More episodes of "The Joy of Why" are coming to KZfaq 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 really enjoy listening Terry Tao diffrent views and deep understanding of math. Thank you😊

I wonder why this wasn’t recommended sooner! I enjoyed listening

i love listening to him, he’s a true genius

I loved it!!

This was so interesting. Well done!

17:22 Freeman Dyson. But I think maybe he was talking about scientists/physicists.

i enjoy him talking very much❤

So enjoyable

I thought this podcast was dead!

Do somebody know a proof assistant like which Terence Tao says?

🎉

Please provide it with video

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 😮🫡

Love Math, The Secret of God is Mathematic. AL PAZA

I know this was probably a mistake but him calling MRI (31:00) medical resonance imaging is cringe for a chemist 😬

"Yeah, no, it's been a pleasure"

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.

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.

never listen to terence tao a 2x....

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

sabka bap me hun 🫣