Thank you for the interview! It was a pleasure to discuss these issues in the philosophy of mathematics with you. Here is a link to the forthcoming book I mention, Proof and the Art of Mathematics: www.amazon.com/Proof-Mathematics-Joel-David-Hamkins/dp/0262539799, now available for pre-order (other buying options at mitpress.mit.edu/books/proof-and-art-mathematics). And incidentally, for those who are interested, here is another interview I had a few years ago with Richard Marshall in his series of diverse philosophers and public thinkers: 316am.site123.me/articles/playing-infinite-chess.
Outline of the conversation: 00:00 Podcast Introduction 00:50 MathOverflow and books in progress 04:08 Mathphobia 05:58 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 Mathematics 16:58 Cantor's Theorem: the real numbers cannot be enumerated 24:22 Russell's Paradox and the Cumulative Hierarchy of Sets 29:20 Hilbert's Program and Godel's Results 35:05 The First Incompleteness Theorem, formal and informal proofs and the connection between mathematical truths and mathematical proofs 40:50 Computer Assisted Proofs and mathematical insight 44:11 Do automated proofs kill the artistic side of Mathematics? 48:50 Infinite Time Turing Machines can settle Goldbach's Conjecture or the Riemann Hypothesis 54:28 Nonstandard models of arithmetic: different conceptions of the natural numbers 1:00:02 The Continuum Hypothesis and related undecidable questions, the Set-Theoretic Multiverse and the quest for new axioms 1:10:31 Minds and computers: Sir Roger Penrose's argument concerning consciousness
@56:0 this is a popular _myth._ Any consistent formal system can be conceived of platonically, even if you only believe it to be consistent but not knowingly so. The problem is one of existence of proofs within the system. That is all Gödel incompleteness says. So it is not the case number theory (or universal geometry) are fuzzy areas of mathematics. They are incomplete only relative to finite proof. It's weird he just admitted a minute before this that infinite proof procedures can decide Gödel undecidable statements. He then seems to go back to finitism to worry about whether number theory and other formal systems are platonic (have the same true statements in every possible world) or not. The beauty in mathematics for us finite creatures is that our minds seem to be creative enough to think of extensions to the axioms that make it possible to search for proofs that would not otherwise be possible, still within limits of finite proof procedures. Another beauty is that we can still practice mathematics without formal proof. Most of everyday mathematics is not proving theorems, it is playful exploration. Greg Chaitin is good on this stuff, he has some interesting takes. The idea you might have an interesting statement that has no proof is in modern light a wonderful thing, it makes mathematics non-boring. For one thing, how about proving there is no proof of your statement within accepted axioms? Also tragically ignored is "spiritual" proof --- what "aught to be" true on principle: this is not acceptable academic mathematics, it even sounds heretical to the high priests of modern academia, and maybe it is not classifiable ever as mathematics, but it is a useful pursuit. Many mathematicians arrive at conjectures with "spiritual" arguments first, a lot turn out to be wrong, but they serve to drive real mathematics forward.
@8:00 the _nature of mathematical object existence_ is not a mathematical problem. It is metaphysical. Everyone so far who has attempted an answer is likely delusional. Metaphysics does not admit anything but trivial answers (the set of metaphysical questions that have logical inferences from known physics, which is a tiny vanishingly small subset of metaphysics).
@1:07:00 it makes a lot of sense that the "set of real numbers" is not yet pinned down. Even the naïve view of "numbers with infinite decimal expansions" is enough to appreciate there can be a lot of statements about such sets that are not decideable. Not the case though with the Natural numbers. For the Naturals it is extremely weird we have not found a definitive axiom schema that can decide what the cardinality of the power set is. I am of the opinion we will eventually find a good (simple and "obvious" in retrospect) axiom, but I am not 100% confident. It does make mathematical foundations exceedingly non-boring at least.
@1:10:00 aha! That's where Hamkins goes wrong. "Different mathematical truth" is a meaningless phrase (a mental disease spread by postmodernists). By studying a different "set" (haha) of set theory axioms you arrive at different theorems, but they are all from the same mathematics. The logical foundations are not altered, the axioms are altered, big difference. The analogy with Riemannian geometry is a good one: hyperbolic geometry truths are not "different mathematics" in a metamathematical sense to Euclidean geometry truths.
For those that are interested in discovering more about Philosophy of Mathematics, please look up Prof. Hamkins’s lectures on his TH-cam channel-they are really good! (They are all based on his recently published book called Lectures on the Philosophy of Mathematics [MIT Press, 2021])
@1:08:00 yes! It was great to hear someone other than Greg Chaitin also expressing a view that there might be a kind of generalized universe of distinct set theories like non-Euclidean geometries. Not much that I know so, but I sure hope so!!! It is fairly natural however to suppose Hamkins' thought there is a good one. There is nothing god-given about the human concept of "set" or "collection". Everyone thinks it is a primitive enough notion to pin down, which only makes a fellow like Gödel laugh out loud (I'd have imagined).
Thank you for the interview! It was a pleasure to discuss these issues in the philosophy of mathematics with you. Here is a link to the forthcoming book I mention, Proof and the Art of Mathematics: www.amazon.com/Proof-Mathematics-Joel-David-Hamkins/dp/0262539799, now available for pre-order (other buying options at mitpress.mit.edu/books/proof-and-art-mathematics).
And incidentally, for those who are interested, here is another interview I had a few years ago with Richard Marshall in his series of diverse philosophers and public thinkers: 316am.site123.me/articles/playing-infinite-chess.
This was a very interesting interview! Keep up the good work!
Outline of the conversation:
00:00 Podcast Introduction
00:50 MathOverflow and books in progress
04:08 Mathphobia
05:58 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 Mathematics
16:58 Cantor's Theorem: the real numbers cannot be enumerated
24:22 Russell's Paradox and the Cumulative Hierarchy of Sets
29:20 Hilbert's Program and Godel's Results
35:05 The First Incompleteness Theorem, formal and informal proofs and the connection between mathematical truths and mathematical proofs
40:50 Computer Assisted Proofs and mathematical insight
44:11 Do automated proofs kill the artistic side of Mathematics?
48:50 Infinite Time Turing Machines can settle Goldbach's Conjecture or the Riemann Hypothesis
54:28 Nonstandard models of arithmetic: different conceptions of the natural numbers
1:00:02 The Continuum Hypothesis and related undecidable questions, the Set-Theoretic Multiverse and the quest for new axioms
1:10:31 Minds and computers: Sir Roger Penrose's argument concerning consciousness
Mathematics is discovered, BUT we invent the way to describe the discoveries.
32:10
Amazing work!! Thank you so much. Great questions and great explanations, quite clear for me despite me not being an expert in the field
Very interesting interview! It is quite different from what we learn in school at Math. This Philosophy of Matematics is great!
Magda, Romania
@56:0 this is a popular _myth._ Any consistent formal system can be conceived of platonically, even if you only believe it to be consistent but not knowingly so. The problem is one of existence of proofs within the system. That is all Gödel incompleteness says. So it is not the case number theory (or universal geometry) are fuzzy areas of mathematics. They are incomplete only relative to finite proof. It's weird he just admitted a minute before this that infinite proof procedures can decide Gödel undecidable statements. He then seems to go back to finitism to worry about whether number theory and other formal systems are platonic (have the same true statements in every possible world) or not.
The beauty in mathematics for us finite creatures is that our minds seem to be creative enough to think of extensions to the axioms that make it possible to search for proofs that would not otherwise be possible, still within limits of finite proof procedures.
Another beauty is that we can still practice mathematics without formal proof. Most of everyday mathematics is not proving theorems, it is playful exploration. Greg Chaitin is good on this stuff, he has some interesting takes. The idea you might have an interesting statement that has no proof is in modern light a wonderful thing, it makes mathematics non-boring. For one thing, how about proving there is no proof of your statement within accepted axioms? Also tragically ignored is "spiritual" proof --- what "aught to be" true on principle: this is not acceptable academic mathematics, it even sounds heretical to the high priests of modern academia, and maybe it is not classifiable ever as mathematics, but it is a useful pursuit. Many mathematicians arrive at conjectures with "spiritual" arguments first, a lot turn out to be wrong, but they serve to drive real mathematics forward.
@8:00 the _nature of mathematical object existence_ is not a mathematical problem. It is metaphysical. Everyone so far who has attempted an answer is likely delusional. Metaphysics does not admit anything but trivial answers (the set of metaphysical questions that have logical inferences from known physics, which is a tiny vanishingly small subset of metaphysics).
@1:07:00 it makes a lot of sense that the "set of real numbers" is not yet pinned down. Even the naïve view of "numbers with infinite decimal expansions" is enough to appreciate there can be a lot of statements about such sets that are not decideable. Not the case though with the Natural numbers. For the Naturals it is extremely weird we have not found a definitive axiom schema that can decide what the cardinality of the power set is. I am of the opinion we will eventually find a good (simple and "obvious" in retrospect) axiom, but I am not 100% confident. It does make mathematical foundations exceedingly non-boring at least.
A gem of an interview
@1:10:00 aha! That's where Hamkins goes wrong. "Different mathematical truth" is a meaningless phrase (a mental disease spread by postmodernists). By studying a different "set" (haha) of set theory axioms you arrive at different theorems, but they are all from the same mathematics. The logical foundations are not altered, the axioms are altered, big difference. The analogy with Riemannian geometry is a good one: hyperbolic geometry truths are not "different mathematics" in a metamathematical sense to Euclidean geometry truths.
Ughhhhg What? You high bro?
How do you make these videos? Are they just zoom recordings uploaded to TH-cam?
Most of them, yes-plus some modest editing. Some episodes, e.g. the William Lane Craig one, were recorded via Riverside instead of Zoom.
For those that are interested in discovering more about Philosophy of Mathematics, please look up Prof. Hamkins’s lectures on his TH-cam channel-they are really good! (They are all based on his recently published book called Lectures on the Philosophy of Mathematics [MIT Press, 2021])
@1:08:00 yes! It was great to hear someone other than Greg Chaitin also expressing a view that there might be a kind of generalized universe of distinct set theories like non-Euclidean geometries. Not much that I know so, but I sure hope so!!! It is fairly natural however to suppose Hamkins' thought there is a good one. There is nothing god-given about the human concept of "set" or "collection". Everyone thinks it is a primitive enough notion to pin down, which only makes a fellow like Gödel laugh out loud (I'd have imagined).