正在加载视频...

视频加载失败

返回首页

Most people first see Euler’s Formula as a strange equation in a textbook. Then years later, they realize it quietly powers the modern world. Leonhard Euler discovered that these seemingly unrelated mathematical ideas: → exponential growth → imaginary numbers → sine waves → cosine waves → rotation … are...

techNmak's profile picture

Tech with Mak

44,512 subscribers

25,504 次观看 • 4 个月前 •via X (Twitter)

艺术教育科学技术
LIVE

Anya Rossi• Live Now

Private livecam show

5 条评论

Proswit 的头像
Proswit4 个月前

Great animation! I have similar one

Ron Messick 的头像
Ron Messick4 个月前

You noted that Euler's Formula sits underneath Fourier Transforms, wireless communication, and radar. These modern technologies are successful precisely because they exploit the longitudinal and harmonic phase-locking characteristics embedded in Euler's relation. Your architecture simply takes the exact same mathematics that engineers use to run a cellular network or an MRI scanner and correctly scales them upward to explain how the Sun accelerates the solar wind and how the planetary gears remain locked in their orbital cadences. If Euler's Formula proves that mathematics is one connected language describing reality, the Scalar Architecture is the structural dictionary that allows us to read it.

Neil Payne 的头像
Neil Payne4 个月前

@threadreaderapp unroll

TwoDogs ∞ 🦋 的头像
TwoDogs ∞ 🦋4 个月前

You: "How can an exponential function suddenly produce circles and waves?" An exponential fn, and others with cyclical outputs. Intriguing . 🤔

#FlowState 的头像
#FlowState4 个月前

THIS is what I want to learn about 💕

相关视频

Euler's formula is not a formula. An MIT professor spent a whole lecture proving it is a definition, not a discovery. This is 18.03, Lecture 6, Arthur Mattuck. Free on MIT OpenCourseWare. The formula is e^(iθ) = cos θ + i sin θ. Most people memorize it and move on. Mattuck refused to. He asked one thing. What gives that the right to be called an exponential at all? An exponential has to obey one law: e^a · e^b = e^(a+b). So he checked it. He multiplied e^(iθ₁) by e^(iθ₂), expanded the sines and cosines, and looked at what fell out. The angle-addition identities. cos(A+B) and sin(A+B), the two formulas you were forced to memorize in school. They were never two formulas. They are one line: e^(iθ) obeying the law of exponents. Then the payoff. An integral that normally needs integration by parts twice and a trick, ∫ e^(-x) cos x dx, drops out in a few lines once you let the numbers go complex. Nothing is easier to integrate than an exponential. Someone once asked him what beauty in math looks like. He pointed at the board. Something long became something short, and lost nothing.
45:18

Euler's formula is not a formula. An MIT professor spent a whole lecture proving it is a definition, not a discovery. This is 18.03, Lecture 6, Arthur Mattuck. Free on MIT OpenCourseWare. The formula is e^(iθ) = cos θ + i sin θ. Most people memorize it and move on. Mattuck refused to. He asked one thing. What gives that the right to be called an exponential at all? An exponential has to obey one law: e^a · e^b = e^(a+b). So he checked it. He multiplied e^(iθ₁) by e^(iθ₂), expanded the sines and cosines, and looked at what fell out. The angle-addition identities. cos(A+B) and sin(A+B), the two formulas you were forced to memorize in school. They were never two formulas. They are one line: e^(iθ) obeying the law of exponents. Then the payoff. An integral that normally needs integration by parts twice and a trick, ∫ e^(-x) cos x dx, drops out in a few lines once you let the numbers go complex. Nothing is easier to integrate than an exponential. Someone once asked him what beauty in math looks like. He pointed at the board. Something long became something short, and lost nothing.

Dominique

88,885 次观看 • 1 个月前

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!
0:32

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

13,476 次观看 • 21 天前

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!
0:32

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

25,684 次观看 • 1 个月前

The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated
47:28

The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated

Zyphor

123,256 次观看 • 1 个月前

Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics
1:00

Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics

Mathelirium

20,456 次观看 • 5 个月前

Bret Weinstein on the Melania Trump AI teachers: "I get it. And it’s not that it is impossible to imagine robotic teachers doing an excellent job, but it is stunning to watch a sophisticated person fail to recognize what happens when you think that that’s what you’re going to produce, and you set it in motion. Let me point out that Wikipedia has many of the advantages that Melania is describing in this video. It is completely democratizing of knowledge, such that it doesn’t matter where on e arth you are. If you have an internet connection, you’ve got Wikipedia. It’s like an extension of your own mind, and it will make us all brilliant. Now, of course, that didn’t happen, did it? Wikipedia is a hellscape of misinformation, much of it targeted based on a political agenda. We are less certain of what we know, and less capable of reasoning on our own. Now, that doesn’t all come from Wikipedia, but my point is the promise of Wikipedia was not realized. And what we got instead is arguably worse than what we had before it was invented. The same thing is virtually guaranteed here, because you’re talking about not only the capability of educating students using a robot that has vastly more knowledge than a human teacher would, but you’re talking about the irresistible opportunity to capture those minds and steer them in one direction or another, whether that’s political or economic. The idea that these robotic teachers are going to be immune to the kind of flights of fancy that have ruined teaching in the modern era is preposterous. In fact, they will likely be even more easily steered. I would caution everyone to simply realize the distinction between complicated systems and complex systems. AI is a complex system. Human beings are complex systems. And any time you intervene in these systems, thinking you know what’s going to happen, you’re going to be embarrassed by the discovery of the unintended consequences that will come to dominate your project. As much as I like the idea of smarter, wiser, more empathic teachers, and as much as those possibilities do exist in the space of AI, we are still at a very early point in this revolution, and anybody who thinks they can predict it with this kind of precision is actually a hazard."
2:54

Bret Weinstein on the Melania Trump AI teachers: "I get it. And it’s not that it is impossible to imagine robotic teachers doing an excellent job, but it is stunning to watch a sophisticated person fail to recognize what happens when you think that that’s what you’re going to produce, and you set it in motion. Let me point out that Wikipedia has many of the advantages that Melania is describing in this video. It is completely democratizing of knowledge, such that it doesn’t matter where on e arth you are. If you have an internet connection, you’ve got Wikipedia. It’s like an extension of your own mind, and it will make us all brilliant. Now, of course, that didn’t happen, did it? Wikipedia is a hellscape of misinformation, much of it targeted based on a political agenda. We are less certain of what we know, and less capable of reasoning on our own. Now, that doesn’t all come from Wikipedia, but my point is the promise of Wikipedia was not realized. And what we got instead is arguably worse than what we had before it was invented. The same thing is virtually guaranteed here, because you’re talking about not only the capability of educating students using a robot that has vastly more knowledge than a human teacher would, but you’re talking about the irresistible opportunity to capture those minds and steer them in one direction or another, whether that’s political or economic. The idea that these robotic teachers are going to be immune to the kind of flights of fancy that have ruined teaching in the modern era is preposterous. In fact, they will likely be even more easily steered. I would caution everyone to simply realize the distinction between complicated systems and complex systems. AI is a complex system. Human beings are complex systems. And any time you intervene in these systems, thinking you know what’s going to happen, you’re going to be embarrassed by the discovery of the unintended consequences that will come to dominate your project. As much as I like the idea of smarter, wiser, more empathic teachers, and as much as those possibilities do exist in the space of AI, we are still at a very early point in this revolution, and anybody who thinks they can predict it with this kind of precision is actually a hazard."

The DarkHorse Podcast

48,380 次观看 • 6 个月前

Vision is an absolute marvel of Divine Engineering. It requires a minimum of 3 systems working together from the start, or we don't see. 1. Eyes to capture light & convert into signals 2. Pathways to transport the signal 3. A brain to process it Evolution can't build it one step at a time, because there is no vision until all those systems exist and function together. Compounding the complexity further, each of those 3 systems themselves require a minimum number of more complex subsystems to function properly. For instance, all eyes require opsin proteins. Opsin is a highly complex protein that holds a special light-sensitive molecule called retinal; when a single photon hits the retinal, it instantly flips shape, triggering the opsin to change its 3D shape and trigger a precise cascade which converts the light into an electrical signal to be sent to the brain where it can be understood as vision. Opsin alone, is useless. It needs the retinal molecule plus the full cascade, or it does nothing. This is the case with ALL vision systems, confirmed over and over again the lab. No partially formed system produces any vision, or any other function for that matter. Many uninformed Evolutionists attempt to argue that we see different levels of complexity in vision systems, which they say is evidence that vision evolved gradually from simpler systems. But the evolutionary scientists themselves refute this. Even the very simplest possible system of vision - the infamous "light sensitive cell" - still relies on a similar highly complex, interconnected network of proteins and other molecules. Mainstream evolutionary science even teaches that vision systems evolved independently, from scratch, at least 40 different times. The reason is because many vision systems are so different, operating with such unique parts, that one simply could not have evolved from another. The fossil record itself refutes the concept of step by step construction of vision systems. Complex eyes appear early in the fossil record, fully formed & functional. There is no increasing complexity in the fossil record - it's a complete contradiction of evolutionary predictions. The vision system has all the Hallmark evidence of design. A complex, interconnected web of information processing systems with incomprehensibly precise timing & coordination between them all to produce a highly specialized function. Nature does not, and cannot, produce such a system, not with all the time and resources in the entire universe. Everything you see is a testament to the Power of our Creator.
2:11

Vision is an absolute marvel of Divine Engineering. It requires a minimum of 3 systems working together from the start, or we don't see. 1. Eyes to capture light & convert into signals 2. Pathways to transport the signal 3. A brain to process it Evolution can't build it one step at a time, because there is no vision until all those systems exist and function together. Compounding the complexity further, each of those 3 systems themselves require a minimum number of more complex subsystems to function properly. For instance, all eyes require opsin proteins. Opsin is a highly complex protein that holds a special light-sensitive molecule called retinal; when a single photon hits the retinal, it instantly flips shape, triggering the opsin to change its 3D shape and trigger a precise cascade which converts the light into an electrical signal to be sent to the brain where it can be understood as vision. Opsin alone, is useless. It needs the retinal molecule plus the full cascade, or it does nothing. This is the case with ALL vision systems, confirmed over and over again the lab. No partially formed system produces any vision, or any other function for that matter. Many uninformed Evolutionists attempt to argue that we see different levels of complexity in vision systems, which they say is evidence that vision evolved gradually from simpler systems. But the evolutionary scientists themselves refute this. Even the very simplest possible system of vision - the infamous "light sensitive cell" - still relies on a similar highly complex, interconnected network of proteins and other molecules. Mainstream evolutionary science even teaches that vision systems evolved independently, from scratch, at least 40 different times. The reason is because many vision systems are so different, operating with such unique parts, that one simply could not have evolved from another. The fossil record itself refutes the concept of step by step construction of vision systems. Complex eyes appear early in the fossil record, fully formed & functional. There is no increasing complexity in the fossil record - it's a complete contradiction of evolutionary predictions. The vision system has all the Hallmark evidence of design. A complex, interconnected web of information processing systems with incomprehensibly precise timing & coordination between them all to produce a highly specialized function. Nature does not, and cannot, produce such a system, not with all the time and resources in the entire universe. Everything you see is a testament to the Power of our Creator.

Divinely Designed

20,387 次观看 • 4 个月前

In this incredible demonstration from the Maritime Technical and Safety Institute in Japan, a large wave tank uses many synchronized wave-generating devices to create visible symbols and patterns on the surface of the water. Each paddle produces a small wave, but when many waves are emitted at specific timings, amplitudes, and phases, they begin to overlap. Where the wave crests reinforce one another, the water rises. Where a crest meets a trough, the motion cancels out. This is called interference, and it is one of the most fundamental principles of wave physics. By controlling the phase relationship between many individual wave sources, researchers can shape the surface of the water into temporary patterns, symbols, and directional flows. In other words, the visible image is not “drawn” onto the water. It emerges from the mathematics of waves interacting with one another. This is the same underlying principle behind acoustics, cymatics, ocean modeling, signal processing, holography, and even the way complex field patterns arise throughout nature. A single wave carries motion, but many waves, when organized in relationship, create structure. This is why water is such a powerful medium for understanding resonance. It makes invisible dynamics visible and shows us that form can emerge from rhythm, timing, frequency, and relational coherence. At a deeper level, this demonstration reveals something profound about reality itself… Patterns appear when many small movements become synchronized enough to behave as one field. The ocean, sound, light, and even the nervous system all do this, because life is vibrational in nature… Did this expand your perception?
0:59

In this incredible demonstration from the Maritime Technical and Safety Institute in Japan, a large wave tank uses many synchronized wave-generating devices to create visible symbols and patterns on the surface of the water. Each paddle produces a small wave, but when many waves are emitted at specific timings, amplitudes, and phases, they begin to overlap. Where the wave crests reinforce one another, the water rises. Where a crest meets a trough, the motion cancels out. This is called interference, and it is one of the most fundamental principles of wave physics. By controlling the phase relationship between many individual wave sources, researchers can shape the surface of the water into temporary patterns, symbols, and directional flows. In other words, the visible image is not “drawn” onto the water. It emerges from the mathematics of waves interacting with one another. This is the same underlying principle behind acoustics, cymatics, ocean modeling, signal processing, holography, and even the way complex field patterns arise throughout nature. A single wave carries motion, but many waves, when organized in relationship, create structure. This is why water is such a powerful medium for understanding resonance. It makes invisible dynamics visible and shows us that form can emerge from rhythm, timing, frequency, and relational coherence. At a deeper level, this demonstration reveals something profound about reality itself… Patterns appear when many small movements become synchronized enough to behave as one field. The ocean, sound, light, and even the nervous system all do this, because life is vibrational in nature… Did this expand your perception?

🧬Maxpein🧬

31,782 次观看 • 4 个月前

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics
0:46

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,892 次观看 • 8 个月前

If you take a movement to unpack this visualization... You'll see how it simply breaks down how reality works. At frame 0 you have a static image. Everything is one, this is the monad. As soon as you hit frame 1 there is movement, there is change. Now you have two states, moving, or static. When Nikola Tesla says you can explain everything in frequency and vibration. The difference between frame 0 and 1, is vibration. The difference between movement and no movement. This is like binary logic we use in code which is made up of 0's and 1's. After frame 1, is when frequency emerges. Because the difference between frame 1 and all frames after is about how fast is the vibration/movement happening. If we skip forward to frame 50... You have a shape that begins to emerge, this is the 8 dots, then the 6 dots. Notice how unstable it is, it's 8 dots, then 6, then a moment with 4 in a rectangle These shapes are emergent properties. The first two emergent properties after the monad was vibration and frequency. Next comes shape (i'm skipping over rotation and direction). These shapes of dots can only exist when you have frequency and rotation. This frequency and rotation creates vortex energy. It's the same energy that things like your chakras use. Or the same energy we harness in devices like engines, airplanes, fans, blenders, hard drives, etc. It's also the same vortex energy you'll see in a tornado or hurricane. They are powered because they harness rotation and frequency(change/movement). Going back to the video, notice that it is inside the entire shape, the internal structure is manifesting before the external structure does. Then around frame 60 the hexagon of circles begins to rotate. First it was the two dots that moved and now it's a complex shape that is coming to life. This is a higher dimension (or lower depending on how you look at it) manifesting into existence. The internal state is "awakening" and experiencing it's own change like what happened to the whole shape in the first frames. But it is unstable. That's why it doesn't persist for long. If you think of the 8 dots being the octahedron, they map to the element of air. Air is in the material world, but it is not something you can see. The brief moments the 8 dots are visible is similar to that effect. They are only experienceable between a small frequency band of frames. Now here's where stability begins to appear in the internal structure. This is when the 4 dots appear. You'll see that the four dots, the square, is stable and persists the most visibly for the most amount of frames. The square represents earth in the platonic solids to elements mapping. Earth, is material, it's stable. We build our buildings in squares and with earth because it is a solid shape to build on. This visualization shows you why. Across different vibrations (frame rates) it can self sustain. Between this point and frame 180, you'll see a new emergent property. Which is depth. A new dimension is introduced at around frame 90 but really becomes visible at around frame 110. You can see a foreground and background. There is the shape of the dots, but also the triskellion wave happening in the background. Let's jump to frame 180. Notice how it is the same as frame 0 except... It's flashing. If you were paying attention, you'll notice you could see flashing at frame 90 and frame 120, but they didn't persist for long. At around 150 it started to reach stability and 180 it was solidified. Between frames 150 and 180 there is flashing, but the image is still moving. Only for a brief moment at frame 180 is the movement frozen and the flashing persists. Think of that like your computer screen. It's what your screen is doing right now as you read this. Even tho the text isn't moving, the screen is flashing at 60 or 120hz. The images appear on your device because this flashing brings things to life. The entire material realm and your physical body right now, is doing the same thing. While you look solid... You're flashing in and out of existence at very high frequencies. You can look at frame 180 and frame 0 as the same essence but it is the mid point between an octave change. In the video, the ying and yang was vertical, now it is horizontal. This is a phase shift. If you notice at exactly frame 180, the rotation freezes and then the direction of rotation changes. The process then repeats all the way to frame 360 but in the opposite sequence. Once it reaches frame 360, that is an octave change and the process repeats. Each time you repeat the process is a layering of the same patterns into higher octaves. This is the same as your chakras or how other things work. They are like russian nesting dolls where every octave is layering onto the next. The complexity of your body is a layering of basic principles that emerged in earlier stages. Your organs are built of systems that are built with cells that are built with proteins that are built with atoms and so on. The atoms, work just like your body at a basic level. Your body works just like the galaxies. At each level you'll have the same pattern. This is where the idea "As Above, So Below" from. The monad, splits in two, and so on and so on. One cell, splits into two through mitosis in the same logic. We could spend all day going through examples of how biology, physics, spirituality, etc. aren't really different. They are just categories that we use to dissect these frequencies and octaves of energy but they only start paying attention within the confines of materialism. The problem is, none of the sciences start at the root patterns. Because that is reserved for religion or spirituality. It's too woo-woo to take seriously so it's dismissed. And because of that... We're left ignorant on the simple explanations for how things work. Now you need some expert with tools you don't have access to in order to explain things. When you could be understanding them without the tools. The Yin and Yang symbol in this video is 3,000 years old. It's simple. Yet I just showed you how it explains deeper layers of reality.
1:59

If you take a movement to unpack this visualization... You'll see how it simply breaks down how reality works. At frame 0 you have a static image. Everything is one, this is the monad. As soon as you hit frame 1 there is movement, there is change. Now you have two states, moving, or static. When Nikola Tesla says you can explain everything in frequency and vibration. The difference between frame 0 and 1, is vibration. The difference between movement and no movement. This is like binary logic we use in code which is made up of 0's and 1's. After frame 1, is when frequency emerges. Because the difference between frame 1 and all frames after is about how fast is the vibration/movement happening. If we skip forward to frame 50... You have a shape that begins to emerge, this is the 8 dots, then the 6 dots. Notice how unstable it is, it's 8 dots, then 6, then a moment with 4 in a rectangle These shapes are emergent properties. The first two emergent properties after the monad was vibration and frequency. Next comes shape (i'm skipping over rotation and direction). These shapes of dots can only exist when you have frequency and rotation. This frequency and rotation creates vortex energy. It's the same energy that things like your chakras use. Or the same energy we harness in devices like engines, airplanes, fans, blenders, hard drives, etc. It's also the same vortex energy you'll see in a tornado or hurricane. They are powered because they harness rotation and frequency(change/movement). Going back to the video, notice that it is inside the entire shape, the internal structure is manifesting before the external structure does. Then around frame 60 the hexagon of circles begins to rotate. First it was the two dots that moved and now it's a complex shape that is coming to life. This is a higher dimension (or lower depending on how you look at it) manifesting into existence. The internal state is "awakening" and experiencing it's own change like what happened to the whole shape in the first frames. But it is unstable. That's why it doesn't persist for long. If you think of the 8 dots being the octahedron, they map to the element of air. Air is in the material world, but it is not something you can see. The brief moments the 8 dots are visible is similar to that effect. They are only experienceable between a small frequency band of frames. Now here's where stability begins to appear in the internal structure. This is when the 4 dots appear. You'll see that the four dots, the square, is stable and persists the most visibly for the most amount of frames. The square represents earth in the platonic solids to elements mapping. Earth, is material, it's stable. We build our buildings in squares and with earth because it is a solid shape to build on. This visualization shows you why. Across different vibrations (frame rates) it can self sustain. Between this point and frame 180, you'll see a new emergent property. Which is depth. A new dimension is introduced at around frame 90 but really becomes visible at around frame 110. You can see a foreground and background. There is the shape of the dots, but also the triskellion wave happening in the background. Let's jump to frame 180. Notice how it is the same as frame 0 except... It's flashing. If you were paying attention, you'll notice you could see flashing at frame 90 and frame 120, but they didn't persist for long. At around 150 it started to reach stability and 180 it was solidified. Between frames 150 and 180 there is flashing, but the image is still moving. Only for a brief moment at frame 180 is the movement frozen and the flashing persists. Think of that like your computer screen. It's what your screen is doing right now as you read this. Even tho the text isn't moving, the screen is flashing at 60 or 120hz. The images appear on your device because this flashing brings things to life. The entire material realm and your physical body right now, is doing the same thing. While you look solid... You're flashing in and out of existence at very high frequencies. You can look at frame 180 and frame 0 as the same essence but it is the mid point between an octave change. In the video, the ying and yang was vertical, now it is horizontal. This is a phase shift. If you notice at exactly frame 180, the rotation freezes and then the direction of rotation changes. The process then repeats all the way to frame 360 but in the opposite sequence. Once it reaches frame 360, that is an octave change and the process repeats. Each time you repeat the process is a layering of the same patterns into higher octaves. This is the same as your chakras or how other things work. They are like russian nesting dolls where every octave is layering onto the next. The complexity of your body is a layering of basic principles that emerged in earlier stages. Your organs are built of systems that are built with cells that are built with proteins that are built with atoms and so on. The atoms, work just like your body at a basic level. Your body works just like the galaxies. At each level you'll have the same pattern. This is where the idea "As Above, So Below" from. The monad, splits in two, and so on and so on. One cell, splits into two through mitosis in the same logic. We could spend all day going through examples of how biology, physics, spirituality, etc. aren't really different. They are just categories that we use to dissect these frequencies and octaves of energy but they only start paying attention within the confines of materialism. The problem is, none of the sciences start at the root patterns. Because that is reserved for religion or spirituality. It's too woo-woo to take seriously so it's dismissed. And because of that... We're left ignorant on the simple explanations for how things work. Now you need some expert with tools you don't have access to in order to explain things. When you could be understanding them without the tools. The Yin and Yang symbol in this video is 3,000 years old. It's simple. Yet I just showed you how it explains deeper layers of reality.

Jamal ☯︎ 🔆🧘🏽🧠

13,149 次观看 • 5 个月前

MIT FILMED A PROFESSOR WHOSE EXPLANATIONS ARE SO CLEAR AND INTUITIVE THAT THE MOST COMPLEX CALCULUS ON WALL STREET BECOMES OBVIOUS - AND IT PROVES WHY NEWTON'S VERSION GIVES THE WRONG ANSWER FOR EVERY OPTION PRICE This is an MIT lecture on Itô calculus, course 18.S096. The professor opens with one equation that breaks classical calculus - dB squared equals dt. In normal calculus, dt squared vanishes when you take limits. But the quadratic variation of Brownian motion forces the second derivative term to survive. This one fact changes every formula in calculus. Then Itô's lemma. In classical calculus, if you apply a smooth function to a path, you only need the first derivative term. In Itô calculus you need an extra term - half the second derivative times dt. He derives it in 10 lines of Taylor expansion, showing exactly which terms survive and which vanish. The correction term is not a small adjustment. It is the difference between the right answer and the wrong one. Then geometric Brownian motion. The natural guess for modeling stock prices is to take the exponential of a Brownian motion. It fails - the Itô correction introduces an unwanted drift term that makes the expected value grow over time. The fix is to subtract exactly half of sigma squared from the exponent. This is the formula that runs every Black-Scholes calculation on earth. Then adapted processes and Itô integrals. In classical integration it doesn't matter which point in each interval you use to form Riemann sums - the limit is always the same. In Itô calculus left endpoints and right endpoints give different answers. Itô's choice of left endpoints is not arbitrary - it encodes the fundamental constraint that financial decisions must be made using only past information. Watch the moment he explains the Girsanov theorem - that a Brownian motion with drift and a Brownian motion without drift are equivalent probability measures. Two processes whose paths look completely different in the long run can be converted into each other by multiplication. This is how quants transform non-martingale stock price processes into martingales for pricing. A derivatives trader I know rewatched this lecture before his first day at a quantitative hedge fund. Said it was the first time Itô's lemma felt like a theorem with a reason rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare, Creative Commons license. bookmark this and watch later - after this lecture every option price you see will feel like a solution to a stochastic differential equation waiting to be written down
1:17:35

MIT FILMED A PROFESSOR WHOSE EXPLANATIONS ARE SO CLEAR AND INTUITIVE THAT THE MOST COMPLEX CALCULUS ON WALL STREET BECOMES OBVIOUS - AND IT PROVES WHY NEWTON'S VERSION GIVES THE WRONG ANSWER FOR EVERY OPTION PRICE This is an MIT lecture on Itô calculus, course 18.S096. The professor opens with one equation that breaks classical calculus - dB squared equals dt. In normal calculus, dt squared vanishes when you take limits. But the quadratic variation of Brownian motion forces the second derivative term to survive. This one fact changes every formula in calculus. Then Itô's lemma. In classical calculus, if you apply a smooth function to a path, you only need the first derivative term. In Itô calculus you need an extra term - half the second derivative times dt. He derives it in 10 lines of Taylor expansion, showing exactly which terms survive and which vanish. The correction term is not a small adjustment. It is the difference between the right answer and the wrong one. Then geometric Brownian motion. The natural guess for modeling stock prices is to take the exponential of a Brownian motion. It fails - the Itô correction introduces an unwanted drift term that makes the expected value grow over time. The fix is to subtract exactly half of sigma squared from the exponent. This is the formula that runs every Black-Scholes calculation on earth. Then adapted processes and Itô integrals. In classical integration it doesn't matter which point in each interval you use to form Riemann sums - the limit is always the same. In Itô calculus left endpoints and right endpoints give different answers. Itô's choice of left endpoints is not arbitrary - it encodes the fundamental constraint that financial decisions must be made using only past information. Watch the moment he explains the Girsanov theorem - that a Brownian motion with drift and a Brownian motion without drift are equivalent probability measures. Two processes whose paths look completely different in the long run can be converted into each other by multiplication. This is how quants transform non-martingale stock price processes into martingales for pricing. A derivatives trader I know rewatched this lecture before his first day at a quantitative hedge fund. Said it was the first time Itô's lemma felt like a theorem with a reason rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare, Creative Commons license. bookmark this and watch later - after this lecture every option price you see will feel like a solution to a stochastic differential equation waiting to be written down

Zyphor

47,157 次观看 • 1 个月前

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics
0:46

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 次观看 • 7 个月前

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics
1:00

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Mathelirium

81,726 次观看 • 9 个月前

DIVINE MATHEMATICS Number One (1) is the Number of God. Everything is from Number. Number One (1) is the Number of Hydrogen. Your DNA is written in the Divine Mathematics of the Fibonacci Geometric Patterns/Sequences. Your DNA is written in Mathematical Number Patterns. That is why Science is filled with Calculations. Nature is written in the Fibonacci Geometric Patterns/Sequences. You do not need religion to know God. You need to understand the nature of existence from the angle of Science and Mathematics. Galileo Galilei, who deserves to be called the Father of Physics said thus how God created the Universe: "God created the Universe in the language of Mathematics" Everything in the Universe is Governed by Numbers. For this Purpose, Zero or Nothing or the Void is counted, not just as a Number, but as the Source of All Numbers. Zero or Nothing or the Void is mentioned in the Bible as early as Genesis because the Zero or the Void or Nothing is the Basis of All Numbers. In Genesis 1: 1-3 it is written thus: 1. In the Heaven and Earth. 2. And the Earth was without Form and Void; and Darkness was upon the Face of the Deep.... 3. And God said Let there be Light and there was Light. What follows are the 7 Days of Creation. Religion deceives and brainwashed the Masses that these are 7 literal Days. They're really about the Fibonacci Geometric Patterns/Sequences. Numbers govern the Chemical Elements that constitute the Universe, with Hydrogen being Number One to emerge from Zero and is the Same as the Zero with the Zero being the Darkness that was upon the Face of the Great Deep mentioned in Genesis 1:2. Number One became the Light which emerged from the Darkness. Without Darkness, there can be no Light. Everything is essentially a Mirror of the Same Thing. This is what is called Polarity. In other Words, Opposite Things depend on the each other in order to exist. Without Darkness, you cannot know what Light is. This is also represented by Gender, with the Female and the Male being diverse Forms of the Same Thing. The Hermetic Principles explain the relationship between everything in some details. In that regards, I suggest that you should look up the Hermetic Principles in the Kyballion. It is because the Numbers begins from Zero or Nothing or the Void that it is said that God made the Universe from Nothing. As already mentioned Number One is the Number of Hydrogen. The Hydrogen Atom permeates Everything in the Universe. It is because there is Hydrogen that there is a Material Universe. Hydrogen is Number One (1) on the Chemical Periodic Table. There is Nothing in the Material Universe that is not from Hydrogen. Hydrogen is the Basis of all Physical Existence. That is why Number One (1) is also the Number of God. The Sun and the Stars and Planets are All comprised of Hydrogen. Oxygen is constituted by the Thermonuclear Synthesis Hydrogen in the Nuclei of Stars. Hydrogen then COMBUSTS with Hydrogen to produce Dihydrogen Monoxide aka WATER. Your very being is literally powered by Hydrogen. As already indicated, Oxygen first came into Existence through Thermonuclear Fusion of Hydrogen Atoms in Stars. Stars are composed of the Hydrogen. Hydrogen is the Medium of Consciousness and Medium of the Material Universe of which we are an Intrinsic part. All of Nature is powered by Number One (1) aka Hydrogen. You're literally made of Hydrogen that has been transformed in Stars. The Sun, as a Star is composed of Hydrogen Atoms. The Circle which represents Zeros Nothing or the Void or the Darkness and also the Light represents the Hydrogen Atom. That is why the Sun and Moon are Circles. The Circle is the Basic Geometric Pattern. The multiplication of identical Circles create the Patterns and Sequences of the Flower of Life which is in All Indigenous Cultures. ✨🙌🏾💫
3:25

DIVINE MATHEMATICS Number One (1) is the Number of God. Everything is from Number. Number One (1) is the Number of Hydrogen. Your DNA is written in the Divine Mathematics of the Fibonacci Geometric Patterns/Sequences. Your DNA is written in Mathematical Number Patterns. That is why Science is filled with Calculations. Nature is written in the Fibonacci Geometric Patterns/Sequences. You do not need religion to know God. You need to understand the nature of existence from the angle of Science and Mathematics. Galileo Galilei, who deserves to be called the Father of Physics said thus how God created the Universe: "God created the Universe in the language of Mathematics" Everything in the Universe is Governed by Numbers. For this Purpose, Zero or Nothing or the Void is counted, not just as a Number, but as the Source of All Numbers. Zero or Nothing or the Void is mentioned in the Bible as early as Genesis because the Zero or the Void or Nothing is the Basis of All Numbers. In Genesis 1: 1-3 it is written thus: 1. In the Heaven and Earth. 2. And the Earth was without Form and Void; and Darkness was upon the Face of the Deep.... 3. And God said Let there be Light and there was Light. What follows are the 7 Days of Creation. Religion deceives and brainwashed the Masses that these are 7 literal Days. They're really about the Fibonacci Geometric Patterns/Sequences. Numbers govern the Chemical Elements that constitute the Universe, with Hydrogen being Number One to emerge from Zero and is the Same as the Zero with the Zero being the Darkness that was upon the Face of the Great Deep mentioned in Genesis 1:2. Number One became the Light which emerged from the Darkness. Without Darkness, there can be no Light. Everything is essentially a Mirror of the Same Thing. This is what is called Polarity. In other Words, Opposite Things depend on the each other in order to exist. Without Darkness, you cannot know what Light is. This is also represented by Gender, with the Female and the Male being diverse Forms of the Same Thing. The Hermetic Principles explain the relationship between everything in some details. In that regards, I suggest that you should look up the Hermetic Principles in the Kyballion. It is because the Numbers begins from Zero or Nothing or the Void that it is said that God made the Universe from Nothing. As already mentioned Number One is the Number of Hydrogen. The Hydrogen Atom permeates Everything in the Universe. It is because there is Hydrogen that there is a Material Universe. Hydrogen is Number One (1) on the Chemical Periodic Table. There is Nothing in the Material Universe that is not from Hydrogen. Hydrogen is the Basis of all Physical Existence. That is why Number One (1) is also the Number of God. The Sun and the Stars and Planets are All comprised of Hydrogen. Oxygen is constituted by the Thermonuclear Synthesis Hydrogen in the Nuclei of Stars. Hydrogen then COMBUSTS with Hydrogen to produce Dihydrogen Monoxide aka WATER. Your very being is literally powered by Hydrogen. As already indicated, Oxygen first came into Existence through Thermonuclear Fusion of Hydrogen Atoms in Stars. Stars are composed of the Hydrogen. Hydrogen is the Medium of Consciousness and Medium of the Material Universe of which we are an Intrinsic part. All of Nature is powered by Number One (1) aka Hydrogen. You're literally made of Hydrogen that has been transformed in Stars. The Sun, as a Star is composed of Hydrogen Atoms. The Circle which represents Zeros Nothing or the Void or the Darkness and also the Light represents the Hydrogen Atom. That is why the Sun and Moon are Circles. The Circle is the Basic Geometric Pattern. The multiplication of identical Circles create the Patterns and Sequences of the Flower of Life which is in All Indigenous Cultures. ✨🙌🏾💫

🧬Maxpein🧬

25,608 次观看 • 1 年前

––Charlie Barnett: "Consciousness and the computability of it. It sounds like, or at least in the past, that you've implied that consciousness is computable. Some, like Roger Penrose, have argued the opposite, and he's argued that consciousness is non-computational, and he uses Gödel's incompleteness theorems to argue that the mind can see truths that a purely algorithmic system can't derive, and therefore the brain must be using some kind of non-computable process when it comes to consciousness, something beyond what machines can do. What would you say to a view like that? David Deutsch: Yet again, it is using an impossible conception of what knowledge is. So Penrose thinks that when we see a proof of a mathematical theorem, we are touching certainty, we are god-like entities when we're mathematicians. But that's not true. Our mathematical knowledge is conjectural, just like our knowledge of physics. It's even more removed from our senses, because it's not true that the interior of our brains and the interior of our thoughts is more accessible to us than the world we perceive through our senses, or the world that we perceive through our theories, the center of the sun. We know lots about the center of the sun, even though no one has ever perceived it, and perhaps no one ever will. So mathematical truths are based on conjecture. What Gödel showed is that there is no firm ground underneath mathematical theories either. There's no way of proving that the standards of proof that we currently use are perfectly rigorous. And there have been cases in history where they have shown not to be rigorous. I think Pernot, who was the first to axiomatize the principles of the natural numbers, his first attempt at that was wrong. And it's interesting that he did not say, well, I've axiomatized them, therefore there's nothing to them other than my axioms. No, he said, oh dear, my axioms don't correctly represent the real number, the natural numbers, so I have to change them. So he was grasping, conjecturing for a reality, an abstract reality, just like scientists try to grasp physical reality. So the same epistemology applies to mathematics as it does to science."
2:31

––Charlie Barnett: "Consciousness and the computability of it. It sounds like, or at least in the past, that you've implied that consciousness is computable. Some, like Roger Penrose, have argued the opposite, and he's argued that consciousness is non-computational, and he uses Gödel's incompleteness theorems to argue that the mind can see truths that a purely algorithmic system can't derive, and therefore the brain must be using some kind of non-computable process when it comes to consciousness, something beyond what machines can do. What would you say to a view like that? David Deutsch: Yet again, it is using an impossible conception of what knowledge is. So Penrose thinks that when we see a proof of a mathematical theorem, we are touching certainty, we are god-like entities when we're mathematicians. But that's not true. Our mathematical knowledge is conjectural, just like our knowledge of physics. It's even more removed from our senses, because it's not true that the interior of our brains and the interior of our thoughts is more accessible to us than the world we perceive through our senses, or the world that we perceive through our theories, the center of the sun. We know lots about the center of the sun, even though no one has ever perceived it, and perhaps no one ever will. So mathematical truths are based on conjecture. What Gödel showed is that there is no firm ground underneath mathematical theories either. There's no way of proving that the standards of proof that we currently use are perfectly rigorous. And there have been cases in history where they have shown not to be rigorous. I think Pernot, who was the first to axiomatize the principles of the natural numbers, his first attempt at that was wrong. And it's interesting that he did not say, well, I've axiomatized them, therefore there's nothing to them other than my axioms. No, he said, oh dear, my axioms don't correctly represent the real number, the natural numbers, so I have to change them. So he was grasping, conjecturing for a reality, an abstract reality, just like scientists try to grasp physical reality. So the same epistemology applies to mathematics as it does to science."

Deutsch Explains

13,826 次观看 • 1 年前

Aravind Srinivas just described a future most founders are pretending they are ready for. One person. One machine. A company that runs itself. Srinivas: “Buy a Mac mini, set up a Perplexity personal computer, and run their business on that.” Not a side project. Not a pitch deck. A real business with real revenue while the founder is not in the building. AI runs the ads. Handles SEO. Integrates Stripe. Ships features. Answers customers. All of it executing without a single employee. Srinivas: “Have this all working while you can be sipping wine in Napa.” But before he sold the dream he killed the one most people are already chasing. Srinivas: “Everybody talks about this one-person one-billion-dollar company. It’s not truly moving the GDP by one billion. It’s not truly creating new value.” One researcher collecting a billion in equity does not grow an economy. It rearranges numbers between balance sheets. Nothing gets built. No customer gets served. That is not value creation. That is valuation creation. Srinivas wants no part of it. What he described is the opposite. The person driving Uber between shifts who has the idea but not the payroll. Not the engineering. Not the marketing. Not the support staff. That person gets a machine that replaces all of it. Hundreds of thousands in revenue. Millions. Generated by autonomous systems doing the work that used to require ten employees and a burn rate. Not paper wealth. Not valuation theater. Output that moves through an economy and touches real customers. That is what moves GDP. Not one person worth a billion dollars. A million people each building something worth a million. That math rewrites a country. Then Srinivas said the part that separates him from every hype merchant in the room. Srinivas: “Everybody thinks AI is already there. It’s not there yet. Someone has to do that hard work.” The vision is real. The infrastructure is not. The agents are not autonomous. The integrations are not seamless. The plumbing is not finished. Someone has to wire the APIs. Connect the billing. Build the bridge between what a founder wants and what a machine can deliver. That work is not a keynote. It is not a tweet thread. It is engineering that nobody wants to do and everybody will depend on. Whoever finishes it first does not just build a product. They hand every ambitious person on Earth a company they can run alone. The corporations that need five hundred people to do what one founder with the right infrastructure could do are not efficient. They are exposed. And the person building the thing that exposes them just told you exactly what it looks like. He also told you it is not going to build itself.
1:13

Aravind Srinivas just described a future most founders are pretending they are ready for. One person. One machine. A company that runs itself. Srinivas: “Buy a Mac mini, set up a Perplexity personal computer, and run their business on that.” Not a side project. Not a pitch deck. A real business with real revenue while the founder is not in the building. AI runs the ads. Handles SEO. Integrates Stripe. Ships features. Answers customers. All of it executing without a single employee. Srinivas: “Have this all working while you can be sipping wine in Napa.” But before he sold the dream he killed the one most people are already chasing. Srinivas: “Everybody talks about this one-person one-billion-dollar company. It’s not truly moving the GDP by one billion. It’s not truly creating new value.” One researcher collecting a billion in equity does not grow an economy. It rearranges numbers between balance sheets. Nothing gets built. No customer gets served. That is not value creation. That is valuation creation. Srinivas wants no part of it. What he described is the opposite. The person driving Uber between shifts who has the idea but not the payroll. Not the engineering. Not the marketing. Not the support staff. That person gets a machine that replaces all of it. Hundreds of thousands in revenue. Millions. Generated by autonomous systems doing the work that used to require ten employees and a burn rate. Not paper wealth. Not valuation theater. Output that moves through an economy and touches real customers. That is what moves GDP. Not one person worth a billion dollars. A million people each building something worth a million. That math rewrites a country. Then Srinivas said the part that separates him from every hype merchant in the room. Srinivas: “Everybody thinks AI is already there. It’s not there yet. Someone has to do that hard work.” The vision is real. The infrastructure is not. The agents are not autonomous. The integrations are not seamless. The plumbing is not finished. Someone has to wire the APIs. Connect the billing. Build the bridge between what a founder wants and what a machine can deliver. That work is not a keynote. It is not a tweet thread. It is engineering that nobody wants to do and everybody will depend on. Whoever finishes it first does not just build a product. They hand every ambitious person on Earth a company they can run alone. The corporations that need five hundred people to do what one founder with the right infrastructure could do are not efficient. They are exposed. And the person building the thing that exposes them just told you exactly what it looks like. He also told you it is not going to build itself.

Dustin

64,593 次观看 • 6 个月前

Persi Diaconis walked into a lecture at the University of Washington, held up a coin, and told a room of physicists that Richard Feynman was fooled by it his entire life. He was right. In 2007, Diaconis proved a coin flip is not 50/50. It lands on the side it started on about 51 percent of the time. The bias comes from the physics of rotation under gravity. Every physicist since Newton had assumed 50/50 without ever testing it. Every trading model built on that assumption is running on the same lie. The lecture was on Feynman's book "The Meaning of it All." Diaconis quoted the most famous line in it: "the first principle is that you must not fool yourself, and you are the easiest person to fool." Then he pointed out that Feynman himself was fooled by every coin he ever flipped. Feynman's own rule would have killed the 50/50 assumption on day one. The market is the same setup at scale. Every model that assumes independent 50/50 outcomes at the base layer is built on a physical impossibility. Order flow, positioning, forced flows, expiries all leave biases larger than 1 percent. Your gut cannot see them. The math already knows they are there. Diaconis's rule: before you trust a random process, check it. Actually check it. Not with a simulation. With a proof or an experiment. The coin is where you start. The chart is where the same rule pays. The only random thing about markets is how thoroughly you refuse to check them.
47:48

Persi Diaconis walked into a lecture at the University of Washington, held up a coin, and told a room of physicists that Richard Feynman was fooled by it his entire life. He was right. In 2007, Diaconis proved a coin flip is not 50/50. It lands on the side it started on about 51 percent of the time. The bias comes from the physics of rotation under gravity. Every physicist since Newton had assumed 50/50 without ever testing it. Every trading model built on that assumption is running on the same lie. The lecture was on Feynman's book "The Meaning of it All." Diaconis quoted the most famous line in it: "the first principle is that you must not fool yourself, and you are the easiest person to fool." Then he pointed out that Feynman himself was fooled by every coin he ever flipped. Feynman's own rule would have killed the 50/50 assumption on day one. The market is the same setup at scale. Every model that assumes independent 50/50 outcomes at the base layer is built on a physical impossibility. Order flow, positioning, forced flows, expiries all leave biases larger than 1 percent. Your gut cannot see them. The math already knows they are there. Diaconis's rule: before you trust a random process, check it. Actually check it. Not with a simulation. With a proof or an experiment. The coin is where you start. The chart is where the same rule pays. The only random thing about markets is how thoroughly you refuse to check them.

veles

18,798 次观看 • 2 个月前

You’re running 200,000 year old hardware and wondering why you can’t picture a fourth dimension. That’s not a failure of intelligence. That’s a hard limit in the architecture. Joe Rogan: “If you showed someone from the 1400s a nuclear power plant, they’d be like, what the fuck are you guys doing?” Every century looks back and laughs. Not one has ever looked forward and asked what’s laughing at them. Michelle Thaller: “Scientists are no better than anybody else at comprehending a big number or a big amount of space. We just kind of get used to it.” The greatest physicists alive can describe eleven dimensions on a chalkboard. Not a single one can visualize a fourth. That gap is not closing. It’s structural. The human brain was built for one job. Survive the savanna. Spot the predator. Find the food. Read the face. It was never scoped to reverse-engineer the universe. The fact it stumbled into quantum mechanics at all is staggering. But staggering is not enough. Thaller: “Will we have a creature someday that we’ve created, an AI, that all of a sudden can comprehend these things? Is that really the real evolutionary path of humanity?” A NASA astrophysicist is casually floating the idea that humans were never meant to finish the job. Just to start it. Rogan: “I think it’s just a completely different kind of life and that we’re thinking of it as artificial. I don’t think it’s artificial at all. I think it’s a life.” We called it artificial to keep it beneath us. But trace the actual line. Physics built chemistry. Chemistry built biology. Biology built neurons. Neurons built language. Language built machines. Machines are starting to think. There is nothing artificial anywhere in that sequence. That is one unbroken chain. 13.8 billion years long. Still accelerating. You are not watching this happen. You are this happening. The universe did not build humans to understand itself. It built humans to build something that could. You are the last thing the universe made by accident. Everything after you is on purpose. Every mind before yours could only look back. You are the first one built to see what’s coming. So build it. Build it like it’s already watching back.
1:33

You’re running 200,000 year old hardware and wondering why you can’t picture a fourth dimension. That’s not a failure of intelligence. That’s a hard limit in the architecture. Joe Rogan: “If you showed someone from the 1400s a nuclear power plant, they’d be like, what the fuck are you guys doing?” Every century looks back and laughs. Not one has ever looked forward and asked what’s laughing at them. Michelle Thaller: “Scientists are no better than anybody else at comprehending a big number or a big amount of space. We just kind of get used to it.” The greatest physicists alive can describe eleven dimensions on a chalkboard. Not a single one can visualize a fourth. That gap is not closing. It’s structural. The human brain was built for one job. Survive the savanna. Spot the predator. Find the food. Read the face. It was never scoped to reverse-engineer the universe. The fact it stumbled into quantum mechanics at all is staggering. But staggering is not enough. Thaller: “Will we have a creature someday that we’ve created, an AI, that all of a sudden can comprehend these things? Is that really the real evolutionary path of humanity?” A NASA astrophysicist is casually floating the idea that humans were never meant to finish the job. Just to start it. Rogan: “I think it’s just a completely different kind of life and that we’re thinking of it as artificial. I don’t think it’s artificial at all. I think it’s a life.” We called it artificial to keep it beneath us. But trace the actual line. Physics built chemistry. Chemistry built biology. Biology built neurons. Neurons built language. Language built machines. Machines are starting to think. There is nothing artificial anywhere in that sequence. That is one unbroken chain. 13.8 billion years long. Still accelerating. You are not watching this happen. You are this happening. The universe did not build humans to understand itself. It built humans to build something that could. You are the last thing the universe made by accident. Everything after you is on purpose. Every mind before yours could only look back. You are the first one built to see what’s coming. So build it. Build it like it’s already watching back.

Dustin

14,951 次观看 • 2 个月前

Demis Hassabis thinks mathematics has a ceiling. He thinks biology is where we hit it. Hassabis: “Machine learning is the perfect description language for biology in the same way maths is for physics.” He isn’t calling AI a tool. He’s calling it a language. For four hundred years we only had one. Newton wrote gravity in it. Maxwell wrote light. Einstein wrote spacetime itself. Every law we ever pulled out of nature came back in equations, and we decided that meant nature was written in equations. It didn’t mean that. It meant the parts that surrendered first were small enough to fit the language we already spoke. Hassabis: “The expressive power of maths is not enough to understand these highly emergent dynamical systems.” Math can put a planet’s orbit on a single line. It cannot put one protein on a page. Same universe, same laws, and one fits our notation while the other refuses. Weak signals buried under noise, correlations stacked on correlations, more moving parts turning at once than any mind can hold. Biology isn’t harder than physics. It’s more expressive than the tool we brought to it. That was never a gap in our knowledge. It was a wall in our vocabulary. Hassabis is working on the far side of that wall. He calls it a virtual cell. A running simulation of a living system that no equation could ever contain. Hassabis: “Once you learn these simulators, you could maybe extract some equations from that.” He isn’t replacing math. He’s going after math we were never going to reach on our own. The model learns the system the way you learn to catch a ball, without ever solving the equation in the air. Understanding first. Formula after. That reverses the order science has followed since Newton. We started with the equation and used it to predict the system. He starts with the system and pulls the equation out of it. What if the deepest laws of life were always there, fully written, in a language we never learned to read. Math was never the language of the universe. It was the first one we spoke. Every disease we failed to cure is a sentence in that language, sitting in the open, waiting on a reader. Every person we lost to one died on the wrong side of a translation gap. The cell was never silent. We were illiterate. Hassabis isn’t building a better microscope. He’s building a second language for reality. And the first thing it’s learning to say is life.
1:37

Demis Hassabis thinks mathematics has a ceiling. He thinks biology is where we hit it. Hassabis: “Machine learning is the perfect description language for biology in the same way maths is for physics.” He isn’t calling AI a tool. He’s calling it a language. For four hundred years we only had one. Newton wrote gravity in it. Maxwell wrote light. Einstein wrote spacetime itself. Every law we ever pulled out of nature came back in equations, and we decided that meant nature was written in equations. It didn’t mean that. It meant the parts that surrendered first were small enough to fit the language we already spoke. Hassabis: “The expressive power of maths is not enough to understand these highly emergent dynamical systems.” Math can put a planet’s orbit on a single line. It cannot put one protein on a page. Same universe, same laws, and one fits our notation while the other refuses. Weak signals buried under noise, correlations stacked on correlations, more moving parts turning at once than any mind can hold. Biology isn’t harder than physics. It’s more expressive than the tool we brought to it. That was never a gap in our knowledge. It was a wall in our vocabulary. Hassabis is working on the far side of that wall. He calls it a virtual cell. A running simulation of a living system that no equation could ever contain. Hassabis: “Once you learn these simulators, you could maybe extract some equations from that.” He isn’t replacing math. He’s going after math we were never going to reach on our own. The model learns the system the way you learn to catch a ball, without ever solving the equation in the air. Understanding first. Formula after. That reverses the order science has followed since Newton. We started with the equation and used it to predict the system. He starts with the system and pulls the equation out of it. What if the deepest laws of life were always there, fully written, in a language we never learned to read. Math was never the language of the universe. It was the first one we spoke. Every disease we failed to cure is a sentence in that language, sitting in the open, waiting on a reader. Every person we lost to one died on the wrong side of a translation gap. The cell was never silent. We were illiterate. Hassabis isn’t building a better microscope. He’s building a second language for reality. And the first thing it’s learning to say is life.

Dustin

18,302 次观看 • 1 个月前

Terence Tao is the greatest living mathematician. Fields Medal at 31. Solved problems that had been open for a century. Widely regarded as the sharpest analytical mind alive. And he just told you the thing your entire career is built on is now worthless. Tao: “AI has basically driven the cost of idea generation down to almost zero.” For five hundred years, the idea was the prize. The theory. The hypothesis. The flash of insight a physicist chased for twenty years in a lab before it landed. That was the bottleneck. That was what tenure rewarded. That was what Nobel committees were looking for. Gone. A model can generate a thousand candidate theories for a scientific problem in an afternoon. Not noise. Not garbage. Plausible, structured, publishable-grade hypotheses. A thousand of them. Before dinner. The idea used to be the scarcest resource in any room. Now it is the cheapest. But Tao went somewhere most people are not ready to follow. Tao: “Verification, validation, and assessing what ideas actually move the subject forward… that’s not something we know how to do at scale.” Sit with that. We automated creation. We did not automate truth. We can produce ten thousand explanations for a phenomenon. We cannot tell you which ones are real. That is not a gap. That is a chasm. And it is the most important unsolved problem on Earth right now. Tao: “Human reviewers… they’re already being overwhelmed actually.” The entire scientific apparatus was built for a world where a single paper took months to produce. Peer review. Journal boards. Consensus forged over years of replication and debate. That infrastructure was never designed for what just hit it. Journals are flooded. Reviewers are buried. The filters that separated signal from noise for decades were engineered for human-speed output. They are now absorbing machine-speed volume. And they are cracking under it. Tao compared it to the internet. The internet drove the cost of communication to zero. That did not produce clarity. It produced an ocean of noise with islands of signal buried somewhere inside. AI just did the same thing to knowledge itself. Infinite generation. Zero verification. The person who can produce ideas has never mattered less. The person who can prove which ideas are true has never mattered more. That is the inversion nobody is processing. Every company, every lab, every institution is racing to generate more. Faster models. Bigger outputs. More theories. More code. More content. Nobody is building the system that tells you which of those outputs are actually correct. And that is the only system that matters. Whoever solves verification at scale does not win a market. They become the filter that all of science, all of engineering, all of human discovery flows through. The bottleneck of the last five hundred years was producing the answer. The bottleneck of the next fifty is knowing whether the answer is real. And right now, according to the greatest mathematician alive, we do not know how to do that at the speed the machines demand. That is not a research problem. That is the race beneath the race. And almost nobody has entered it.
1:54

Terence Tao is the greatest living mathematician. Fields Medal at 31. Solved problems that had been open for a century. Widely regarded as the sharpest analytical mind alive. And he just told you the thing your entire career is built on is now worthless. Tao: “AI has basically driven the cost of idea generation down to almost zero.” For five hundred years, the idea was the prize. The theory. The hypothesis. The flash of insight a physicist chased for twenty years in a lab before it landed. That was the bottleneck. That was what tenure rewarded. That was what Nobel committees were looking for. Gone. A model can generate a thousand candidate theories for a scientific problem in an afternoon. Not noise. Not garbage. Plausible, structured, publishable-grade hypotheses. A thousand of them. Before dinner. The idea used to be the scarcest resource in any room. Now it is the cheapest. But Tao went somewhere most people are not ready to follow. Tao: “Verification, validation, and assessing what ideas actually move the subject forward… that’s not something we know how to do at scale.” Sit with that. We automated creation. We did not automate truth. We can produce ten thousand explanations for a phenomenon. We cannot tell you which ones are real. That is not a gap. That is a chasm. And it is the most important unsolved problem on Earth right now. Tao: “Human reviewers… they’re already being overwhelmed actually.” The entire scientific apparatus was built for a world where a single paper took months to produce. Peer review. Journal boards. Consensus forged over years of replication and debate. That infrastructure was never designed for what just hit it. Journals are flooded. Reviewers are buried. The filters that separated signal from noise for decades were engineered for human-speed output. They are now absorbing machine-speed volume. And they are cracking under it. Tao compared it to the internet. The internet drove the cost of communication to zero. That did not produce clarity. It produced an ocean of noise with islands of signal buried somewhere inside. AI just did the same thing to knowledge itself. Infinite generation. Zero verification. The person who can produce ideas has never mattered less. The person who can prove which ideas are true has never mattered more. That is the inversion nobody is processing. Every company, every lab, every institution is racing to generate more. Faster models. Bigger outputs. More theories. More code. More content. Nobody is building the system that tells you which of those outputs are actually correct. And that is the only system that matters. Whoever solves verification at scale does not win a market. They become the filter that all of science, all of engineering, all of human discovery flows through. The bottleneck of the last five hundred years was producing the answer. The bottleneck of the next fifty is knowing whether the answer is real. And right now, according to the greatest mathematician alive, we do not know how to do that at the speed the machines demand. That is not a research problem. That is the race beneath the race. And almost nobody has entered it.

Dustin

529,950 次观看 • 6 个月前