Great conversation!

🎯 Key Takeaways for quick navigation: 00:00 🤖 The Nature of Existence in Mathematics - Mathematics deals with abstract objects. - Abstract existence in mathematics is distinct from physical existence. - The challenge of causal interaction with abstract mathematical concepts. 04:52 🤯 Causality and Abstract Existence - The problem of causal interaction with the abstract realm. - The debate over whether causality flows between the physical and abstract realms. - Barbara Montero's perspective on the issue. 07:23 🌌 Abstract vs. Physical Existence - The concept that physical existence is more mysterious and less clear than abstract existence. - The idea that physical existence, when examined deeply, becomes more incoherent and mysterious. - Satisfactory accounts of abstract existence compared to the nature of physical existence. 09:18 🤔 Intuition and Objective Truth - The role of intuition in determining the truth value of a statement. - The challenge of mistaking intuitive truth for objective truth. - The connection between abstract and intuitive truth. 12:16 📚 Understanding the Notion of Proof - Different views on what constitutes a mathematical proof. - The distinction between formal proofs and informal arguments. - The use of formal proofs to understand the nature of proof itself. 19:10 🧩 Turing Machines and Understanding Computation - Turing machines as a theoretical model for understanding computation. - The utility of studying Turing machines for understanding the nature of computation. - The parallel between Turing machines and formal proofs in mathematics. 22:20 📜 From Turing Machines to Completeness Theorem - Gödel's Completeness Theorem and its connection to formal theories. - The concept of logical consequences and the role of proof. - The profound implications of the Completeness Theorem for verifiability in mathematics. 24:25 🧠 Philosophy of mathematics and the Completeness Theorem - Joel discusses the Completeness Theorem's profound implications for mathematics. - The Completeness Theorem states that if a statement is true in all mathematical structures where a particular theory holds, there is a proof for it. - Completeness Theorem connects formal proofs to semantic models. 29:24 🤔 Perceptions of Incompleteness - The discussion shifts to the Incompleteness Theorem and how it is perceived. - Some may view incompleteness as a problem, but Joel sees it as a fundamental feature of mathematical reality. - Incompleteness leads to the creation of a hierarchy of theories, expanding mathematical exploration. 36:26 🏗️ Building Hierarchy of Consistency Theories - Joel explains the hierarchy of consistency theories and its significance. - This hierarchy reveals that even foundational theories like Peano arithmetic can't prove their own consistency. - Mathematics has become highly specialized and too vast for any single person to master it entirely. 47:54 🧮 The Axiom of Choice in Mathematics - Axiom of choice, a fundamental concept in mathematics. - Discussion about the role of axioms and their intrinsic and extrinsic justifications. 50:09 🤔 Intrinsic and Extrinsic Justifications for Axioms - Exploring the distinction between intrinsic and extrinsic justification of axioms. - Intrinsic justification based on fundamental intuition, extrinsic justification based on consequences. 58:00 🤷‍♂️ Choosing Areas of Research in Pure Mathematics - The consideration of personal interest and curiosity as the primary criteria for mathematical research. - The discussion of mathematicians' choices of topics in pure mathematics and their intrinsic appeal. 01:08:07 🌟 Pursuing Unconventional Mathematical Questions - The importance of following personal interest even in non-mainstream or quirky areas of mathematics. - Examples of pursuing unconventional mathematical questions, leading to valuable discoveries. - Encouragement to ignore advice against pursuing topics of interest. 01:10:10 🌟 The Value of Abstract Mathematical Questions - Abstract mathematical questions may lack immediate practical applications. - Mathematicians often pursue such questions because of their intrinsic value. - The progress in understanding these questions contributes to cultural advancement. 01:15:37 🧠 Motivations and Evolution of Human Minds - Human minds are naturally drawn to abstract questions due to their cognitive constitution. - The pursuit of abstract mathematical questions can be motivated by curiosity and play. - The analogy of children drawn to a candy aisle illustrates the impulsive nature of our interests. 01:19:31 📚 Serialization of "The Book of Infinity" - Joel started a substack and serialized a book titled "The Book of Infinity" for an undergraduate class. - The book covers various mathematical conundrums, paradoxes, puzzles, and historical examples. - Joel also mentions other book projects he's serializing on his substack, such as "Panorama of Logic" and "Math for Seven-Year-Olds." 01:22:04 🌌 AI Superintelligence and Choosing a Representative - Discussing the concept of AI superintelligence and selecting a representative for humanity. - Emphasizing that the smartest person may not necessarily be the best representative. - Reflecting on a humorous story illustrating the importance of asking the right question. Made with HARPA AI

@sheidak.2347

4 күн бұрын

Thank you, this is very helpful!

In about two minutes and 20 seconds, the speaker mentions an author, but I wasn’t able to understand the name. Does the channel know what it is?

@joeldavidhamkins5484

9 ай бұрын

It was Paul Benacerraf and his famous articles on the philosophy of mathematics: "What Numbers Could Not Be" (1965) and "Mathematical Truth" (1973).

@MatthewGeleta

9 ай бұрын

I've added this to the show notes: www.jstor.org/stable/2183530

@jakecarlo9950

9 ай бұрын

@@joeldavidhamkins5484 Thanks very much sir!

wow 😮😮😮

Do numbers live and then die? How many times? Do they mutate over time? Can one conceive of time without number or math at the same time?

@James-ll3jb

8 күн бұрын

Sure. Why not?

If you look at proofs solely as the results that they prove, then the "tower of consistency" does indeed look a lot like hole plugging. However, when we do this, we are really investigating the nature of consequence/derivation itself and not so much the direct results we get from any individual "plug".

@James-ll3jb

8 күн бұрын

Inevitably. The "metaplug"😊

Humans didn't evolve to do mathematics? Disagree quite strongly. We evolved. We do mathematics. For whatever reason, our brain structure allows us to do calculus. So does a dragonfly's. Don't put humanity on a pedestal.

@MatthewGeleta

9 ай бұрын

You are correct and I modify my framing as: our minds are not well optimised for doing maths

@James-ll3jb

8 күн бұрын

Humans didn't evolve to have appendicitis by possessing a useless appendix. But some do get appendicitis so we did evolve to possess a useless appendix. (Duh.)

@James-ll3jb

8 күн бұрын

​@@MatthewGeletaI don't think that is his actual intention Matthew, which is to assert (albeit lamely) that our brains HAVE evolved sufficiently to perform a feat e.g. calculus. (I leave the facticity of assertions like "dragonflys can 'do' calculus" to the comical "Glorys Of Science" propagandists😅