Formal Verification, AI Hallucinations, and Mathematical Truth | Tudor Achim, Co-Founder, Harmonic
The AI ReportShow Notes
In this episode, Tudor Achim, Co-Founder and CEO of Harmonic, the AI lab behind Aristotle, a mathematical reasoning system that won gold at the International Math Olympiad, makes the case that AI hallucinations aren't the problem with today's models. The real problem is that nobody can verify whether a hallucination is right or wrong. Tudor explains why his team bakes formal, computer-checkable verification (using a language called Lean) directly into how Aristotle reasons, so instead of trusting an AI's word, you can mathematically prove it's correct.
Liam and Tudor go deep on what "truth" actually means in mathematics versus the real world, why Andrew Wiles's famous proof of Fermat's Last Theorem had a two-year hidden flaw, and why Tudor believes math is in the middle of its first fundamental shift in 4,000 years, moving from proofs written in English to proofs written in verifiable code. They also get into a spirited debate about the U.S. education system, what Harmonic actually looks for when hiring (hint: it's not the résumé), and why Tudor thinks AI will never be trusted to grade its own homework.
Key Topics Covered
What "truth" means in mathematics versus science, and why logical reasoning is really just a simple form of math
Why the proof of Fermat's Last Theorem had a hidden flaw for two years, even after being announced
Why hallucinations are actually necessary for AI reasoning, and what separates a good hallucination from a bad one
How Harmonic uses Lean and formal verification to make Aristotle's math proofs checkable step by step, like reviewing code
Why Tudor doesn't think any AI will ever be trusted to fully verify its own output
Why Harmonic gives away the Aristotle API for free right now, and where the business model is headed
The "phase transition" Tudor believes is happening in math for the first time in 4,000 years: from English proofs to machine-verified code
Why open-sourcing formal math matters more to Harmonic than keeping a competitive edge
Harmonic's five-year goal: contributing to solving a Millennium Prize Problem by 2028
A debate on whether the U.S. education system actually teaches critical thinking, and what AI should (and shouldn't) change about how kids learn
What Harmonic actually looks for when hiring, and why résumés carry almost no signal anymore
Tudor's take on inequality, taxation, and what a healthy AI-driven economy could look like
Episode Timestamps 00:00 Intro and welcome
00:33 What is truth, and is logic just math?
07:59 Why the company is called harmonic.fun
08:48 The real problem with LLMs and truth
12:57 How formal verification works inside Aristotle
16:43 Who else is building in this space
19:48 Can AI ever be verified with 100% accuracy?
23:10 Why AI can't fully verify its own answers
26:46 Open-source math vs a venture-backed business
29:39 The five-year goal: a Millennium Prize Problem by 2028
31:51 Has Harmonic's vision changed?
34:28 AI, formal verification and the future of education
44:34 Debating the US education system
45:09 How Harmonic hires and what they look for
50:21 Testing for trust and honesty in interviews
54:14 Harmonic's biggest challenge right now
58:19 The future of working with AI
59:55 Family, kids and optimism about the future
1:01:38 Inequality, capitalism and AI's role in the economy
1:04:19 Why Tudor does what he does
1:05:19 Where to find Tudor
Connect with Tudor on LinkedIn: https://www.linkedin.com/in/tudorachim/
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