What does it actually mean for a number to be “big”? In this episode of Breaking Math, Autumn chats with mathematician Richard Elwes to explore how huge numbers reveal the limits of human intuition, language, and even mathematics itself. The discussion moves from exponential growth in pandemics and finance to numbers larger than the universe itself, emerging in games like chess and abstract possibility spaces. Finally, it reaches one of the most profound ideas in modern mathematics: that there are true statements about numbers that can never be proven. This episode challenges how we think about scale, complexity, and the systems we rely on to make sense of reality.

Key Topics

Limits of ancient numeral systems like Roman numerals

Mathematical logic and the concept of huge numbers

Evolution of number notation from Roman to Hindu-Arabic systems

The significance of place value in expressing large numbers

The Mayan long count and its implications for understanding time scales

Chapters

00:00 Introduction and Inspiration for the Book

01:39 Redefining Big Numbers

01:55 Limits of Numerical Systems

05:33 Evolution of Number Sense

10:02 Language and Numerical Understanding

11:53 Cultural Influences on Numerical Systems

14:18 Hacks in Ancient Number Systems

16:55 Archimedes and the Concept of Infinity

22:01 The Importance of Place Value

25:45 Mayan Cosmology and Time Scales

31:55 Exponential Growth and Its Dangers

32:20 Understanding Exponential Growth

36:14 The Dangers of Exponential Growth

37:23 Limits of Exponential Growth in the Physical World

39:42 Exploring Possibility Space

45:38 Goodstein's Theorem and Mathematical Logic

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