Breaking Math Podcast

Mathematician, mental-math performer, and backgammon grandmaster Arthur Benjamin joins Noah and Autumn to talk about his book How to Win. Together they explore what mathematics knows about winning that human intuition gets wrong: coincidences, the birthday paradox, gambler's ruin, and why the house wins even when you reach your goal more than half the time.

Timeline:

00:00 Can math ruin the fun of games?

00:38 Why people play for the challenge, not the answer

02:35 Mental arithmetic and the human element

03:30 Calculators, AI, and the future of coding

05:47 What does math know about winning that intuition gets wrong?

06:02 Coincidences and the birthday paradox

10:16 Why games are a microcosm for pattern-seeking

16:40 What games teach that the classroom doesn't

16:52 Gambler's ruin: how to get to the concert

20:30 The paradox: why casinos aren't afraid

22:30 What does "winning" really mean?

23:29 Wordle streaks vs. average guesses

27:07 Rubik's Cube, blackjack, and why some games never get old

35:38 Does math take the mystery out of games?

Follow Arthur Benjamin on his website:

(https://www.arthurbenjamin.info/)

Subscribe for more on math, AI, technology, and the systems running the world. Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

"I'm just not a math person." If you've ever said that about yourself, this episode is for you.

Dr. Geillan Aly, author of Compassionate Math, joins Breaking Math to break down the real difference between math anxiety and math trauma — and why mathematics, more than almost any other subject, is uniquely built to make one bad experience feel like permanent proof of failure.

In this episode we cover:

00:00 – What is math trauma, and why is it different from anxiety?

02:35 – Why math specifically (vs. history, reading, or other subjects)?

07:28 – Teaching concepts vs. teaching procedures

13:32 – The "sixes" story: a failing student's transformative breakthrough

20:15 – Math anxiety vs. math trauma, explained

28:56 – Whose fault is math trauma — students, teachers, or society?

36:21 – How compassionate math scales beyond the classroom

39:21 – One piece of advice for students, and one for teachers

Follow Geillan Aly on her website:

(https://www.compassionatemath.com/)

Subscribe for more on math, AI, technology, and the systems running the world. Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

Professional forecaster Molly Hickman breaks down what it really means to assign a probability to the future — and why she believes generalists often out-forecast subject-matter experts. This episode explores the art and science of forecasting, from techniques to ethical considerations, and how AI and prediction markets are shaping our understanding of the future.

Key Topics

The definition of forecasting and its importance

Techniques for starting in forecasting

The role of AI and large language models in forecasting

How to interpret probabilities and conditional forecasts

Forecasting in complex systems like climate and geopolitics

Ethical boundaries and red lines in prediction markets

The impact of AI bots on forecasting accuracy and decision making

Chapters

03:06 Getting Started with Forecasting: Tools and Techniques

06:15 Beginning Forecasting as a Beginner

07:31 Gut Feelings vs Market Wisdom

08:33 The Delphi Loop and Group Forecasting

09:39 Measuring Forecast Accuracy and Skill

11:17 Forecasting Long-Term and Uncertain Events

12:40 Extrapolating Trends and Model Limitations

14:19 AI Bots in Forecasting and Their Performance

18:16 Prediction Markets as Collective Wisdom

19:19 The Future of Prediction Markets and Society

24:06 The Meaning of Probabilities and Risk Assessment

27:20 Dealing with Chaos and Unpredictability

32:52 Combining Models and Expert Opinions

36:31 Forecasting and Expertise in Science and Policy

39:01 Forecasting AI Risks and Ethical Boundaries

Follow Molly Hickman on X (https://x.com/celloMolly)

Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

What does it mean to be alive? In this episode of Breaking Math, Autumn and Noah speak with Melanie Challenger, author of Alive, about one of the most profound questions in science and philosophy: how do we define life?

Challenger argues that life is not simply a machine-like process or a bundle of genetic instructions. Living beings are embodied, purposeful agents. From single-celled organisms to sequoia seeds, from animals to human beings, life is marked by an astonishing capacity to work to keep itself alive.

Chapters

08:12 The concept of purpose in living beings

09:14 The scientific view of purpose and agency

11:52 The importance of purpose and meaning in life

13:19 The danger of ignoring organism agency in science

14:34 Living beings as purposeful agents

15:35 Comparing purpose in a Roomba and a single-celled organism

18:03 Autopoetic vs allopoetic systems

20:03 Free will, agency, and the universe

23:24 The physical basis of life and energy

28:38 Aristotle's concept of psyche and purpose

33:46 The importance of understanding what life truly is

37:56 Material integration and the difference between machines and living beings

38:15 The concept of self and embodiment in life

41:09 The whole body as the agent, not just the brain

Follow Melanie Challenger on her website:

(https://www.melaniechallenger.com/) Subscribe for more on math, AI, technology, and the systems running the world. Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

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

Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

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

Follow Vivienne Ming on X (https://x.com/neuraltheory) Get Vivienne's book, Robot Proof: (https://amzn.to/3Tz21aP)

Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

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

Follow Marc on X [https://x.com/MarcDunkelman]

Get Marc's book, Why Nothing Works: https://amzn.to/4pbFvAB]

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

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)

Follow Julia Angwin on

X (https://x.com/JuliaAngwin/)

Bluesky (https://bsky.app/profile/juliaangwin.com)

Proof News (https://www.proofnews.org/)

Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Instagram (https://www.instagram.com/1autumnleaf/)

email: [email protected]

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)

Follow Kevin Hartnett on

X (https://x.com/KSHartnett)

Bluesky (https://bsky.app/profile/kevinhartnett.bsky.social)

Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]

What happens when the evidence of injustice is buried in messy, redacted, or inaccessible data? Mathematician and data scientist Chad Topaz joins Breaking Math to discuss his book Unlocking Justice. Together, we explore policing, sentencing, public records, Rikers Island, algorithmic risk, and the limits of quantifying human lives. This is a conversation about math, power, transparency, and the small acts of hope that can change systems.

Chapters

00:00 Introduction and Context of the Conversation

01:11 Chad's Journey from Mathematics to Social Justice

03:50 The Personal Nature of Chad's Book

04:40 Challenges in Data Collection and Access

08:03 The Impact of Data on Policing and Surveillance

09:51 Humorous Yet Tragic Data Collection Experiences

12:55 The Importance of Data Preparation and Cleaning

14:40 Navigating Imperfect Data and Its Consequences

17:48 The Balance Between Quantification and Human Stories

22:25 Incarceration and Public Health: The Rikers Island Case Study

31:36 Mathematics and Social Justice: Secrets of the Elite

39:03 Hope and Action: A Personal Journey in Data for Justice

Follow Chad Topaz on

Bluesky(https://bsky.app/profile/chadtopaz.bsky.social) Book (https://amzn.to/3S21pKb)

Follow Breaking Math on

Substack (https://breakingmath.substack.com/)

X (https://x.com/breakingmathpod)

Instagram (https://www.instagram.com/breakingmathmedia/)

Bluesky (https://bsky.app/profile/breakingmath.bsky.social)

Website (https://www.breakingmath.io/)

YouTube (https://www.youtube.com/@BreakingMathPod)

Follow Noah on

Instagram (https://www.instagram.com/profnoahgian/)

X (https://x.com/ProfNoahGian)

Bluesky (https://bsky.app/profile/profnoahgian.bsky.social)

Follow Autumn on

X (https://x.com/1autumn_leaf)

Bluesky (https://bsky.app/profile/1autumnleaf.bsky.social)

Instagram (https://www.instagram.com/1autumnleaf/)

Substack (https://substack.com/@1autumnleaf)

email: [email protected]