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?

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 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

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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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