@snowboat84: Today, let's discuss something hardcore. One question: what level of mathematics does AI use? From the perspective of tools and models themselves, the mathematics used by AI has an average age of 150 years, with most being from before the mid-19th century: matrix multiplication, gradient descent, chain rule, Fourier transform, inner product, probability — mostly content from the first two years of undergraduate studies. But some phenomena emerging from AI...

Let’s talk about something hardcore today. One question: how much math is actually used in AI? From the perspective of tools and models themselves, the mathematics behind AI has an average age of about 150 years—most of it existed before the mid-19th century: matrix multiplication, gradient descent, the chain rule, Fourier transforms, inner products, probability. For the most part, it’s content from the first two years of an undergraduate degree.

But some phenomena that have emerged from AI cannot be explained by even the most advanced mathematics available today. Here are a few of the most notable:

• Scaling Law: As you scale up model size, data, and compute, the model’s loss decreases along an extraordinarily clean power-law curve—nearly a straight line on a log-log plot. A massive system with hundreds of billions of parameters and deep internal nonlinearities behaves so regularly at the macro level. Why this regularity exists? No one knows.

• Emergent Abilities: Capabilities like three-digit addition, multi-step reasoning, and writing code are almost nonexistent in small models. But once the model parameter count crosses a certain threshold and the model becomes large enough, it suddenly acquires all these abilities. In physics, this is the same kind of phenomenon as water turning into steam—a phase transition. But while phase transitions in water have a complete theoretical framework, there is no model at all for AI’s “capability phase transitions.”

• Double Descent: Traditional statistical learning theory tells us: the larger the model, the easier it overfits; test error should first decrease and then increase. But the observed error curve actually goes: down, up, then down again, eventually falling to a point lower than the optimal predicted by classical theory. The entire body of statistical learning theory has been overturned by large models. Why? No widely accepted explanation exists.

• In-Context Learning: A phenomenon that emerged after GPT-3. Give the model a few examples, and it can complete a new task without updating any parameters. By definition, “learning” requires changing parameters, yet large models can learn on the fly during inference. This means the model hides some kind of “meta-learning” inside that we cannot see. What is this mathematically? No one can clearly explain it.

• Representation Geometry: What exactly has the model internalized? Interpretability research from A company found a strange phenomenon: a single neuron encodes several unrelated concepts at once. For example, the same neuron might respond to “Golden Gate Bridge,” “Japanese language,” and “DNA sequence.” Intuitively, one dimension should encode one thing, but neural networks seem to have found some kind of “superposition” trick, cramming far more features into limited dimensions than the number of dimensions would suggest.

Why does this happen? No mathematics can explain it.

To draw an analogy with physics: at the end of the 19th century, physics mainly relied on calculus and was good enough. But a few “clouds” hung in the sky: blackbody radiation and the speed-of-light experiment—phenomena that contemporary theory could not explain. Those clouds later exploded into quantum mechanics and relativity, forcing the development of the newest mathematics of the 20th century (functional analysis, differential geometry, mathematical structures).

AI today is in a situation very similar to physics in 1900: the tools are as old as can be, the results are better than expected, but a pile of unexplained phenomena remains, and existing mathematical tools are completely helpless.

If history rhymes, these “AI clouds” might just be the trigger for the next great leap in 21st-century mathematics.