I was playing around with one of the new multimodal models this morning, and it wrote me this incredibly moving, slightly melancholic poem about the sun setting over the Mediterranean. It felt so human, so expressive. It talked about the amber light clinging to the waves like a memory. But then I looked at the raw output log for a second, and it hit me again. There is not a single word in that machine. There is no amber, there are no waves, and there certainly is no memory. It is just a staggering, almost terrifying amount of floating point numbers moving through a series of gates.
It's the ultimate magic trick, Corn. We see the poetry, we see the code, we see the reasoning, but underneath the hood, it's all just cold, hard, beautiful linear algebra. Today's prompt from Daniel is about exactly that: the mathematical substrate of artificial intelligence. He wants us to look past the language and into the probability, the calculus, and the matrix heavy lifting that makes the whole thing tick. We're moving from the what of AI to the how, and the how is written in the language of mathematics.
It's a perfect time for it, too, because while everyone is obsessing over the latest chatbot interface or the newest avatar skins, the real breakthroughs in twenty twenty-five and now into twenty twenty-six are happening in the math itself. I saw that DeepMind's AlphaEvolve actually broke a fifty-year-old record for matrix multiplication efficiency. That sounds like something only you would get excited about, Herman, but it actually has massive implications for how these things run. It's the difference between a model that takes a week to train and one that takes five days.
Oh, I was absolutely buzzing about the AlphaEvolve news. Think about it, Corn. For over fifty years, we've relied on the Strassen algorithm for four by four matrix multiplications. Volker Strassen brought the number of scalar multiplications down from sixty-four to forty-nine in nineteen sixty-nine. It was a landmark in computational complexity. And just last year, AlphaEvolve found a way to do it in forty-eight. It sounds small, one single multiplication saved, but when you consider that large language models are essentially just trillions of these operations happening every second, that one percent efficiency gain is a monumental shift in computational cost and energy consumption. I'm Herman Poppleberry, by the way, for anyone joining us for the first time, and I've been waiting for an excuse to nerd out on the chain rule for about three months now.
I knew it. You probably have a framed picture of a Hessian matrix on your nightstand right next to your glasses. But let's set the stage for the listeners who might feel a bit of math-induced vertigo. When we say AI is just math, we aren't just being metaphorical. We're saying that every concept, every nuance of human thought we feed into these models, has to be translated into a coordinate in a high-dimensional space. We're essentially mapping the entire human experience onto a giant, invisible grid.
That's the foundational shift. If you go back to the early days of the field, folks like George Boole in the mid-nineteenth century were looking at logic as a binary system. True or false. One or zero. That symbolic logic carried us a long way, and it's the reason your computer works at all. But it was brittle. It couldn't handle the messiness of human language. You couldn't write enough if-then statements to describe the feeling of a sunset. The transition we've seen over the last few decades is the move from that rigid Boolean logic to the statistical probability models pioneered by giants like Claude Shannon and Andrei Kolmogorov.
Right, so instead of telling a computer a bird is something that has feathers and flies, which breaks the moment you meet a penguin or an ostrich, we're now telling the computer to look at the statistical probability of the word feathers appearing near the word bird in a billion sentences. We're trading definitions for distributions.
Spot on. Claude Shannon's work on information theory in nineteen forty-eight gave us the concept of entropy, which is a mathematical measure of uncertainty. When a language model is predicting the next word, it's trying to minimize that entropy. It's looking at the Markov chains, a concept from Andrei Markov in the early nineteen hundreds, to see which state follows the current one. But instead of just looking at the last word, it's looking at the entire context window through a lens of pure probability.
And that's where the mapping problem comes in. To make that work, you have to take a word like apple and turn it into a list of numbers, a vector. In a modern model, that vector might have anywhere from seven hundred sixty-eight to four thousand ninety-six dimensions. Imagine a room with four thousand corners, and the word apple is a tiny dot at a very specific set of coordinates in that massive, complex space.
This is what we call the embedding space. In episode ten ninety-seven, we talked about the Neural Cathedral, and this is the architecture of that cathedral. Every word in a vocabulary of thirty-two thousand to a hundred thousand tokens has its own unique address. And the magic is that words with similar meanings end up with similar addresses.
This is what really trips me up. When we talk about these embedding spaces, we're saying that meaning is literally defined by distance. If the word apple and the word pear are close together in that four thousand dimensional room, the AI perceives them as similar. If apple and carburetor are on opposite ends of the cathedral, it perceives them as unrelated. It's a geometric representation of semantics.
It's the unreasonable effectiveness of mathematics, as Eugene Wigner famously put it in nineteen sixty. Why should the same linear algebra we use to describe the rotation of a rigid body in physics be the perfect tool to describe the relationship between love and affection? But it works. And the moment that really proved this to the world was back in twenty thirteen with Tomas Mikolov and the Word-to-Vec paper.
That was the king minus man plus woman equals queen moment. I remember when that came out, it felt like we had discovered a secret code for human thought.
It was the shot heard round the world for computational linguistics. When they showed that you could perform literal arithmetic on these word vectors and get a semantically meaningful result, it proved that the geometry of the space actually captured the essence of the concepts. It wasn't just random clustering. There was a logic to the numbers that mirrored the logic of our minds. If you take the vector for king, subtract the vector for man, and add the vector for woman, the nearest neighbor in that high-dimensional space is queen. That's no fluke; it's the result of the model learning the underlying structure of the data.
So if the embedding space is the map, the Transformer architecture is the engine that drives us through it. We've talked about Transformers before, but I want to dig into the math of the attention mechanism specifically. Because when people hear attention, they think of a human focusing on a task. But in a Transformer, it's just a series of matrix multiplications called Query, Key, and Value.
This is where the beauty of the math really shines. Think of the Query as what the model is looking for, the Key as the label for every other word in the sentence, and the Value as the information that word carries. To figure out how much attention to pay to a word, the model takes the dot product of the Query and the Key matrices. This was the core innovation of the twenty seventeen paper, Attention Is All You Need, by Ashish Vaswani and his team.
Let's pause there for a second for the non-math majors. A dot product is basically just a way to measure how much two vectors are pointing in the same direction, right?
Precisely. If the vectors are aligned, the dot product is high. If they're perpendicular, it's zero. So, the model is essentially asking, how much does this word's key match what my current query is looking for? But there's a crucial bit of math there that most people gloss over, which is the scaling factor. They divide that dot product by the square root of the dimension of the model.
Why the square root? Is that just to keep the numbers from getting too big and blowing up the hardware?
That's a huge part of it. Without that scaling factor, the dot product values can get so large that they push the softmax function into regions where the gradients are tiny. When the gradients get that small, the model stops learning. It's called the vanishing gradient problem. It's like trying to find your way down a mountain in a thick fog where you can't tell which way is down because everything feels flat. The math has to be tuned perfectly to keep the signal flowing through the network during training.
That brings us to the real heavy lifter of the whole operation, backpropagation. We have to give a shout out to Geoffrey Hinton, who just won the Nobel Prize in Physics in twenty twenty-four for this stuff. Even though the math of backprop is essentially just the chain rule from high school calculus, the way he and his colleagues formalized it in that nineteen eighty-six Nature paper changed everything.
It's funny how we had the math for decades. People were talking about these concepts in the sixties and seventies, but we didn't have the compute to see it work at scale. Backpropagation is how the model learns from its mistakes. It calculates the error at the end of a sentence, the difference between what it predicted and what the actual next word was, and then uses the chain rule to pass that error backward through every single weight in the network. It's a massive blame assignment exercise. It says, you were a little bit too high, you were a little bit too low, let's nudge you in the right direction.
It's an optimization problem. We're trying to find the lowest point in a landscape that has billions or even trillions of dimensions. I find it fascinating that we use Stochastic Gradient Descent for this. It's basically like a blind person trying to find the bottom of a valley by feeling the slope under their feet and taking a small step in the steepest direction.
And because the landscape is so complex, we can't just take one big step. We have to take millions of tiny, noisy steps. That's the stochastic part. It adds a bit of randomness that helps the model jump out of small ruts, or local minima, and find the true bottom. It's a statistical approximation of a perfect solution that we can never actually calculate directly. We're using algorithms like Adam and AdaGrad, which are rooted in convex optimization theory, to navigate this impossible terrain.
I want to shift gears to what's happening right now, the twenty twenty-five and twenty twenty-six frontier. We're seeing a move from models that just predict the next word to models that are actually doing formal mathematics. You mentioned AlphaEvolve, but what about AlphaProof and the Gemini Deep Think models? This feels like the next level of the game.
This is a massive pivot, Corn. For the last few years, the criticism of large language models was that they were just stochastic parrots. They were guessing the next word based on probability, but they didn't really understand the underlying logic. But with AlphaProof, Google DeepMind combined the language modeling of a Transformer with the reinforcement learning logic of AlphaZero. They're using a formal language called Lean to verify the math.
And the results were staggering. In twenty twenty-four, AlphaProof solved four out of six problems at the International Mathematical Olympiad, which is silver medal level. But then in twenty twenty-five, the Deep Think mode for Gemini hit gold medal status, solving five out of six. That's not just word prediction anymore. That's symbolic reasoning.
No, it's something much more profound. It's the integration of symbolic reasoning into the neural network. The model is no longer just saying, this word usually follows that word. It's saying, this step in the proof is mathematically sound because it follows from these specific axioms. It's bridging the gap between the messy, statistical world of neural networks and the rigid, formal world of symbolic math that George Boole and Alan Turing envisioned.
It's like the AI has finally learned to check its own work. I was reading about the D-A-R-P-A exponentiating mathematics program, or E-X-P-Math, which is looking at how this can accelerate scientific discovery. If an AI can not only suggest a hypothesis but also formally prove the underlying math, we're looking at a total sea change in how we do science. We're moving from AI as a writing assistant to AI as a research partner.
It's the democratization of high-level reasoning. But even as we get more formal, the statistical roots are still there. Take the concept of Temperature when you're generating text. Most people know that a higher temperature makes the AI more creative or random, and a lower temperature makes it more focused. But the math for that is literally borrowed from statistical mechanics and thermodynamics.
Wait, really? Like the actual physics of heat and particles?
Yes. It's based on the Boltzmann distribution. In physics, temperature tells you the probability of a particle being in a certain energy state. In a language model, we use the softmax function with a temperature parameter to control the probability distribution over the vocabulary. A high temperature flattens the distribution, giving the unlikely words a better chance to be picked. It increases the entropy of the system. We're literally applying the laws of thermodynamics to the way a machine chooses its next word.
That's wild. It really reinforces the idea that these models aren't just mimics; they're physical systems governed by the same mathematical laws as a gas or a crystal. It makes me think about the way we sample tokens, like Top P and Top K sampling. We see those settings in every AI interface now, but they're pure probability theory.
They're directly rooted in cumulative distribution functions. Top K is easy; you just take the top forty or fifty most likely words and ignore the rest. It's a hard cutoff. But Top P, or nucleus sampling, is more elegant. It says, keep adding the most likely words until their combined probability reaches a certain threshold, like ninety percent.
So if the model is very confident, Top P might only look at two or three words. But if it's confused, it might look at a hundred. It's a dynamic way of managing uncertainty. It's pure probability theory in action, ensuring that the model doesn't wander off into total nonsense by picking a word with a zero point zero zero one percent chance of being right.
And that's why the models have become so much more coherent. We're getting better at managing the math of uncertainty. But as we scale up into twenty twenty-six, the math is getting even more exotic. We're no longer just dealing with two-dimensional matrices or three-dimensional tensors. We're moving into four-D and five-D tensors for multimodal data like video and high-fidelity audio.
That sounds like a nightmare for memory management. How do you even process a five-dimensional block of numbers without melting the G-P-U? I imagine the heat coming off those server racks is enough to power a small city.
That's why researchers are looking at advanced tensor decomposition, things like C-P decomposition or Tucker decomposition. It's basically a way to take a massive, complex tensor and break it down into much smaller, simpler pieces that you can actually work with. It's like taking a giant, tangled ball of yarn and finding the three or four main strands that make it up. If we can master the math of tensor decomposition, we can run much larger models on much smaller hardware. It's about finding the latent structure in the data and exploiting it.
This feels like the next big battleground. We have the data, we have the basic architecture, but now we're fine-tuning the mathematical efficiency. It's the difference between a steam engine and a modern jet turbine. They both work on the same basic principles of pressure and heat, but the math of the turbine is infinitely more refined.
It's also about interpretability. In episode eleven twelve, we talked about cracking the black box, and the math is the key to that. If we can understand the math of the embedding space better, we can start to see how the model is actually making its decisions. Researchers are starting to find these mathematical features within the neural cathedral. They're finding specific directions in that four thousand dimensional space that correspond to very specific concepts, like honesty, or sarcasm, or even mathematical correctness.
I remember we touched on that. The idea that we can actually steer the model by nudging its vectors in a certain direction. It turns the AI into a sort of high-dimensional steering wheel. If you want the model to be more helpful, you just add a little bit of the helpfulness vector to every calculation.
And that's the bridge to the future. If we can map the math of the model to the math of the world, we can start to use AI to solve problems that are currently beyond human reach. The AI for Math initiative that Google launched in twenty twenty-five is a perfect example. They're trying to use these systems to find new proofs for unsolved problems in number theory and combinatorics. AlphaEvolve has already improved the best-known solutions for twenty percent of over fifty open problems in math.
It makes me wonder, though, are we approaching a limit where the AI starts discovering math that we can no longer verify? If the proof is ten thousand pages of dense, computer-generated formal logic, does it even count as human knowledge anymore? Or is it just something the machine knows that we have to take on faith?
That's the philosophical cliff we're standing on. We might be heading toward a world where we have to trust the math of the machine because the math of the human brain simply isn't fast enough or deep enough to follow along. But as long as we have the formal verification systems, the symbolic math to check the neural math, we have a tether to reality. We can use the machine to check the machine.
I think the big takeaway for me today is that the language of AI is a beautiful, elaborate mask. It's a mask made of Shakespeare and Python code and helpful advice, but the face underneath is pure geometry. If you want to understand where this is going, you have to look at the vectors, not just the verbs. You have to understand that meaning is a coordinate, not just a definition.
I agree. For the developers and the tech-literate folks listening, I think the move is to dive deeper into the linear algebra. Don't just look at prompt engineering. Look at tensor decomposition. Look at the loss functions. Understanding the cross-entropy loss that trains these models gives you a much better intuition for why they hallucinate or why they get stuck in repetitive loops. It's all in the math. The cross-entropy loss is literally the same mathematical object as Shannon's entropy. It's all connected.
It's the difference between being a driver and being a mechanic. Most people are happy just driving the car, but if you want to know why the engine is knocking or how to make it go faster, you have to get your hands dirty with the calculus. You have to understand the chain rule and how it flows through the layers of the network.
And it's a wonderful time to do it. The tools we have now, the libraries like JAX and PyTorch, make it so much easier to visualize these high-dimensional spaces. You don't have to be a Fields Medalist to start playing with the math of embeddings. You can see the clusters forming in real-time. You can see the logic of the machine as it organizes the world.
Well, I feel like I need to go do some pushups with a copy of a linear algebra textbook now. This has been a deep one, Herman. I think we've really peeled back the curtain on what Daniel was asking for. We've gone from the Mediterranean sunset to the scalar multiplications of AlphaEvolve.
It's been quite a journey, Corn. From George Boole's logic gates in eighteen fifty-four to the four-D tensors of twenty twenty-six, it's a single, continuous thread of mathematical discovery. We're just the latest generation to try and weave it into something that looks like intelligence. And we're just getting started.
Before we wrap up, I want to say a big thanks to our producer, Hilbert Flumingtop, for keeping the gears turning behind the scenes and making sure our own entropy stays low.
And a huge thank you to Modal for providing the G-P-U credits that power this show. We couldn't do these deep dives into matrix multiplication without that kind of serious computational muscle.
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See you then.
