Audio Edition: ‘Reverse Mathematics’ Illuminates Why Hard Problems Are Hard

In 1852, the mathematician Francis Guthrie was coloring in a map of English counties when he noticed that he needed only four colors. Was this always true, he wondered? The question captured the imagination of hobbyists and professionals alike — and in 1976, the mathematicians Kenneth Appel and Wolfgang Haken proved it, rather controversially using computers. Recently, mathematicians have revisited the four-color theorem and found a more efficient way to color maps with just four colors.On this episode of The Quanta Podcast, host Samir Patel speaks with writer Greg Barber about the “four-color disease,” its history of false starts and dashed hopes, and what modern mathematicians have learned about graphs by revisiting this famous problem.This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor-in-chief Samir Patel and Senior Editor Hannah Waters speak with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.

Researchers have used metamathematical techniques to show that certain theorems that look superficially distinct are in fact logically equivalent.The article ‘Reverse Mathematics’ Illuminates Why Hard Problems Are Hard first appeared on Quanta Magazine.

In popular culture, computer science is practically synonymous with programming. There are obviously branches of computer science in which computer hardware and software are essential, such as in the design of operating systems or the study of memory management. But in the theoretical side of the field, researchers need not ever touch a real, physical computer. Do computers play an essential conceptual role in that kind of theoretical work? On this episode of The Quanta Podcast, host Samir Patel wades into an age-old debate with writer Ben Brubaker: If computer science isn’t about computers, then what is it about?This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor-in-chief Samir Patel speaks with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.The audio coda clip comes from the Computer History Museum. The use of this audio clip shall not be used to show or imply CHM endorsements of any commercial product or enterprise, or to indicate that the CHM concurs with the opinions expressed in the podcast.

If you are well rested and full, your brain tells you that a bruised overripe apple is a piece of subpar fruit to be passed over. However, if you are exhausted and hungry, an overripe apple is categorized as much-needed food. But how is the brain able to categorize that same apple in these situations? On this episode of The Quanta Podcast, co-host Hannah Waters speaks with contributing writer Conor Feehly about how the brain manages to categorize similar sets of sensory signals in radically different ways. This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, hear the people behind the award-winning publication navigate through some of the most important and mind-expanding questions in science and math.

The newly described microbe represents a world of parasitic, intercellular biodiversity only beginning to be revealed by genome sequencing.The article A Cell So Minimal That It Challenges Definitions of Life first appeared on Quanta Magazine.

Combinatorial discrepancy theory is a branch of mathematics concerned with balance. If you’re making two teams — sports, trivia, you name it — how can you make sure that they’re perfectly balanced and fair, skill-wise? You start with vectors. On this episode of The Quanta Podcast, host Samir Patel speaks with award-winning science journalist and regular contributor Max Levy on an “absurd,” “holy grail” conjecture, and the major breakthrough made in fall 2025 by two theoretical computer scientists. This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor in chief Samir Patel and Senior Editor Hannah Waters speak with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.

The Navier-Stokes equations are enormously successful at predicting how fluids flow and swirl. But they fail to account for the microscopic bits that make up matter. But now math based on symmetries offers a new conception of fluids and why they behave the way that they do.On this episode of The Quanta Podcast, host Samir Patel speaks with writer Charlie Wood on the longstanding effort to rebuild the theory of fluids from the bottom up, including a strange connection to black holes and models that simultaneously run time forward and backward.This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor in chief Samir Patel and Senior Editor Hannah Waters speak with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.

Mathematicians have broken through a long-standing barrier in the study of “minimizing surfaces,” which play an important role in both math and physics.The article New Proofs Probe Soap-Film Singularities first appeared on Quanta Magazine.

Reasoning comes in many technically defined forms, but the basic procedure is easily recognizable: arriving at a sound conclusion by linking together intermediate steps that logically follow from each other. But when it comes to artificial intelligence, it appears AI can either solve famous mathematical research problems in one shot — or experience a “complete accuracy collapse” under the simplest conditions. So which is it: Can AI reason reliably or not?On this episode of The Quanta Podcast, host Samir Patel speaks with writer John Pavlus on the intellectual whiplash surrounding artificial intelligence’s ability to reason.This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor in chief Samir Patel and Senior Editor Hannah Waters speak with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.

Why are only 10% of people left-handed? Geneticists, developmental biologists, neuroscientists, and evolutionary biologists have all sought explanations, resulting in an astronomical amount of research into the mystery of handedness. No one has put all the pieces together yet, but over the last few years, some major new clues have emerged.On this episode of The Quanta Podcast, host Hannah Waters speaks with writer Natalie Wolchover and traces these new clues to uncover the mystery of human handedness.This topic was covered in a recent story for Quanta Magazine.  Each week on The Quanta Podcast, Quanta Magazine editor in chief Samir Patel and Senior Editor Hannah Waters speak with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.