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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In this episode we sit down with Sheamus McGovern, founder of the Open Data Science Conference (ODSC AI), to unpack what AI actually looks like. Sheamus shares what’s really happening behind the scenes of the AI boom and why the biggest shift isn’t job loss, but a complete transformation of skills. From explaining why AI is reshaping—not replacing—jobs, to breaking down the gap between hype and real-world applications, this conversation explores how early algorithmic trading foreshadowed today’s AI revolution, why open-source tools like TensorFlow and PyTorch changed everything, what the “AI Skill Flip” means for your career, and why even data scientists are questioning their future. Along the way, the biggest mistake people make when trying to learn AI, and why the smartest approach isn’t to learn everything—but to start intentionally and build from there.

Timestamps

00:00 – The biggest misconception about AI

02:00 – Algorithmic trading and the origins of AI in finance

05:00 – The birth of ODSC AI and the data science movement

09:30 – Breakthrough moments in AI

16:30 – Democratization of AI and open-source tools

19:00 –The AI Skill Flip

24:00 – The truth about AI replacing jobs

27:00 – Real-world AI success stories

32:30 – How to actually start learning AI today

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In this episode, mathematician and award-winning magician Matt Baker explores the fascinating intersection of mathematics and magic, revealing how mathematical principles, intuition, and storytelling create unforgettable illusions. From a mind-reading trick based on the classic concept of “casting out nines” to his groundbreaking research on the Riemann–Roch theorem for graphs, Baker demonstrates how creativity and pattern recognition drive both mathematical discovery and magical performance.

Chapters

00:00 The Interplay of Math and Magic

04:58 Aesthetic Connections in Math and Magic

08:57 Balancing Family, Math, and Magic

12:34 The Impact of Magic on Mathematical Thinking

16:32 The Art of Clarity in Communication

16:44 A Live Magic Demonstration

25:14 Intuition and Pattern Recognition in Math

30:03 The Riemann-Roch Theorem for Graphs

41:42 The Role of AI in Mathematics and Magic

50:21 The Art of Communicating Mathematics

50:47 The Magic of Math and Performance

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In this episode, Thomas Plümper and Eric Neumayer explore the hidden challenges in modern science, from outright fraud to the subtler practice of “tweaking” data that distorts results. They examine why the self-correcting nature of science often falls short, how incentives and academic pressure drive misconduct, and the double-edged role of AI in both enabling and detecting fraud. The conversation also tackles debates around p-values and statistical reasoning, shares cautionary case studies, and proposes solutions like greater data transparency and stronger verification standards.

Chapters

00:00 Introduction to Fraud in Research

06:21 The Nature of Fraud Detection

08:56 Incentives and Motivations for Fraud

10:43 Self-Correction in Science

12:13 Understanding Statistical Significance

13:04 The Role of Replication in Research

14:32 Bayesian vs Frequentist Approaches

23:09 Understanding Bayesian Statistics and Its Implications

26:24 The Humility of Empirical Science

27:16 Concrete Examples of Scientific Fraud

32:52 Proposed Solutions to Scientific Fraud

34:50 The Reality of Scientific Fraud and Human Nature

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This Women in History Mini-Series episode with Dr. Victoria Bateman explores the inspiring story of Mary T. Washington Wylie, the first African-American woman CPA, her challenges, achievements, and legacy in breaking racial and gender barriers in the early 20th century. Mathematics is supposed to be objective—but access to it has never been equal.

Chapters

00:00 Introduction to Mary T. Washington Wylie

00:48 Early Life and Challenges

02:58 Breaking Barriers in Accountancy

05:25 Pioneering a Path for Others

07:21 Legacy and Impact

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In this episode of Breaking Math, we sit down with Dr. Hortensia Soto, former president of the Mathematical Association of America, to explore the role of mathematics in a world shaped by AI, education, and access. We discuss how debates in math are often really about opportunity and inclusion, how educators can navigate the rapid pace of AI, and why communication and human connection matter more than ever. The conversation also highlights the importance of financial literacy, embodied learning, and collaboration, reframing mathematics not as a set of answers, but as a way of thinking grounded in curiosity, effort, and relationships.

Chapters

00:00 Introduction to Hortensia Soto and the Math Community

02:48 The Role of AI in Mathematics

05:17 Access to Mathematics and Its Political Nature

07:34 The Importance of Financial Literacy in Math Education

10:19 Communication Skills for Mathematicians

13:06 The Culture of the Mathematical Association of America

15:29 Reflections on Leadership in the Math Community

25:01 Innovative Approaches to Mathematics Education

25:50 Recognizing Math Identity in Students

27:02 Nurturing Student Potential

35:31 The Role of AI in Learning

38:26 The Human Element in Mathematics

39:51 Mathematics Beyond Symbols and Procedures

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This Women in History Mini-Series episode with Dr. Victoria Bateman explores the groundbreaking contributions of Florence Nightingale, highlighting her use of data and statistics to revolutionize healthcare and public health during the 19th century. Discover how her innovative use of data visualization and her perseverance against gender biases transformed modern nursing and public health policy.

Chapters

00:00 Introduction to Florence Nightingale's Legacy

02:21 The Crimean War and Nightingale's Impact

05:18 Data Collection and Analysis in Healthcare

07:18 Overcoming Gender Bias in Medicine

09:23 Innovations in Data Visualization

11:59 Nightingale's Lasting Influence and Conclusion

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Mathematics quietly shapes some of the most important decisions in public life, from redistricting and congressional apportionment to federal research funding and AI policy. In this episode, Autumn and Noah speak with Dr. Karen Saxe, Senior Vice President of Government Relations at the American Mathematical Society, about how mathematical ideas influence representation, fairness, education, and the future of research. From gerrymandering and geometric compactness to life inside the U.S. Senate and the growing policy debates around AI, Karen reveals how deeply math is woven into the systems that govern everyday life.

Chapters

00:00 Introduction to the Conversation

01:15 The Hot Tea in DC

01:24 Gerrymandering and Mathematics

03:42 Understanding Gerrymandering and Redistricting

08:07 The Role of Mathematicians in Politics

12:19 Experiences in the Senate with Al Franken

19:32 Government Relations and the Role of Mathematics

23:01 The Impact of AI on Mathematics and Policy

28:41 Community Readiness for AI Transformations

29:22 Diversity in Education and Its Challenges

29:40 Bridging Mathematics and Politics

29:58 Career Pathways: Academia to Policy

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This Women in History Mini-Series episode with Dr. Victoria Bateman explores the groundbreaking work of Anna Schwartz, a pioneering economist who transformed macroeconomics through data-driven research. Discover how her meticulous analysis of monetary history shaped economic policy and the legacy she left for future generations.

Chapters

00:00 Introduction to Anna Schwartz and Her Impact

01:45 The Historical Context of Economic Data

04:10 Challenges Faced by Women in Economics

06:03 A Monetary History of the United States

09:04 The Methodology of Anna Schwartz

11:46 Legacy and Personal Insights on Anna Schwartz

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In this episode, Lauren Williams, professor of mathematics at Harvard University and a 2025 MacArthur Fellow, speaks about the surprising and often messy reality of mathematical research. The conversation begins with a turbulent moment in academia, when federal grants supporting her work were suddenly canceled—only months before she received the MacArthur “Genius Grant,” an unexpected recognition that allowed her to continue her research. Williams explains her work in algebraic combinatorics, illustrating how abstract mathematics can connect to real-world systems. The discussion also explores the human side of discovery, from collaborations that span continents to the strange coincidence of research papers and babies arriving the same week. Finally, the episode dives into one of the most intriguing experiments in modern mathematics: the First Proof project, which tests whether artificial intelligence can produce genuine mathematical proofs, revealing both the promise and the current limitations of AI-generated reasoning.

Chapters

01:27 Winning the MacArthur Genius Grant

01:43 Becoming a Woman in Mathematics at Harvard

04:25 Research Applications

10:04 The Human Side of Research

12:20 The First Proof Project

18:29 Advice for Young Mathematicians

22:51 The Intersection of Mathematics and AI

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This Women in History Mini-Series with Dr. Victoria Bateman explores the life and contributions of Priscilla Wakefield, a revolutionary figure in financial literacy and women's empowerment during the Industrial Revolution. Wakefield's work in establishing savings banks and community insurance schemes for women highlights her belief in the practical application of mathematics for everyday life. The discussion also addresses the challenges women faced in finance during her time and her lasting impact on feminist economics.

Takeaways

• Priscilla Wakefield taught ordinary people how to use numbers.
• She established England's first savings bank for women and children.
• Wakefield's work was pivotal during the British Industrial Revolution.
• She recognized the need for financial education among women.
• Her community insurance scheme empowered women financially.
• Wakefield's approach to mathematics was practical and accessible.
• She published influential works on women's rights and economics.
• Her philosophy emphasized the importance of financial literacy.

Chapters

00:00 Introduction to Priscilla Wakefield

01:19 Priscilla Wakefield: A Revolutionary Mathematician

04:28 The Financial Landscape of Georgian Britain

06:34 Groundbreaking Contributions to Banking and Finance

07:41 Fun Facts and Legacy of Priscilla Wakefield

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