network can be a linear function.

@@satishkpradhan The activation function of each layer (sigmoid, tanh, relu, etc) is usually where the non-linearity is introduced.

@@PedroContipelli2 analog artificial intelligence

I stopped at the "nobody knows how neural networks work" and "billions of hidden layers" sentence. MY GOD, why did they have some moron who has no idea what he's talking about write this? And another one read it? MY GOD.

Seems not as any great or surprising breakthrough then.

thanks for single handedly eradicating the breakthrough level of that paper 😅

Not for dnns though

what useful?

@@zfyl Deep neural nets and quantum physics/gravity.

I learned about kernels back in 2017 when using SVM... How are kernels breakthroughs? If you have inputs that are not activated in 1 dimension, exploding to a higher dimension will not include them... So it's fine to skip the work, DUH

@@seenaman96 I'm not a mathematician and I don't know anything about kernels, but the video wasn't saying that kernels are the breakthrough. It's saying they are the old, easily computible thing that neural nets can be mapped to. The mapping is the breakthrough.

Yea, I agree. I know enough set theory (and it ain't much) to know that this is a bunch of hogwash.

Yes. I agree. Set theory basics are easy enough to understand for undergraduates so it's the most approachable subject among all in these videos but hearing how wrong their explanation is I now must wonder how wrong are their explanations of the other discoveries.

@FriedIcecreamIsAReality I think you make a good point, but I don't think many people would understand "likely true" as "intuitively making sense". That's just not what "true" means.

@FriedIcecreamIsAReality I'm still just a mathematics student, so I'm not in the best position to judge whether it really is used this way, but this video isn't aimed at professors, so I think the phrasing is at least misleading

Dimension in this context just means additional parameters from my understanding. EG: For a light, one dimension might be wavelength, one might be frequency and another might be luminosity

even i thought so... but as i saw all comments of people in amazment i was confused. Thank God someone else also think so ... else I thought to reread everything I had learned... or revisit my analytical thinking.

It is in fact (part of) what my Master's thesis was about and I am quite confused because indeed this has been known for some time already

It’s about dnn here

@@David-rb9lh I did a bit of digging and it turns out that the paper which introduces the result (wide deep neural networks are equivalent to kernel machines) has in fact been written in 2017. Now don't get me wrong, this is a very nice result, but by no means a 2021 breakthrough unfortunately.

@@StratosFair I’m agree with you . I’ve not digged to much into details to be honest .

Yes the continuum hypotheses is known to be undecidable in ZFC since Gödel. It has also been known for a while that if you were to add either of the axioms MM++ or Woodin's axiom (*) to ZFC, then the Continuum hypotheses would be false. Now, the paper by David Asperó and Ralf Schindler proves that (*) is weaker than MM++. This ofcourse has no bearing on the continuum hypotheses at all unless you consider either of them an axiom. How the video chooses to present this is quite odd. I guess the point they are trying to make is that since they were always considered rival axioms and we now know that one actually implies the other we might just add M++ as an axiom to ZFC. Woodin stated something along the lines that we shouldn't accept MM++ or (*) an axiom because MM++ is incompatible with the natural strengthening of (*). Regardless of what that actually means it at least should be clear that there are objections to simply accepting MM++ as an axiom.

@@hunterdjohny4427 Thank you. I also found it weird how they framed it in the video. At least to me it came across that they were implying that the results could also be used in ZFC alone. Hence my question.

@FriedIcecreamIsAReality But isn't that just creating new problems? If I remember Gödel correctly, every sufficiently powerful system of axioms will run into similar problems as the continuum hypothesis. My issue is that the video, at least to how I perceived it, framed the issue in a way that implies that result leads to a result which is "more true". But the notion of true solely depends on the axioms we choose and is subjective to a certain extent.

​@@dEntz88 Adding an axiom to ZFC wouldn't create new problems. Every theorem that was previously provable (or refutable) is still provable (or refutable), and some that were previously undecidable may now be provable (or refutable). So by adding an axiom your theory gets more 'specific'. What Gödel showed is that this process of adding axioms can never lead to a system of mathematics in which every statement is provable (or refutable), unless you add many many axioms in such a way that your set of axioms loses it's recursiveness. This is hardly desirable, since the set of axioms being non-recursive means that if I write down a statement you have no way of telling whether it is an axiom or not, neither will you be able to tell whether a given proof is valid or not. Our only option is to accept that any decent theory of mathematics (decent as in powerful enough to express basic arithmetic) can't be complete. Your issue with the video is correct of course, they pretend statements have an absolute truth value regardless of the system of axioms worked in. What is said at 6:33 is especially bizarre: [MM++ and (*) are both likely true] makes no sense whatsoever since both axioms are independent of ZFC.

@@hunterdjohny4427 Thank you for your explanation. I only have a somewhat superficial knowledge of that area of maths and was actually thinking about the issues you elaborated.

No one's attributing it to this Google employee