i^2 = j^2 = k^2 = ijk = -1. This deceptively simple formula, discovered by Irish mathematician William Rowan Hamilton in 1843, led to a revolution in the way 19th century mathematicians and scientists thought about vectors and rotation. This formula, which extends the complex numbers, allows us to talk about certain three-dimensional problems with more ease. So what are quaternions? Where are they still used? And what is inscribed on Broom Bridge? All of this and more on this episode of Breaking Math.

This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.
The theme for this episode was written by Elliot Smith.
[Featuring: Sofía Baca, Meryl Flaherty]

---

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60: HAMILTON! [But Not the Musical] (Quaternions)

Hosted by Gabriel Hesch and Autumn Phaneuf, who have advanced degrees in EE and industrial engineering/operations research respectively, come together to discuss mathematics as a pure field al in its own as well as how it describes the language of science, engineering, and even creativity. &nbsp;

Breaking Math brings you the absolute best in interdisciplinary science discussions - &nbsp;bringing together experts in varying fields including artificial intelligence, neuroscience, evolutionary biology, physics, chemistry and materials-science, and more - &nbsp;to discuss where humanity is headed. website: &nbsp;breakingmath.io&nbsp; linktree: &nbsp;linktree.com/breakingmathmedia email: &nbsp;breakingmathpodcast@gmail.com

Science

Mathematics

Summary:  The episode discusses the 10,000 year dilemma, which is a thought experiment on how to deal with nuclear waste in the future.  Today's episode is hosted by guest host David Gibson, who is the founder of the Ray Kitty Creation Workshop [https://www.youtube.com/@RayKitty]. (Find out more about the Ray Kitty Creation Workshop by clicking here [https://mrdave.raykitty.com/]).  

Gabriel and Autumn are out this week, but will be returning in short order with 3 separate interviews with authors of some fantastic popular science and math books including:


 * The Gravity of Math:  How Geometry Rules the Universe [https://www.hachettebookgroup.com/titles/steve-nadis/the-gravity-of-math/9781541604292/] by Dr. Shing-Tung Yau and Steve Nadis.    This book is all about the history of our understanding of gravity from the theories of Isaac Newton to Albert Einstein and beyond, including gravitational waves, black holes, as well as some of the current uncertainties regarding a precise definition of mass.  On sale now!  
 * EVERYTHING IS PREDICTABLE: How Bayesian Statistics Explain Our World [https://www.netgalley.com/widget/516523/redeem/23dc460e436c2d0582e559b4d84ee909cf774accba78999f2291a8f97813bae0] by Tom Chivers.  Published by Simon and Schuster.   This book explains the importance of Baye's Theorem in helping us to understand why  highly accurate screening tests can lead to false positives, a phenomenon we saw during the Covid-19 pandemic; How a failure to account for Bayes' Theorem has put innocent people in jail; How military strategists using the theorem can predict where an enemy will strike next, and how Baye's Theorem is helping us to understang machine learning processes - a critical skillset to have in the 21st century.
   Available 05/07/2024
 * A CITY ON MARS: Can we settle space, should we settle space, and have we really thought this through? [https://www.penguinrandomhouse.com/books/639449/a-city-on-mars-by-kelly-and-zach-weinersmith/]  by authors Dr. Kelly and Zach Weinersmith.  Zach Weinersmith is the artist and creator of the famous cartoon strip Saturday Morning Breaking Cereal [https://www.smbc-comics.com/]!  
   
   We've got a lot of great episodes coming up!  Stay tuned.

Ep. 94  The 10,000 Year Problem (feat. David Gibson of Ray Kitty Creation Workship)

An interview with Prof. Marcus du Sautoy about his book Around the Wold in Eighty Games . . . .a Mathematician Unlocks the Secrets of the World's Greatest Games.  

Topics covered in Today's Episode: 

1. Introduction to Professor Marcus du Sautoy and the Role of Games

- Impact of games on culture, strategy, and learning

- The educational importance of games throughout history

2. Differences in gaming cultures across regions like India and China

3. Creative Aspects of Mathematics

4. The surprising historical elements and banned games by Buddha

5. Historical and geographical narratives of games rather than rules

6. Game Theory and Education

7.  Unknowable questions like thermodynamics and universe's infinity

8. Professor du Sautoy's Former Books and Collections

9.  A preview of his previous books and their themes

10. Gaming Cultures and NFTs in Blockchain

11. Gamification in Education

12. The Role of AI in Gaming

13. Testing machine learning in mastering games like Go

14. Alphago's surprising move and its impact on Go strategies

15 . The future of AI in developing video game characters, plots, and environments

16. Conclusion and Giveaway Announcement

*Free Book Giveaway of Around The World in 88 Games . . .  by Professor Marcus Du Sautory!  Follow us on our socials for details:  

Follow us on X:  @BreakingMathPod [https://twitter.com/breakingmathpod]

Follow us on Instagram:  @Breaking Math Media [https://www.instagram.com/breakingmathmedia/]

Email us:  BreakingMathPodacst@gmail.com

92. The Mathematical Heart of Games Explored with Prof. du Sautoy

Summary

Brain Organelles, A.I. and Defining Intelligence in  Nature- 

In this episode, we continue our fascinating interview with GT, a science content creator on TikTok [https://www.tiktok.com/@bearbaitofficial] and YouTube [https://www.youtube.com/@bearbaitofficial] known for their captivating - and sometimes disturbing science content.

GT can be found on the handle '@bearBaitOfficial' on most social media channels.  

In this episode, we resume our discussion on Brain Organelles -  which are grown from human stem cells - how they are being used to learn about disease, how they may be integrated in A.I.  as well as eithical concerns with them.

We also ponder what constitutes intelligence in nature, and even touch on the potential risks of AI behaving nefariously.

You won't want to miss this thought-provoking and engaging discussion.

30% Off ZenCastr Discount

Use My Special Link to save e 30%  Off Your First Month of Any ZenCastr Paid Plan [https://zen.ai/1e7eBWWMLcSL_G10VxiSlQ]

91.  Brain Organelles, AI, and Other Scary Science - An Interview with GT (Part 2)

This episode is inspired by a correspondence the Breaking Math Podcast had with the editors of Digital Discovery, a journal by the Royal Society of Chemistry.  In this episode the hosts review a paper [https://pubs.rsc.org/en/content/articlelanding/2024/dd/d3dd00077j] about how the Lean Interactive Theorem Prover, which is usually used as a tool in creating mathemtics proofs, can be used to create rigorous and robust models in physics and chemistry.  

Also -  we have a brand new member of the Breaking Math Team!  This episode is the debut episode for Autumn, CEO of Cosmo Labs, occasional co-host / host of the Breaking Math Podcast, and overall contributor who has been working behind the scenes on the podcast on branding and content for the last several months. Welcome Autumn!  

Autumn and Gabe discuss how the paper explores the use of interactive theorem provers to ensure the accuracy of scientific theories and make them machine-readable. The episode discusses the limitations and potential of interactive theorem provers and highlights the themes of precision and formal verification in scientific knowledge.  This episode also provide resources (listed below) for listeners interested in learning more about working with the LEAN interactive theorem prover.  

Takeaways

 * Interactive theorem provers can revolutionize the way scientific theories are formulated and verified, ensuring mathematical certainty and minimizing errors.
 * Interactive theorem provers require a high level of mathematical knowledge and may not be accessible to all scientists and engineers.
 * Formal verification using interactive theorem provers can eliminate human error and hidden assumptions, leading to more confident and reliable scientific findings.
 * Interactive theorem provers promote clear communication and collaboration across disciplines by forcing explicit definitions and minimizing ambiguities in scientific language. Lean Theorem Provers enable scientists to construct modular and reusable proofs, accelerating the pace of knowledge acquisition.
 * Formal verification presents challenges in terms of transforming informal proofs into a formal language and bridging the reality gap.
 * Integration of theorem provers and machine learning has the potential to enhance creativity, verification, and usefulness of machine learning models.
 * The limitations and variables in formal verification require rigorous validation against experimental data to ensure real-world accuracy.
 * Lean Theorem Provers have the potential to provide unwavering trust, accelerate innovation, and increase accessibility in scientific research.
 * AI as a scientific partner can automate the formalization of informal theories and suggest new conjectures, revolutionizing scientific exploration.
 * The impact of Lean Theorem Provers on humanity includes a shift in scientific validity, rapid scientific breakthroughs, and democratization of science. 

Help Support The Podcast by clicking on the links below:

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 * Breaking Math Website [http://breakingmath.io/]Email us for copies of the transcript!

90. LEAN Theorem Provers used to model Physics and Chemistry

This conversation explores the topic of brain organoids and their integration with robots. The discussion covers the development and capabilities of brain organoids, the ethical implications of their use, and the differences between sentience and consciousness. The conversation also delves into the efficiency of human neural networks compared to artificial neural networks, the presence of sleep in brain organoids, and the potential for genetic memories in these structures. The episode concludes with an invitation to part two of the interview and a mention of the podcast's Patreon offering a commercial-free version of the episode.



Takeaways

 * Brain organoids are capable of firing neural signals and forming structures similar to those in the human brain during development.
 * The ethical implications of using brain organoids in research and integrating them with robots raise important questions about sentience and consciousness.
 * Human neural networks are more efficient than artificial neural networks, but the reasons for this efficiency are still unknown.
 * Brain organoids exhibit sleep-like patterns and can undergo dendrite growth, potentially indicating learning capabilities.
 * Collaboration between scientists with different thinking skill sets is crucial for advancing research in brain organoids and related fields.

Chapters

 1.  00:00 Introduction: Brain Organoids and Robots
 2.  00:39 Brain Organoids and Development
 3.  01:21 Ethical Implications of Brain Organoids
 4.  03:14 Summary and Introduction to Guest
 5.  03:41 Sentience and Consciousness in Brain Organoids
 6.  04:10 Neuron Count and Pain Receptors in Brain Organoids
 7.  05:00 Unanswered Questions and Discomfort
 8.  05:25 Psychological Discomfort in Brain Organoids
 9.  06:21 Early Videos and Brain Organoid Learning
 10. 07:20 Efficiency of Human Neural Networks
 11. 08:12 Sleep in Brain Organoids
 12. 09:13 Delta Brainwaves and Brain Organoids
 13. 10:11 Creating Brain Organoids with Specific Components
 14. 11:10 Genetic Memories in Brain Organoids
 15. 12:07 Efficiency and Learning in Human Brains
 16. 13:00 Sequential Memory and Chimpanzees
 17. 14:18 Different Thinking Skill Sets and Collaboration
 18. 16:13 ADHD and Hyperfocusing
 19. 18:01 Ethical Considerations in Brain Research
 20. 19:23 Understanding Genetic Mutations
 21. 20:51 Brain Organoids in Rat Bodies
 22. 22:14 Dendrite Growth in Brain Organoids
 23. 23:11 Duration of Dendrite Growth
 24. 24:26 Genetic Memory Transfer in Brain Organoids
 25. 25:19 Social Media Presence of Brain Organoid Companies
 26. 26:15 Brain Organoids Controlling Robot Spiders
 27. 27:14 Conclusion and Invitation to Part 2

References:

Muotri Labs (Brain Organelle piloting Spider Robot) [https://pediatrics.ucsd.edu/research/faculty-labs/muotri-lab/index.html]

Cortical Labs (Brain Organelle's trained to play Pong) [https://corticallabs.com/]

*For a copy of the episode transcript, email us at breakingmathpodcast@gmail.com 

Help Support The Podcast by clicking on the links below:

 * Start YOUR podcast on ZenCastr!  
   Use my special link  ZenCastr Discount [https://zen.ai/1e7eBWWMLcSL_G10VxiSlQ] to save 30% off your first month of any Zencastr paid plan
   
 * Visit our Patreon [http://www.patreon.com/breakingmath]

Summary:

60: HAMILTON! [But Not the Musical] (Quaternions)

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