The Rest Is Science: Zero, Singularities, and the Topology of Holes

Overview

In this episode of The Rest Is Science, the hosts question the idea that math breaks at zero, discuss singularities in fluid dynamics and at cosmological scales, and argue that the fix is better modeling rather than abandoning zero. The conversation then pivots to a playful topology lesson using a straw, a glass, and a T-shirt to dissect holes and boundaries, leading into an evolution tale of the gut and a humorous debate about orientation and symmetry in human design. The show closes with reflections on scientific publishing, AI, and the limits of knowledge, framed by a lively, accessible exchange between presenters.

Introduction and Zero The Problem

The episode opens with a teaser about an object related to the human body and a straw, signaling a playful foray into how we think about holes. The hosts address a question from Kevin about whether scientists should try to invent a math without zero, a tongue in cheek way to critique the notion that mathematics breaks when division by zero occurs. They discuss the everyday teaching of division by zero and contrast it with high level mathematics where singularities arise, such as in the Navier–Stokes equations used to model fluid flow. The central argument is that mathematics does not fail; rather, the models fail when they reach limits of applicability, and zeros can signal where new physics or new modeling is needed.

Zeros, Singularities, and the Limits of Equations

The conversation broadens to how singularities appear when terms in equations blow up as denominators approach zero. They emphasize that the breakdown is not a failure of math per se but a reminder that the equations rely on assumptions that hold only within certain scales or conditions. They touch on gravitational contexts like black holes and Planck-scale physics, explaining that at extreme densities or times near the Big Bang, our current theories lose their predictive power. The takeaway is that rather than discarding zero, scientists should pursue more robust ways to connect mathematics with reality, including better physical theories and perhaps new mathematical frameworks for extreme regimes.

Scale, Constants, and the Peer-Review Landscape

Beyond pure math and physics, the hosts critique how science is published and vetted. They discuss a Sunday Times article about peer review, retractions, and the potential impact of AI on scientific writing. They caution that the drive to publish, secure grants, and gain recognition can create incentives that occasionally skew data or emphasis, and they argue for a more reflective approach to knowledge that accepts limits to what we can know. They also entertain the possibility that future science could embrace certain epistemic limits as part of the human condition rather than a failing to be overcome.

Topology in Everyday Objects

Shifting gears, the episode introduces topology through tangible examples. A lively debate about how many holes a straw has becomes a gateway to talking about the distinction between boundaries and tunnels. The topologist would view a straw as having a single through hole on the surface, while a 3D intuition might suggest two boundary circles. The hosts use the balloon analogy and the notion of a three-dimensional object to illustrate how a straw can be thought of as either two boundaries or one tunnel, depending on perspective. They apply the same reasoning to a T-shirt, concluding there are three holes in total when viewed as a through-body topology, and four boundary circles if stretched into a balloon. This exploration reveals how human perception, language, and mathematical formalism intersect in topology.

Evolutionary Perspective: The Butthole and Digestive Architecture

The discussion then traverses the evolution of body plans. They recount the historical moment around 550 million years ago when a gut-wrenching innovation occurred: the evolution of a second hole that formed a through tunnel, enabling more complex digestion and a more efficient body plan. This topology shift from a single opening to a through tunnel is tied to why animals grew larger and more complex. The hosts highlight how the brain, senses, and digestive tract co-evolved around a central tube, creating the body plan we recognize. They also reflect on how modern anatomy is adapted to route air and food through distinct channels to minimize choking risk, illustrating how topology shapes physiology as well as behavior.

Centering, Symmetry, and Human Uniqueness

In a lighter segment, the hosts revisit the poster centered in the office and link symmetry and right angles to human evolution. They argue that our affinity for centered, symmetric, and rule-bound structures reflects social and cognitive evolution that favored cooperation, planning, and mate selection. They argue that symmetric axes and right angles are not mere aesthetic preferences but indicators of skill and cognitive architecture that aided survival and social coordination. The conversation ends by acknowledging piercings and other modifications that add holes, underscoring how topology still applies in human variation.

Wrap-Up and Call to Curiosity

The episode closes with an invitation to send questions and to stay curious, reinforcing the show’s mission to blend humor with robust science across physics, mathematics, biology, and the philosophy of science.