Revisiting Mathematical Foundations with Generative AI to Uncover Hidden Structure

A century ago, mathematics was shaped by foundational debates. Logicism (Frege, Russell), intuitionism (Brouwer), formalism (Hilbert), and structuralism (Bourbaki) all wrestled with the nature of mathematical truth.

For over half a century, foundational exploration in mathematics has been in retreat. After Gödel, the spotlight shifted: from questions about what math is to questions about what math can prove. Most work today operates within inherited systems—rings, fields, groups—without questioning their assumptions.

As a result, foundational exploration became niche—relegated to logicians and category theorists, rarely intersecting with core mathematical practice.

Why It’s Time to Revisit Foundations?

With today’s computational tools, maybe it’s time to return to foundations.

At a recent Simons Institute and SLMath workshop (April 2025), researchers explored how AI tools—proof assistants, symbolic engines, generative models—could augment mathematical reasoning. Not automate it. Extend it.

We now have systems that can:

- Prototype alternative algebraic structures,

- Translate intuition into formal syntax,

- Search over huge relational spaces,

- Generate examples and abstractions at scale.

This opens a door. Not just to check proofs faster—but to explore consistency of entirely new systems that are too brittle or strange to build by hand.

An Example Axiomatic Structure to Search for: Anti-Rings

Suppose we reimagine the natural numbers under a new system where Multiplication is fully closed but addition is left partially closed under certain structural constraints.

A sort of anti-ring.

- Closed under multiplication.

- Partially defined under addition.

- Additive combinations are selectively permitted—conditioned on some deeper structural symmetry.

This isn’t just a thought experiment. It’s a concrete use case for returning to foundational design, now powered by computational exploration.

We’re no longer boxed into proving theorems inside legacy frameworks.

We can now ask:

What structures might explain what existing systems obscure?

Anti Rings and the Twin Prime Conjecture

What are Twin Primes

Pairs of primes that differ by 2: (3, 5), (11, 13), (29, 31)...

The Twin Prime Conjecture asks are there infinitely many?

We still don’t know.

Techniques like sieve theory, the Hardy–Littlewood conjecture, and connections to the Riemann Hypothesis have narrowed the gap—but not cracked it.

At its core, the twin prime conjecture reveals a fundamental asymmetry in classical arithmetic:

• Primes are multiplicative—they generate the integers, and their behavior is tightly connected to factorization.
• But the twin prime conjecture is additive—it asks whether primes maintain a rigid difference (of 2) under addition.

Rings, formalized by David Hilbert, the classical algebraic setting for number theory, expect addition to be closed, associative, and distributive. But prime numbers are not closed under addition adding two primes typically gives a composite. This makes rings and fields ill-suited to model the delicate additive patterns of primes.

An anti-ring could provide that structure. Not by smoothing over exceptions, but by preserving them.

Most algebra flattens exceptionality. In a ring, everything is defined. But maybe that’s the problem.

By explicitly breaking additive closure, we can:

- Make rare patterns structurally visible.

- Represent twin primes as edges in a graph—where nodes are numbers and edges are conditionally permitted sums.

- Build topologies over these sparse constraints.

- Search for emergent structure in what current systems treat as noise.

From Poker to Proofs

Until recently, crafting such a framework would have been speculative at best. But we're now entering an era where generative AI systems and strategic learning algorithms (like those used in poker, Go, and AlphaFold) are becoming potent tools not just for modeling problems, but for discovering new mathematical ideas.

In poker, AI systems like DeepStack and Pluribus don't rely on fixed rules—they reason under uncertainty, model hidden variables, and iteratively refine strategies. In AlphaFold, neural architectures learned the language of protein folding—an unsolved scientific puzzle—by internalizing constraints from physics, chemistry, and evolutionary biology.

Now imagine applying the same philosophy to mathematics:

• Uncertainty: Instead of deducing from fixed axioms, allow the system to explore axiomatic “moves” that are only partially defined.
• Constraints: Encode known truths about primes as ground rules.
• Optimization: Train models to construct and evaluate candidate axiomatic systems that replicate prime-like behavior.

AI would not just search for proofs—it would help construct the system in which a proof becomes possible.

We finally have tools that let us:

- Build partial algebra systems at scale,

- Formalize new axiom sets and test their consistency,

- Search for structure across spaces we used to explore only by hand.

This doesn’t replace proofs. It changes how we search for the fundamental assumptions behind what we prove.

Imagine a generative AI model that learns:

• When partial addition operations make sense within a structure mimicking the primes.
• How to define consistency conditions so the additive and multiplicative components don’t conflict.
• How to evaluate new systems for internal coherence and external alignment with known number-theoretic phenomena.

Large language models (LLMs) trained on formalized mathematics, algebraic topology, and number theory could generate plausible candidate axioms and test them against constraints—just as AlphaFold evaluates folds against molecular stability.

But LLMs alone are not enough. To ensure these systems are consistent and meaningful, we must incorporate formal proof assistants—Lean, Coq, Isabelle, Metamath—as a layer of validation. These tools rigorously test the coherence of axiomatic systems, just as compilers verify the logic of code.

Together, this forms a pipeline:

Generate → Formalize → Validate → Iterate

Just like in game-playing AI, the space of axiomatic possibilities becomes a strategic environment to explore—where consistent systems, not moves, are the objects of play.

Let’s reopen the door to foundations—not out of nostalgia, but because new tools are finally available to attack these old questions from new angles.

This may or may not unlock the twin prime conjecture but it can provide us with a new way to think about math—and that alone is worth the effort.

If you're working at the intersection of number theory, logic, or computational discovery, I'd love to hear your thoughts. Have you explored similar paths? What kind of datasets or models would you build to expand on these ideas. Let me know in the comments.

ABOUT THE AUTHOR

Aaron (Ari) Bornstein is a Principal AI researcher with a passion for history, engaging with new technologies and computational medicine. As Applied Machine Learning and Data Science lead for Microsoft Healthcare in Israel, he works on developing disruptive technologies for the Healthcare industry using advanced NLP algorithms. Previously he worked in positions in which collaborated with the Machine Learning Community and Israeli Start Up Ecosystem.

Disclaimer: All opinions in the above article are my own and should be treated as such. I’m not a mathematician. But I’m deeply interested in structure and using computational tools to prototype new formal systems—especially in domains where human intuition seems to stall. This post is more about probing than proving and if anything I say doesn't make sense please let me know in the comments.