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Can AI do math? Three stages and who checks the proof
Why does a chatbot sometimes miscalculate a restaurant tip while solving hard Olympiad problems? Three stages of mathematics, a prime-number trap and how formal proof checkers like Lean verify every step.
Tip-calculation errors and Olympiad gold
Anyone using AI in daily life knows the paradox: a chatbot can stumble over a long receipt total while headlines report that new models solve hard mathematics competitions or assist with research proofs.
The resolution is that "Can AI do math?" mixes up three very different tasks: everyday arithmetic, multi-step competition problems and mathematical proofs covering infinitely many cases.
Stages 1 and 2: Token prediction, calculators and thinking time
A pure language model is not a pocket calculator: it predicts the next likely word or digit piece step by step (Stage 1). Across long chains of numbers without tools, a single mispredicted digit can throw off the result. Once the system hands the calculation to a code tool (such as Python), an exact interpreter computes the arithmetic and returns the result.
At Stage 2 (competition benchmarks such as AIME 2025 or FrontierMath), a single formula is not enough. Reasoning models break the problem into intermediate steps, test approaches and backtrack from dead ends. While models now reach high scores on AIME 2025, FrontierMath Tiers 1 to 4 show that many research-grade problems remain unsolved.
Stage 3: Why forty hits still do not make a proof
Higher mathematics rarely asks about a single number; it asks about statements covering infinitely many numbers. A classic teaching example shows why testing examples is not enough:
Consider the formula n² + n + 41 for nonnegative whole numbers n:
- For n = 0, it gives 41 (prime).
- For n = 1, it gives 43 (prime).
- For n = 2, it gives 47 (prime).
In fact, from n = 3 all the way to n = 39, the formula produces nothing but prime numbers. Yet the statement "this formula always produces a prime" is false: at n = 40, we get 40² + 40 + 41 = 1,681 = 41 × 41. A single counterexample refutes forty successful tests.
Contrast that with the claim: The difference between two consecutive square numbers is always odd. Here, the algebraic identity (n + 1)² − n² = n² + 2n + 1 − n² = 2n + 1 provides a general proof for every integer n, because 2n is always even and 2n + 1 is always odd.
How Lean and formal verification enter the picture
When AI systems such as AlphaProof or research models work on mathematical proofs (such as OpenAI's September 2026 paper on a variant of the Navier–Stokes problem with a smooth external force), a crucial question arises: who verifies a multi-page proof that sounds plausible?
That is where a formal proof assistant like Lean comes in:
- What it is needed for: Checking mathematical statements and proofs without gaps against strict logical rules.
- What it receives: Exact definitions, assumptions, the theorem statement and every proof step written in formal syntax.
- What it does: Its small checking kernel verifies step by step whether every transformation follows from the rules and no assumption is missing.
- What comes out: A formally verified proof, or the exact line where a deduction fails.
Humans still need to verify whether the formalized statement actually captures the intended question and which boundary assumptions apply.
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