Synthetic vs Analytic Mathematics

We often think of Large Language Models (LLMs) as all-knowing, but as the team reveals, they still struggle with the logic of a second-grader. Why can’t ChatGPT reliably add large numbers? Why does it "hallucinate" the laws of physics? The answer lies in the architecture. This episode explores how *Category Theory* —an ultra-abstract branch of mathematics—could provide the "Periodic Table" for neural networks, turning the "alchemy" of modern AI into a rigorous science.

In this deep-dive exploration, *Andrew Dudzik*, *Petar Velichkovich*, *Taco Cohen*, *Bruno Gavranović*, and *Paul Lessard* join host *Tim Scarfe* to discuss the fundamental limitations of today’s AI and the radical mathematical framework that might fix them.

TRANSCRIPT:

https://app.rescript.info/public/share/LMreunA-BUpgP-2AkuEvxA7BAFuA-VJNAp2Ut4MkMWk

---

Key Insights in This Episode:

* *The "Addition" Problem:* *Andrew Dudzik* explains why LLMs don't actually "know" math—they just recognize patterns. When you change a single digit in a long string of numbers, the pattern breaks because the model lacks the internal "machinery" to perform a simple carry operation.

* *Beyond Alchemy:* deep learning is currently in its "alchemy" phase—we have powerful results, but we lack a unifying theory. Category Theory is proposed as the framework to move AI from trial-and-error to principled engineering. [00:13:49]

* *Algebra with Colors:* To make Category Theory accessible, the guests use brilliant analogies—like thinking of matrices as *magnets with colors* that only snap together when the types match. This "partial compositionality" is the secret to building more complex internal reasoning. [00:09:17]

* *Synthetic vs. Analytic Math:* *Paul Lessard* breaks down the philosophical shift needed in AI research: moving from "Analytic" math (what things are made of) to "Synthetic" math [00:23:41]

---

Why This Matters for AGI

If we want AI to solve the world's hardest scientific problems, it can't just be a "stochastic parrot." It needs to internalize the rules of logic and computation. By imbuing neural networks with categorical priors, researchers are attempting to build a future where AI doesn't just predict the next word—it understands the underlying structure of the universe.

---

TIMESTAMPS:

00:00:00 The Failure of LLM Addition & Physics

00:01:26 Tool Use vs Intrinsic Model Quality

00:03:07 Efficiency Gains via Internalization

00:04:28 Geometric Deep Learning & Equivariance

00:07:05 Limitations of Group Theory

00:09:17 Category Theory: Algebra with Colors

00:11:25 The Systematic Guide of Lego-like Math

00:13:49 The Alchemy Analogy & Unifying Theory

00:15:33 Information Destruction & Reasoning

00:18:00 Pathfinding & Monoids in Computation

00:20:15 System 2 Reasoning & Error Awareness

00:23:31 Analytic vs Synthetic Mathematics

00:25:52 Morphisms & Weight Tying Basics

00:26:48 2-Categories & Weight Sharing Theory

00:28:55 Higher Categories & Emergence

00:31:41 Compositionality & Recursive Folds

00:34:05 Syntax vs Semantics in Network Design

00:36:14 Homomorphisms & Multi-Sorted Syntax

00:39:30 The Carrying Problem & Hopf Fibrations

Petar Veličković (GDM)

https://petar-v.com/

Paul Lessard

https://www.linkedin.com/in/paul-roy-lessard/

Bruno Gavranović

https://www.brunogavranovic.com/

Andrew Dudzik (GDM)

https://www.linkedin.com/in/andrew-dudzik-222789142/

---

REFERENCES:

Model:

[00:01:05] Veo

https://deepmind.google/models/veo/

[00:01:10] Genie

https://deepmind.google/blog/genie-3-a-new-frontier-for-world-models/

Paper:

[00:04:30] Geometric Deep Learning Blueprint

https://arxiv.org/abs/2104.13478

https://www.youtube.com/watch?v=bIZB1hIJ4u8

[00:16:45] AlphaGeometry

https://arxiv.org/abs/2401.08312

[00:16:55] AlphaCode

https://arxiv.org/abs/2203.07814

[00:17:05] FunSearch

https://www.nature.com/articles/s41586-023-06924-6

[00:37:00] Attention Is All You Need

https://arxiv.org/abs/1706.03762

[00:43:00] Categorical Deep Learning

https://arxiv.org/abs/2402.15332