You Are Not So Smart – Episode 328: "Shape" with Jordan Ellenberg (Rebroadcast)

Date: December 8, 2025
Host: David McRaney
Guest: Jordan Ellenberg, mathematician and author

Episode Overview

In this wide-ranging discussion, host David McRaney interviews Jordan Ellenberg, math professor at the University of Wisconsin-Madison and author of Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. The episode explores how geometry—and more broadly, mathematical reasoning—is deeply woven into human intuition, perception, and even everyday debates, revealing how abstract math concepts inform our understanding of stories, games, society, and ourselves. The conversation is lively, accessible, and often poetic, focusing on the human side of mathematical discovery.

Key Discussion Points & Insights

1. Introduction: What is Mathematics, Really?

• Math as Naming:

  • [01:37]
    Ellenberg: "It's calling different things by the same name, places on the globe."
  • McRaney invokes Poincaré:
    [01:53] "Mathematics is the art of giving the same name to different things."

• Intuiting Math Before Formalization:

  • [07:09]
    Ellenberg: "Are we inarticulately recognizing the thing and then articulating it and making a word for it, or are we in some sense actually bringing it into existence by naming it?"

2. Math, Geometry, and Human Experience

• The Formal and the Intuitive Sides of Geometry:

  • [08:07]
    Ellenberg: "Geometry is built into us... When we do the subject in school, it's presented as formal and abstract... but part of the charm is that both aspects are there. Geometry is built into our bodies, but the leap into formality allows us to explore beyond physical dimensions."

• The Primal Layer of Perception:

  • [10:01]
    Book quote: "Ayahuasca drinkers have a similar take: the drug reboots the brain and lifts the mind above the tortured labyrinth it thinks it’s stuck in."
  • Math Inspired by Intuition:
    [10:39]
    McRaney on mathematical joy: "Whenever I have felt the most love and passion and obsession for mathematical concepts, it came from something intuitive in this way."

3. Simple Rules, Infinite Complexity: The Game of Life

• John Conway’s Game of Life as Metaphor:

  • [12:03]
    Ellenberg:
    "A very simple mathematical construct... The rule is very simple... it produces the most unimaginably baroque patterns... a metaphor for geometry itself: all this richness from a very small set of initial rules."
  • McRaney:
    [14:24]
    "I can't get enough of the concept that a very simple set of rules then put in motion in frames, so every frame, something changes..."

• Perpetual Near-Understanding:

  • [15:59]
    Ellenberg:
    "That experience of 'I kind of sort of get it, something I can't quite describe'—that's like every day of your life as a research mathematician..."

4. Geometry’s Place in History, Human Development, and Reason

• Lincoln and Euclid:

  • [21:52]
    Ellenberg:
    "Lincoln's fondness for Euclid... as he told it, he was troubled... 'What is a demonstration?' He realized he needed to go back to Euclid to understand what a demonstration was."
  • Integrity & Geometry:
    [33:11]
    Book quote: "The ultimate reason for teaching kids to write a proof is not that the world is full of proofs, it's that the world is full of non proofs and grownups need to know the difference."

• The Human Side of Math:

  • [28:09]
    Ellenberg:
    "Mathematics is a human activity. Every single formalism was created by people, to solve a problem that they had, and they were less confused after..."

5. The Power & Play of Definitions: "How Many Holes Does a Straw Have?"

• A Gateway to Deep Reasoning and Argument:

  • [40:07]
    Ellenberg:
    "It's one of those things where people think the answer is obvious, and then they're absolutely stunned to find out... it's not the same answer."
  • Definitional Debates:
    [41:27]
    McRaney:
    "Are we playing a language game or a mathematical game? And I love it."
  • Sense and Sense-Making:
    [41:56]
    Ellenberg:
    "They're recognizing that there is an actual mathematical issue. They may not use those words, but it exactly speaks to my contention that math sense is in us all and we react when something touches that nerve."

• Multiple Vantage Points & Topology:

  • [45:04]
    Ellenberg:
    "If there's two compelling answers that both seem mathematically right, usually the right conclusion is that we have to understand from which vantage they're both correct."

6. Symmetry, Transformations & the "Scranch" Plane

• Generalizing Symmetry:

  • [47:32]
    Ellenberg:
    "Just as you can imagine flipping something, you can imagine expanding it, or stretching it in one direction, which I called a 'scranch.'"
  • [49:09]
    McRaney:
    "You just did what Kant did. You made a word... he made up the word 'angst.'"

• Symmetries in Physics and Relativity:

  • [49:32]
    Ellenberg:
    "In the history of physics, you eventually have to accept that these kinds of symmetries are the ones that spacetime actually has... that's how space is. It’s just not as we thought."

7. Trees, Games, and Abstraction

• Tree Structures in Nature, Games, and Reasoning:

  • [57:02]
    Ellenberg:
    "The geometry of a tree... is so fundamental and appears in so many places... The geometry of a tree describes what happens when you play a game with specified rules."
  • [59:15]
    "A game like Chess or Tic Tac Toe... they’re really different only in size. There’s an actual answer to the question of whether a perfect chess player... would always win, lose, or draw."

• Games as Models for AI and Decision-Making:

  • [60:23]
    McRaney:
    "A game is... first, to understand the geometry of the board, then the rules... then the geometry of gameplay, the metagame. This mimics how brains prefer to make sense of things."

8. Lessons and Human Meaning

• Learning from History and Unknowing:
  • [32:08]
    Ellenberg:
    "Both [writing and teaching] require you to imagine your way into the state of not understanding the thing... and to put yourself into the mind and situation of students learning for the first time."
• On Discovering Truth Beyond Authority:
  • [68:25]
    Ellenberg:
    "It’s a very powerful moment... to understand they can make knowledge by themselves and the authority is themselves and their own insight... Geometry should still be seen as dangerous."

Notable Quotes & Memorable Moments

• [01:53] Poincaré’s aphorism:
  “Mathematics is the art of giving the same name to different things.”

• [10:01] On Ayahuasca and perception:
  “Ayahuasca drinkers have a similar take: the drug reboots the brain and lifts the mind above the tortured labyrinth it thinks it’s stuck in.” — Ellenberg (quoting his own book)

• [15:59] The mathematician’s frontier:
  “That experience... that's like every day of your life as a research mathematician... those things that you sort of can't quite talk about, but you can sense they're there, is like, that's where you live.” — Ellenberg

• [33:11] On the purpose of proofs:
  “The ultimate reason for teaching kids to write a proof is not that the world is full of proofs, it's that the world is full of non proofs and grownups need to know the difference.” — Ellenberg

• [45:19] On perspectives and categorization:
  “Imagine... a sterile argument to be like, well, which is it? Is it actually round or actually square? The question is really like, why do I have this ability to perceive it as round and also as square?” — Ellenberg

• [62:48] Decision rules and similarity:
  “It's not so much I want a situation that's identical with a situation I've seen before. I want to be able to measure if a situation is in some way similar to one I've encountered, and then maybe do the same thing... The moment you introduce similarity, you're being geometric.” — Ellenberg

• [65:21] Why we play games:
  “People did not stop playing checkers, even though the game is solved... because the point of a game is not to win. The point... is to play the game.” — Ellenberg

• [68:25] On mathematical authority:
  “It’s a very powerful moment for a child or an adult to understand they can make knowledge by themselves and the authority is themselves and their own insight. Geometry should still be seen as dangerous. Let's show some respect.”

Timestamps for Key Segments

• [01:21] — Math as naming and Poincare's aphorism
• [08:07] — Geometry as embodied intuition and abstraction
• [12:03] — Conway's Game of Life as a metaphor for complexity
• [21:52] — Abraham Lincoln, Euclid, and the habit of demonstration
• [40:07] — "How many holes does a straw have?" and the deeper meaning of argument
• [47:32] — Scranch planes and generalizing symmetry
• [57:02] — Trees in geometry, games, and abstraction
• [62:48] — Similarity, decision making, and the geometry of choices
• [65:21] — Checkers, chess, AI, and the meaning of games
• [68:25] — Discovering truths that transcend authority

Summary Flow and Tone

The conversation is energetic, nerdily enthusiastic, and deeply welcoming of philosophical rabbit holes. Ellenberg’s language is generous, witty, and poetic, while McRaney plays both curious novice and sharp commentator, drawing connections between math, psychology, and everyday reasoning. The discussion blends rigorous mathematical insights with approachable metaphors, personal anecdotes, and playful but genuinely deep questions (like holes in straws).

The episode is perfect for anyone interested in not just learning about mathematics or geometry but understanding how such abstract tools make us more deeply human, curious, and wise.

Recommended If You Like:

• Math and science explained through stories
• Connections between abstract reasoning and everyday experience
• Lively, funny, and philosophical discussions

Listen for:

• Insights on the human drive to categorize and model reality
• How simple mathematical rules generate immense complexity
• The beauty, and even danger, of reasoning from first mathematical principles