Podcasts about Set theory

Dr. Andreas Schlatter is a classically trained physicist (EPFL, Princeton) with a decidedly heretical approach to physics. Though deeply mathematical in his approach, he dispenses with the purely field-based approach to understanding the building blocks of nature, and asks far deeper question about what the mathematics is telling us about the hidden structures of nature. Rather than take the positivist approach, which suggests that anything that cannot be experimentally encountered is not worth considering, Schlatter follows in the tradition of Gödel and the other mid 20th century logicians, who believed that a layer of the universe beyond the visible is available to us if we can reason our way to it. By following this path, Schlatter has reached the conclusion that the only viable interpretation of quantum mechanics is the transactional one. Unlike the other transnational theorists we've had on the show, Schlatter has gone one step further to propose that there is a transactional interpretation of gravity just as is there is for quantum mechanics. He calls it entropic gravity, and in this episode we explore the convoluted path he took to physics, how he found the transactionalists, and how he and Ruth Kastner formulated an entropic explanation for spacetime. PATREON: get episodes early + join our weekly Patron Chat https://bit.ly/3lcAasB MERCH: Rock some DemystifySci gear : https://demystifysci.myspreadshop.com/ AMAZON: Do your shopping through this link for Caver Mead's Collective Electrodynamics: https://amzn.to/4e01Slj (00:00) Go! (00:05:28) Andreas Schlatter's Academic Journey (00:10:39) Exploration of Mathematics in Physics (00:25:51) The Vienna Circle and Logical Positivism (00:30:04) Einstein's Transition in Theoretical Approach (00:37:37) Philosophical Inquiry in Physics Education (00:41:08) The Quest for Understanding in Logic and Set Theory (00:48:02) Transition from Academia to Finance (00:56:02) Challenges of Financial Modeling (01:09:59) Trust and Economic Stability (01:16:10) Light and Gravity Intersect (01:23:02) Entropy and Information Theory (01:31:07) Absorption and Entropy Dynamics (01:37:22) Exploration of Quantum Transactions (01:46:30) Transactional Approach to Gravity (01:56:31) Light Clocks and the Nature of Time (02:04:13) Multiverses and Quantum Realms #Physics, #QuantumMechanics, #Mathematics, #PhilosophyOfScience, #LogicalPositivism, #EmpiricalScience, #TheoreticalPhysics, #Einstein, #Newton, #QuantumReality, #Entropy, #Cosmology, #Multiverse, #GravityTheory, #EconomicStability, #TransactionalInterpretation, #ScienceEducation, #Philosophy, #QuantumGravity, #FinanceAndPhysics, #ScientificUnderstanding #sciencepodcast, #longformpodcast Check our short-films channel, @DemystifySci: https://www.youtube.com/c/DemystifyingScience AND our material science investigations of atomics, @MaterialAtomics https://www.youtube.com/@MaterialAtomics Join our mailing list https://bit.ly/3v3kz2S PODCAST INFO: Anastasia completed her PhD studying bioelectricity at Columbia University. When not talking to brilliant people or making movies, she spends her time painting, reading, and guiding backcountry excursions. Shilo also did his PhD at Columbia studying the elastic properties of molecular water. When he's not in the film studio, he's exploring sound in music. They are both freelance professors at various universities. - Blog: http://DemystifySci.com/blog - RSS: https://anchor.fm/s/2be66934/podcast/rss - Donate: https://bit.ly/3wkPqaD - Swag: https://bit.ly/2PXdC2y SOCIAL: - Discord: https://discord.gg/MJzKT8CQub - Facebook: https://www.facebook.com/groups/DemystifySci - Instagram: https://www.instagram.com/DemystifySci/ - Twitter: https://twitter.com/DemystifySci MUSIC: -Shilo Delay: https://g.co/kgs/oty671

Soccer. Set Theory. TV show reviews. Other stuff.

Fay Dowker is Professor of Theoretical Physics at Imperial College London, where she works broadly on quantum gravity, and more particularly on an approach called causal set theory that takes the most basic pieces of the universe to be atoms of spacetime. In this episode, Robinson and Fay begin by discussing her studies with Stephen Hawking and their work on wormholes before turning to quantum gravity and causal set theory. Fay is also a faculty member at the John Bell Institute for the Foundations of Physics. If you're interested in the foundations of physics—which you absolutely should be—then please check out the JBI, which is devoted to providing a home for research and education in this important area. Any donations are immensely helpful at this early stage in the institute's life. The John Bell Institute: https://www.johnbellinstitute.org OUTLINE 00:00 In This Episode… 00:50 Introduction 04:49 How Do Physicists Think of Wormholes? 15:56 Stephen Hawking, Philosophy, and Quantum Gravity 26:00 Causal Set Theory and The Problem of Quantum Gravity 43:45 What is the Path Integral? 54:43 Is Spacetime Discrete? 57:40 Causal Set Theory and Black Holes 01:14:27 Lorentz Symmetry, Non-Locality, and Phenomenology in Causal Set Theory Robinson's Website: http://robinsonerhardt.com Robinson Erhardt researches symbolic logic and the foundations of mathematics at Stanford University. Join him in conversations with philosophers, scientists, weightlifters, artists, and everyone in-between.  --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

Episode: 2961 In which we struggle to give meaning to the word infinity.  Today, infinity.

John Burgess is John N. Woodhull Professor of Philosophy at Princeton University, where he works in mathematical and philosophical logic and the philosophy of mathematics. In this episode, Robinson and John discuss realism in the philosophy of mathematics, and while the nature of this question is itself disputed, it can be roughly described as concerning the extent to which we should be committed to the mind-independent truth of mathematical theorems, or to the existence of the objects they apparently describe. Robinson and John begin by addressing the nuances of this question, and they then turn to various developments in mathematics that have been historically associated with realism—set theory, in particular—as well as specific philosophical positions associated with realism (such as Platonism) and anti-realism (such as conventionalism). John's most recent book is Set Theory (Cambridge, 2022). Set Theory: https://a.co/d/cF305wf OUTLINE 00:00 In This Episode… 00:22 Introduction 03:17 Mathematics or Philosophy? 08:06 What is Realism in the Philosophy of Mathematics? 14:11 Objectivity and Mathematics 24:34 What Is Set Theory? 47:29 Platonism and the Continuum Problem 01:15:42 Conventionalism 01:22:06 Finitism 01:31:17 A Cap on Infinity? Robinson's Website: http://robinsonerhardt.com Robinson Erhardt researches symbolic logic and the foundations of mathematics at Stanford University. Join him in conversations with philosophers, scientists, weightlifters, artists, and everyone in-between.  --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

It's the Season 10 finale of the Elixir Wizards podcast! José Valim, Guillaume Duboc, and Giuseppe Castagna join Wizards Owen Bickford and Dan Ivovich to dive into the prospect of types in the Elixir programming language! They break down their research on set-theoretical typing and highlight their goal of creating a type system that supports as many Elixir idioms as possible while balancing simplicity and pragmatism. José, Guillaume, and Giuseppe talk about what initially sparked this project, the challenges in bringing types to Elixir, and the benefits that the Elixir community can expect from this exciting work. Guillaume's formalization and Giuseppe's "cutting-edge research" balance José's pragmatism and "Guardian of Orthodoxy" role. Decades of theory meet the needs of a living language, with open challenges like multi-process typing ahead. They come together with a shared joy of problem-solving that will accelerate Elixir's continued growth. Key Topics Discussed in this Episode: Adding type safety to Elixir through set theoretical typing How the team chose a type system that supports as many Elixir idioms as possible Balancing simplicity and pragmatism in type system design Addressing challenges like typing maps, pattern matching, and guards The tradeoffs between Dialyzer and making types part of the core language Advantages of typing for catching bugs, documentation, and tooling The differences between typing in the Gleam programming language vs. Elixir The possibility of type inference in a set-theoretic type system The history and development of set-theoretic types over 20 years Gradual typing techniques for integrating typed and untyped code How José and Giuseppe initially connected through research papers Using types as a form of "mechanized documentation" The risks and tradeoffs of choosing syntax Cheers to another decade of Elixir! A big thanks to this season's guests and all the listeners! Links and Resources Mentioned in this Episode: Bringing Types to Elixir | Guillaume Duboc & Giuseppe Castagna | ElixirConf EU 2023 (https://youtu.be/gJJH7a2J9O8) Keynote: Celebrating the 10 Years of Elixir | José Valim | ElixirConf EU 2022 (https://youtu.be/Jf5Hsa1KOc8) OCaml industrial-strength functional programming https://ocaml.org/ ℂDuce: a language for transformation of XML documents http://www.cduce.org/ Ballerina coding language https://ballerina.io/ Luau coding language https://luau-lang.org/ Gleam type language https://gleam.run/ "The Design Principles of the Elixir Type System" (https://www.irif.fr/_media/users/gduboc/elixir-types.pdf) by G. Castagna, G. Duboc, and J. Valim "A Gradual Type System for Elixir" (https://dlnext.acm.org/doi/abs/10.1145/3427081.3427084) by M. Cassola, A. Talagorria, A. Pardo, and M. Viera "Programming with union, intersection, and negation types" (https://www.irif.fr/~gc/papers/set-theoretic-types-2022.pdf), by Giuseppe Castagna "Covariance and Contravariance: a fresh look at an old issue (a primer in advanced type systems for learning functional programmers)" (https://www.irif.fr/~gc/papers/covcon-again.pdf) by Giuseppe Castagna "A reckless introduction to Hindley-Milner type inference" (https://www.lesswrong.com/posts/vTS8K4NBSi9iyCrPo/a-reckless-introduction-to-hindley-milner-type-inference) Special Guests: Giuseppe Castagna, Guillaume Duboc, and José Valim.

Joan Bagaria is ICREA Research Professor in the Department of Experimental Sciences and Mathematics at the University of Barcelona. He is a mathematical logician who works in set theory, which is the branch of mathematics that not only specializes in the investigation of infinity but serves as the foundation for the rest of mathematics—what this means, and its implications, are explored in the episode. Joan and Robinson discuss all things set theory, beginning with its origins in the mind of Georg Cantor, its development in the 20th century, some philosophical questions, and some current outstanding problems. They also briefly touch on Catalan independence, a topic dear to Joan's heart. Joan's Twitter: ⁠https://twitter.com/BagariaJoan⁠ Set Theory: https://plato.stanford.edu/entries/set-theory/ The Early Development of Set Theory: https://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=settheory-early OUTLINE 00:00 In This Episode… 01:01 Introduction 06:18 Joan and Set Theory 09:11 The Development of Set Theory 21:08 Naive Set Theory and Axiomatic Set Theory 30:52 Zermelo-Fraenkel Set Theory with Choice 46:35 Metaphysics and Epistemology 01:03:06 Set Theory as the Foundation of Mathematics 01:09:48 The Continuum Problem 01:16:13 Settling the Continuum Problem 01:35:21 Alternative Set Theories 01:43:37 Alternative Foundations 01:47:53 Catalan Independence Robinson's Website: ⁠http://robinsonerhardt.com⁠ Robinson Erhardt researches symbolic logic and the foundations of mathematics at Stanford University. Join him in conversations with philosophers, scientists, weightlifters, artists, and everyone in-between. --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

Covering Part 8 of Alain Badiou's Being and Event on “Theory of the Subject,” Alex and Andrew discuss the theory of subject and the event, and Badiou's wider work. Guest Andrei Rodin contextualizes Badiou's project through its relation to the wider philosophy of mathematics. Rodin is a mathematician and philosopher with affiliations in France, including the University of Lorraine and the University Paris-Cité, and in Russia at the Russian Academy of Sciences, Saint-Petersburgh State University, as well as the Russian Society for History and Philosophy of Science. He is the author of Axiomatic Method and Category Theory. Concepts related to the Theory of the Subject Badiou's Theory of the Subject, the Future Anterior of Truth, Paul Cohen's Forcing, Comments on Lacan, Event versus Language, Subject, The Outside, The Undocumented Family, State as Preventing the Event, Decolonize Badiou. Recap of Being and Event (Parts 1-3) normal and natural, being qua being, entities multiples sets void, ordinal chains, infinity (natural and real), being is the state and state of situation (form through set theory) (Part 4) turning point, there will always be sites that are presented but whose members are represented, gap, normal and abnormal, un- in- ex-, (Second Half of the Book) how things work, fidelity as a procedure that assigning belonging (temporal), quasi existentialism of the decision, against a construction which is an internal model that grinds through itself, construction always hits an impasse (errancy of the excess of the situation), external model, excess (End of the Book), fidelity to the event, not an act of construction, subtraction, the subtractive procedure is forcing (Cohen), the generic is a product of forcing (Cohen), the four truth procedures (love, art, science, politics) are for subjects, the subject is local configuration of event, fidelity, force, generic. Further Reading Manifesto for Philosophy (BE Explainer), Number and Numbers (math notes for BE), Conditions (Four Truth Procedures); BE Trilogy: (1) BE is both abstract and set theoretical, (2) LW is in the world and takes the perspective from world that truth interrupts, and IT (3) takes the perspective of truth to asks where everything else comes from (in favor of infinite against finite); Logic of Worlds is less heroic, undoes the eureka theory of event, more temporality and history, subjectivity as process, phenomenology, additional math theories, category theory; Immanence of Truths, back to set theory, transfinite mathematics and large cardinals, in the Gödel-Cohen debate “I choose Cohen” Interview with Andrei Rodin WVO Quine, Set Theory, Meta-Mathematics, Category Theory, Computation, ZFC and Paul Cohen, Constructivist Mathematics, Infinities and Georg Cantor, Euclid and Numbers, Big Numbers, Non-Countable Sets, Axioms, David Hilbert, Generic, Forcing Links Rodin page, http://philomatica.org/ Rodin papers, https://varetis.academia.edu/AndreiRodin Rodin texts, http://philomatica.org/my-stuff/my-texts/ Rodin, Review of Badiou's “Mathematics of the Transcendental,” http://philomatica.org/wp-content/uploads/2013/01/braspublished.pdf Rodin, Axiomatic Method and Category Theory, https://link.springer.com/book/10.1007/978-3-319-00404-4

Frank Jackson is Emeritus Professor at the Australian National University. He is best known for the knowledge argument and Mary's Room—its accompanying thought experiment—but has published widely in the philosophy of mind, epistemology, metaphysics, and the philosophy of language. Graham Priest is a Distinguished Professor in the philosophy department at the CUNY Graduate Center. Like Frank, he is one of the most influential philosophers of the past fifty years, and has done important work on a wide range of topics, ranging from the philosophy of mathematics to logic and eastern philosophy. In this episode, Robinson, Frank, and Graham talk about David Lewis and his immense legacy in the philosophical world. They cover his character—Frank and Graham were friends with him for many years—as well as some of his work, ranging from the thesis of modal realism to Humean supervenience and the philosophy of set theory. David Lewis: ⁠https://plato.stanford.edu/entries/david-lewis/⁠ Graham's Website: ⁠https://grahampriest.net⁠ OUTLINE 00:00 In This Episode… 01:17 Introduction 07:54 David Lewis as a Friend and Philosopher 24:12 Australian Philosophy 28:53 Lewisian Themes 34:30 Modal Realism 52:43 Kripke and Lewis on Possible Worlds 58:07 Making Use of Possible Worlds 01:23:29 Humean Supervenience 01:38:19 Set Theory and Mereology 01:45:19 Final Thoughts Robinson's Website: ⁠http://robinsonerhardt.com⁠ Robinson Erhardt researches symbolic logic and the foundations of mathematics at Stanford University. Join him in conversations with philosophers, scientists, weightlifters, artists, and everyone in-between. --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

Covering Part 3 of Alain Badiou's Being and Event on “Nature & Infinity,” Alex and Andrew complete the "arithmetic, natural story" that constitutes Badiou's presentation of being within the book so far. Guest Sarah Pourciau explores the history and philosophy of set theory, while also scrutinizing the conclusions Badiou tries to draw from it. Pourciau is a professor of German Studies at Duke University. Her expertise includes 19th Century German thought, including both philosophy and mathematics (Dedekind, Cantor). She is the author of the book The Writing of Spirit: Soul, System, and the Roots of Language Science. Concepts on Nature and Infinity Political Modernism, Math as the Difference between Real and Natural Numbers, Martin Heidegger's Poetic Ontology, Jacques Lacan's Matheme, Physis, Nature, Natural Multiples, the Non-existence of Nature, Cardinality and Ordinality, Ordinal Chain, Infinity and Finitude, Arithmetic and Natural Infinity, Georg Cantor and Richard Dedekind, Five Critiques of GWF Hegel's Notion of Infinity. Interview with Sarah Pourciau Digital Ocean, Richard Dedekind, Platonic Eidos, Georg Cantor and the Abyss, Gender and “The Feminine,” Kantian Intuition, Logos and the Origin of Set Theory, Politics, Naming and Numbers, Spontaneity, Différance, Alan Turing and Kurt Gödel, Computability. Links Pourciau profile, https://scholars.duke.edu/person/sarah.pourciau Pourciau, The Writing of Spirit: Soul, System, and the Roots of Language Science, https://www.fordhampress.com/9780823275632/the-writing-of-spirit/ Pourciau, "A/logos: An Anomalous Episode in the History of Number," https://muse.jhu.edu/article/728110 Pourciau, "On the Digital Ocean," https://www.journals.uchicago.edu/doi/abs/10.1086/717319

Alexander R. Galloway and Andrew Culp join Acid Horizon to discuss their new podcast series on Alain Badiou's 1988 work Being and Event. We discuss Badiou's mathematical ontology and its roots in Cantor's Set Theory and Cohen's theory of the Generic. We also trace the roots of his militant arithmetic in philosophers of the French Resistance such as Cavailles, and his revolutionary Marxist Anti-Statism.There will be an online launch event for their new podcast series where people can learn more here http://cultureandcommunication.org/BeingAndEvent/And you can listen to the first two episodes now! On Spotify, Apple Podcasts, and many more platforms! https://open.spotify.com/show/7skR7GRkElz3crZlQYHjhRSupport the podcast:Linktree: https://linktr.ee/acidhorizonAcid Horizon on Patreon: https://www.patreon.com/acidhorizonpodcastZer0 Books and Repeater Media Patreon: https://www.patreon.com/zer0repeaterMerch: http://www.crit-drip.comOrder 'The Philosopher's Tarot': https://repeaterbooks.com/product/the-philosophers-tarot/Subscribe to us on Apple Podcasts: https://tinyurl.com/169wvvhiHappy Hour at Hippel's (Adam's blog): https://happyhourathippels.wordpress.com​Revolting Bodies (Will's Blog): https://revoltingbodies.com​Split Infinities (Craig's Substack): https://splitinfinities.substack.com/​Music: https://sereptie.bandcamp.com/ and https://thecominginsurrection.bandcamp.com/Support the show

Covering Part 1 of Alain Badiou's Being and Event on the topic of “Being,” Alex and Andrew introduce some foundational concepts and address Badiou's relation to other philosophers. Guest Knox Peden outlines where Badiou fits within the intellectual history of French philosophy, Marxism, and science. Peden is author of Spinoza Contra Phenomenology: French Rationalism from Cavaillès to Deleuze (published in 2014). Knox has also worked as an editor and translator including collaborations on Cahiers pour l'Analyse (published as Concept and Form, volumes 1 and 2) and On Logic and the Theory of Science by Jean Cavaillès. Schools of Philosophy Math and the Philosophy of Mathematics, a Mathematic Ontology based in Set Theory, Being Qua Being, Martin Heidegger and Badiou's Critique of Poetic Ontology, Post-Cartesian Theories of the Subject from Karl Marx, Sigmund Freud, and Jacques Lacan, Logical Positivism and the Vienna Circle. Key Thinkers and Concepts Jean Cavaillès, Albert Lautman, Georg Cantor, and Kurt Gödel, Axiomatic Set Theory (Axiom of Extensionality, Power Sets, Axiom of Union, Axiom of Separation, Axioms of Replacement and Substitution), The Count, The One, Void, ∅ (Mark Naught), Nature, Name, Cardinality. Interview with Knox Peden French Marxism, Marxist Science and Ideology, Rationalism, Empiricism, Phenomenology and Edmund Husserl, Gaston Bachelard and Philosophy of Science, Truth, Cahiers pour l'Analyse including Jacques-Alain Miller and Jean-Claude Milner, “Mark and Lack,” the Subject, Suture. Links Knox Peden profile, https://hass.uq.edu.au/profile/7697/knox-peden Peden, Spinoza Contra Phenomenology: French Rationalism from Cavaillès to Deleuze, https://www.sup.org/books/title/?id=22793 Hallward and Peden, Concept and Form, two volumes dedicated to Cahiers pour l'Analyse, https://www.versobooks.com/series_collections/35-concept-and-form Cahiers pour l'Analyse(electronic edition) http://cahiers.kingston.ac.uk/ Cavaillès, On Logic and the Theory of Science, translated by Peden and Mackay, https://www.urbanomic.com/book/logic-theory-science/

Dr. Richard Sternberg speaks on his mathematical/logical work showing the difficulty of identifying genes purely with material phenomena. Source

Joel David Hamkins is the O'Hara Professor of Philosophy and Mathematics at the University of Notre Dame, where he recently moved from the University of Oxford. Joel is one of the world's leading set theorists and philosophers of mathematics. Graham Priest is a Distinguished Professor in the philosophy department at the CUNY Graduate Center. He is one of the most influential philosophers of the past fifty years, and has done important work on a wide range of topics, ranging from the philosophy of mathematics (his doctorate is in mathematics from the London School of Economics) to logic and eastern philosophy. Robinson, Graham, and Joel discuss two topics—the liar paradox and the set-theoretic multiverse. More particularly, they address how solutions to the former revolve around questions of logical pluralism (is there more than one “correct” logic, and if so, how should we determine which to use in any given situation?), and regarding the latter, they address the metaphysics of the multiverse, how the multiverse theory squares with its monist alternative, and how it relates to the age-old question: Is mathematics created or discovered? Some resources for background information are included below. Check out Joel's current project, The Book of Infinity, which is an accessible text on paradoxes and infinity. Joel has made the novel move of serializing it on Substack, so you can participate in its creation by checking out the link below, and otherwise see what he's thinking about and working on through Twitter, MathOverflow, and his blog. You can keep up with Graham and his ever-growing, immense body of work through his website. Graham's Website: https://grahampriest.net Joel's Blog: http://jdh.hamkins.org Joel's MathOverflow: https://mathoverflow.net/users/1946/joel-david-hamkins Joel's Substack: https://joeldavidhamkins.substack.com Joel's Twitter: https://twitter.com/JDHamkins Background: The Liar Paradox on the SEP: https://plato.stanford.edu/entries/liar-paradox/ Set Theory on the SEP: https://plato.stanford.edu/entries/set-theory/ Robinson's Website: http://robinsonerhardt.com OUTLINE: 00:00 In This Episode… 1:12 Introduction 11:16 Graham's History with the Liar Paradox 12:51 An Explication of the Liar 15:03 Paraconsistent Logic and the Liar 32:32 A Deflationary Account of Truth and the Liar 34:51 Joel's Approach to the Liar 38:37 Hartry Field and the Liar 41:18 The Yablo Paradox 48:22 When to Change the Logic 56:24 A Difference in Opinion on Logic? 1:01:44 The Set-Theoretic Multiverse 1:14:43 Monism and Pluralism About the Set-Theoretic Universe 1:35:35 Philosophical Answers to Mathematical Questions 1:39:16 On Woodin's Program 1:46:12 Logical Pluralism and the Set-Theoretic Multiverse 1:58:13 The Metaphysics of the Set-Theoretic Multiverse 2:09:42 Is Mathematics Created or Discovered? 2:16:59 The Continuity From Ancient To Contemporary Mathematics Robinson Erhardt researches symbolic logic and the foundations of mathematics at Stanford University. Join him in conversations with philosophers, scientists, weightlifters, artists, and everyone in-between. --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

Joel David Hamkins is the O'Hara Professor of Philosophy and Mathematics at the University of Notre Dame, where he recently moved from the University of Oxford. Joel is one of the leading set theorists and philosophers of mathematics in the world, and he and Robinson discuss a lot—Hilbert's Hotel, the continuum hypothesis, the set-theoretic multiverse, and even Joel's dapper hat collection—but the main subject is his upcoming book, The Book of the Infinite, which is an accessible text on paradoxes and infinity. Joel has made the novel move of serializing it on Substack, so you can participate in its creation by checking out the link below, and otherwise see what he's thinking about and working on through Twitter, MathOverflow, and his blog. The conversation grows technical from 1:10:26-2:00:25, but for those to whom that doesn't appeal there are timestamps to navigate around this portion of the show. Substack: https://joeldavidhamkins.substack.com Twitter: https://twitter.com/JDHamkins MathOverflow: https://mathoverflow.net/users/1946/joel-david-hamkins Joel's Blog: http://jdh.hamkins.org OUTLINE: 00:00 Introduction 2:52 Is Joel a Mathematician or a Philosopher? 6:13 The Philosophical Influence of Hugh Woodin 10:29 The Intersection of Set Theory and Philosophy of Math 16:29 Serializing the Book of the Infinite 20:05 Zeno of Elea, Continuity, and Geometric Series 39:39 Infinite Games and the Chocolatier 53:35 Hilbert's Hotel 1:10:26 Cantor's Theorem 1:31:37 The Continuum Hypothesis 1:43:02 The Set-Theoretic Multiverse 2:00:25 Berry's Paradox and Large Numbers 2:16:15 Skolem's Paradox and Indescribable Numbers 2:28:41 Pascal's Wager and Reasoning Around Remote Events 2:49:35 MathOverflow 3:04:40 Joel's Impeccable Fashion Sense Linktree: https://linktr.ee/robinsonerhardt Twitter: https://twitter.com/robinsonerhardt Instagram: https://www.instagram.com/robinsonerhardt/ Twitch (Robinson Eats): https://www.twitch.tv/robinsonerhardt YouTube (Robinson Eats): youtube.com/@robinsoneats TikTok: https://www.tiktok.com/@robinsonerhardt --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

We're getting deep into the Battle of Hoth today! This is minutes 26-30 of The Empire Strikes Back with composer/orchestrator Dominic Sewell. The quickie topic is a primer on the anatomy and function of a score.  To follow along with the visuals in this episode, I recommend checking out the video version (or just spot checking the areas you want to see): https://youtu.be/oOvHkaciOVk  Timestamps: 00:00 - Hello there! 2:10 - What stands out to you about the Battle of Hoth? 8:26 - Who is the greatest 20th century composer? 10:30 - Quickie topic: Anatomy and function of a score. 16:05 - Start of these minutes. 18:51 - Contrapuntal fudge. 28:18 - Frank Lehman's breakdown of the Battle of Hoth themes from a-z. 32:46 - Cool things about octatonic scales and John Williams's use of them. 45:42 - Military preparation theme (e) 48:00 - "One is okay, two is memorable, three is too much." 54:05 - Optimistic Walton-esque theme (h) 1:01:07 - Increasing rhythm, momentum, texture before big crescendo. 1:03:35 - Start of 3M3 "The Snow Battle" 1:10:30 - Pitch class sets, Forte numbers, using the pitch-class set calculator. 4-19, 6-z19, 5-32, 8-28, etc... 1:26:29 - “Fingers are not to be despised: they are the great inspirers, and, in contact with a musical instrument, often give birth to subconscious ideas which might otherwise never come to life.” -Stravinsky 1:29:10 - Alpha chords. 1:38:47 - What JW wrote about this cue in the liner notes ("bizarre, mechanical, brutal...") 1:49:10 - "The rub" + Why are minor seconds and major sevenths "the same?" 2:00:14 - "Into the Maw" from Solo: A Star Wars Story. 2:06:13 - Cluster splurge. 2:13:33 - Diverging outline. 2:19:01 - Filigree, frills, and trills. 2:23:41 - 6-20 classic hexatonic set. 2:31:40 - Philosophical question: how do you decide the boundaries of the set you're analyzing? 2:36:46 - The Hoth sequence contains nearly all of John Williams's action hallmarks. 2:45:38 - SWMM Questionnaire References: Complete Catalogue of the Musical Themes of Star Wars (by Frank Lehman): https://franklehman.com/starwars/. Analysis Through Composition - Principles of the Classical Style (book by Nicholas Cook) - https://searchworks.stanford.edu/view/3425258 Vaughan Williams: Fantasia on a theme by Thomas Tallis Orb and Sceptre (William Walton) - https://youtu.be/v6qjUdaDE_Q Crown Imperial March (William Walton) - https://youtu.be/1M9xZlA2zn8 Dominic's video intro to pitch class set theory: https://youtu.be/Am2KLFMGuvw Pitch-Class Set Calculator: https://www.mta.ca/pc-set/calculator/pc_calculate.html List of set classes: https://en.wikipedia.org/wiki/List_of_set_classes Petrushka (Stravinsky) - https://youtu.be/jeSC0vtdn3g David Matthews piece: The Golden Kingdom song cycle, "Spell of Creation" movement - https://www.fabermusic.com/shop/the-golden-kingdom-p3081 Mussorgsky: Night on Bald Mountain - https://youtu.be/SLCuL-K39eQ Mickey Mouse (2013) series, music by Christopher Willis - https://www.youtube.com/watch?v=yNdhDq-DF8k&list=PLIUzyJZLGeBM2mB9MaXtfzCxHRZ6s8PIl Mark Richards's film scoring courses: https://filmmusicnotes.com/all-courses/ https://en.wikipedia.org/wiki/Mickey_Mouse_(TV_series) The Music of The Lord of the Rings Films: A Comprehensive Account of Howard Shore's Scores (by Doug Adams) - https://www.goodreads.com/en/book/show/8882617 Check out Dominic Sewell's channel to see more analysis with video walkthroughs of these cues: 3M2 Leia's Instructions (final segment) - https://youtu.be/DThcEWb_CoE 3M3 The Snow Battle - https://youtu.be/fIHn1e521DY 3M4/4M1 Luke's First Crash - https://youtu.be/dfKjTX7gpNs  Cues in these minutes: 3M2 "Leia's Instructions" 3M3 "The Snow Battle" 3M4/4M1 "Luke's First Crash" Musical Themes: 2. Rebel Fanfare 18) Descending Heroic Tetrachords 1a. Main Theme (A Section) 13. Droids Where are we in the soundtrack(s)?: "The Battle Of Hoth (Ion Cannon/Imperial Walkers/Beneath the" --------------- STAR WARS MUSIC MINUTE QUESTIONNAIRE: 1. In exactly 3 words, what does Star Wars sound like? Transporting. Transforming. Transcendental. 2. What's something related to Star Wars music or sound that you want to learn more about? Dom would like to talk to John Williams about what his teachers taught him. 3. What's a score or soundtrack you're fond of besides anything Star Wars? Lord of the Rings - Howard Shore Doctor Who (TV series) - Murray Gold The Orville (TV series) - Bruce Broughton (main theme), Joel McNeely, Andrew Cottee, John Debney, and Kevin Kaska Death of Stalin - Christopher Willis --------------- Guest: Dominic Sewell Website: https://dominicsewell.co.uk YouTube: https://www.youtube.com/c/DominicSewellMusic Twitter: https://twitter.com/dominicsewell Patreon: https://www.patreon.com/DominicSewellMusic ------------------ If you want to support the show and join the Discord server, consider becoming a patron!  https://patreon.com/chrysanthetan Leave a voice message, and I might play it on the show...   https://starwarsmusicminute.com/comlink Where else to find SWMM: Twitter: https://twitter.com/StarWarsMusMin Spotify: https://smarturl.it/swmm-spotify Apple Podcasts: https://smarturl.it/swmm-apple YouTube: https://youtube.com/starwarsmusicminute TikTok: https://www.tiktok.com/@starwarsmusicminute? Instagram: https://instagram.com/starwarsmusicminute Email: podcast@starwarsmusicminute.com Buy Me A Coffee: https://buymeacoffee.com/starwarsmusmin

Haim Gaifman is a professor of philosophy at Columbia university in New York City. He is also a mathematician and probability theorist. In this episode (Haim's fourth appearance), Robinson and Haim discuss the origins of set theory as the mathematical discipline developed to study the infinite, as well as its relation to Richard's paradox. Instagram: @robinsonerhardt TikTok: @robinsonerhardt Twitch (Robinson Eats): @robinsonerhardt YouTube (Robinson Eats): youtube.com/@robinsoneats --- Support this podcast: https://podcasters.spotify.com/pod/show/robinson-erhardt/support

Are our future sins forgiven in the Atonement? Dr. Craig answers this and questions on Theistic Evolution, The Logos, and Set Theory.

set theory for dumb dumbz

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: On learning difficult things, published by So8res on the LessWrong. I have been autodidacting quite a bit lately. You may have seen my reviews of books on the MIRI course list. I've been going for about ten weeks now. This post contains my notes about the experience thus far. Much of this may seem obvious, and would have seemed obvious if somebody had told me in advance. But nobody told me in advance. As such, this is a collection of things that were somewhat surprising at the time. Part of the reason I'm posting this is because I don't know a lot of autodidacts, and I'm not sure how normal any of my experiences are. (Though on average, I'd guess they're about average.) As always, keep in mind that I am only one person and that your mileage may vary. Pair up When I began my quest for more knowledge, I figured that in this modern era, a well-written textbook and an account on math.stackexchange would be enough to get me through anything. And I was right. sort of. But not really. The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I've either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words. "Dude. Disjunction. Disjunction." These are the things that eat my days. In principle, places like stackexchange can get me unstuck, but they're an awkward tool for the job. First of all, my stupid mistakes are heavily contextualized. A full context dump is necessary before I can even ask my question, and this takes time. Furthermore, I feel dumb asking stupid questions on stackexchange-type sites. My questions are usually things that I can figure out with a close re-read (except, I'm not sure which part needs a re-read). I usually opt for a close re-read of everything rather than asking for help. This is even more time consuming. The infuriating thing is that answering these questions usually doesn't require someone who already knows the answers: it just requires someone who didn't make exactly the same mistakes as me. I lose hours on little mistakes that could have been fixed within seconds if I was doing this with someone else. That's why my number one piece of advice for other people attempting to learn on their own is do it with a friend. They don't need to be more knowledgeable than you to answer most of the questions that come up. They just need to make different misunderstandings, and you'll be able to correct each other as you go along. The thing I miss most about college is tight feedback loops while learning. When autodidacting, the feedback loop can be long. I still haven't managed to follow my own advice here. I'm writing this advice in part because it should motivate me to actually pair up. Unfortunately, there is nobody in my immediate circle who has the time or patience to read along with me, but there are a number of resources I have not yet explored (the LessWrong study hall, for example, or soliciting to actual mathematicians). It's on my list of things to do. Read, reread, rereread Reading Model Theory was one of the hardest things I've done. Not necessarily because the content was hard, but because it was the first time I actually learned something that was way outside my comfort zone. The short version is that Basic Category Theory and Naïve Set Theory left me somewhat overconfident, and that I should have read a formal logic textbook before diving in. I had basic familiarity with logic, but no practice. Turns out practice is important. Anyway, it's not like Model Theory was impossible just because I skipped my logic exercises. It was just hard. There are a number of little misconceptions you have when you're familiar with something but you've never applied it, and I found myself having to clean t...

Today's solo episode covers: - Peter Boghossian's Resignation Letter and the Evolution of the Academy away from Truth Seeking to Ideological Rigidity - How to create layers of communities for truth and peace - The latest news in smart toilets, promise of health plus security concerns - The set theory principle of inclusion and exclusion in set theory and how the mathematical jump from "union" to "addition" is a difficult one

Jakob and Todd talk about set theory, its historical origins, Georg Cantor, trigonometric series, cardinalities of number systems, the continuum hypothesis, cardinalities of infinite sets, set theory as a foundation for mathematics, Cantor's paradox, Russell's paradox, axiomatization, the Zermelo–Fraenkel axiomatic system, the axiom of choice, and the understanding of mathematical objects as "sets with structure".

Dorothea Rockburne (Canadian, b.1932) is a painter and a draughtswoman, as well as a mixed media and installation artist. Born in Montréal, Quebec, Rockburne began classic training in Painting, Drawing, and Sculpture in 1942 at the Ecole des Beaux-Arts, Paris, France, where she studied under the Abstract artist Paul Emile Borduas (Canadian, 1905–1960). After winning a scholarship, Rockburne studied at the Montréal Museum School, where she began to distance her artistic style from the classical manner she had been studying since a young age. Moses Martin Reinblatt (Canadian, 1917–1979), one of Rockburne’s teachers at the museum school, convinced her to apply to Black Mountain College in Asheville, NC, which was known for being the radical art school of the time. Rockburne attended Black Mountain College from 1950–1955, studying a variety of subjects including Painting, Music, Dance, Math, Theater, Linguistics, Philosophy, Literature, Writing, Poetry, and Photography. Rockburne moved to New York, NY after she graduated. Although she won the Walter Gutman Emerging Artist Award in 1957, Rockburne struggled with her art, and so she turned to dance and performance art for several years. During this time she took on some side jobs to support herself, including a bookkeeping job at the Metropolitan Museum of Art, NY, where she catalogued the Egyptian Antiquities collection. Rockburne took a great interest in the art of ancient Egypt from a young age, and she later incorporated this interest into her works entitled Egyptian Paintings (1979–1980). In 1963, Rockburne began assisting her friend and former schoolmate Robert Rauschenberg (American, 1925–2008). For the next five years Rockburne worked in Rauschenberg’s studio; she participated in various performances with other artists, including Claes Oldenberg (b.1929) in a work entitled Washes (1965) at Al Roon’s Heath Club in New York City. A year later, Rockburne was working in her own studio again. She incorporated mathematics into her art, inspired by dance and how the body moves through space. Rockburne produced her Set Theory installations, which were first shown in 1970 at the Bykert Gallery, NY, with this new inspiration. In 1972, Rockburne received a Guggenheim Fellowship and traveled to Italy, where she continued her studies in Italian Art, and began to merge her classical training into her work. In the early 1990s, Rockburne began to study Astronomy and frescoes, combining these interests to create a major fresco secco for SONY headquarters in New York City entitled Northern Sky, Southern Sky (1992–1993). In 2001, Rockburne participated in the comprehensive exhibition The Universe: Contemporary Art and the Cosmos, combining her knowledge and skill in Art, Music, Science, and Astronomy. She has received many awards and honors during her successful career including the National Endowment for the Arts grant (1974), the Witowsky Prize for Painting (1976), participation at the 1980 Venice Biennale, and a membership in the Department of Art at the American Academy of Arts and Letters (2001). https://www.dorothearockburne.com @tipntell tipntellpodcast@gmail.com Host & Cover Art: Cydney Williams @cydneywilliamsstudio Sound & Music: Ian Eckstein @ian_eckstein Listen on Breaker, Google Podcasts, Pocket Casts, Radiopublic, Spotify, Copy RSS, Anchor, Apple Podcasts, Youtube, & IGTV

CONTACT ME: @SpazPhoenix @SpazPhoenix1 Check out my friends: @blackcatfeline @Guapo_504 @CountdownEnded @OKayFabe On Youtube! https://www.youtube.com/user/SpazPhoenix Search "Spaz Phoenix Podcast" on: -Anchor.fm -Apple Podcasts (iTunes) -Spotify -Stitcher -Google Podcasts -iHeartRadio -TuneIn -Breaker -Castbox -RadioPublic -Pocket Casts -Spreaker -Castro -Podcast Addict -Overcast -Player.fm -Listen Notes #WWE #NXT #AEW

Mathematical relations are an important part of the Set Theory. Also known as the Subsets of the Cartesian product of sets, we have a variety of Relations which are defined on different sets.

The true value of any friendship is what occurs in the words that are unspoken. This is the case with myself and my colleague Dr. Steven Yalkowsky. While we are both psychologists and are likely to share similar bloodlines all the way back to Abraham, we could not be more different politically. Observe the miracle of two people in sweet disagreement ... agreeing. Hear the music of Set Theory. Nothing is endorsed. That I remember. Not a political party. Nothing. A brother and friend. --- This episode is sponsored by · Anchor: The easiest way to make a podcast. https://anchor.fm/app

Set theory & Types of sets | Mathematics | Radio badi | Online classes | E FM | Eenadu

Study of Subsets is an integral part of the Set Theory. Let's hope we can understand the concept in a better way.

Joel David Hamkins is an American Mathematician who is currently Professor of Logic at the University of Oxford. He is well known for his important contributions in the fields of Mathematical Logic, Set Theory and Philosophy of Mathematics. Moreover, he is very popular in the mathematical community for being the highest rated user on MathOverflow. Outline of the conversation:00:00 Podcast Introduction00:50 MathOverflow and books in progress04:08 Mathphobia05:58 What is mathematics and what sets it apart?08:06 Is mathematics invented or discovered (more at 54:28)09:24 How is it the case that Mathematics can be applied so successfully to the physical world?12:37 Infinity in Mathematics16:58 Cantor's Theorem: the real numbers cannot be enumerated24:22 Russell's Paradox and the Cumulative Hierarchy of Sets29:20 Hilbert's Program and Godel's Results35:05 The First Incompleteness Theorem, formal and informal proofs and the connection between mathematical truths and mathematical proofs40:50 Computer Assisted Proofs and mathematical insight44:11 Do automated proofs kill the artistic side of Mathematics?48:50 Infinite Time Turing Machines can settle Goldbach's Conjecture or the Riemann Hypothesis54:28 Nonstandard models of arithmetic: different conceptions of the natural numbers1:00:02 The Continuum Hypothesis and related undecidable questions, the Set-Theoretic Multiverse and the quest for new axioms1:10:31 Minds and computers: Sir Roger Penrose's argument concerning consciousnessTwitter: https://twitter.com/tedynenu

This particular podcast covers the material in Chapter 2 of my new book “Statistical Machine Learning: A unified framework” with expected publication date May 2020. In this episode we discuss Chapter 2 of my new book, which discusses how to represent knowledge using set theory notation. Chapter 2 is titled “Set Theory for Concept Modeling”.

Episode: 1829 Multiple authors, multiple problems: Whose work is this?  Today, whose work is this?

Es gibt zwei Möglichkeiten, eine Menge zu notieren: die aufzählende und die beschreibende Mengenschreibweise. In diesem Video erfährst du, was die aufzählende Mengenschreibweise ist. Dieses Video steht unter CC-BY-SA 4.0.

In diesem Video erkläre ich dir, was Verknüpfungen zwischen Mengen sind. Es ist nämlich möglich, Mengen zu neuen Mengen zusammenzufassen. Dieses Video steht unter CC-BY-SA 4.0.

Um Mengen zu visualisieren gibt es in der Mathematik zwei Tools: Venn-Diagramme und Eulerdiagramme. Dieses Video erklärt beide Arten von Diagrammen anhand einiger Beispiele. Dieses Video steht unter CC-BY-SA 4.0.

Example

  Far too often, we evaluate math ability in high schoolers solely on the basis of grades and level of math learned. A more accurate assessment of a student’s potential on challenging math tasks--including those posed on tests like the SAT and ACT--should consider mathematical maturity.  Amy and Mike invited author and test prep professional Dr. Steve Warner to define what this means and explain the link between mathematical maturity and test success.  What are five things you will learn in this episode? What is mathematical maturity? How can you determine your "level" of mathematical maturity? Can mathematical maturity be improved? How does mathematical maturity relate to standardized test scores? What steps can students seeking higher levels of mathematical maturity take? MEET OUR GUEST Dr. Steve Warner, a New York native, earned his Ph.D. at Rutgers University in Pure Mathematics in May 2001. After Rutgers, Dr. Warner joined the Penn State Mathematics Department as an Assistant Professor and in September 2002, he returned to New York to accept an Assistant Professor position at Hofstra University. By September 2007, Dr. Warner had received tenure and was promoted to Associate Professor. He has taught undergraduate and graduate courses in Precalculus, Calculus, Linear Algebra, Differential Equations, Mathematical Logic, Set Theory, and Abstract Algebra. From 2003 – 2008, Dr. Warner participated in a five-year NSF grant, “The MSTP Project,” to study and improve mathematics and science curriculum in poorly performing junior high schools. He also published several articles in scholarly journals, specifically on Mathematical Logic. Dr. Warner has nearly two decades of experience in general math tutoring and tutoring for standardized tests such as the SAT, ACT, GRE, GMAT, and AP Calculus exams. He has tutored students both individually and in group settings. In February 2010 Dr. Warner released his first SAT prep book “The 32 Most Effective SAT Math Strategies,” and in 2012 founded Get 800 Test Prep. Since then Dr. Warner has written books for the SAT, ACT, SAT Math Subject Tests, AP Calculus exams, and GRE. In 2018 Dr. Warner released his first pure math book called “Pure Mathematics for Beginners.” Since then he has released several more books, each one addressing a specific subject in pure mathematics. Dr. Steve Warner can be reached at steve@SATPrepGet800.com LINKS Gaining Mathematical Maturity Dr. Warner’s extensive catalog of math prep books ABOUT THIS PODCAST Tests and the Rest is THE college admissions industry podcast. Explore all of our episodes on the show page.

Iván Ongay Valverde is a graduate student at the University of Wisconsin-Madison doing research in mathematical logic under the supervision of Kenneth Kunen. His research is in the field of Set Theory, in particular, he studies subsets of the Real Numbers and their behavior under different axioms and forcing extensions.His website can be found here: https://www.math.wisc.edu/~ongay/We'd like to thank Iván for being on our show "Meet a Mathematician" and for sharing his stories and perspective with us!www.sensemakesmath.comPODCAST: http://sensemakesmath.buzzsprout.com/TWITTER: @SenseMakesMathPATREON: https://www.patreon.com/sensemakesmathFACEBOOK: https://www.facebook.com/SenseMakesMathSTORE: https://sensemakesmath.storenvy.comSupport the show (https://www.patreon.com/sensemakesmath)

Graham Priest (CUNY and St Andrews) gives a talk at the Conference on Paraconsistent Reasoning in Science and Mathematics (11-13 June, 2014) titled "Models of Paraconsistent Set Theory". Abstract: Any adequate paraconsistent set theory must be able to validate at least a major part of the standard results of orthodox set theory. One way to achieve this is to take the universe or universes of sets to be such as to validate not only the naïve principles, but also all the theorems of Zermelo Fraenkel set theory. In this talk I will discuss various constructions of models of set theory which do just this.

Bienvenidos a ERA Magazine, el podcast de la música independiente española. En el capítulo de hoy, conoceremos el lado más salvaje, enigmático y oscuro de la música con el grupo Captains. Buenos días, antes de nada, dejadme hablar un poco de ERA Magazine y de la financiación de este podcast. No tenemos ninguna empresa detrás ni ningún patrocinador. Lo hacemos porque nos gusta y apasiona la música independiente de nuestro país, sus grupos, sus discográficas, sus festivales, sus salas de conciertos… ¿Y cómo pretendemos seguir? Gracias a ti, a los oyentes de ERA Magazine. Visita eramagazine.fm/mecenas, dale al botón azul que pone Apoyar, y desde solo 1,49 euros al mes, nos ayudas a que sigamos descubriendo propuestas muy interesantes. Sé un mecenas de ERA Magazine y participa en este red de podcast, que poco a poco incorpora muchos más programas. Esencia rock De la unión de la astu-alemana Fee Reega y el productor David Baldo nos llevamos una grata sorpresa el año pasado con su álbum de debut. Ahora, Captains publican Pure Pleasure, (Set Theory, 2018), nueve canciones donde las guitarras siguen siendo las protagonistas, apoyadas por una salvaje voz de Fee, que te eriza el vello en cada canción. Lo llaman post-punk, hasta grunge he leído, pero la verdad es que a mí me suena a esencia rock en su estado más crudo. Vámonos hasta el centro de Madrid para conocer cómo se ha gestado este proyecto. “Pure Pleasure”. “Broken Body”. “Mysterious Pretty Cowboy Sunrise”. “Brand New Girl”. Con esta canción nos despedimos por hoy. También recuerda, que si quieres ayudar a este podcast, y seguir disfrutando de la música de muchos más grupos, visita eramagazine.fm/mecenas, dale al botón Apoyar y desde 1,49 euros al mes contribuyes a que sigamos descubriendo más propuestas emergentes. Sé un mecenas de ERA Magazine. Porque recuerda: a la gente le encanta la música indie, pero todavía no lo sabe. Adiós. Captains Pure Pleasure (Set Theory, 2018)Facebook | YouTube | Bandcamp | Instagram La entrada ERA MAGAZINE #358 Captains, guitarras crudas se publicó primero en ERA Magazine.

Sets are everywhere! If you've worked with relational databases, made a venn diagram, maybe touched some relational algebra, then you've already worked with sets. We talk about why they're so common, how well they perform (time for some Big O Notation!), and how they're actually implemented. Based on Vaidehi Joshi's blog post, "Set Theory: the Method To Database Madness".

Set theory might sound like a scary, super-math thing, but it's not! Well, it is a math thing, but it doesn't have to be super scary. In fact, if you already know how venn diagrams work, then you basically already know set theory. We'll walk you through it all and show you how it connects back to computer science with the help of our favorite foods. Based on Vaidehi Joshi's blog post, "Set Theory: the Method To Database Madness".

In episode 3, I quickly review the iPhone X (spoiler: it's nice), talk about my efforts to benchmark the user-facing speed of iPhones, and conclude with a discussion of how Swift types relate to set theory and what that means for structs and enums. Audio editing for this episode was by @josevazquez.

  Stream it from here: It is important to think “What is the big goal that I am trying to do?” – Jouko Väänänen How often should you apply for grants? How much time are you supposed to use on grant applications? What is the single most important thing you need to have in order […]

Mauritius-born New Zealandian-Indian artist Nandita Kumar is all over the place – in a good way. She is on a quest to find connections between art, science and technology through her own art as well as community projects to shift consciousness of people – in her own words. I met Nandita at the exhibition opening of ARS17 in […]

Nicolas Monod teaches at the École polytechnique fédérale in Lausanne and leads the Ergodic and Geometric Group Theory group there. In May 2016 he was invited to give the Gauß lecture of the German Mathematical Society (DMV) at the Technical University in Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski Paradox. The morning after his lecture we met to talk about paradoxes and hidden assumptions our mind makes in struggling with geometrical representations and measures. A very well-known game is Tangram. Here a square is divided into seven pieces (which all are polygons). These pieces can be rearranged by moving them around on the table, e.g.. The task for the player is to form given shapes using the seven pieces – like a cat etc.. Of course the Tangram cat looks more like a flat Origami-cat. But we could take the Tangram idea and use thousands or millions of little pieces to build a much more realistic cat with them – as with pixels on a screen. In three dimensions one can play a similar game with pieces of a cube. This could lead to a LEGO-like three-dimensional cat for example. In this traditional Tangram game, there is no fundamental difference between the versions in dimension two and three. But in 1914 it was shown that given a three-dimensional ball, there exists a decomposition of this ball into a finite number of subsets, which can then be rearranged to yield two identical copies of the original ball. This sounds like a magical trick – or more scientifically said – like a paradoxical situation. It is now known under the name Banach-Tarski paradox. In his lecture, Nicolas Monod dealt with the question: Why are we so surprised about this result and think of it as paradoxical? One reason is the fact that we think to know deeply what we understand as volume and expect it to be preserved under rearrangements (like in the Tangram game, e.g.).Then the impact of the Banach-Tarski paradox is similar for our understanding of volume to the shift in understanding the relation between time and space through Einstein's relativity theory (which is from about the same time). In short the answer is: In our every day concept of volume we trust in too many good properties of it. It was Felix Hausdorff who looked at the axioms which should be valid for any measure (such as volume). It should be independent of the point in space where we measure (or the coordinate system) and if we divide objects, it should add up properly. In our understanding there is a third hidden property: The concept "volume" must make sense for every subset of space we choose to measure. Unfortunately, it is a big problem to assign a volume to any given object and Hausdorff showed that all three properties cannot all be true at the same time in three space dimensions. Couriously, they can be satisfied in two dimensions but not in three. Of course, we would like to understand why there is such a big difference between two and three space dimensions, that the naive concept of volume breaks down by going over to the third dimension. To see that let us consider motions. Any motion can be decomposed into translations (i.e. gliding) and rotations around an arbitrarily chosen common center. In two dimensions the order in which one performs several rotations around the same center does not matter since one can freely interchange all rotations and obtains the same result. In three dimensions this is not possible – in general the outcomes after interchanging the order of several rotations will be different. This break of the symmetry ruins the good properties of the naive concept of volume. Serious consequences of the Banach-Tarski paradox are not that obvious. Noone really duplicated a ball in real life. But measure theory is the basis of the whole probability theory and its countless applications. There, we have to understand several counter-intuitive concepts to have the right understanding of probabilities and risk. More anecdotally, an idea of Bruno Augenstein is that in particle physics certain transformations are reminiscent of the Banach-Tarski phenomenon. Nicolas Monod really enjoys the beauty and the liberty of mathematics. One does not have to believe anything without a proof. In his opinion, mathematics is the language of natural sciences and he considers himself as a linguist of this language. This means in particular to have a closer look at our thought processes in order to investigate both the richness and the limitations of our models of the universe. References: F. Hausdorff: Bemerkung über den Inhalt von Punktmengen. Math. Ann. 75 (3), 428–433, 1914. S. Banach and A.Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae 6, 244–277, 1924. J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta Mathematicae 13, 73–116, 1929. S. Wagon: The Banach–Tarski Paradox. Cambridge University Press, 1994. B.W. Augenstein: Links Between Physics and Set Theory, Chaos, Solitons and Fractals, 7 (11), 1761–1798, 1996. N. Monod: Groups of piecewise projective homeomorphisms, PNAS 110 (12), 4524-4527, 2013. Vsauce-Video on the Banach-Tarksi Paradox

Nicolas Monod teaches at the École polytechnique fédérale in Lausanne and leads the Ergodic and Geometric Group Theory group there. In May 2016 he was invited to give the Gauß lecture of the German Mathematical Society (DMV) at the Technical University in Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski Paradox. The morning after his lecture we met to talk about paradoxes and hidden assumptions our mind makes in struggling with geometrical representations and measures. A very well-known game is Tangram. Here a square is divided into seven pieces (which all are polygons). These pieces can be rearranged by moving them around on the table, e.g.. The task for the player is to form given shapes using the seven pieces – like a cat etc.. Of course the Tangram cat looks more like a flat Origami-cat. But we could take the Tangram idea and use thousands or millions of little pieces to build a much more realistic cat with them – as with pixels on a screen. In three dimensions one can play a similar game with pieces of a cube. This could lead to a LEGO-like three-dimensional cat for example. In this traditional Tangram game, there is no fundamental difference between the versions in dimension two and three. But in 1914 it was shown that given a three-dimensional ball, there exists a decomposition of this ball into a finite number of subsets, which can then be rearranged to yield two identical copies of the original ball. This sounds like a magical trick – or more scientifically said – like a paradoxical situation. It is now known under the name Banach-Tarski paradox. In his lecture, Nicolas Monod dealt with the question: Why are we so surprised about this result and think of it as paradoxical? One reason is the fact that we think to know deeply what we understand as volume and expect it to be preserved under rearrangements (like in the Tangram game, e.g.).Then the impact of the Banach-Tarski paradox is similar for our understanding of volume to the shift in understanding the relation between time and space through Einstein's relativity theory (which is from about the same time). In short the answer is: In our every day concept of volume we trust in too many good properties of it. It was Felix Hausdorff who looked at the axioms which should be valid for any measure (such as volume). It should be independent of the point in space where we measure (or the coordinate system) and if we divide objects, it should add up properly. In our understanding there is a third hidden property: The concept "volume" must make sense for every subset of space we choose to measure. Unfortunately, it is a big problem to assign a volume to any given object and Hausdorff showed that all three properties cannot all be true at the same time in three space dimensions. Couriously, they can be satisfied in two dimensions but not in three. Of course, we would like to understand why there is such a big difference between two and three space dimensions, that the naive concept of volume breaks down by going over to the third dimension. To see that let us consider motions. Any motion can be decomposed into translations (i.e. gliding) and rotations around an arbitrarily chosen common center. In two dimensions the order in which one performs several rotations around the same center does not matter since one can freely interchange all rotations and obtains the same result. In three dimensions this is not possible – in general the outcomes after interchanging the order of several rotations will be different. This break of the symmetry ruins the good properties of the naive concept of volume. Serious consequences of the Banach-Tarski paradox are not that obvious. Noone really duplicated a ball in real life. But measure theory is the basis of the whole probability theory and its countless applications. There, we have to understand several counter-intuitive concepts to have the right understanding of probabilities and risk. More anecdotally, an idea of Bruno Augenstein is that in particle physics certain transformations are reminiscent of the Banach-Tarski phenomenon. Nicolas Monod really enjoys the beauty and the liberty of mathematics. One does not have to believe anything without a proof. In his opinion, mathematics is the language of natural sciences and he considers himself as a linguist of this language. This means in particular to have a closer look at our thought processes in order to investigate both the richness and the limitations of our models of the universe. References: F. Hausdorff: Bemerkung über den Inhalt von Punktmengen. Math. Ann. 75 (3), 428–433, 1914. S. Banach and A.Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae 6, 244–277, 1924. J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta Mathematicae 13, 73–116, 1929. S. Wagon: The Banach–Tarski Paradox. Cambridge University Press, 1994. B.W. Augenstein: Links Between Physics and Set Theory, Chaos, Solitons and Fractals, 7 (11), 1761–1798, 1996. N. Monod: Groups of piecewise projective homeomorphisms, PNAS 110 (12), 4524-4527, 2013. Vsauce-Video on the Banach-Tarksi Paradox