What Makes for ‘Good’ Math? | Podcast: The Joy of Why

Terence Tao, who has been called the “Mozart of Mathematics,” wrote an essay in 2007 about the common ingredients in “good” mathematical research. In this episode, the Fields Medalist joins Steven Strogatz to revisit the topic. S3EP01 Originally Published February 1, 2024

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“The Joy of Why” is a Quanta Magazine podcast about curiosity and the pursuit of knowledge. The mathematician and author Steven Strogatz and the astrophysicist and author Janna Levin take turns interviewing leading researchers about the great scientific and mathematical questions of our time. The Joy of Why is produced by PRX Productions

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[QuantaScienceChannel]

More episodes of "The Joy of Why" are coming to YouTube soon. In the meantime, you can subscribe wherever you get your podcasts or explore past episodes on the Quanta website.

[IcECreAm-svqv]

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

[noahgilbertson]

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

[jabalatiwari]

I really enjoy listening Terry Tao diffrent views and deep understanding of math. Thank you😊

[benjaminandersson]

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

[Suigin.]

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

[AbhinavLal]

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.

[Stacee-jxyz]

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.

[austinhaider]

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

[GPSPYHGPSPYH-dsgu]

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

[modrypotucek]

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?

[fahimuddin]

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

[liijio]

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