This "Star Talk Cosmic Queries" episode features mathematician and computer scientist Grant Sanderson of the popular "3Blue1Brown" YouTube channel, delving into the fascinating world of unsolved, unsolvable, and conceptually challenging mathematical problems.
The discussion covers a wide range of topics, including the 19th-century revolution in abstract algebra that proved the impossibility of general formulas for solving polynomial equations of degree five or higher, the philosophical implications of "dividing by zero," and the "million-dollar" Clay Millennium Problems (like the Riemann Hypothesis and Navier-Stokes equations). Sanderson clarifies the practical utility of "imaginary" numbers for modeling cyclical phenomena and explains how mathematicians conceptualize "higher dimensions" as abstract frameworks for data rather than physical realities. The episode concludes with an exploration of chaos theory through the lens of the "three-body problem," revealing that some problems are inherently "unknowable," highlighting math's power to define the limits of predictability.
Star Talk Cosmic Queries: Exploring the Depths of Math with Grant Sanderson
0:00 A Revolution in Math: The episode opens discussing a 19th-century revolution in mathematics, specifically the realization that general formulas for solving polynomial equations become impossible for degrees five and higher, leading to the birth of abstract algebra.
0:47 Introducing Grant Sanderson & 3Blue1Brown: Neil deGrasse Tyson and Chuck Nice welcome Grant Sanderson, creator of the highly successful YouTube channel "3Blue1Brown" (with 7 million subscribers), known for making complex mathematical concepts accessible and engaging.
3:03 Unsolved Math Problems - The Clay Millennium Problems: The hosts and guest explore the Clay Math Institute's seven million-dollar unsolved problems, noting their abstract and often hard-to-describe nature.
5:00 "Celebrity" Unsolved Problems: Discussions shift to more relatable "celebrity" unsolved problems, such as the Twin Prime Conjecture (infinitely many prime numbers two apart) and the theoretical challenges posed by the Navier-Stokes equations for fluid flow.
7:55 The Limits of Solvability: The conversation delves into the history of solving polynomial equations, from the well-known quadratic formula to the complex, but still existing, formulas for cubic and quartic equations.
10:35 The Impossibility Proof: A key takeaway is the profound discovery that polynomial equations of degree five or higher are literally impossible to solve with general formulas using basic operations (addition, subtraction, multiplication, division, and roots). This was a monumental achievement by young mathematicians like Évariste Galois, whose work, famously recorded before his death in a duel, laid the foundation for modern abstract algebra.
14:04 The Enigma of Division by Zero: The fundamental question "Why can't we divide by zero?" is addressed. It's explained as leading to an incoherent mathematical concept in general contexts, though specialized areas like projective geometry introduce concepts like "points at infinity" to handle it.
16:23 Math in Cosmology - Singularities: Neil deGrasse Tyson discusses singularities in cosmology (e.g., black holes) where equations imply division by zero, suggesting a limitation in our physical theories, not a flaw in math itself. He highlights how non-Euclidean geometry (pioneered by Riemann in the 1800s) was crucial for understanding curved spacetime.
19:18 The Riemann Hypothesis: A Personal Favorite: Grant Sanderson shares his favorite unsolved problem: the Riemann Hypothesis, one of the million-dollar Clay problems. It concerns a function (the Riemann zeta function) whose zeros, if understood, would reveal the distribution of prime numbers.
21:18 Reimagining "Imaginary" Numbers: The misnomer "imaginary numbers" (square root of -1) is critiqued. They are reframed as "clock numbers" that naturally model cyclical processes, explaining their utility in quantum mechanics and electrical engineering. Their historical origin lies in the cubic formula, where they emerged as a necessary intermediate step to find real solutions.
25:52 Circle Inversion & Conformal Geometry: Circle inversion is explained as a geometric transformation where a circle acts as a "mirror," reflecting points to simplify complex geometry problems, providing a "vibe" for conformal geometry.
28:16 Understanding Tensor Products: Tensors are described as multi-dimensional arrays of numbers (beyond 1D vectors and 2D matrices), useful for organizing complex data in fields like general relativity (describing spacetime curvature) and machine learning (representing words in high-dimensional spaces).
31:36 Modern Mathematical Revolutions: Despite a common perception, new revolutionary paradigms in math continue to emerge, such as algebraic geometry (seeking unified theories across different math branches) and category theory (a new language for proving mathematical statements), though these are primarily tools for research and not typically seen in K-12 education.
35:10 The Irrationality of Pi and Root 2: The discussion clarifies why Pi (the ratio of a circle's circumference to its diameter) is an irrational number. This is illustrated with a classic proof by contradiction for the irrationality of the square root of two, a common method for proving mathematical impossibilities.
40:02 Exploring Flatland and Higher Dimensions: The classic book "Flatland" is discussed, where 2D beings struggle to comprehend 3D, serving as an analogy for our difficulty visualizing higher dimensions. Mathematicians use "higher dimensions" as abstract frameworks for lists of numbers or complex data, not necessarily as physical realities.
46:58 Mobius Strips and Klein Bottles: Examples of non-orientable surfaces (no clear inside/outside) are given: the Mobius strip (with one edge) and the Klein bottle (a 4D analogue that is a closed surface with no edge, requiring self-intersection in 3D).
49:55 The Three-Body Problem and Chaos Theory: The "three-body problem" (predicting the motion of three celestial bodies) is discussed not as an unsolved problem due to missing math, but as one inherently defined by chaos theory. Even with perfect equations, tiny initial errors lead to unpredictable long-term outcomes, meaning its solution is that it is effectively "unknowable" due to extreme sensitivity to initial conditions.
53:19 The Power of Mathematics: Neil deGrasse Tyson reflects on math as a profound model of reality, allowing scientists to explore the universe conceptually and make predictions, thus strengthening the confidence in the fundamental unity of mathematics and the cosmos.
54:51 Connect with Grant Sanderson: Listeners are encouraged to explore Grant Sanderson's YouTube channel, "3Blue1Brown" (or 3B1B), and his social media for visual and accessible explanations of mathematical concepts.