Joel David Hamkins: Philosophy of mathematics and truth

Joel is a renowned mathematician and philosopher whose work covers a wide range of important topics, including logic, computability theory, game theory, the philosophy of infinity, and more.

Joel is the author of several books including Lectures on the Philosophy of Mathematics, and the The Book of Infinity, which he’s publishing in a serialised form on his Substack, Infinitely More.

#truth #maths #logic #philosophy #infinity

Topics:
- the concept of truth in maths and elsewhere
- the nature of proof in mathematics
- the acclaimed completeness and incompleteness theorems
- the relationship between mathematical thinking and the human mind
... and other topics

Timestamps:
0:00 Intro
0:17: Truth
7:38 Intuition vs objective truth
12:15 Proof
19:39 Completeness
29:18 Incompleteness
36:20 Is completeness a 'problem'?
42:07 Hierarchies of logical systems
47:44 Axioms and where they come from
1:02:50 Motivations for studying pure mathematics
1:18:57 Joel's books
1:21:58 Who should represent humanity to an AI superintelligence?

Joel' links:

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🎯 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

[johnbova]

I really wanted the discussion at 36:20 to be, "Is *completeness* a problem?" Answer: yes! (Alas, it's just a misprint for, "Is incompleteness a problem?")

[jcsahnwaldt]

Minor correction: 20:24 The "wobbly table theorem" is about a table with four legs (not three), the legs must have equal length, and the surface has to be continuous (but not necessarily flat).

[whitb]

Best interview I’ve seen of Hamkins, who I really like. Great questions.

[jakecarlo]

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?

[oldsachem]

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?

[almightysapling]

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.

[almightysapling]

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".