More episodes of "The Joy of Why" are coming to TH-cam soon. In the meantime, you can subscribe wherever you get your podcasts or explore past episodes on the Quanta website. 🎧 Listen and subscribe: www.quantamagazine.org/joy/ 📑 Explore our archive of transcripts: www.quantamagazine.org/podcasts/
I remember there was one with shapes or knots, where mathematicians limited the possibilities down, and then used the computer to run through a very large number of all the possibilities, and it found the remaining shapes/knots.
1) Calculus Foundations Contradictory: Newtonian Fluxional Calculus dx/dt = lim(Δx/Δt) as Δt->0 This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale. Non-Contradictory: Leibnizian Infinitesimal Calculus dx = ɛ, where ɛ is an infinitesimal dx/dt = ɛ/dt Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities. 2) Foundations of Mathematics Contradictory Paradoxes: - Russell's Paradox, Burali-Forti Paradox - Banach-Tarski "Pea Paradox" - Other Set-Theoretic Pathologies Non-Contradictory Possibilities: Algebraic Homotopy ∞-Toposes a ≃ b ⇐⇒ ∃n, Path[a,b] in ∞Grpd(n) U: ∞Töpoi → ∞Grpds (univalent universes) Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations. 3) Foundational Paradoxes in Arithmetic Contradictory: - Russell's Paradox about sets/classes - Berry's Paradox about definability - Other set-theoretic pathologies These paradoxes revealed fundamental inconsistencies in early naive attempts to formalize arithmetic foundations. Non-Contradictory Possibility: Homotopy Type Theory / Univalent Foundations a ≃ b ⇐⇒ α : a =A b (Equivalence as paths in ∞-groupoids) Arithmetic ≃ ∞-Topos(A) (Numbers as objects in higher toposes) Representing arithmetic objects categorically as identifications in higher homotopy types and toposes avoids the self-referential paradoxes. 4) The Foundations of Arithmetic Contradictory: Peano's Axioms contain implicit circularity, while naive set theory axiomatizations lead to paradoxes like Russell's Paradox about the set of all sets that don't contain themselves. Non-Contradictory Possibility: Homotopy Type Theory / Univalent Foundations N ≃ W∞-Grpd (Natural numbers as objects in ∞-groupoids) S(n) ≃ n = n+1 (Successor is path identification) Let Z ≃ Grpd[N, Π1(S1)] (Integers from N and winding paths) Defining arithmetic objects categorically using homotopy theory and mapping into higher toposes avoids the self-referential paradoxes.
I learnt recently, that to enjoy life, you must stop asking why. Or in other words, stop asking why, and enjoy life. And here Quanta has a podcast called the "Joy of Why"? wewewew.
Sure, there is always a truth to the saying “ignorance is bliss”. But there can be so much joy in the pursuit of why. The issue is that many people become so fixated on the answer that they fail to enjoy the journey. Personally I find great satisfaction in knowing there are always problems waiting to be solved. Isn’t it incredible that even with 8 billion of us on Earth, we don’t know why we dream? We don’t know why we yawn? We don’t know why we exist? It’s amazing to think we might one day unlock the answers to these questions
I was skeptical about mr. terence idea , especially in his words where if someone has this credit , then they can make some "theories" that gauge some sort of belief in it ? I think mathematics is a rigorous field , not the one based on imagination and thought ideas
Interesting and nice. He is bit "young" and a lot rich, but yes, mathematics have to reflect reality, or stay on the ground. And would be mathematics like some wisdom?
More episodes of "The Joy of Why" are coming to TH-cam soon. In the meantime, you can subscribe wherever you get your podcasts or explore past episodes on the Quanta website.
🎧 Listen and subscribe: www.quantamagazine.org/joy/
📑 Explore our archive of transcripts: www.quantamagazine.org/podcasts/
i love listening to him, he’s a true genius
Philosophy & Math... Inexorably linked.
I wonder why this wasn’t recommended sooner! I enjoyed listening
I really enjoy listening Terry Tao diffrent views and deep understanding of math. Thank you😊
This is a gem. It took me quite some time to click on the recommendation. Perhaps TH-cam knows me better than I believe it does.
this is so interesting! listen to their conversation almost made me want to switch my career to a professional mathematician!
Go ahead:D
17:22 Freeman Dyson. But I think maybe he was talking about scientists/physicists.
I thought this podcast was dead!
This was so interesting. Well done!
I loved it!!
i enjoy him talking very much❤
So enjoyable
Do somebody know a proof assistant like which Terence Tao says?
Yes, the most prominent ones are Coq and Lean. There's a full article on Formal proof assistants on Wikipedia, you may check it out.
I remember there was one with shapes or knots, where mathematicians limited the possibilities down, and then used the computer to run through a very large number of all the possibilities, and it found the remaining shapes/knots.
1) Calculus Foundations
Contradictory:
Newtonian Fluxional Calculus
dx/dt = lim(Δx/Δt) as Δt->0
This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale.
Non-Contradictory:
Leibnizian Infinitesimal Calculus
dx = ɛ, where ɛ is an infinitesimal
dx/dt = ɛ/dt
Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.
2) Foundations of Mathematics
Contradictory Paradoxes:
- Russell's Paradox, Burali-Forti Paradox
- Banach-Tarski "Pea Paradox"
- Other Set-Theoretic Pathologies
Non-Contradictory Possibilities:
Algebraic Homotopy ∞-Toposes
a ≃ b ⇐⇒ ∃n, Path[a,b] in ∞Grpd(n)
U: ∞Töpoi → ∞Grpds (univalent universes)
Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations.
3) Foundational Paradoxes in Arithmetic
Contradictory:
- Russell's Paradox about sets/classes
- Berry's Paradox about definability
- Other set-theoretic pathologies
These paradoxes revealed fundamental inconsistencies in early naive attempts to formalize arithmetic foundations.
Non-Contradictory Possibility:
Homotopy Type Theory / Univalent Foundations
a ≃ b ⇐⇒ α : a =A b (Equivalence as paths in ∞-groupoids)
Arithmetic ≃ ∞-Topos(A) (Numbers as objects in higher toposes)
Representing arithmetic objects categorically as identifications in higher homotopy types and toposes avoids the self-referential paradoxes.
4) The Foundations of Arithmetic
Contradictory:
Peano's Axioms contain implicit circularity, while naive set theory axiomatizations lead to paradoxes like Russell's Paradox about the set of all sets that don't contain themselves.
Non-Contradictory Possibility:
Homotopy Type Theory / Univalent Foundations
N ≃ W∞-Grpd (Natural numbers as objects in ∞-groupoids)
S(n) ≃ n = n+1 (Successor is path identification)
Let Z ≃ Grpd[N, Π1(S1)] (Integers from N and winding paths)
Defining arithmetic objects categorically using homotopy theory and mapping into higher toposes avoids the self-referential paradoxes.
So you think everything can be fixed with infinity topoi?
I learnt recently, that to enjoy life, you must stop asking why. Or in other words, stop asking why, and enjoy life. And here Quanta has a podcast called the "Joy of Why"? wewewew.
Sure, there is always a truth to the saying “ignorance is bliss”. But there can be so much joy in the pursuit of why. The issue is that many people become so fixated on the answer that they fail to enjoy the journey. Personally I find great satisfaction in knowing there are always problems waiting to be solved. Isn’t it incredible that even with 8 billion of us on Earth, we don’t know why we dream? We don’t know why we yawn? We don’t know why we exist? It’s amazing to think we might one day unlock the answers to these questions
Please provide it with video
I know this was probably a mistake but him calling MRI (31:00) medical resonance imaging is cringe for a chemist 😬
never listen to terence tao a 2x....
?
BPRP has the same thing...
Im doing that right now.
2x is fine, 3x is pushing it.
That probably corresponds to listening to a normal person at 400 %
🎉
Love Math, The Secret of God is Mathematic. AL PAZA
I was skeptical about mr. terence idea , especially in his words where if someone has this credit , then they can make some "theories" that gauge some sort of belief in it ? I think mathematics is a rigorous field , not the one based on imagination and thought ideas
Imagination for a mathematician is as important as rigor.
Interesting and nice. He is bit "young" and a lot rich, but yes, mathematics have to reflect reality, or stay on the ground. And would be mathematics like some wisdom?
wow nice 😮🫡
"Yeah, no, it's been a pleasure"
sabka bap me hun 🫣