AfterMath

In the third episode of his new series, AfterMath, Edward Frenkel dives into the question of whether computers can ever reach the cognitive capacity of the human mind -- specifically, in the realm of mathematics. While computers excel at number crunching, they struggle with general mathematical statements. Edward introduces the concept of the Turing Wall, a fundamental limit to what computers can do with numbers, named in honor of Alan Turing, the father of modern computing. Frenkel further explores insights from Alfred Tarski's Undefinability Theorem (closely connected to Kurt Gödel's Incompleteness Theorems). And he also touches upon the limitations of Large Language Models, such as ChatGPT, in handling mathematical truths. At the end of the episode, Frenkel goes back to the 3D sphere he talked about in Episode #2. He gives a 4D spacetime demonstration of it, using... a balloon. Edward Frenkel is a professor of mathematics at UC Berkeley, member of the American Academy of Arts and Sciences, winner of the Hermann Weyl Prize and the Euler Book Prize, and author of the international bestseller “Love and Math” which has been published in 20 languages. LINKS: •⁠ ⁠Edward Frenkel's Official Website: https://edwardfrenkel.com • ⁠Frenkel's X/Twitter: https://x.com/edfrenkel •⁠ Edward Frenkel's LinkedIn https://www.linkedin.com/in/edfrenkel/ •⁠ Edward Frenkel's Instagram: https://www.instagram.com/edfrenkel •⁠ Edward Frenkel's Facebook: https://www.facebook.com/edfrenkel/ •⁠ Edward Frenkel’s SoundCloud (DJ Moonstein): https://soundcloud.com/moonstein Tarski's Undefinability Theorem: https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem Gödel's Incompleteness Theorems: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems Alan Turing's quote is from his report "Proposed Electronic Calculator" submitted to the Executive Committee of the National Physical Laboratory in February 1946, published as Chapter 20 of the book "Alan Turing's Automatic Computing Engine: The Master Codebreaker's Struggle to build the Modern Computer," ed. B. Jack Copeland, Oxford University Press, 2008. Michael Atiyah's quote is from page 1 of the book "The Unravelers: Mathematical Snapshots" edited by Jean-François Dars, Annick Lesne, and Anne Papillault, translated by Vivienne Méla, A K Peters, 2008. It is included in the article "Sir Michael Atiyah, a Knight Mathematician" by Alain Connes and Joseph Kouneiher, published in the Notices of AMS, vol. 66, No. 10, pp. 1660-1671, 2019. Eric Weinstein's X/Twitter: https://x.com/EricRWeinstein We used clips from the following films: The Imitation Game, directed by Morten Tyldum, 2014: https://www.imdb.com/title/tt2084970/ A Few Good Men, directed by Rob Reiner, 1992: https://www.imdb.com/title/tt0104257/ I Heart Huckabees, directed by David O. Russell, 2004: https://www.imdb.com/title/tt0356721/ Dreamy image at the end (homage to Marc Chagall): https://www.craiyon.com/fr/image/kOKc1OCVSa6WwXVF8MJ0hQ Edward Frenkel’s book “Love and Math”: https://amzn.to/4evbBkS CREDITS: Production: Anna Fedorova Editing: Didi Kayling Animation: Ross Flat Pack FX For all business inquiries please contact frenkelmath@gmail.com © 2025 by Edward Frenkel

Dive into the fascinating intersection of mathematics and psychology as Edward Frenkel explores Carl Jung's Collective Unconscious. Discover how mathematical concepts are tied to archetypes, especially the intriguing archetype of Flat Space. Frenkel delves into the shadow integration process, urging self-awareness and accountability. The discussion weaves through historical insights and the relevance of non-Euclidean geometry, making a compelling case for how understanding these dimensions can enrich our perception of reality.

This intriguing discussion dives into how mathematics, quantum physics, and psychology intertwine to reveal a deeper connection among our minds. The contrasting permanence of math and the evolving nature of physical theories are explored. Concepts like Euclidean and non-Euclidean geometry demonstrate unity, while Einstein's ideas introduce the notion of curved space-time. Schrödinger’s insights on consciousness hint at a shared experience among individuals, raising thought-provoking questions about reality and existence.