Black holes are objects that seem exotic to us because they have properties that boggle our comparatively mild-mannered minds. These are objects that light cannot escape from, yet glow with the energy they have captured until they evaporate out all of their mass. They thus have temperature, but Einstein's general theory of relativity predicts a paradoxically smooth form. And perhaps most mind-boggling of all, it seems at first glance that they have the ability to erase information. So what is black hole thermodynamics? How does it interact with the fabric of space? And what are virtual particles?

The idea of something that is inescapable, at first glance, seems to violate our sense of freedom. This sense of freedom, for many, seems so intrinsic to our way of seeing the universe that it seems as though such an idea would only beget horror in the human mind. And black holes, being objects from which not even light can escape, for many do beget that same existential horror. But these objects are not exotic: they form regularly in our universe, and their role in the intricate web of existence that is our universe is as valid as the laws that result in our own humanity. So what are black holes? How can they have information? And how does this relate to the edge of the universe?

In the United States, the fourth of July is celebrated as a national holiday, where the focus of that holiday is the war that had the end effect of ending England’s colonial influence over the American colonies. To that end, we are here to talk about war, and how it has been influenced by mathematics and mathematicians. The brutality of war and the ingenuity of war seem to stand at stark odds to one another, as one begets temporary chaos and the other represents lasting accomplishment in the sciences. Leonardo da Vinci, one of the greatest western minds, thought war was an illness, but worked on war machines. Feynman and Von Neumann held similar views, as have many over time; part of being human is being intrigued and disgusted by war, which is something we have to be aware of as a species. So what is warfare? What have we learned from refining its practice? And why do we find it necessary?

The fabric of the natural world is an issue of no small contention: philosophers and truth-seekers universally debate about and study the nature of reality, and exist as long as there are observers in that reality. One topic that has grown from a curiosity to a branch of mathematics within the last century is the topic of cellular automata. Cellular automata are named as such for the simple reason that they involve discrete cells (which hold a (usually finite and countable) range of values) and the cells, over some field we designate as "time", propagate to simple automatic rules. So what can cellular automata do? What have we learned from them? And how could they be involved in the future of the way we view the world?

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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The spectre of disease causes untold mayhem, anguish, and desolation. The extent to which this spectre has yielded its power, however, has been massively curtailed in the past century. To understand how this has been accomplished, we must understand the science and mathematics of epidemiology. Epidemiology is the field of study related to how disease unfolds in a population. So how has epidemiology improved our lives? What have we learned from it? And what can we do to learn more from it?

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

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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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