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The Cartesian Cafe: Daily Summaries Delivered
by Timothy Nguyen
The Cartesian Cafe is the podcast where an expert guest and Timothy Nguyen map out scientific and mathematical subjects in detail. This collaborative journey with other experts will have us writing down formulas, drawing pictures, and reasoning about them together on a whiteboard. If you’ve been longing for a deeper dive into the intricacies of scientific subjects, then this is the podcast for you. Topics covered include mathematics, physics, machine learning, artificial intelligence, and computer science.Content also viewable on YouTube: www.youtube.com/timothynguyen and Spotify.Timothy Nguyen is a mathematician and AI researcher working in industry. Homepage: www.timothynguyen.com, Twitter: @IAmTimNguyenPatreon: www.patreon.com/timothynguyen
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- Dec 6, 20242h 34m
Justin Clarke-Doane | Mathematics, Reality, and Morality
Justin Clarke-Doane is a professor of philosophy at Columbia University, whose interests span metaethics, epistemology, and the philosophy of logic & mathematics. In this thought provoking-discussion, Justin and I go deep into topics that are typically neglected by most mathematicians and scientists, namely the philosophy of mathematics and morality. Justin has contributed to both these areas via his book Morality and Mathematics, which takes the view that the standard position of being both a mathematical realist and moral antirealist is incoherent. Perhaps the most novel aspect of Justin's work is the treatment of the philosophy of mathematics and morality side-by-side, showing how these two topics, which are usually thought of as being unrelated, in fact have strong analogies. Along the way, we discuss many other foundational topics in epistemology and ethics, with elements of set theory, metaphysics, and logic sprinkled in. Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen Part I. Introduction 00:00 : Preview 01:56 : Naturalism & Mathematical vs Moral Realism 05:34 : Outline of the Discussion Part II. Philosophy of Mathematics 13:25 : Mathematical Realism 18:36 : The Reality of Numbers 27:58 : Anti-Realist Positions in Mathematics 41:49 : Fictionalism in Mathematics 44:06 : Distinguishing Metaphysics from Epistemology 45:39 : The Role of Naturalism and Fictionalism Part III. Philosophy of Morality (vs Mathematics) 50:24 : Moral Realism and Anti-Realism 58:31 : Analogies Between Mathematical and Moral Realism 01:05:30 : Kant's Constructivism and Ethical Contextualism 01:10:40 : Error Theory in Ethics 01:16:02 : Mathematical Realism and Moral Anti-Realism 01:17:22 : Contextualism and Moral Realism Part IV. Select Topics from Justin's Book 01:19:11 : Justification and Self-Evidence 01:21:24 : The Practice of Axiomatization: Mathematics vs Ethics 01:24:51 : Pushback: Is there really controversy in math? 01:30:24 : Justification and Belief: Quinean Empiricism and Harman's Thesis 01:41:44 : Observations, Explanations, and Moral Facts 01:48:41 : Supervenience and High-Level Descriptions 02:00:43 : Justification vs Truth: Reliability Challenge in Mathematics and Morality 02:03:53 : 2+2 not equaling 4: Accidental Truth vs Truth per se 02:13:10 : Pluralism in Mathematics and Ethics 02:31:27 : Concluding Thoughts 02:32:49 : Correction: "relativism" should be "realism" Further reading: Justin Clarke-Doane. Morality and Mathematics. X: @iamtimnguyen Webpage: http://www.timothynguyen.org
- Oct 2, 20242h 59m
Jay McClelland | Neural Networks: Artificial and Biological
Jay McClelland is a pioneer in the field of artificial intelligence and is a cognitive psychologist and professor at Stanford University in the psychology, linguistics, and computer science departments. Together with David Rumelhart, Jay published the two volume work Parallel Distributed Processing, which has led to the flourishing of the connectionist approach to understanding cognition. In this conversation, Jay gives us a crash course in how neurons and biological brains work. This sets the stage for how psychologists such as Jay, David Rumelhart, and Geoffrey Hinton historically approached the development of models of cognition and ultimately artificial intelligence. We also discuss alternative approaches to neural computation such as symbolic and neuroscientific ones. Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen Part I. Introduction 00:00 : Preview 01:10 : Cognitive psychology 07:14 : Interdisciplinary work and Jay's academic journey 12:39 : Context affects perception 13:05 : Chomsky and psycholinguists 8:03 : Technical outline Part II. The Brain 00:20:20 : Structure of neurons 00:25:26 : Action potentials 00:27:00 : Synaptic processes and neuron firing 00:29:18 : Inhibitory neurons 00:33:10 : Feedforward neural networks 00:34:57 : Visual system 00:39:46 : Various parts of the visual cortex 00:45:31 : Columnar organization in the cortex 00:47:04 : Colocation in artificial vs biological networks 00:53:03 : Sensory systems and brain maps Part III. Approaches to AI, PDP, and Learning Rules 01:12:35 : Chomsky, symbolic rules, universal grammar 01:28:28 : Neuroscience, Francis Crick, vision vs language 01:32:36 : Neuroscience = bottom up 01:37:20 : Jay’s path to AI 01:43:51 : James Anderson 01:44:51 : Geoff Hinton 01:54:25 : Parallel Distributed Processing (PDP) 02:03:40 : McClelland & Rumelhart’s reading model 02:31:25 : Theories of learning 02:35:52 : Hebbian learning 02:43:23 : Rumelhart’s Delta rule 02:44:45 : Gradient descent 02:47:04 : Backpropagation 02:54:52 : Outro: Retrospective and looking ahead Image credits: http://timothynguyen.org/image-credits/ Further reading: Rumelhart, McClelland. Parallel Distributed Processing. McClelland, J. L. (2013). Integrating probabilistic models of perception and interactive neural networks: A historical and tutorial review Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org
- Jul 19, 20242h 52m
Michael Freedman | A Fields Medalist Panorama
Michael Freedman is a mathematician who was awarded the Fields Medal in 1986 for his solution of the 4-dimensional Poincare conjecture. Mike has also received numerous other awards for his scientific contributions including a MacArthur Fellowship and the National Medal of Science. In 1997, Mike joined Microsoft Research and in 2005 became the director of Station Q, Microsoft’s quantum computing research lab. As of 2023, Mike is a Senior Research Scientist at the Center for Mathematics and Scientific Applications at Harvard University. Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen In this wide-ranging conversation, we give a panoramic view of Mike’s extensive body of work over the span of his career. It is divided into three parts: early, middle, and present day, which respectively include his work on the 4-dimensional Poincare conjecture, his transition to topological physics, and finally his recent work in applying ideas from mathematics and philosophy to social economics. Our conversation is a blend of both the nitty-gritty details and the anecdotal story-telling that can only be obtained from a living legend. I. Introduction 00:00 : Preview 01:34 : Fields Medalist working in industry 03:24 : Academia vs industry 04:59 : Mathematics and art 06:33 : Technical overview II. Early Mike: The Poincare Conjecture (PC) 08:14 : Introduction, statement, and history 14:30 : Three categories for PC (topological, smooth, PL) 17:09 : Smale and PC for d at least 5 17:59 : Homotopy equivalence vs homeomorphism 22:08 : Joke 23:24 : Morse flow 33:21 : Whitney Disk 41:47 : Casson handles 50:24 : Manifold factors and the Whitehead continuum 1:00:39 : Donaldson’s results in the smooth category 1:04:54 : (Not) writing up full details of the proof then and now 1:08:56 : Why Perelman succeeded II. Mid Mike: Topological Quantum Field Theory (TQFT) and Quantum Computing (QC) 1:10:54: Introduction 1:11:42: Cliff Taubes, Raoul Bott, Ed Witten 1:12:40 : Computational complexity, Church-Turing, and Mike’s motivations 1:24:01 : Why Mike left academia, Microsoft’s offer, and Station Q 1:29:23 : Topological quantum field theory (according to Atiyah) 1:34:29 : Anyons and a theorem on Chern-Simons theories 1:38:57 : Relation to QC 1:46:08 : Universal TQFT 1:55:57 : Witten: Donalson theory cannot be a unitary TQFT 2:01:22 : Unitarity is possible in dimension 3 2:05:12 : Relations to a theory of everything? 2:07:21 : Where topological QC is now III. Present Mike: Social Economics 2:11:08 : Introduction 2:14:02 : Lionel Penrose and voting schemes 2:21:01 : Radical markets (pun intended) 2:25:45 : Quadratic finance/funding 2:30:51 : Kant’s categorical imperative and a paper of Vitalik Buterin, Zoe Hitzig, Glen Weyl 2:36:54 : Gauge equivariance 2:38:32 : Bertrand Russell: philosophers and differential equations IV: Outro 2:46:20 : Final thoughts on math, science, philosophy 2:51:22 : Career advice Some Further Reading: Mike’s Harvard lecture on PC4: https://www.youtube.com/watch?v=TSF0i6BO1Ig Behrens et al. The Disc Embedding Theorem. M. Freedman. Spinoza, Leibniz, Kant, and Weyl. arxiv:2206.14711 Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org
- May 10, 20243h 1m
Marcus Hutter | Universal Artificial Intelligence and Solomonoff Induction
Marcus Hutter is an artificial intelligence researcher who is both a Senior Researcher at Google DeepMind and an Honorary Professor in the Research School of Computer Science at Australian National University. He is responsible for the development of the theory of Universal Artificial Intelligence, for which he has written two books, one back in 2005 and one coming right off the press as we speak. Marcus is also the creator of the Hutter prize, for which you can win a sizable fortune for achieving state of the art lossless compression of Wikipedia text. Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen In this technical conversation, we cover material from Marcus’s two books “Universal Artificial Intelligence” (2005) and “Introduction to Universal Artificial Intelligence” (2024). The main goal is to develop a mathematical theory for combining sequential prediction (which seeks to predict the distribution of the next observation) together with action (which seeks to maximize expected reward), since these are among the problems that intelligent agents face when interacting in an unknown environment. Solomonoff induction provides a universal approach to sequence prediction in that it constructs an optimal prior (in a certain sense) over the space of all computable distributions of sequences, thus enabling Bayesian updating to enable convergence to the true predictive distribution (assuming the latter is computable). Combining Solomonoff induction with optimal action leads us to an agent known as AIXI, which in this theoretical setting, can be argued to be a mathematical incarnation of artificial general intelligence (AGI): it is an agent which acts optimally in general, unknown environments. The second half of our discussion concerning agents assumes familiarity with the basic setup of reinforcement learning. I. Introduction 00:38 : Biography 01:45 : From Physics to AI 03:05 : Hutter Prize 06:25 : Overview of Universal Artificial Intelligence 11:10 : Technical outline II. Universal Prediction 18:27 : Laplace’s Rule and Bayesian Sequence Prediction 40:54 : Different priors: KT estimator 44:39 : Sequence prediction for countable hypothesis class 53:23 : Generalized Solomonoff Bound (GSB) 57:56 : Example of GSB for uniform prior 1:04:24 : GSB for continuous hypothesis classes 1:08:28 : Context tree weighting 1:12:31 : Kolmogorov complexity 1:19:36 : Solomonoff Bound & Solomonoff Induction 1:21:27 : Optimality of Solomonoff Induction 1:24:48 : Solomonoff a priori distribution in terms of random Turing machines 1:28:37 : Large Language Models (LLMs) 1:37:07 : Using LLMs to emulate Solomonoff induction 1:41:41 : Loss functions 1:50:59 : Optimality of Solomonoff induction revisited 1:51:51 : Marvin Minsky III. Universal Agents 1:52:42 : Recap and intro 1:55:59 : Setup 2:06:32 : Bayesian mixture environment 2:08:02 : AIxi. Bayes optimal policy vs optimal policy 2:11:27 : AIXI (AIxi with xi = Solomonoff a priori distribution) 2:12:04 : AIXI and AGI. Clarification: ASI (Artificial Super Intelligence) would be a more appropriate term than AGI for the AIXI agent. 2:12:41 : Legg-Hutter measure of intelligence 2:15:35 : AIXI explicit formula 2:23:53 : Other agents (optimistic agent, Thompson sampling, etc) 2:33:09 : Multiagent setting 2:39:38 : Grain of Truth problem 2:44:38 : Positive solution to Grain of Truth guarantees convergence to a Nash equilibria 2:45:01 : Computable approximations (simplifying assumptions on model classes): MDP, CTW, LLMs 2:56:13 : Outro: Brief philosophical remarks Further Reading: M. Hutter, D. Quarrel, E. Catt. An Introduction to Universal Artificial Intelligence M. Hutter. Universal Artificial Intelligence S. Legg and M. Hutter. Universal Intelligence: A Definition of Machine Intelligence Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org
- Feb 2, 20242h 5m
Richard Borcherds | Monstrous Moonshine: From Group Theory to String Theory
Richard Borcherds is a mathematician and professor at University of California Berkeley known for his work on lattices, group theory, and infinite-dimensional algebras. His numerous accolades include being awarded the Fields Medal in 1998 and being elected a fellow of the American Mathematical Society and the National Academy of Sciences. Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen In this episode, Richard and I give an overview of Richard's most famous result: his proof of the Monstrous Moonshine conjecture relating the monster group on the one hand and modular forms on the other. A remarkable feature of the proof is that it involves vertex algebras inspired from elements of string theory. Some familiarity with group theory and representation theory are assumed in our discussion. I. Introduction 00:25: Biography 02:51 : Success in mathematics 04:04 : Monstrous Moonshine overview and John Conway 09:44 : Technical overview II. Group Theory 11:31 : Classification of finite-simple groups + history of the monster group 18:03 : Conway groups + Leech lattice 22:13 : Why was the monster conjectured to exist + more history 28:43 : Centralizers and involutions 32:37: Griess algebra III. Modular Forms 36:42 : Definitions 40:06 : The elliptic modular function 48:58 : Subgroups of SL_2(Z) IV. Monstrous Moonshine Conjecture Statement 57:17: Representations of the monster 59:22 : Hauptmoduls 1:03:50 : Statement of the conjecture 1:07:06 : Atkin-Fong-Smith's first proof 1:09:34 : Frenkel-Lepowski-Meurman's work + significance of Borcherd's proof V. Sketch of Proof 1:14:47: Vertex algebra and monster Lie algebra 1:21:02 : No ghost theorem from string theory 1:25:24 : What's special about dimension 26? 1:28:33 : Monster Lie algebra details 1:32:30 : Dynkin diagrams and Kac-Moody algebras 1:43:21 : Simple roots and an obscure identity 1:45:13: Weyl denominator formula, Vandermonde identity 1:52:14 : Chasing down where modular forms got smuggled in 1:55:03 : Final calculations VI. Epilogue 1:57:53 : Your most proud result? 2:00:47 : Monstrous moonshine for other sporadic groups? 2:02:28 : Connections to other fields. Witten and black holes and mock modular forms. Further reading: V Tatitschef. A short introduction to Monstrous Moonshine. https://arxiv.org/pdf/1902.03118.pdf Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org
- Jan 9, 20241 min
Announcements for 2024 and a Message to Viewers
Thought I'd share some exciting news about what's happening at The Cartesian Cafe in 2024 and also a personal message to viewers on how they can support the cafe. Patreon: https://www.patreon.com/timothynguyen
- Dec 1, 20232h 41m
Tim Maudlin | Bell’s Theorem and Beyond: Nobody Understands Quantum Mechanics
Tim Maudlin is a philosopher of science specializing in the foundations of physics, metaphysics, and logic. He is a professor at New York University, a member of the Foundational Questions Institute, and the founder and director of the John Bell Institute for the Foundations of Physics. Patreon (bonus materials + video chat): https://www.patreon.com/timothynguyen In this very in-depth discussion, Tim and I probe the foundations of science through the avenues of locality and determinism as arising from the Einstein-Poldosky-Rosen (EPR) paradox and Bell's Theorem. These issues are so intricate that even the Nobel Prize committee incorrectly described the significance of Bell's work in their press release for the 2022 prize in physics. Viewers motivated enough to think deeply about these ideas will be rewarded with a conceptually proper understanding of the nonlocal nature of physics and its manifestation in quantum theory. I. Introduction 00:00 : 00:25: Biography 05:26: Interdisciplinary work 11:54 : Physicists working on the wrong things 16:47 : Bell's Theorem soft overview 24:14: Common misunderstanding of "God does not play dice." 25:59: Technical outline II. EPR Paradox / Argument 29:14 : EPR is not a paradox 34:57 : Criterion of reality 43:57 : Mathematical formulation 46:32 : Locality: No spooky action at a distance 49:54 : Bertlmann's socks 53:17 : EPR syllogism summarized 54:52 : Determinism is inferred not assumed 1:02:18 : Clarifying analogy: Coin flips 1:06:39 : Einstein's objection to determinism revisited III. Bohm Segue 1:11:05 : Introduction 1:13:38: Bell and von Neumann's error 1:20:14: Bell's motivation: Can I remove Bohm's nonlocality? IV. Bell's Theorem and Related Examples 1:25:13 : Setup 1:27:59 : Decoding Bell's words: Locality is the key! 1:34:16 : Bell's inequality (overview) 1:36:46 : Bell's inequality (math) 1:39:15 : Concrete example of violation of Bell's inequality 1:49:42: GHZ Example V. Miscellany 2:06:23 : Statistical independence assumption 2:13:18: The 2022 Nobel Prize 2:17:43: Misconceptions and hidden variables 2:22:28: The assumption of local realism? Repeat: Determinism is a conclusion not an assumption. VI. Interpretations of Quantum Mechanics 2:28:44: Interpretation is a misnomer 2:29:48: Three requirements. You can only pick two. 2:34:52: Copenhagen interpretation? Further Reading: J. Bell. Speakable and Unspeakable in Quantum Mechanics T. Maudlin. Quantum Non-Locality and Relativity Wikipedia: Mermin's device, GHZ experiment Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org
- Sep 27, 20232h 34m
Antonio Padilla | Fantastic Numbers, Naturalness, and Anthropics in Physics
Antonio (Tony) Padilla is a theoretical physicist and cosmologist at the University of Nottingham. He serves as the Associate Director of the Nottingham Centre of Gravity, and in 2016, Tony shared the Buchalter Cosmology Prize for his work on the cosmological constant. Tony is also a star of the Numberphile YouTube channel, where his videos have received millions of views and he is also the author of the book Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity. Patreon: https://www.patreon.com/timothynguyen This episode combines some of the greatest cosmological questions together with mathematical imagination. Tony and I go through the math behind some oft-quoted numbers in cosmology and calculate the age, size, and number of atoms in the universe. We then stretch our brains and consider how likely it would be to find your Doppelganger in a truly large universe, which takes us on a detour through black hole entropy. We end with a discussion of naturalness and the anthropic principle to round out our discussion of fantastic numbers in physics. Part I. Introduction 00:00 : Introduction 01:06 : Math and or versus physics 12:09 : Backstory behind Tony's book 14:12 : Joke about theoreticians and numbers 16:18 : Technical outline Part II. Size, Age, and Quantity in the Universe 21:42 : Size of the observable universe 22:32 : Standard candles 27:39 : Hubble rate 29:02 : Measuring distances and time 37:15 : Einstein and Minkowski 40:52 : Definition of Hubble parameter 42:14 : Friedmann equation 47:11 : Calculating the size of the observable universe 51:24 : Age of the universe 56:14 : Number of atoms in the observable universe 1:01:08 : Critical density 1:03:16: 10^80 atoms of hydrogen 1:03:46 : Universe versus observable universe Part III. Extreme Physics and Doppelgangers 1:07:27 : Long-term fate of the universe 1:08:28 : Black holes and a googol years 1:09:59 : Poincare recurrence 1:13:23 : Doppelgangers in a googolplex meter wide universe 1:16:40 : Finitely many states and black hole entropy 1:25:00 : Black holes have no hair 1:29:30 : Beckenstein, Christodolou, Hawking 1:33:12 : Susskind's thought experiment: Maximum entropy of space 1:42:58 : Estimating the number of doppelgangers 1:54:21 : Poincare recurrence: Tower of four exponents. Part IV: Naturalness and Anthropics 1:54:34 : What is naturalness? Examples. 2:04:09 : Cosmological constant problem: 10^120 discrepancy 2:07:29 : Interlude: Energy shift clarification. Gravity is key. 2:15:34 : Corrections to the cosmological constant 2:18:47 : String theory landscape: 10^500 possibilities 2:20:41 : Anthropic selection 2:25:59 : Is the anthropic principle unscientific? Weinberg and predictions. 2:29:17 : Vacuum sequestration Further reading: Antonio Padilla. Fantastic Numbers and Where to Find Them: A Cosmic Quest from Zero to Infinity Twitter: @iamtimnguyen Webpage: http://www.timothynguyen.org
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About The Cartesian Cafe
The Cartesian Cafe is the podcast where an expert guest and Timothy Nguyen map out scientific and mathematical subjects in detail. This collaborative journey with other experts will have us writing down formulas, drawing pictures, and reasoning about them together on a whiteboard. If you’ve been longing for a deeper dive into the intricacies of scientific subjects, then this is the podcast for you. Topics covered include mathematics, physics, machine learning, artificial intelligence, and computer science.Content also viewable on YouTube: www.youtube.com/timothynguyen and Spotify.Timothy Nguyen is a mathematician and AI researcher working in industry. Homepage: www.timothynguyen.com, Twitter: @IAmTimNguyenPatreon: www.patreon.com/timothynguyen
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