Podcasts about theorems

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In this episode of Chatting with Channing, we sit down with renowned author, journalist, and TV producer and the speaker at our annual STEM lecture Simon Singh, along with sixth form students Alisa and Natalia, who are aspiring mathematicians. Simon discusses his journey from particle physics to becoming a celebrated science communicator, exploring the fascinating worlds of mathematics, cryptography, and even The Simpsons. Together, they uncover the magic of maths, the importance of challenging oneself, and the value of pursuing passions. Don't miss Simon's advice on balancing creativity and discipline, and hear how his groundbreaking books continue to inspire.Channing School Online: Facebook: www.facebook.com/channingschool LinkedIn: www.linkedin.com/company/channing-school Twitter: twitter.com/ChanningSchool

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: A simple model of math skill, published by Alex Altair on July 21, 2024 on LessWrong. I've noticed that when trying to understand a math paper, there are a few different ways my skill level can be the blocker. Some of these ways line up with some typical levels of organization in math papers: Definitions: a formalization of the kind of objects we're even talking about. Theorems: propositions on what properties are true of these objects. Proofs: demonstrations that the theorems are true of the objects, using known and accepted previous theorems and methods of inference. Understanding a piece of math will require understanding each of these things in order. It can be very useful to identify which of type of thing I'm stuck on, because the different types can require totally different strategies. Beyond reading papers, I'm also trying to produce new and useful mathematics. Each of these three levels has another associated skill of generating them. But it seems to me that the generating skills go in the opposite order. This feels like an elegant mnemonic to me, although of course it's a very simplified model. Treat every statement below as a description of the model, and not a claim about the totality of doing mathematics. Understanding Understanding these more or less has to go in the above order, because proofs are of theorems, and theorems are about defined objects. Let's look at each level. Definitions You might think that definitions are relatively easy to understand. That's usually true in natural languages; you often already have the concept, and you just don't happen to know that there's already a word for that. Math definitions are sometimes immediately understandable. Everyone knows what a natural number is, and even the concept of a prime number isn't very hard to understand. I get the impression that in number theory, the proofs are often the hard part, where you have to come up with some very clever techniques to prove theorems that high schoolers can understand (Fermat's last theorem, the Collatz conjecture, the twin primes conjecture). In contrast, in category theory, the definitions are often hard to understand. (Not because they're complicated per se, but because they're abstract.) Once you understand the definitions, then understanding proofs and theorems can be relatively immediate in category theory. Sometimes the definitions have an immediate intuitive understanding, and the hard part is understanding exactly how the formal definition is a formalization of your intuition. In a calculus class, you'll spend quite a long time understanding the derivative and integral, even though they're just the slope of the tangent and the area under the curve, respectively. You also might think that definitions were mostly in textbooks, laid down by Euclid or Euler or something. At least in the fields that I'm reading papers from, it seems like most papers have definitions (usually multiple). This is probably especially true for papers that are trying to help form a paradigm. In those cases, the essential purpose of the paper is to propose the definitions as the new paradigm, and the theorems are set forth as arguments that those definitions are useful. Theorems Theorems are in some sense the meat of mathematics. They tell you what you can do with the objects you've formalized. If you can't do anything meaty with an object, then you're probably holding the wrong object. Once you understand the objects of discussion, you have to understand what the theorem statement is even saying. I think this tends to be more immediate, especially because often, all the content has been pushed into the definitions, and the theorem will be a simpler linking statement, like "all As are Bs" or "All As can be decomposed into a B and a C". For example, the fundamental theorem of calculus...

Link to original articleWelcome 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: A simple model of math skill, published by Alex Altair on July 21, 2024 on LessWrong. I've noticed that when trying to understand a math paper, there are a few different ways my skill level can be the blocker. Some of these ways line up with some typical levels of organization in math papers: Definitions: a formalization of the kind of objects we're even talking about. Theorems: propositions on what properties are true of these objects. Proofs: demonstrations that the theorems are true of the objects, using known and accepted previous theorems and methods of inference. Understanding a piece of math will require understanding each of these things in order. It can be very useful to identify which of type of thing I'm stuck on, because the different types can require totally different strategies. Beyond reading papers, I'm also trying to produce new and useful mathematics. Each of these three levels has another associated skill of generating them. But it seems to me that the generating skills go in the opposite order. This feels like an elegant mnemonic to me, although of course it's a very simplified model. Treat every statement below as a description of the model, and not a claim about the totality of doing mathematics. Understanding Understanding these more or less has to go in the above order, because proofs are of theorems, and theorems are about defined objects. Let's look at each level. Definitions You might think that definitions are relatively easy to understand. That's usually true in natural languages; you often already have the concept, and you just don't happen to know that there's already a word for that. Math definitions are sometimes immediately understandable. Everyone knows what a natural number is, and even the concept of a prime number isn't very hard to understand. I get the impression that in number theory, the proofs are often the hard part, where you have to come up with some very clever techniques to prove theorems that high schoolers can understand (Fermat's last theorem, the Collatz conjecture, the twin primes conjecture). In contrast, in category theory, the definitions are often hard to understand. (Not because they're complicated per se, but because they're abstract.) Once you understand the definitions, then understanding proofs and theorems can be relatively immediate in category theory. Sometimes the definitions have an immediate intuitive understanding, and the hard part is understanding exactly how the formal definition is a formalization of your intuition. In a calculus class, you'll spend quite a long time understanding the derivative and integral, even though they're just the slope of the tangent and the area under the curve, respectively. You also might think that definitions were mostly in textbooks, laid down by Euclid or Euler or something. At least in the fields that I'm reading papers from, it seems like most papers have definitions (usually multiple). This is probably especially true for papers that are trying to help form a paradigm. In those cases, the essential purpose of the paper is to propose the definitions as the new paradigm, and the theorems are set forth as arguments that those definitions are useful. Theorems Theorems are in some sense the meat of mathematics. They tell you what you can do with the objects you've formalized. If you can't do anything meaty with an object, then you're probably holding the wrong object. Once you understand the objects of discussion, you have to understand what the theorem statement is even saying. I think this tends to be more immediate, especially because often, all the content has been pushed into the definitions, and the theorem will be a simpler linking statement, like "all As are Bs" or "All As can be decomposed into a B and a C". For example, the fundamental theorem of calculus...

Today I want to discuss mathematics! I'm sure some of you love math and see not only its usefulness but also its beauty. I have not counted myself among that number. Ever since bouts with trigonometry and calculus, I have steered as clear as possible from math. But as the executive director of two museums devoted to teaching STEM, I feel hypocritical in not embracing the last part of that acronym. So I calculated that by talking to experts on our podcast, AMSEcast, about this topic, I would find a new appreciation of math and that was indeed the case when I spoke to Raphael Rosen about his book, Math Geek: From Klein Bottles to Chaos Theory, a Guide to the Nerdiest Math Facts, Theorems, and Equations.

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.

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: Explaining “Hell is Game Theory Folk Theorems”, published by electroswing on May 5, 2023 on LessWrong. I, along with many commenters, found the explanation in Hell is Game Theory Folk Theorems somewhat unclear. I am re-explaining some of the ideas from that post here. Thanks to jessicata for writing a post on such an interesting topic. 1-shot prisoner's dilemma. In a 1-shot prisoner's dilemma, defecting is a dominant strategy. Because of this, (defect, defect) is the unique Nash equilibrium of this game. Which kind of sucks, since (cooperate, cooperate) would be better for both players. Nash equilibrium. Nash equilibrium is just a mathematical formalism. Consider a strategy profile, which is a list of which strategy each player chooses. A strategy profile is a Nash equilibrium if no player is strictly better off switching their strategy, assuming everyone else continues to play their strategy listed in the strategy profile. Notably, Nash equilibrium says nothing about: What if two or more people team up and deviate from the Nash equilibrium strategy profile? What if people aren't behaving fully rationally? (see bounded rationality) Nash equilibria may or may not have predictive power. It depends on the game. Much work in game theory involves refining equilibrium concepts to have more predictive power in different situations (e.g. subgame perfect equilibrium to handle credible threats, trembling hand equilibrium to handle human error). n-shot prisoner's dilemma. OK, now what if people agree to repeat a prisoner's dilemma n=10 times? Maybe the repeated rounds can build trust among players, causing cooperation to happen? Unfortunately, the theory says that (defect, defect) is still the unique Nash equilibrium. Why? Because in the 10th game, players don't care about their reputation anymore. They just want to maximize payoff, so they may as well defect. So, it is common knowledge that each player will defect in the 10th game. Now moving to the 9th game, players know their reputation doesn't matter in this game, because everyone is going to defect in the 10th game anyway. So, it is common knowledge that each player will defect in the 9th game. And so on. This thought process is called backwards induction. This shows that the unique Nash equilibrium is still (defect, defect), even if the number of repetitions is large. Why might this lack predictive power? In the real world there might be uncertainty about the number of repetitions. (again) people might not behave fully rationally—backwards induction is kind of complicated! Probably the simplest way to model “uncertainty about the number of repetitions” is by assuming an infinite number of repetitions. infinitely repeated prisoner's dilemma. OK, now assume the prisoner's dilemma will be repeated forever. Turns out, now, there exists a Nash equilibrium which involves cooperation! Here is how it goes. Each player agrees to play cooperate, indefinitely. Whenever any player defects, the other player responds by defecting for the rest of all eternity (“punishment”). technical Q: Hold on a second. I thought Nash equilibrium was “static” in the sense that it just says: given that everybody is playing a Nash equilibrium strategy profile, if a single person deviates (while everyone else keeps playing according to the Nash equilibrium strategy profile), then they will not be better off from deviating. This stuff where players choose to punish other players in response to bad behavior seems like a stronger equilibrium concept not covered by Nash. A: Nope! A (pure) strategy profile is a list of strategies for each player. In a single prisoner's dilemma, this is just a choice of “cooperate” or “defect”. In a repeated prisoner's dilemma, this is much more complicated. A strategy is a complete contingency plan of what the player plans to...

Link to original articleWelcome 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: Explaining “Hell is Game Theory Folk Theorems”, published by electroswing on May 5, 2023 on LessWrong. I, along with many commenters, found the explanation in Hell is Game Theory Folk Theorems somewhat unclear. I am re-explaining some of the ideas from that post here. Thanks to jessicata for writing a post on such an interesting topic. 1-shot prisoner's dilemma. In a 1-shot prisoner's dilemma, defecting is a dominant strategy. Because of this, (defect, defect) is the unique Nash equilibrium of this game. Which kind of sucks, since (cooperate, cooperate) would be better for both players. Nash equilibrium. Nash equilibrium is just a mathematical formalism. Consider a strategy profile, which is a list of which strategy each player chooses. A strategy profile is a Nash equilibrium if no player is strictly better off switching their strategy, assuming everyone else continues to play their strategy listed in the strategy profile. Notably, Nash equilibrium says nothing about: What if two or more people team up and deviate from the Nash equilibrium strategy profile? What if people aren't behaving fully rationally? (see bounded rationality) Nash equilibria may or may not have predictive power. It depends on the game. Much work in game theory involves refining equilibrium concepts to have more predictive power in different situations (e.g. subgame perfect equilibrium to handle credible threats, trembling hand equilibrium to handle human error). n-shot prisoner's dilemma. OK, now what if people agree to repeat a prisoner's dilemma n=10 times? Maybe the repeated rounds can build trust among players, causing cooperation to happen? Unfortunately, the theory says that (defect, defect) is still the unique Nash equilibrium. Why? Because in the 10th game, players don't care about their reputation anymore. They just want to maximize payoff, so they may as well defect. So, it is common knowledge that each player will defect in the 10th game. Now moving to the 9th game, players know their reputation doesn't matter in this game, because everyone is going to defect in the 10th game anyway. So, it is common knowledge that each player will defect in the 9th game. And so on. This thought process is called backwards induction. This shows that the unique Nash equilibrium is still (defect, defect), even if the number of repetitions is large. Why might this lack predictive power? In the real world there might be uncertainty about the number of repetitions. (again) people might not behave fully rationally—backwards induction is kind of complicated! Probably the simplest way to model “uncertainty about the number of repetitions” is by assuming an infinite number of repetitions. infinitely repeated prisoner's dilemma. OK, now assume the prisoner's dilemma will be repeated forever. Turns out, now, there exists a Nash equilibrium which involves cooperation! Here is how it goes. Each player agrees to play cooperate, indefinitely. Whenever any player defects, the other player responds by defecting for the rest of all eternity (“punishment”). technical Q: Hold on a second. I thought Nash equilibrium was “static” in the sense that it just says: given that everybody is playing a Nash equilibrium strategy profile, if a single person deviates (while everyone else keeps playing according to the Nash equilibrium strategy profile), then they will not be better off from deviating. This stuff where players choose to punish other players in response to bad behavior seems like a stronger equilibrium concept not covered by Nash. A: Nope! A (pure) strategy profile is a list of strategies for each player. In a single prisoner's dilemma, this is just a choice of “cooperate” or “defect”. In a repeated prisoner's dilemma, this is much more complicated. A strategy is a complete contingency plan of what the player plans to...

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: Hell is Game Theory Folk Theorems, published by jessicata on May 1, 2023 on LessWrong. [content warning: simulated very hot places; extremely bad Nash equilibria] (based on a Twitter thread) Rowan: "If we succeed in making aligned AGI, we should punish those who committed cosmic crimes that decreased the chance of an positive singularity sufficiently." Neal: "Punishment seems like a bad idea. It's pessimizing another agent's utility function. You could get a pretty bad equilibrium if you're saying agents should be intentionally harming each others' interests, even in restricted cases." Rowan: "In iterated games, it's correct to defect when others defect against you; that's tit-for-tat." Neal: "Tit-for-tat doesn't pessimize, though, it simply withholds altruism sometimes. In a given round, all else being equal, defection is individually rational." Rowan: "Tit-for-tat works even when defection is costly, though." Neal: "Oh my, I'm not sure if you want to go there. It can get real bad. This is where I pull out the game theory folk theorems." Rowan: "What are those?" Neal: "They're theorems about Nash equilibria in iterated games. Suppose players play normal-form game G repeatedly, and are infinitely patient, so they don't care about their positive or negative utilities being moved around in time. Then, a given payoff profile (that is, an assignment of utilities to players) could possibly be the mean utility for each player in the iterated game, if it satisfies two conditions: feasibility, and individual rationality." Rowan: "What do those mean?" Neal: "A payoff profile is feasible if it can be produced by some mixture of payoff profiles of the original game G. This is a very logical requirement. The payoff profile could only be the average of the repeated game if it was some mixture of possible outcomes of the original game. If some player always receives between 0 and 1 utility, for example, they can't have an average utility of 2 across the repeated game." Rowan: "Sure, that's logical." Neal: "The individual rationality condition, on the other hand, states that each player must get at least as much utility in the profile as they could guarantee getting by min-maxing (that is, picking their strategy assuming other players make things as bad as possible for them, even at their own expense), and at least one player must get strictly more utility." Rowan: "How does this apply to an iterated game where defection is costly? Doesn't this prove my point?" Neal: "Well, if defection is costly, it's not clear why you'd worry about anyone defecting in the first place." Rowan: "Perhaps agents can cooperate or defect, and can also punish the other agent, which is costly to themselves, but even worse for the other agent. Maybe this can help agents incentivize cooperation more effectively." Neal: "Not really. In an ordinary prisoner's dilemma, the (C, C) utility profile already dominates both agents' min-max utility, which is the (D, D) payoff. So, game theory folk theorems make mutual cooperation a possible Nash equilibrium." Rowan: "Hmm. It seems like introducing a punishment option makes everyone's min-max utility worse, which makes more bad equilibria possible, without making more good equilibria possible." Neal: "Yes, you're beginning to see my point that punishment is useless. But, things can get even worse and more absurd." Rowan: "How so?" Neal: "Let me show you my latest game theory simulation, which uses state-of-the-art generative AI and reinforcement learning. Don't worry, none of the AIs involved are conscious, at least according to expert consensus." Neal turns on a TV and types some commands into his laptop. The TV shows 100 prisoners in cages, some of whom are screaming in pain. A mirage effect appears across the landscape, as if the area is very hot. Rowan: "Wow, tha...

Link to original articleWelcome 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: Hell is Game Theory Folk Theorems, published by jessicata on May 1, 2023 on LessWrong. [content warning: simulated very hot places; extremely bad Nash equilibria] (based on a Twitter thread) Rowan: "If we succeed in making aligned AGI, we should punish those who committed cosmic crimes that decreased the chance of an positive singularity sufficiently." Neal: "Punishment seems like a bad idea. It's pessimizing another agent's utility function. You could get a pretty bad equilibrium if you're saying agents should be intentionally harming each others' interests, even in restricted cases." Rowan: "In iterated games, it's correct to defect when others defect against you; that's tit-for-tat." Neal: "Tit-for-tat doesn't pessimize, though, it simply withholds altruism sometimes. In a given round, all else being equal, defection is individually rational." Rowan: "Tit-for-tat works even when defection is costly, though." Neal: "Oh my, I'm not sure if you want to go there. It can get real bad. This is where I pull out the game theory folk theorems." Rowan: "What are those?" Neal: "They're theorems about Nash equilibria in iterated games. Suppose players play normal-form game G repeatedly, and are infinitely patient, so they don't care about their positive or negative utilities being moved around in time. Then, a given payoff profile (that is, an assignment of utilities to players) could possibly be the mean utility for each player in the iterated game, if it satisfies two conditions: feasibility, and individual rationality." Rowan: "What do those mean?" Neal: "A payoff profile is feasible if it can be produced by some mixture of payoff profiles of the original game G. This is a very logical requirement. The payoff profile could only be the average of the repeated game if it was some mixture of possible outcomes of the original game. If some player always receives between 0 and 1 utility, for example, they can't have an average utility of 2 across the repeated game." Rowan: "Sure, that's logical." Neal: "The individual rationality condition, on the other hand, states that each player must get at least as much utility in the profile as they could guarantee getting by min-maxing (that is, picking their strategy assuming other players make things as bad as possible for them, even at their own expense), and at least one player must get strictly more utility." Rowan: "How does this apply to an iterated game where defection is costly? Doesn't this prove my point?" Neal: "Well, if defection is costly, it's not clear why you'd worry about anyone defecting in the first place." Rowan: "Perhaps agents can cooperate or defect, and can also punish the other agent, which is costly to themselves, but even worse for the other agent. Maybe this can help agents incentivize cooperation more effectively." Neal: "Not really. In an ordinary prisoner's dilemma, the (C, C) utility profile already dominates both agents' min-max utility, which is the (D, D) payoff. So, game theory folk theorems make mutual cooperation a possible Nash equilibrium." Rowan: "Hmm. It seems like introducing a punishment option makes everyone's min-max utility worse, which makes more bad equilibria possible, without making more good equilibria possible." Neal: "Yes, you're beginning to see my point that punishment is useless. But, things can get even worse and more absurd." Rowan: "How so?" Neal: "Let me show you my latest game theory simulation, which uses state-of-the-art generative AI and reinforcement learning. Don't worry, none of the AIs involved are conscious, at least according to expert consensus." Neal turns on a TV and types some commands into his laptop. The TV shows 100 prisoners in cages, some of whom are screaming in pain. A mirage effect appears across the landscape, as if the area is very hot. Rowan: "Wow, tha...

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: Abstracts should be either Actually Short™, or broken into paragraphs, published by Raemon on March 24, 2023 on LessWrong. It looks to me like academia figured out (correctly) that it's useful for papers to have an abstract that makes it easy to tell-at-a-glance what a paper is about. They also figured out that abstract should be about a paragraph. Then people goodharted on "what paragraph means", trying to cram too much information in one block of text. Papers typically have ginormous abstracts that should actually broken into multiple paragraphs. I think LessWrong posts should probably have more abstracts, but I want them to be nice easy-to-read abstracts, not worst-of-all-worlds-goodharted-paragraph abstracts. Either admit that you've written multiple paragraphs and break it up accordingly, or actually streamline it into one real paragraph. Sorry to pick on the authors of this particular post, but my motivating example today was bumping into the abstract for the Natural Abstractions: Key claims, Theorems, and Critiques. It's a good post, it's opening summary happened to be written in an academic-ish style that exemplified the problem. It opens with: TL;DR: John Wentworth's Natural Abstraction agenda aims to understand and recover “natural” abstractions in realistic environments. This post summarizes and reviews the key claims of said agenda, its relationship to prior work, as well as its results to date. Our hope is to make it easier for newcomers to get up to speed on natural abstractions, as well as to spur a discussion about future research priorities. We start by summarizing basic intuitions behind the agenda, before relating it to prior work from a variety of fields. We then list key claims behind John Wentworth's Natural Abstractions agenda, including the Natural Abstraction Hypothesis and his specific formulation of natural abstractions, which we dub redundant information abstractions. We also construct novel rigorous statements of and mathematical proofs for some of the key results in the redundant information abstraction line of work, and explain how those results fit into the agenda. Finally, we conclude by critiquing the agenda and progress to date. We note serious gaps in the theoretical framework, challenge its relevance to alignment, and critique John's current research methodology. There are 179 words. They blur together, I have a very hard time parsing it. If this were anything other than an abstract I expect you'd naturally write it in about 3 paragraphs: TL;DR: John Wentworth's Natural Abstraction agenda aims to understand and recover “natural” abstractions in realistic environments. This post summarizes and reviews the key claims of said agenda, its relationship to prior work, as well as its results to date. Our hope is to make it easier for newcomers to get up to speed on natural abstractions, as well as to spur a discussion about future research priorities. We start by summarizing basic intuitions behind the agenda, before relating it to prior work from a variety of fields. We then list key claims behind John Wentworth's Natural Abstractions agenda, including the Natural Abstraction Hypothesis and his specific formulation of natural abstractions, which we dub redundant information abstractions. We also construct novel rigorous statements of and mathematical proofs for some of the key results in the redundant information abstraction line of work, and explain how those results fit into the agenda. Finally, we conclude by critiquing the agenda and progress to date. We note serious gaps in the theoretical framework, challenge its relevance to alignment, and critique John's current research methodology. If I try to streamline this without losing info, it's still hard to get it into something less than 3 paragraphs (113 words) We review John Wentwor...

Link to original articleWelcome 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: Abstracts should be either Actually Short™, or broken into paragraphs, published by Raemon on March 24, 2023 on LessWrong.It looks to me like academia figured out (correctly) that it's useful for papers to have an abstract that makes it easy to tell-at-a-glance what a paper is about. They also figured out that abstract should be about a paragraph. Then people goodharted on "what paragraph means", trying to cram too much information in one block of text. Papers typically have ginormous abstracts that should actually broken into multiple paragraphs.I think LessWrong posts should probably have more abstracts, but I want them to be nice easy-to-read abstracts, not worst-of-all-worlds-goodharted-paragraph abstracts. Either admit that you've written multiple paragraphs and break it up accordingly, or actually streamline it into one real paragraph.Sorry to pick on the authors of this particular post, but my motivating example today was bumping into the abstract for the Natural Abstractions: Key claims, Theorems, and Critiques. It's a good post, it's opening summary happened to be written in an academic-ish style that exemplified the problem. It opens with:TL;DR: John Wentworth's Natural Abstraction agenda aims to understand and recover “natural” abstractions in realistic environments. This post summarizes and reviews the key claims of said agenda, its relationship to prior work, as well as its results to date. Our hope is to make it easier for newcomers to get up to speed on natural abstractions, as well as to spur a discussion about future research priorities. We start by summarizing basic intuitions behind the agenda, before relating it to prior work from a variety of fields. We then list key claims behind John Wentworth's Natural Abstractions agenda, including the Natural Abstraction Hypothesis and his specific formulation of natural abstractions, which we dub redundant information abstractions. We also construct novel rigorous statements of and mathematical proofs for some of the key results in the redundant information abstraction line of work, and explain how those results fit into the agenda.Finally, we conclude by critiquing the agenda and progress to date. We note serious gaps in the theoretical framework, challenge its relevance to alignment, and critique John's current research methodology.There are 179 words. They blur together, I have a very hard time parsing it. If this were anything other than an abstract I expect you'd naturally write it in about 3 paragraphs:TL;DR: John Wentworth's Natural Abstraction agenda aims to understand and recover “natural” abstractions in realistic environments. This post summarizes and reviews the key claims of said agenda, its relationship to prior work, as well as its results to date. Our hope is to make it easier for newcomers to get up to speed on natural abstractions, as well as to spur a discussion about future research priorities.We start by summarizing basic intuitions behind the agenda, before relating it to prior work from a variety of fields. We then list key claims behind John Wentworth's Natural Abstractions agenda, including the Natural Abstraction Hypothesis and his specific formulation of natural abstractions, which we dub redundant information abstractions. We also construct novel rigorous statements of and mathematical proofs for some of the key results in the redundant information abstraction line of work, and explain how those results fit into the agenda.Finally, we conclude by critiquing the agenda and progress to date. We note serious gaps in the theoretical framework, challenge its relevance to alignment, and critique John's current research methodology.If I try to streamline this without losing info, it's still hard to get it into something less than 3 paragraphs (113 words)We review John Wentwor...

Link to original articleWelcome 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: Abstracts should be either Actually Short™, or broken into paragraphs, published by Raemon on March 24, 2023 on LessWrong. It looks to me like academia figured out (correctly) that it's useful for papers to have an abstract that makes it easy to tell-at-a-glance what a paper is about. They also figured out that abstract should be about a paragraph. Then people goodharted on "what paragraph means", trying to cram too much information in one block of text. Papers typically have ginormous abstracts that should actually broken into multiple paragraphs. I think LessWrong posts should probably have more abstracts, but I want them to be nice easy-to-read abstracts, not worst-of-all-worlds-goodharted-paragraph abstracts. Either admit that you've written multiple paragraphs and break it up accordingly, or actually streamline it into one real paragraph. Sorry to pick on the authors of this particular post, but my motivating example today was bumping into the abstract for the Natural Abstractions: Key claims, Theorems, and Critiques. It's a good post, it's opening summary happened to be written in an academic-ish style that exemplified the problem. It opens with: TL;DR: John Wentworth's Natural Abstraction agenda aims to understand and recover “natural” abstractions in realistic environments. This post summarizes and reviews the key claims of said agenda, its relationship to prior work, as well as its results to date. Our hope is to make it easier for newcomers to get up to speed on natural abstractions, as well as to spur a discussion about future research priorities. We start by summarizing basic intuitions behind the agenda, before relating it to prior work from a variety of fields. We then list key claims behind John Wentworth's Natural Abstractions agenda, including the Natural Abstraction Hypothesis and his specific formulation of natural abstractions, which we dub redundant information abstractions. We also construct novel rigorous statements of and mathematical proofs for some of the key results in the redundant information abstraction line of work, and explain how those results fit into the agenda. Finally, we conclude by critiquing the agenda and progress to date. We note serious gaps in the theoretical framework, challenge its relevance to alignment, and critique John's current research methodology. There are 179 words. They blur together, I have a very hard time parsing it. If this were anything other than an abstract I expect you'd naturally write it in about 3 paragraphs: TL;DR: John Wentworth's Natural Abstraction agenda aims to understand and recover “natural” abstractions in realistic environments. This post summarizes and reviews the key claims of said agenda, its relationship to prior work, as well as its results to date. Our hope is to make it easier for newcomers to get up to speed on natural abstractions, as well as to spur a discussion about future research priorities. We start by summarizing basic intuitions behind the agenda, before relating it to prior work from a variety of fields. We then list key claims behind John Wentworth's Natural Abstractions agenda, including the Natural Abstraction Hypothesis and his specific formulation of natural abstractions, which we dub redundant information abstractions. We also construct novel rigorous statements of and mathematical proofs for some of the key results in the redundant information abstraction line of work, and explain how those results fit into the agenda. Finally, we conclude by critiquing the agenda and progress to date. We note serious gaps in the theoretical framework, challenge its relevance to alignment, and critique John's current research methodology. If I try to streamline this without losing info, it's still hard to get it into something less than 3 paragraphs (113 words) We review John Wentwor...

Jim Stein, Professor Emeritus of CSULS, returns to complete his (admittedly subjective) list of the ten greatest math theorems of all time, with fascinating insights and anecdotes for each. Last time he did the runners up and numbers 8, 9 and 10. Here he completes numbers 1 through 7, again ranging over geometry, trig, calculus, probability, statistics, primes and more. --- Send in a voice message: https://anchor.fm/the-art-of-mathematics/message

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: There are no coherence theorems, published by Dan H on February 20, 2023 on The AI Alignment Forum. [Written by EJT as part of the CAIS Philosophy Fellowship. Thanks to Dan for help posting to the Alignment Forum] Introduction For about fifteen years, the AI safety community has been discussing coherence arguments. In papers and posts on the subject, it's often written that there exist coherence theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. Despite the prominence of these arguments, authors are often a little hazy about exactly which theorems qualify as coherence theorems. This is no accident. If the authors had tried to be precise, they would have discovered that there are no such theorems. I'm concerned about this. Coherence arguments seem to be a moderately important part of the basic case for existential risk from AI. To spot the error in these arguments, we only have to look up what cited ‘coherence theorems' actually say. And yet the error seems to have gone uncorrected for more than a decade. More detail below. Coherence arguments Some authors frame coherence arguments in terms of ‘dominated strategies'. Others frame them in terms of ‘exploitation', ‘money-pumping', ‘Dutch Books', ‘shooting oneself in the foot', ‘Pareto-suboptimal behavior', and ‘losing things that one values' (see the Appendix for examples). In the context of coherence arguments, each of these terms means roughly the same thing: a strategy A is dominated by a strategy B if and only if A is worse than B in some respect that the agent cares about and A is not better than B in any respect that the agent cares about. If the agent chooses A over B, they have behaved Pareto-suboptimally, shot themselves in the foot, and lost something that they value. If the agent's loss is someone else's gain, then the agent has been exploited, money-pumped, or Dutch-booked. Since all these phrases point to the same sort of phenomenon, I'll save words by talking mainly in terms of ‘dominated strategies'. With that background, here's a quick rendition of coherence arguments: There exist coherence theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. Sufficiently-advanced artificial agents will not pursue dominated strategies. So, sufficiently-advanced artificial agents will be ‘coherent': they will be representable as maximizing expected utility. Typically, authors go on to suggest that these expected-utility-maximizing agents are likely to behave in certain, potentially-dangerous ways. For example, such agents are likely to appear ‘goal-directed' in some intuitive sense. They are likely to have certain instrumental goals, like acquiring power and resources. And they are likely to fight back against attempts to shut them down or modify their goals. There are many ways to challenge the argument stated above, and many of those challenges have been made. There are also many ways to respond to those challenges, and many of those responses have been made too. The challenge that seems to remain yet unmade is that Premise 1 is false: there are no coherence theorems. Cited ‘coherence theorems' and what they actually say Here's a list of theorems that have been called ‘coherence theorems'. None of these theorems state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue dominated strategies. Here's what the theorems say: The Von Neumann-Morgenstern Expected Utility Theorem: The Von Neumann-Morgenstern Expected Utility Theorem is as follows: An agent can be represented as maximizing expected utility if...

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: There are no coherence theorems, published by Dan H on February 20, 2023 on LessWrong. [Written by EJT as part of the CAIS Philosophy Fellowship. Thanks to Dan for help posting to the Alignment Forum] Introduction For about fifteen years, the AI safety community has been discussing coherence arguments. In papers and posts on the subject, it's often written that there exist 'coherence theorems' which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. Despite the prominence of these arguments, authors are often a little hazy about exactly which theorems qualify as coherence theorems. This is no accident. If the authors had tried to be precise, they would have discovered that there are no such theorems. I'm concerned about this. Coherence arguments seem to be a moderately important part of the basic case for existential risk from AI. To spot the error in these arguments, we only have to look up what cited ‘coherence theorems' actually say. And yet the error seems to have gone uncorrected for more than a decade. More detail below. Coherence arguments Some authors frame coherence arguments in terms of ‘dominated strategies'. Others frame them in terms of ‘exploitation', ‘money-pumping', ‘Dutch Books', ‘shooting oneself in the foot', ‘Pareto-suboptimal behavior', and ‘losing things that one values' (see the Appendix for examples). In the context of coherence arguments, each of these terms means roughly the same thing: a strategy A is dominated by a strategy B if and only if A is worse than B in some respect that the agent cares about and A is not better than B in any respect that the agent cares about. If the agent chooses A over B, they have behaved Pareto-suboptimally, shot themselves in the foot, and lost something that they value. If the agent's loss is someone else's gain, then the agent has been exploited, money-pumped, or Dutch-booked. Since all these phrases point to the same sort of phenomenon, I'll save words by talking mainly in terms of ‘dominated strategies'. With that background, here's a quick rendition of coherence arguments: There exist coherence theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. Sufficiently-advanced artificial agents will not pursue dominated strategies. So, sufficiently-advanced artificial agents will be ‘coherent': they will be representable as maximizing expected utility. Typically, authors go on to suggest that these expected-utility-maximizing agents are likely to behave in certain, potentially-dangerous ways. For example, such agents are likely to appear ‘goal-directed' in some intuitive sense. They are likely to have certain instrumental goals, like acquiring power and resources. And they are likely to fight back against attempts to shut them down or modify their goals. There are many ways to challenge the argument stated above, and many of those challenges have been made. There are also many ways to respond to those challenges, and many of those responses have been made too. The challenge that seems to remain yet unmade is that Premise 1 is false: there are no coherence theorems. Cited ‘coherence theorems' and what they actually say Here's a list of theorems that have been called ‘coherence theorems'. None of these theorems state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue dominated strategies. Here's what the theorems say: The Von Neumann-Morgenstern Expected Utility Theorem: The Von Neumann-Morgenstern Expected Utility Theorem is as follows: An agent can be represented as maximizing expected utility if and only i...

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: There are no coherence theorems, published by EJT on February 20, 2023 on The Effective Altruism Forum. Introduction For about fifteen years, the AI safety community has been discussing coherence arguments. In papers and posts on the subject, it's often written that there exist 'coherence theorems' which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. Despite the prominence of these arguments, authors are often a little hazy about exactly which theorems qualify as coherence theorems. This is no accident. If the authors had tried to be precise, they would have discovered that there are no such theorems. I'm concerned about this. Coherence arguments seem to be a moderately important part of the basic case for existential risk from AI. To spot the error in these arguments, we only have to look up what cited ‘coherence theorems' actually say. And yet the error seems to have gone uncorrected for more than a decade. More detail below. Coherence arguments Some authors frame coherence arguments in terms of ‘dominated strategies'. Others frame them in terms of ‘exploitation', ‘money-pumping', ‘Dutch Books', ‘shooting oneself in the foot', ‘Pareto-suboptimal behavior', and ‘losing things that one values' (see the Appendix for examples). In the context of coherence arguments, each of these terms means roughly the same thing: a strategy A is dominated by a strategy B if and only if A is worse than B in some respect that the agent cares about and A is not better than B in any respect that the agent cares about. If the agent chooses A over B, they have behaved Pareto-suboptimally, shot themselves in the foot, and lost something that they value. If the agent's loss is someone else's gain, then the agent has been exploited, money-pumped, or Dutch-booked. Since all these phrases point to the same sort of phenomenon, I'll save words by talking mainly in terms of ‘dominated strategies'. With that background, here's a quick rendition of coherence arguments: There exist coherence theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. Sufficiently-advanced artificial agents will not pursue dominated strategies. So, sufficiently-advanced artificial agents will be ‘coherent': they will be representable as maximizing expected utility. Typically, authors go on to suggest that these expected-utility-maximizing agents are likely to behave in certain, potentially-dangerous ways. For example, such agents are likely to appear ‘goal-directed' in some intuitive sense. They are likely to have certain instrumental goals, like acquiring power and resources. And they are likely to fight back against attempts to shut them down or modify their goals. There are many ways to challenge the argument stated above, and many of those challenges have been made. There are also many ways to respond to those challenges, and many of those responses have been made too. The challenge that seems to remain yet unmade is that Premise 1 is false: there are no coherence theorems. Cited ‘coherence theorems' and what they actually say Here's a list of theorems that have been called ‘coherence theorems'. None of these theorems state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue dominated strategies. Here's what the theorems say: The Von Neumann-Morgenstern Expected Utility Theorem: The Von Neumann-Morgenstern Expected Utility Theorem is as follows: An agent can be represented as maximizing expected utility if and only if their preferences satisfy the following four axioms: Completeness: For all lotteries X and Y, X...

Jim Stein, Professor Emeritus of CSULB, presents his very subjective list of what he believes are the ten most important theorems, with several runners up. These theorems cover a broad range of mathematics--geometry, calculus, foundations, combinatorics and more. Each is accompanied by background on the problems they solve, the mathematicians who discovered them, and a couple personal stories. We cover all the runners up and numbers 10, 9 and 8. Next month we'll learn about numbers 1 through 7.  --- Send in a voice message: https://anchor.fm/the-art-of-mathematics/message

Marlawn's Podcast NetworkMarlawn Heavenly VII 307 Cañon Ave #123Manitou Springs, CO 80829 ------------------Cash App: $Marlawn7PayPal: SportyNerd@ymail.com Venmo: Marlawn7 www.Marlawn.comMy Store/Closet: https://posh.mk/yDWgHKjARub Marlawn's Podcast Network (On All Platforms)They Shooting! How Not to Get Murdered https://podcasts.apple.com/us/podcast/they-shooting-how-not-to-get-murdered/id1631954669Mirror of Questionshttps://podcasts.apple.com/us/podcast/mirror-of-questions/id1614039454The Cult of The Individual https://podcasts.apple.com/us/podcast/the-cult-of-the-individual/id1614040017Colorado Rocky Motives https://podcasts.apple.com/us/podcast/colorado-rocky-motives/id1614310570Elon Musk Fail… https://podcasts.apple.com/us/podcast/elon-musk-fail/id1614038329The Black Briefcase https://podcasts.apple.com/us/podcast/the-black-briefcase/id1614040001Hello I Am… Marlawn A Genius! https://podcasts.apple.com/us/podcast/hello-i-am-marlawn-a-genius/id1621007594Billions Black Breakdown https://podcasts.apple.com/us/podcast/billions-black-breakdown/id1609891799Braincell Mateshttps://podcasts.apple.com/us/podcast/braincell-mates/id1609838577Sherlock Homeboy https://podcasts.apple.com/us/podcast/sherlock-homeboy/id1614039938Marlawn's Brief History in Rhyme https://podcasts.apple.com/us/podcast/marlawns-brief-history-of-rhyme/id1614039274Emoji-Less Words https://podcasts.apple.com/us/podcast/emoji-less-words/id1614038618I Hate I love Boxing https://podcasts.apple.com/us/podcast/i-hate-i-love-boxing/id1615090078Internal Negro Affairs https://podcasts.apple.com/us/podcast/internal-negro-affairs/id1614039121Culture Blueshttps://podcasts.apple.com/us/podcast/culture-blues/id1614030193The United State of Vanilla Sky https://podcasts.apple.com/us/podcast/the-united-state-of-vanilla-sky/id1634695175

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: Prize and fast track to alignment research at ALTER, published by Vanessa on September 18, 2022 on The Effective Altruism Forum. Cross-posted from the AI Alignment Forum. On behalf of ALTER and Superlinear, I am pleased to announce a prize of at least 50,000 USD, to be awarded for the best substantial contribution to the learning-theoretic AI alignment research agenda among those submitted before October 1, 2023. Depending on the quality of submissions, the winner(s) may be offered a position as a researcher in ALTER (similar to this one), to continue work on the agenda, if they so desire. Submit here. Topics The research topics eligible for the prize are: Studying the mathematical properties of the algorithmic information-theoretic definition of intelligence. Building and analyzing formal models of value learning based on the above. Pursuing any of the future research directions listed in the article on infra-Bayesian physicalism. Studying infra-Bayesian logic in general, and its applications to infra-Bayesian reinforcement learning in particular. Theoretical study of the behavior of RL agents in population games. In particular, understand to what extent infra-Bayesianism helps to avoid the grain-of-truth problem. Studying the conjectures relating superrationality to thermodynamic Nash equilibria. Studying the theoretical properties of the infra-Bayesian Turing reinforcement learning setting. Developing a theory of reinforcement learning with traps, i.e. irreversible state transitions. Possible research directions include studying the computational complexity of Bayes-optimality for finite state policies (in order to avoid the NP-hardness for arbitrary policies) and bootstrapping from a safe baseline policy. New topics might be added to this list over the year. Requirements The format of the submission can be either a LessWrong post/sequence or an arXiv paper. The submission is allowed to have one or more authors. In the latter case, the authors will be considered for the prize as a team, and if they win, the prize money will be split between them either equally or according to their own internal agreement. For the submission to be eligible, its authors must not include: Anyone employed or supported by ALTER. Members of the board of directors of ALTER. Members of the panel of the judges. First-degree relatives or romantic partners of judges. In order to win, the submission must be a substantial contribution to the mathematical theory of one of the topics above. For this, it must include at least one of: A novel theorem, relevant to the topic, which is difficult to prove. A novel unexpected mathematical definition, relevant to the topic, with an array of natural properties. Some examples of known results which would be considered substantial at the time: Theorems 1 and 2 in "RL with imperceptible rewards". Definition 1.1 in "infra-Bayesian physicalism", with the various theorems proved about it. Theorem 1 in "Forecasting using incomplete models". Definition 7 in "Basic Inframeasure Theory", with the various theorems proved about it. Evaluation The evaluation will consist of two phases. In the first phase, I will select 3 finalists. In the second phase, each of the finalists will be evaluated by a panel of judges comprising of: Adam Shimi Alexander Appel Daniel Filan Vanessa Kosoy (me) Each judge will score the submission on a scale of 0 to 4. These scores will be added to produce a total score between 0 and 16. If no submission achieves a score of 12 or more, the main prize will not be awarded. If at least one submission achieves a score of 12 or more, the submission with the highest score will be the winner. In case of a tie, the money will be split between the front runners. The final winner will be announced publicly, but the scores received by various submissions...

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: Prize and fast track to alignment research at ALTER, published by Vanessa Kosoy on September 17, 2022 on The AI Alignment Forum. On behalf of ALTER and Superlinear, I am pleased to announce a prize of at least 50,000 USD, to be awarded for the best substantial contribution to the learning-theoretic AI alignment research agenda among those submitted before October 1, 2023. Depending on the quality of submissions, the winner(s) may be offered a position as a researcher in ALTER (similar to this one), to continue work on the agenda, if they so desire. Submit here. Topics The research topics eligible for the prize are: Studying the mathematical properties of the algorithmic information-theoretic definition of intelligence. Building and analyzing formal models of value learning based on the above. Pursuing any of the future research directions listed in the article on infra-Bayesian physicalism. Studying infra-Bayesian logic in general, and its applications to infra-Bayesian reinforcement learning in particular. Theoretical study of the behavior of RL agents in population games. In particular, understand to what extent infra-Bayesianism helps to avoid the grain-of-truth problem. Studying the conjectures relating superrationality to thermodynamic Nash equilibria. Studying the theoretical properties of the infra-Bayesian Turing reinforcement learning setting. Developing a theory of reinforcement learning with traps, i.e. irreversible state transitions. Possible research directions include studying the computational complexity of Bayes-optimality for finite state policies (in order to avoid the NP-hardness for arbitrary policies) and bootstrapping from a safe baseline policy. New topics might be added to this list over the year. Requirements The format of the submission can be either a LessWrong post/sequence or an arXiv paper. The submission is allowed to have one or more authors. In the latter case, the authors will be considered for the prize as a team, and if they win, the prize money will be split between them either equally or according to their own internal agreement. For the submission to be eligible, its authors must not include: Anyone employed or supported by ALTER. Members of the board of directors of ALTER. Members of the panel of the judges. First-degree relatives or romantic partners of judges. In order to win, the submission must be a substantial contribution to the mathematical theory of one of the topics above. For this, it must include at least one of: A novel theorem, relevant to the topic, which is difficult to prove. A novel unexpected mathematical definition, relevant to the topic, with an array of natural properties. Some examples of known results which would be considered substantial at the time: Theorems 1 and 2 in "RL with imperceptible rewards". Definition 1.1 in "infra-Bayesian physicalism", with the various theorems proved about it. Theorem 1 in "Forecasting using incomplete models". Definition 7 in "Basic Inframeasure Theory", with the various theorems proved about it. Evaluation The evaluation will consist of two phases. In the first phase, I will select 3 finalists. In the second phase, each of the finalists will be evaluated by a panel of judges comprising of: Adam Shimi Alexander Appel Daniel Filan Vanessa Kosoy (me) Each judge will score the submission on a scale of 0 to 4. These scores will be added to produce a total score between 0 and 16. If no submission achieves a score of 12 or more, the main prize will not be awarded. If at least one submission achieves a score of 12 or more, the submission with the highest score will be the winner. In case of a tie, the money will be split between the front runners. The final winner will be announced publicly, but the scores received by various submissions will not. Fast Track If the prize is awar...

August 17, 2022Torah Smash! The Podcast for Nerdy JewsEpisode 9 - Finding the Value of WhyThere are distinct similarities between the subject of Mathematics and the study of Judaism. We won't ask you to come up to the chalkboard to prove it, but make sure you have your pocket protectors and TI-85 ready for today's class…I mean episode. 00:00:38 - Barak's start with Mathematics00:04:52 - Judaism can be found in everything00:06:08 - The Torah doesn't change, I do00:08:58 - Letters, Numbers, Theorems, and Talmud00:18:19 - Favorite Midrash Proofs00:28:40 - Recognizing that it's all around us00:38:50 - Favorite math subjectsThis episode is supported in part by Sinai and Synapses, which offers people a worldview that is both scientifically grounded and spiritually uplifting. They provide tools and language for learning and living to those who see science as their ally as they pursue personal growth and the repair of our world. Sinai and Synapses helps to equip scientists, clergy and dedicated lay people with knowledge and skills to become role models, ambassadors and activists for grappling with the biggest and most important questions we face. They believe that in order to enhance ourselves and our world, we need both religion and science as sources of wisdom, as the spark for new questions, and as inspiration and motivation.Through classes, seminars, lectures, videos and writings, they help create a vision of religion that embraces critical thinking and scientific inquiry, and at the same time, gives meaning to people's lives and helps them make a positive impact on society.Sinai and Synapses is incubated at Clal – The National Jewish Center for Learning and Leadership, which links Jewish wisdom with innovative scholarship to deepen civic and spiritual participation in American life. You can learn more at www.sinaiandsynapses.org or @SinaiSynapses on twitter.Share this episode with a friend: https://www.torahsmash.com/post/episode-9-finding-the-value-of-whyConnect with us online, email us directly, and more at www.torahsmash.com

The “singularity theorems” of the 1960s demonstrated that large enough celestial bodies, or collections of such bodies, would, collapse gravitationally, to “singularities”, where the equations and assumptions of Einstein's general relativity cannot be mathematically continued. Such singularities are expected to lie deep within what we now call black holes. Similar arguments (largely by Stephen Hawking) apply also to the “Big-Bang” picture of the origin of the universe, but whose singularity has a profound structural difference, resulting in the 2nd law of thermodynamics, whereby “randomness” in the universe increases with time. It is hard to see how any ordinary procedures of “quantization” of Einstein's theory can resolve this contrasting singularity conundrum,Yet, a deeper understanding of the special nature of the Big Bang is obtained from the perspective of conformal geometry, removing the distinction between “big” and “small, and whereby the Big-Bang singularity, unlike those in black holes, becomes non-singular, and can be regarded as the conformal continuation of a previous “cosmic aeon”, leading to the picture of conformal cyclic cosmology (CCC) according to which the entire universe consists of a succession of such cosmic aeons, each of whose big bang is the conformal continuation of the remote future of a previous aeon. Some recently observed effects provide some remarkable support for this CCC picture.A lecture by Sir Roger Penrose.The transcript and downloadable versions of the lecture are available from the Gresham College website:https://www.gresham.ac.uk/watch-now/thomas-gresham-22Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 2,000 lectures free to access or download from the website.Website: http://www.gresham.ac.ukTwitter: http://twitter.com/GreshamCollegeFacebook: https://www.facebook.com/greshamcollegeInstagram: http://www.instagram.com/greshamcollege

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: How Do Selection Theorems Relate To Interpretability?, published by johnswentworth on June 9, 2022 on The AI Alignment Forum. One pretty major problem with today's interpretability methods (e.g. work by Chris Olah & co) is that we have to redo a bunch of work whenever a new net comes out, and even more work when a new architecture or data modality comes along (like e.g. transformers and language models). This breaks one of the core load-pillars of scientific progress: scientific knowledge is cumulative (approximately, in the long run). Newton saw far by standing on the shoulders of giants, etc. We work out the laws of mechanics/geometry/electricity/chemistry/light/gravity/computation/etc, and then reuse those laws over and over again in millions of different contexts without having to reinvent the core principles every time. Economically, that's why it makes sense to pump lots of resources into scientific research: it's a capital investment. The knowledge gained will be used over and over again far into the future. But for today's interpretability methods, that model fails. Some principles and tools do stick around, but most of the knowledge gained is reset when the next hot thing comes out. Economically, that means there isn't much direct profit incentive to invest in interpretability; the upside of the research is mostly the hope that people will figure out a better approach. One way to frame the Selection Theorems agenda is as a strategy to make interpretability transferable and reusable, even across major shifts in architecture. Basic idea: a Selection Theorem tells us what structure is selected for in some broad class of environments, so operating such theorems “in reverse” should tell us what environmental features produce observed structure in a trained net. Then, we run the theorems “forward” to look for the analogous structure produced by the same environmental features in a different net. How would Selection Theorems allow reuse? An example story for how I expect this sort of thing might work, once all the supporting pieces are in place: While probing a neural net, we find a circuit implementing a certain filter. We've (in this hypothetical) previously shown that a broad class of neural nets learn natural abstractions, so we work backward through those theorems and find that the filter indeed corresponds to a natural abstraction in the dataset/environment. We can then work forward through the theorems to identify circuits corresponding to the same natural abstraction in other nets (potentially with different architectures) trained on similar data. In this example, the hypothetical theorem showing that a broad class of neural nets learn natural abstractions would be a Selection Theorem: it tells us about what sort of structure is selected for in general when training systems in some class of environments. More generally, the basic use-case of Selection-Theorem-based interpretability would be: Identify some useful interpretable structure in a trained neural net (similar to today's work). Work backward through some Selection Theorems to find out what features of the environment/architecture selected for that structure. Work forward through the theorems to identify corresponding internal structures in other nets. Assuming that the interpretable structure shows up robustly, we should expect a fairly broad class of environments/architectures will produce similar structure, so we should be able to transfer structure to a fairly broad class of nets - even other architectures which embed the structure differently. (And if the interpretable structure doesn't show up robustly, i.e. the structure is just an accident of this particular net, then it probably wasn't going to be useful for future nets anyway.) Other Connections This picture connects closely to the converge...

How well do old poker theorems still apply in today's games? Should you continue using Zeebo Theorem or Baluga Theorem in 2022? In this episode, coach “w34z3l” takes an in-depth look at 5 poker theories popularized in the 2p2 heyday, breaks them all down, and lets you know if you should stash or trash them. Enjoy! Upgrade To PRO Today: https://redchippoker.com/membership C-Betting When You Miss The Flop: https://redchippoker.com/continuation-betting-bluffs-on-flops/ Understanding Poker Ranges: https://www.splitsuit.com/poker-ranges-reading Be sure to join our free Discord as well to post hands, ask questions, and join a community of players who love poker as much as you do: https://redchippoker.com/discord

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: The No Free Lunch theorems and their Razor, published by Adrià Garriga-alonso on May 24, 2022 on The AI Alignment Forum. The No Free Lunch (NFL) family of theorems contains some of the most misunderstood theorems of machine learning. They apply to learning[1] and optimization[2] and, in rough terms, they state: All algorithms for learning [respectively, optimization] do equally well at generalization performance [cost of the found solution] when averaged over all possible problems. This has some counterintuitive consequences. For example, consider a learning algorithm that chooses a hypothesis with the highest accuracy on a training set. This algorithm generalizes to a test set just as well as the algorithm which chooses the hypothesis with the lowest accuracy on training! Randomly guessing for every new instance is also just as good as these two. The NFL theorems thus seem to show that designing an algorithm that learns from experience is impossible. And yet, there exist processes (e.g. you, the reader of this sentence) which successfully learn from their environment and extrapolate to circumstances they never experienced before. How is this possible? The answer is that problems encountered in reality are not uniformly sampled from the set of all possible problems: the world is highly regular in specific ways. Successful learning and optimization processes (e.g. the scientific method, debate, evolution, gradient descent) exploit these regularities to generalize. If NFL theorems don't usually apply in reality, why should you care about them? My central claim is that they are essential to to think about why and when learning processes work. Notably, when analyzing some process, it is important to state the assumptions under which it can learn or optimize. Sometimes it is possible to test some of these assumptions, but ultimately unjustifiable assumptions will always remain. NFL theorems also allow us to quickly discard explanations for learning as incorrect or incomplete, if they make no reference to the conditions under which they apply. I call this the No Free Lunch Razor for theories of learning. In this post, I will: State a simple NFL theorem in detail and informally prove it. I describe some variants of it and the link between NFL theorems and Hume's problem of induction. Discuss when the NFL theorems are and aren't applicable in practice, and what they tell us about analyzing processes for generating knowledge that extrapolate (or interpolate). Introduce the NFL razor and use it to discard a simple theory of deep learning, as well as re-argue that the value loading problem is hard. Discuss PAC-Bayes generalization bounds as a possible way to pay for lunch in some worlds. NFL theorem statement and proof We will focus on a very simple NFL theorem about learning. The reasoning for other NFL theorems is pretty similar[1:1]. Suppose we have a finite set X of possible training points, as well as an arbitrary distribution D over the input space X. We are concerned with learning a function f:X→{0,1} that classifies elements of X into two classes. We are interested in algorithms A which, given a training set of input-output pairs (x1,f(x1)),(y2,f(y2)),.(yN,f(yN)), output a candidate hypothesis h which classifies elements of X. The training data are sampled independently from the distribution D. There are at least two ways in which we can measure the accuracy of a learning algorithm: The overall error or empirical risk, the proportion of possible inputs for which the learning algorithm makes a mistake. Its expression is R(h)=1/|X|∑x∈X[f(x)≠h(x)], and its value is between 0 and 1. This is the traditional measure in learning theory. The out-of-sample (OOS) generalization error, which is like the error but only for elements of X unseen during training. Its expression is ROO...

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In this episode Cody Roux talks about the Gödel's Incompleteness Theorems. We go through it's underlying historical context, Hilbert's Program, how it relates with Turing, Church, Von Neumann, Termination and more. Links Cody's website Cody's dblp The Lady or the Tiger? - Short Story The Lady or the Tiger? - Amazon Logicomix An Introduction to Gödel's Theorems Jeremy Avigad's Lecture Notes

In this episode Cody Roux talks about the Gödel's Incompleteness Theorems. We go through it's underlying historical context, Hilbert's Program, how it relates with Turing, Church, Von Neumann, Termination and more. Links Cody's website Cody's dblp The Lady or the Tiger? - Short Story The Lady or the Tiger? - Amazon Logicomix An Introduction to Gödel's Theorems Jeremy Avigad's Lecture Notes

In this episode Cody Roux talks about the Gödel's Incompleteness Theorems. We go through it's underlying historical context, Hilbert's Program, how it relates with Turing, Church, Von Neumann, Termination and more. Links Cody's website Cody's dblp The Lady or the Tiger? - Short Story The Lady or the Tiger? - Amazon Logicomix An Introduction to Gödel's Theorems Jeremy Avigad's Lecture Notes

In diesem Podcast beschreibt Romina Achatz unterschiedlichste Körper des Politischen im Werk des italienischen Poeten, Schriftstellers, Malers, Redakteurs, Theater- und Filmregisseurs Pier Paolo Pasolini. Sie beschreibt verschiedene Körper, die Pasolini in das Scheinwerferlicht seiner Werke stellt: in den literarischen sind es die friulanischen BäuernInnen, kommunistischen LandarbeiterInnen, PartisanenInnen, Antonio Gramsci und vor allem die römischen SubproletarierInnen, die der Autor zu Heiligenfiguren erhebt. Sie sind Träger politischer Utopien und Niederlagen. Die erste Schaffenszeit seiner Filme erzählt die Geschichte eines gewissen Körpertypus, nämlich die des „subproletarischen Körpers“ als Passionsfigur, der im Film Il Vangelo secondo Matteoseine Matrix findet. Romina Achatz beschreibt die Gramsci Phase, die 1964 mit einem Film endet, der vom weltweiten Zerfall der Staatsmarxismen und vom Ende des italienischen Kommunismus handelt. Pasolini gibt an, dass er bis zu dem Tod Palmiro Togliattis aus dem Drang einer politischen Bewusstseinsbildung heraus Filme gemacht habe. Danach habe es große gesellschaftlich Veränderungen gegeben-Pasolini beschreibt sie als anthropologische Mutation. Pasolini schreibt zwischen 1973 und 1975 politische Pamphlete für die Mailänder Tageszeitung Corriere della sera. Diese Zeitungsartikel fungieren als Grundlage des Theorems für seinen letzten Film Saló o le centoventi giornate di Sodoma. Er handelt von den Folgen der neuen Religion des Kapitalismus beziehungsweise Konsumismus auf die Menschen: von der Verdinglichung und Homogenisierung der Körper, dem Verlust der Sinne, der Werte, der Dialekte, des Glaubens an Gott, Kirche und den eigenen Körper.

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: Project Intro: Selection Theorems for Modularity, published by TheMcDouglas on April 4, 2022 on LessWrong. Introduction - what is modularity, and why should we care? It's a well-established meme that evolution is a blind idiotic process, that has often resulted in design choices that no sane systems designer would endorse. However, if you are studying simulated evolution, one thing that jumps out at you immediately is that biological systems are highly modular, whereas neural networks produced by genetic algorithms are not. As a result, the outputs of evolution often look more like something that a human might design than do the learned weights of those neural networks. Humans have distinct organs, like hearts and livers, instead of a single heartliver. They have distinct, modular sections of their brains that seem to do different things. They consist of parts, and the elementary neurons, cells and other building blocks that make up each part interact and interconnect more amongst each other than with the other parts. Neural networks evolved with genetic algorithms, in contrast, are pretty much uniformly interconnected messes. A big blob that sort of does everything that needs doing all at once. Again in contrast, networks in the modern deep learning paradigm apparently do exhibit some modular structure. Top: Yeast Transcriptional Regulatory Modules - clearly modular Bottom: Circuit diagram evolved with genetic algorithms - non-modular mess Why should we care about this? Well, one reason is that modularity and interpretability seem like they might be very closely connected. Humans seem to mentally subpartition cognitive tasks into abstractions, which work together to form the whole in what seems like a modular way. Suppose you wanted to figure out how a neural network was learning some particular task, like classifying an image as either a cat or a dog. If you were explaining to a human how to do this, you might speak in terms of discrete high-level concepts, such as face shape, whiskers, or mouth. How and when does that come about, exactly? It clearly doesn't always, since our early networks built by genetic algorithms work just fine, despite being an uninterpretable non-modular mess. And when networks are modular, do the modules correspond to human understandable abstractions and subtasks? Ultimately, if we want to understand and control the goals of an agent, we need to answer questions like “what does it actually mean to explicitly represent a goal”, “what is the type structure of a goal”, or “how are goals connected to world models, and world models to abstractions?” It sure feels like we humans somewhat disentangle our goals from our world models and strategies when we perform abstract reasoning, even as they point to latent variables of these world models. Does that mean that goals inside agents are often submodules? If not, could we understand what properties real evolution, and some current ML training exhibit that select for modularity and use those to make agents evolve their goals as submodules, making them easier to modify and control? These questions are what we want to focus on in this project, started as part of the 2022 AI Safety Camp. The current dominant theory in the biological literature for what the genetic algorithms are missing is called modularly varying goals (MVG). The idea is that modular changes in the environment apply selection pressure for modular systems. For example, features of the environment like temperature, topology etc might not be correlated across different environments (or might change independently within the same environment), and so modular parts such as thermoregulatory systems and motion systems might form in correspondence with these features. This is outlined in a paper by Kashtan and Alon from 2005, where they aim to...

Link to original articleWelcome 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: Project Intro: Selection Theorems for Modularity, published by TheMcDouglas on April 4, 2022 on LessWrong. Introduction - what is modularity, and why should we care? It's a well-established meme that evolution is a blind idiotic process, that has often resulted in design choices that no sane systems designer would endorse. However, if you are studying simulated evolution, one thing that jumps out at you immediately is that biological systems are highly modular, whereas neural networks produced by genetic algorithms are not. As a result, the outputs of evolution often look more like something that a human might design than do the learned weights of those neural networks. Humans have distinct organs, like hearts and livers, instead of a single heartliver. They have distinct, modular sections of their brains that seem to do different things. They consist of parts, and the elementary neurons, cells and other building blocks that make up each part interact and interconnect more amongst each other than with the other parts. Neural networks evolved with genetic algorithms, in contrast, are pretty much uniformly interconnected messes. A big blob that sort of does everything that needs doing all at once. Again in contrast, networks in the modern deep learning paradigm apparently do exhibit some modular structure. Top: Yeast Transcriptional Regulatory Modules - clearly modular Bottom: Circuit diagram evolved with genetic algorithms - non-modular mess Why should we care about this? Well, one reason is that modularity and interpretability seem like they might be very closely connected. Humans seem to mentally subpartition cognitive tasks into abstractions, which work together to form the whole in what seems like a modular way. Suppose you wanted to figure out how a neural network was learning some particular task, like classifying an image as either a cat or a dog. If you were explaining to a human how to do this, you might speak in terms of discrete high-level concepts, such as face shape, whiskers, or mouth. How and when does that come about, exactly? It clearly doesn't always, since our early networks built by genetic algorithms work just fine, despite being an uninterpretable non-modular mess. And when networks are modular, do the modules correspond to human understandable abstractions and subtasks? Ultimately, if we want to understand and control the goals of an agent, we need to answer questions like “what does it actually mean to explicitly represent a goal”, “what is the type structure of a goal”, or “how are goals connected to world models, and world models to abstractions?” It sure feels like we humans somewhat disentangle our goals from our world models and strategies when we perform abstract reasoning, even as they point to latent variables of these world models. Does that mean that goals inside agents are often submodules? If not, could we understand what properties real evolution, and some current ML training exhibit that select for modularity and use those to make agents evolve their goals as submodules, making them easier to modify and control? These questions are what we want to focus on in this project, started as part of the 2022 AI Safety Camp. The current dominant theory in the biological literature for what the genetic algorithms are missing is called modularly varying goals (MVG). The idea is that modular changes in the environment apply selection pressure for modular systems. For example, features of the environment like temperature, topology etc might not be correlated across different environments (or might change independently within the same environment), and so modular parts such as thermoregulatory systems and motion systems might form in correspondence with these features. This is outlined in a paper by Kashtan and Alon from 2005, where they aim to...

Join Eric, @TimAndrewsHere, @Autopritts, @JaredYamamoto, @EnglishNick, and Greg as they chat about media ignorance, slime tutorials, Eric's duck tail and much more! “Brought to you by Findlay Roofing”

In this episode I gather with two good friends Eric and Nitin to randomly talk random subjects that pops up. Among them we talked about POPL, Scala, Isabelle, Parametricity, Dependent Object Types (DOT, for short) and more! Links Nitin Twitter @NitinJohnRaj2 Eric Twitter @EricBond10 Collection of links on logical relations Theorems for Free Reynolds Paper Practical Foundations for Programming Languages

Peter Kuchment; Texas A&M University 23 January 2007 – 16:00 to 17:00

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 Highly Advanced Epistemology 101 for Beginners, Part 13: Godel's Completeness and Incompleteness Theorems, published by Eliezer Yudkowsky. Followup to: Standard and Nonstandard Numbers So... last time you claimed that using first-order axioms to rule out the existence of nonstandard numbers - other chains of numbers besides the 'standard' numbers starting at 0 - was forever and truly impossible, even unto a superintelligence, no matter how clever the first-order logic used, even if you came up with an entirely different way of axiomatizing the numbers. "Right." How could you, in your finiteness, possibly know that? "Have you heard of Godel's Incompleteness Theorem?" Of course! Godel's Theorem says that for every consistent mathematical system, there are statements which are true within that system, which can't be proven within the system itself. Godel came up with a way to encode theorems and proofs as numbers, and wrote a purely numerical formula to detect whether a proof obeyed proper logical syntax. The basic trick was to use prime factorization to encode lists; for example, the ordered list could be uniquely encoded as: 23 37 51 74 And since prime factorizations are unique, and prime powers don't mix, you could inspect this single number, 210,039,480, and get the unique ordered list back out. From there, going to an encoding for logical formulas was easy; for example, you could use the 2 prefix for NOT and the 3 prefix for AND and get, for any formulas Φ and Ψ encoded by the numbers #Φ and #Ψ: ¬Φ = 22 3#Φ Φ ∧ Ψ = 23 3#Φ 5#Ψ It was then possible, by dint of crazy amounts of work, for Godel to come up with a gigantic formula of Peano Arithmetic [](p, c) meaning, 'P encodes a valid logical proof using first-order Peano axioms of C', from which directly followed the formula []c, meaning, 'There exists a number P such that P encodes a proof of C' or just 'C is provable in Peano arithmetic.' Godel then put in some further clever work to invent statements which referred to themselves, by having them contain sub-recipes that would reproduce the entire statement when manipulated by another formula. And then Godel's Statement encodes the statement, 'There does not exist any number P such that P encodes a proof of (this statement) in Peano arithmetic' or in simpler terms 'I am not provable in Peano arithmetic'. If we assume first-order arithmetic is consistent and sound, then no proof of this statement within first-order arithmetic exists, which means the statement is true but can't be proven within the system. That's Godel's Theorem. "Er... no." No? "No. I've heard rumors that Godel's Incompleteness Theorem is horribly misunderstood in your Everett branch. Have you heard of Godel's Completeness Theorem?" Is that a thing? "Yes! Godel's Completeness Theorem says that, for any collection of first-order statements, every semantic implication of those statements is syntactically provable within first-order logic. If something is a genuine implication of a collection of first-order statements - if it actually does follow, in the models pinned down by those statements - then you can prove it, within first-order logic, using only the syntactical rules of proof, from those axioms." I don't see how that could possibly be true at the same time as Godel's Incompleteness Theorem. The Completeness Theorem and Incompleteness Theorem seem to say diametrically opposite things. Godel's Statement is implied by the axioms of first-order arithmetic - that is, we can see it's true using our own mathematical reasoning - "Wrong." What? I mean, I understand we can't prove it within Peano arithmetic, but from outside the system we can see that - All right, explain. "Basically, you just committed the equivalent of saying, 'If all kittens are little, and some little things ar...

Link to original articleWelcome 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 Highly Advanced Epistemology 101 for Beginners, Part 13: Godel's Completeness and Incompleteness Theorems, published by Eliezer Yudkowsky. Followup to: Standard and Nonstandard Numbers So... last time you claimed that using first-order axioms to rule out the existence of nonstandard numbers - other chains of numbers besides the 'standard' numbers starting at 0 - was forever and truly impossible, even unto a superintelligence, no matter how clever the first-order logic used, even if you came up with an entirely different way of axiomatizing the numbers. "Right." How could you, in your finiteness, possibly know that? "Have you heard of Godel's Incompleteness Theorem?" Of course! Godel's Theorem says that for every consistent mathematical system, there are statements which are true within that system, which can't be proven within the system itself. Godel came up with a way to encode theorems and proofs as numbers, and wrote a purely numerical formula to detect whether a proof obeyed proper logical syntax. The basic trick was to use prime factorization to encode lists; for example, the ordered list could be uniquely encoded as: 23 37 51 74 And since prime factorizations are unique, and prime powers don't mix, you could inspect this single number, 210,039,480, and get the unique ordered list back out. From there, going to an encoding for logical formulas was easy; for example, you could use the 2 prefix for NOT and the 3 prefix for AND and get, for any formulas Φ and Ψ encoded by the numbers #Φ and #Ψ: ¬Φ = 22 3#Φ Φ ∧ Ψ = 23 3#Φ 5#Ψ It was then possible, by dint of crazy amounts of work, for Godel to come up with a gigantic formula of Peano Arithmetic [](p, c) meaning, 'P encodes a valid logical proof using first-order Peano axioms of C', from which directly followed the formula []c, meaning, 'There exists a number P such that P encodes a proof of C' or just 'C is provable in Peano arithmetic.' Godel then put in some further clever work to invent statements which referred to themselves, by having them contain sub-recipes that would reproduce the entire statement when manipulated by another formula. And then Godel's Statement encodes the statement, 'There does not exist any number P such that P encodes a proof of (this statement) in Peano arithmetic' or in simpler terms 'I am not provable in Peano arithmetic'. If we assume first-order arithmetic is consistent and sound, then no proof of this statement within first-order arithmetic exists, which means the statement is true but can't be proven within the system. That's Godel's Theorem. "Er... no." No? "No. I've heard rumors that Godel's Incompleteness Theorem is horribly misunderstood in your Everett branch. Have you heard of Godel's Completeness Theorem?" Is that a thing? "Yes! Godel's Completeness Theorem says that, for any collection of first-order statements, every semantic implication of those statements is syntactically provable within first-order logic. If something is a genuine implication of a collection of first-order statements - if it actually does follow, in the models pinned down by those statements - then you can prove it, within first-order logic, using only the syntactical rules of proof, from those axioms." I don't see how that could possibly be true at the same time as Godel's Incompleteness Theorem. The Completeness Theorem and Incompleteness Theorem seem to say diametrically opposite things. Godel's Statement is implied by the axioms of first-order arithmetic - that is, we can see it's true using our own mathematical reasoning - "Wrong." What? I mean, I understand we can't prove it within Peano arithmetic, but from outside the system we can see that - All right, explain. "Basically, you just committed the equivalent of saying, 'If all kittens are little, and some little things ar...

Author of Math Geek: From Klein Bottles to Chaos Theory, a Guide to the Nerdiest Math Facts, Theorems, and Equations, Rosen is a science communications associate at the Princeton Plasma Physics Laboratory and was inspired to write while working at the San Francisco Exploratorium.

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: Selection Theorems: A Program For Understanding Agents, published by johnswentworth on the AI Alignment Forum. What's the type signature of an agent? For instance, what kind-of-thing is a “goal”? What data structures can represent “goals”? Utility functions are a common choice among theorists, but they don't seem quite right. And what are the inputs to “goals”? Even when using utility functions, different models use different inputs - Coherence Theorems imply that utilities take in predefined “bet outcomes”, whereas AI researchers often define utilities over “world states” or “world state trajectories”, and human goals seem to be over latent variables in humans' world models. And that's just goals. What about “world models”? Or “agents” in general? What data structures can represent these things, how do they interface with each other and the world, and how do they embed in their low-level world? These are all questions about the type signatures of agents. One general strategy for answering these sorts of questions is to look for what I'll call Selection Theorems. Roughly speaking, a Selection Theorem tells us something about what agent type signatures will be selected for (by e.g. natural selection or ML training or economic profitability) in some broad class of environments. In inner/outer agency terms, it tells us what kind of inner agents will be selected by outer optimization processes. We already have many Selection Theorems: Coherence and Dutch Book theorems, Good Regulator and Gooder Regulator, the Kelly Criterion, etc. These theorems generally seem to point in a similar direction - suggesting deep unifying principles exist - but they have various holes and don't answer all the questions we want. We need better Selection Theorems if they are to be a foundation for understanding human values, inner agents, value drift, and other core issues of AI alignment. The quest for better Selection Theorems has a lot of “surface area” - lots of different angles for different researchers to make progress, within a unified framework, but without redundancy. It also requires relatively little ramp-up; I don't think someone needs to read the entire giant corpus of work on alignment to contribute useful new Selection Theorems. At the same time, better Selection Theorems directly tackle the core conceptual problems of alignment and agency; I expect sufficiently-good Selection Theorems would get us most of the way to solving the hardest parts of alignment. Overall, I think they're a good angle for people who want to make useful progress on the theory of alignment and agency, and have strong theoretical/conceptual skills. Outline of this post: More detail on what “type signatures” and “Selection Theorems” are Examples of existing Selection Theorems and what they prove (or assume) about agent type signatures Aspects which I expect/want from future Selection Theorems How to work on Selection Theorems What's A Type Signature Of An Agent? We'll view the “type signature of an agent” as an answer to three main questions: Representation: What “data structure” represents the agent - i.e. what are its high-level components, and how can they be represented? Interfaces: What are the “inputs” and “outputs” between the components - i.e. how do they interface with each other and with the environment? Embedding: How does the abstract “data structure” representation relate to the low-level system in which the agent is implemented? A selection theorem typically assumes some parts of the type signature (often implicitly), and derives others. For example, coherence theorems show that any non-dominated strategy is equivalent to maximization of Bayesian expected utility. Representation: utility function and probability distribution. Interfaces: both the utility function and distribution take in “bet ...

The boys discuss some of the most chilling conspiracy theories that they know.

Containing Matters which pertain to Brevity in approaching vars. Aspects of the mathematical Discipline, including novel Solutions to Problems, ethical Concerns and Methods of Discovery. Timestamps: Edward Page Mitchell - "The Tachypomp" (1874) (0:00) Henry A. Hering - "Silas P. Cornu's Dry Calculator" (1898) (17:03) Mary E. Wilkins Freeman - "An Old Arithmetician" (1885) (35:25) Nikolai Georgievich Mikhailovsky - "The Genius" (1901) (1:03:10) Elizabeth Wormeley Latimer - "The Sirdar's Chess-Board" (1885) (1:18:06) Bibliography: Carper, Steve. Flying Cars and Food Pills. commentary on "Silas Cornu's Dry Calculator". https://www.flyingcarsandfoodpills.com/silas-p-cornu-s-dry-calculator Fundamental Electronic Library. "Garin, N." http://www.feb-web.ru/feb/kle/kle-abc/ke2/ke2-0663.htm?cmd=p&istext=1 (in Russian) Garin, Mikhailovsky, N.G. "Collected works in five volumes." (in Russian) Gorky, Maxim. "About Garin Mikhailovsky" https://web.archive.org/web/20060516223250/http://www.maximgorkiy.narod.ru/garin-m.htm (in Russian) Kasman, Alex. "The Sirdar's Chess Board" https://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf630 Kaylin, Jeff. Works of Mary E. Wilkins Freeman. http://wilkinsfreeman.info/

Crypto's home for live, breaking real time cryptocurrency news. Covering Bitcoin, Ethereum, ICO's and Blockchain Technology along with current prices. https://www.globalcryptopress.com/2021/07/axel-brings-industry-leading-data.html AXELAXEL GoTheorem AXELAXEL GoDiscordTwitterTelegramLinkedInTheoremPracticeSupport@axel.orgBreaking Crypto News

On today's pod-meal, Myq discusses Gödel's Incompleteness Theorem (not completely!), wise words from old friend the Dalai Lama, and probably more who knows! (Answer: you if you listen!) Alternate titles: Thing Us a Song You're the Podcast Man I've Got a Singing Feeling

Transcript http://nav.al/prove

Rathinavani 90.8 Community Radio | Special Podcast about World Homeopathy Day | And Siblings Day | Special Talk by Triveni M.sc Psychology (clinical) on Existing Siblings Psychology Theorems & Studies Theme: This year the theme of the day in India is, “Homeopathy- Roadmap for Integrative Medicine”.

This episode is also available as a blog post: https://garycgibson.wordpress.com/2016/11/06/g-r-and-godels-incompleteness-theorems-for-others/ --- Support this podcast: https://anchor.fm/garrison-clifford-gibson/support

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

Gresham Professor of Geometry, Raymond Flood, begins his series 'Great Mathematicians, Great Mathematics' with Pierre de Fermat:http://www.gresham.ac.uk/lectures-and-events/fermats-theoremsThe seventeenth century mathematician Pierre de Fermat is mainly remembered for contributions to number theory even though he often stated his results without proof and published very little. He is particularly remembered for his 'last theorem' which was only proved in the mid-1990s by Andrew Wiles. He also stated other influential results, in particular Fermat's 'Little Theorem' about certain large numbers which can be divided by primes. His 'Little Theorem' is the basis of important recent work in cryptography and internet security.The transcript and downloadable versions of the lecture are available from the Gresham College Website: http://www.gresham.ac.uk/lectures-and-events/fermats-theoremsGresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,500 lectures free to access or download from the website.Website: http://www.gresham.ac.ukTwitter: http://twitter.com/GreshamCollegeFacebook: https://www.facebook.com/greshamcollege

Melvyn Bragg and guests discuss an iconic piece of 20th century maths - Gödel's Incompleteness Theorems. In 1900, in Paris, the International Congress of Mathematicians gathered in a mood of hope and fear. The edifice of maths was grand and ornate but its foundations, called axioms, had been shaken. They were deemed to be inconsistent and possibly paradoxical. At the conference, a young man called David Hilbert set out a plan to rebuild the foundations of maths – to make them consistent, all encompassing and without any hint of a paradox. Hilbert was one of the greatest mathematicians that ever lived, but his plan failed spectacularly because of Kurt Gödel. Gödel proved that there were some problems in maths that were impossible to solve, that the bright clear plain of mathematics was in fact a labyrinth filled with potential paradox. In doing so Gödel changed the way we understand what mathematics is and the implications of his work in physics and philosophy take us to the very edge of what we can know.With Marcus du Sautoy, Professor of Mathematics at Wadham College, University of Oxford; John Barrow, Professor of Mathematical Sciences at the University of Cambridge and Gresham Professor of Geometry and Philip Welch, Professor of Mathematical Logic at the University of Bristol.