Neel Somani Explores the Role of Proof Systems in AI and Mathematical Reasoning

Neel Somani approaches artificial intelligence and mathematical reasoning from a systems-first perspective grounded in formal methods, proof systems, and mechanistic understanding. His work sits at the intersection of machine learning research, structured reasoning, and verification, where questions of abstraction, internal consistency, and proof are not philosophical curiosities but practical constraints. 

In examining AI’s role in mathematical reasoning, Somani emphasizes that progress depends less on surface-level problem solving and more on how well systems can assist humans in constructing, validating, and formalizing arguments within rigorous frameworks.

That distinction is why recent discussions around AI-assisted mathematics are worth careful consideration. The question is not whether AI can “solve math problems” in isolation, but whether it can meaningfully assist humans in constructing, validating, and formalizing mathematical arguments. This difference is subtle but fundamental.

“The real challenge isn’t whether a model can output a correct answer,” says Neel Somani. “It’s whether the reasoning that leads to that answer can be inspected, constrained, and trusted under formal assumptions.” This framing shifts the conversation away from isolated problem-solving and toward collaboration between human insight and machine verification.

A recent discussion on ErdosProblems provides a useful case study. The thread focused on a combinatorial identity involving binomial coefficients and the construction of infinitely many distinct-index solutions. The interest was not novelty or computation, but structure: why the construction works, how it generalizes, and how it can be verified. The example illustrates both the promise and the current limits of AI-assisted reasoning in mathematics.

A Concrete Mathematical Example

The identity under discussion involves products of binomial coefficients with carefully chosen parameters. By selecting values that preserve symmetry while maintaining distinct indices, it becomes possible to generate infinitely many valid solutions. One explicit instance demonstrates how a general parameterization reduces to a clean numerical identity.

What makes this example meaningful is not computational difficulty. Any modern system can evaluate binomial coefficients efficiently. The challenge lies in reasoning about structure: ensuring indices remain distinct, ensuring the identity holds generically rather than accidentally, and ensuring the construction scales as parameters grow.

These steps are conceptual. They depend on understanding why the identity holds, not merely confirming that it evaluates correctly in isolated cases. Framing the construction, selecting parameters, and recognizing generalization all require human mathematical insight.

At the same time, AI tools can contribute once that framing exists. They can help explore candidate constructions, check internal consistency, and test boundary cases. Used correctly, they act less like autonomous problem solvers and more like accelerants for structured reasoning.

What AI Contributes Today

In mathematical contexts, AI is most effective when constrained. Given a clear problem statement and a proposed structure, models can assist by identifying overlooked assumptions, checking algebraic relationships, or suggesting alternate parameterizations consistent with the original logic.

This role differs sharply from discovery. AI does not currently originate deep mathematical insight without structure. Instead, it explores the implications of the ideas humans introduce. In that sense, AI functions as a high-speed collaborator operating within a bounded domain.

Somani argues that this bounded role is not a limitation but a design principle: “Mathematics already operates inside strict constraints. AI becomes most useful when it respects those constraints rather than trying to bypass them.” In this view, models function best as tools for exploration and verification, not as sources of unstructured mathematical insight.

This limitation is not a weakness. Mathematics itself is built on constraints. Definitions, axioms, and proof systems exist to restrict ambiguity. AI systems that operate within these boundaries are more useful than systems that attempt to bypass them.

The ErdosProblems example reflects this dynamic clearly. The reasoning behind the construction is human-driven. AI’s contribution lies in verification, exploration, and consistency checking. The result is not automation of mathematics, but augmentation of mathematical work.

Reasoning About Programs, Not Just Results

This distinction mirrors a broader theme in AI research: understanding systems well enough to reason about their behavior under constraints. In recent writing on mechanistic interpretability, I’ve argued that explanation alone is insufficient. What matters is whether we can localize behavior, intervene meaningfully, and certify outcomes within bounded domains.

Mathematics offers a uniquely clean environment to test these ideas. Correctness is binary. Structure is explicit. There is no ambiguity about whether a proof holds. In this sense, AI-assisted mathematical reasoning is not a novelty but an early proving ground for what reliable reasoning systems might look like more generally.

Viewed this way, mathematical arguments resemble programs. They take inputs, apply transformations, and produce outputs that must satisfy strict invariants. Reasoning about these systems requires more than intuition. It requires guarantees.

Proof Systems and Formal Verification

This naturally connects to proof assistants and formal verification systems such as Lean. These tools occupy a critical position between human reasoning and machine verification. They demand that proofs be expressed with exact precision, eliminating ambiguity while increasing the burden of formalization.

Autoformalization, the translation of informal reasoning into fully formal proofs, remains an open challenge. While progress has been made, it is still difficult to automate completely. However, examples like the one discussed suggest that certain classes of arguments are becoming increasingly amenable to partial automation.

Reflecting on proof systems and formal verification, Somani notes that “Generating plausible arguments is easy. Certifying correctness across all relevant assumptions is the hard part, and that’s where formal methods still matter most.” This distinction underscores why verification, rather than generation, remains the central bottleneck in AI-assisted mathematics.

The significance lies not in replacing mathematicians, but in reducing friction. If more reasoning steps can be checked or formalized automatically, researchers can focus on conceptual questions rather than bookkeeping. Over time, this could change how mathematical knowledge is validated and shared.

This focus on verification over surface-level explanation reflects a broader argument I’ve made elsewhere about the limits of interpretability without guarantees. In The Endgame for Mechanistic Interpretability, I outline why true understanding requires systems that can be reasoned about formally, not merely described intuitively.

Verification, not generation, is the bottleneck. Generating plausible arguments is easy. Proving they are correct under all relevant assumptions is not. Proof systems excel at the latter, and AI may help bridge the gap between informal insight and formal proof.

Where Researchers Disagree

Despite these advances, researchers remain divided on how far AI can go in mathematical reasoning. Some view current progress as incremental, arguing that genuine understanding remains uniquely human. Others believe hybrid workflows combining AI and formal systems could scale in ways previously impractical.

Skepticism is justified. Mathematics has a long history of tools that promised automation but delivered modest gains. At the same time, dismissing current developments outright risks missing meaningful shifts in how research is conducted.

The most defensible position lies between extremes. AI is neither a replacement for mathematical insight nor a trivial convenience. It is a tool whose impact depends on how precisely it is used and how well it integrates with formal frameworks.

Implications Going Forward

The broader implication of examples like this one is methodological. They suggest a future where mathematical work increasingly blends conceptual reasoning, computational assistance, and formal verification. Each component reinforces the others.

For AI researchers, mathematics provides a rigorous benchmark. Success here requires more than scale or fluency. It requires systems that respect structure, constraints, and proof. For mathematicians, AI offers new ways to test ideas, explore constructions, and reduce verification overhead.

The ErdosProblems discussion does not signal a breakthrough in isolation. Instead, it reflects a gradual convergence of tools and techniques. As AI systems improve and proof assistants become more accessible, the boundary between informal reasoning and formal proof may continue to narrow.

Thoughtful use of AI within formal, constrained frameworks has the potential to improve how mathematical knowledge is constructed and verified. More importantly, it offers a glimpse of what it might mean to reason about complex systems with guarantees rather than heuristics, a theme that extends well beyond mathematics.