09-05-2026, 09:53 PM
Quanta Magazine’s special series “The Evolving Foundations of Math” examines how the basic assumptions underlying mathematics have repeatedly been questioned, repaired and rebuilt from the late 19th century to the present. Beginning with Cantor’s revolutionary work on infinity and the development of rigorous set theory, the series traces the tension between intuition, formal proof and logical consistency through figures such as Cantor, Zermelo, Gödel and Grothendieck. It explores controversial alternatives such as ultrafinitism, the paradoxes created by infinity, the growing use of computer-verified proofs, and modern attempts to reconstruct areas such as topology and geometry from more fundamental concepts. The central message is that mathematics is not built on an immutable foundation: its definitions, axioms, methods of proof and even ideas about what mathematical objects should exist continue to evolve as mathematicians encounter paradoxes, new technologies and deeper structural questions.
Articles in the series
Chapter 2 — Logic Versus Proof
Chapter 3 — Cut to the Core
7. Mathematicians Want to Banish Infinity. What Might They Gain? — Gregory Barber
An exploration of ultrafinitism, which questions whether infinite mathematical objects should exist at all.
8. Why Math’s Final Axiom Proved So Controversial — Gregory Barber
The difficult historical development of Zermelo-Fraenkel set theory and its axioms.
9. Gödel’s Incompleteness Proof, Explained — Natalie Wolchover
An accessible account of Gödel’s incompleteness theorems and the fundamental limits they place on formal mathematical systems.
10. How Infinity Leads to One of Math’s Strangest Paradoxes — Max G. Levy
The Banach-Tarski paradox and the counterintuitive consequences of infinity.
Chapter 4 — A Revolution Begins
11. Two Researchers Are Rebuilding Mathematics From the Ground Up — Konstantin Kakaes
Peter Scholze and Dustin Clausen’s ambitious program to rethink fundamental concepts in topology and number theory.
12. How Alexander Grothendieck Revolutionized 20th-Century Mathematics — Konstantin Kakaes
An overview of Grothendieck’s transformative ideas in algebraic geometry.
13. Inside the Fight to Fix Geometry’s Foundations — Kevin Hartnett
A debate over whether fundamental arguments in modern geometry can truly be trusted and how they should be rebuilt.
Overall, the collection moves roughly from Cantor and infinity → rigor and formal proof → axioms and Gödel → Grothendieck and new foundations, making it essentially a compact history of how mathematicians have repeatedly reconsidered what mathematics is built from and what counts as a valid proof.
ARTICLES
Articles in the series
Chapter 1 — Original Sin
- Long-Lost Letters Reveal the Lies That Gave Rise To Modern Math — Joseph HowlettCantor, the discovery of different sizes of infinity, and newly uncovered evidence concerning the origins of his famous 1874 result.
- How Can Infinity Come in Many Sizes? A Visual Investigation — Mark Belan & Jordana CepelewiczA visual explanation of countable and uncountable infinities.
- ‘Fantastic’ Proof Implies the Existence of In-Between Infinities — Natalie WolchoverNew developments surrounding the continuum problem and possible intermediate sizes of infinity.
Chapter 2 — Logic Versus Proof
4. In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far? — Leila Sloman
The history of mathematical rigor and the modern movement toward formalizing proofs with systems such as Lean.
5. How Writing Changes Mathematical Thought — John Pavlus
An interview exploring how mathematical notation influences the way mathematicians think.
6. The Jagged, Monstrous Function That Broke Calculus — Solomon Adams
How pathological functions challenged 19th-century intuition and forced mathematicians to make calculus more rigorous.
Chapter 3 — Cut to the Core
7. Mathematicians Want to Banish Infinity. What Might They Gain? — Gregory Barber
An exploration of ultrafinitism, which questions whether infinite mathematical objects should exist at all.
8. Why Math’s Final Axiom Proved So Controversial — Gregory Barber
The difficult historical development of Zermelo-Fraenkel set theory and its axioms.
9. Gödel’s Incompleteness Proof, Explained — Natalie Wolchover
An accessible account of Gödel’s incompleteness theorems and the fundamental limits they place on formal mathematical systems.
10. How Infinity Leads to One of Math’s Strangest Paradoxes — Max G. Levy
The Banach-Tarski paradox and the counterintuitive consequences of infinity.
Chapter 4 — A Revolution Begins
11. Two Researchers Are Rebuilding Mathematics From the Ground Up — Konstantin Kakaes
Peter Scholze and Dustin Clausen’s ambitious program to rethink fundamental concepts in topology and number theory.
12. How Alexander Grothendieck Revolutionized 20th-Century Mathematics — Konstantin Kakaes
An overview of Grothendieck’s transformative ideas in algebraic geometry.
13. Inside the Fight to Fix Geometry’s Foundations — Kevin Hartnett
A debate over whether fundamental arguments in modern geometry can truly be trusted and how they should be rebuilt.
Overall, the collection moves roughly from Cantor and infinity → rigor and formal proof → axioms and Gödel → Grothendieck and new foundations, making it essentially a compact history of how mathematicians have repeatedly reconsidered what mathematics is built from and what counts as a valid proof.
ARTICLES
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