Breaking Math Podcast

Neuroscientist Stuart Firestein (Columbia University) joins Breaking Math to make an extravagant claim: uncertainty isn't a weakness in science — it's the defining feature that makes progress possible. In this episode, we break down why the "one right answer" myth is one of the most damaging ideas in science, why real experts are often the most uncertain people in the room, and why authority and expertise pull in opposite directions, covering two fundamentally different kinds of probability, why Darwin never erased a 300-year-old classification system built on an assumption he disproved, why AI is exceptional at prediction but not built for causation, and why pseudoscience always has a confident answer while real science rarely does — plus the philosophical difference between hope and optimism, and why Voltaire had to invent the word "optimism" in 1759 to describe it.

Chapters
03:00 Predictability and the sea of uncertainties
04:08 Science as a search for probabilities and multiple solutions
06:16 Biological classification and the dynamic nature of species
09:10 The optimistic view of a branching universe
12:41 Probability as the language of optimism
16:48 Two types of probability and their roles
17:50 AI, probabilistic models, and the future of certainty
21:40 Science and the creation of better ignorance
23:21 The importance of asking questions over giving answers
27:21 Authority versus knowledge in science
30:04 Pluralism and multiple solutions in science
32:46 Science in the gray area of uncertainty
35:39 The brain and randomness in thought
39:44 Science as a source of hope and optimism

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Neuroscientist, entrepreneur, and author Dr. Vivienne Ming joins Autumn and Noah to make the case that if we want better AI, we need to build better people first. We get into why AI tutors that hand students answers make learning worse, not better; what her research on "hybrid intelligence" reveals about the human traits — not the AI model — that predict elite human-AI collaboration; a wild experiment running Dungeons & Dragons with Claude and Gemini as dungeon masters to expose the gap between knowing and understanding; her case for "fiduciary AI," legal duty-of-care standards for tutors, hiring tools, and diagnostic models; and the real story of a hiring algorithm that learned to discriminate against women after every explicit gender marker was stripped out.

Chapters
02:20 Why build this book now? The importance of human qualities
04:16 AI in education and the concept of robot-proofing
06:37 The median student and AI personalization
09:31 The limitations of AI understanding and theory of mind
11:30 Building better people with AI and human interaction
14:23 Hybrid intelligence and the role of human-AI collaboration
23:56 Case study: AI in Dungeons & Dragons
30:42 AI's strengths and limitations in understanding and cognition
37:34 The science of purpose and its impact on life and society
44:44 The collective intelligence of humans versus AI
46:54 Key takeaway: Build better people for better

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In this episode, Historian and author Marc Dunkelman to explain why the 19th-century fight over railroad power is the exact fight we're about to have over algorithms and AI. Drawing on his acclaimed book Why Nothing Works: Who Killed Progress — and How to Bring It Back (a Best Book of the Year in the Financial Times and The Economist), Marc unpacks the two competing tools America has always used against concentrated power — antitrust vs. regulation — and why our government's "endemic diffusion of authority" now means nobody can decide anything, from congestion pricing to clean-energy transmission lines to AI safety.
CHAPTERS
04:52 — When private projects come back to the public: Warp Speed, DARPA, CHIPS
08:55 — Two ways to fight concentrated power: break them up vs. regulate
10:52 — Railroads, island communities & the birth of regulation
12:29 — The railroad = algorithm parallel
20:33 — Why nothing gets built: the diffusion of authority
27:30 — "A voice without a veto" and the AI moment
32:53 — Where should government draw the line on new tech?
37:20 — Dunkelman the pragmatist: there is no simple answer
38:21 — Where math and AI can genuinely help public policy
40:46 — The lesson we keep overlooking

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In this conversation we chat with Julia Angwin — Pulitzer Prize-winning journalist, founder of Proof News, and former Wall Street Journal and ProPublica reporter — to make the case that journalism should function more like mathematical proof than anecdote.
We cover how Angwin's team at The Markup used a decision-tree model to prove Amazon was favoring its own products in search results by an 8-to-1 margin — a finding the House Antitrust Committee later cited when referring Amazon to the DOJ for possible perjury. We dig into her "ingredients label" approach to reporting at Proof News (hypothesis, sample size, techniques, limitations), the difference between mathematical proof and the scientific method, and why she thinks control over algorithmic media is now the central battleground for authoritarian power. She also unpacks her new book on resisting authoritarianism, built from interviews with dissidents worldwide, including the "Swiss cheese" model of personal security and why perfectionism is dangerous in a crisis.
Chapters
09:50 Proof News: A New Era in Journalism
19:56 Data-Driven Investigations: A Case Study
30:02 The Future of Journalism and AI
32:53 The Evolution of Search Rankings
35:06 The Role of Algorithms in Information Access
36:41 Fighting Authoritarianism Through Journalism
44:52 Community Resistance Against Authoritarianism
48:33 The Dangers of Perfectionism in Resistance
51:26 Declaring a Position in Journalism
56:25 The Importance of Math in Modern Society

Julia Angwin's book, “On Courage” (https://amzn.to/448G8kY)
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In this episode, Autumn and Noah talk with Kevin Hartnett about why mathematicians are willing to spend years reducing an idea to a level of detail a machine can check, whether formal verification can catch an AI that's technically correct but fundamentally misaligned, the cold-start problem that kept earlier theorem-provers niche, and what it means for the future of mathematical trust once AI can generate proofs faster than any human community can read them.
Timeline:
00:00 Introduction to Lean and Its Significance
03:18 The Journey of Writing the Book
05:13 Human Element in Mathematical Formalization
06:57 Understanding Formal Proofs in Mathematics
11:21 The Origins of Lean and Its Purpose
13:03 Misalignment in Software Specifications
14:39 Building Mathematical Libraries in Lean
17:23 Ensuring Accuracy in Mathematical Foundations
22:00 Overcoming the Cold Start Problem in Lean Adoption
24:36 The Future of Mathematical Proofs
30:26 AI's Role in Mathematics
38:29 Expanding Beyond Mathematics
41:40 The Long-Term Impact of Lean

The Proof in the Code is out now from Quanta Books. (https://amzn.to/3SuNlJm)
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