The Cartesian Cafe

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

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

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

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

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