Breaking Math Podcast

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This episode reviewed the article Formalizing chemical physics using the Lean theorem prover with the journal Digital Discovery, a journal with the Royal Society of Chemistry.  

The website for Digital Discovery can be found here

Special thanks to the editorial staff at Digital Discovery for answering our countless emails asking questions about this article as well as their mission as a journal.  

Special thanks as well to Professors Tyler Josephson and lead author of the paper Max Bobbin for giving us some of their time in clarifying some of the concepts in their paper.  

Summary

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 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.  

The paper is titled Formalizing chemical physics using the Lean Theorem prover and can be found in Digital Discovery, a journal with the Royal Society of Chemistry.  

Also -  we have a brand new member of the Brekaing 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 intersted 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.Continuous expansion of mathematical libraries in Lean Theorem Provers contributes to the codification of human knowledge.Resources are available for learning Lean Theorem Proving, including textbooks, articles, videos, and summer programs.
Resources / Links:

Professor Tyler Josephson, one of the authors of the article, sent us several links to learn more about LEAN which we have included below.  
Email Professor Tyler Josephson about summer REU undergraduate opportunities at the University of Maryland Baltimore (or online!) at [email protected].  The Natural Number Game:  Start in a world without math, unlock tactics and collect theorems until you can beat a 'boss' level and prove that 2+2=4, and go further.  Free LEAN Texbook and CourseProfessor Josephson's most-recommended resource for beginners learning Lean - a free online course and textbook from Prof. Heather Macbeth at Fordham University. Quanta Magazine articles on LeanProf. Kevin Buzzard of Imperial College London's lecture on LEAN interactive theorem prover and the future of mathematics.

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Summary
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
00:00 Introduction: Brain Organoids and Robots00:39 Brain Organoids and Development01:21 Ethical Implications of Brain Organoids03:14 Summary and Introduction to Guest03:41 Sentience and Consciousness in Brain Organoids04:10 Neuron Count and Pain Receptors in Brain Organoids05:00 Unanswered Questions and Discomfort05:25 Psychological Discomfort in Brain Organoids06:21 Early Videos and Brain Organoid Learning07:20 Efficiency of Human Neural Networks08:12 Sleep in Brain Organoids09:13 Delta Brainwaves and Brain Organoids10:11 Creating Brain Organoids with Specific Components11:10 Genetic Memories in Brain Organoids12:07 Efficiency and Learning in Human Brains13:00 Sequential Memory and Chimpanzees14:18 Different Thinking Skill Sets and Collaboration16:13 ADHD and Hyperfocusing18:01 Ethical Considerations in Brain Research19:23 Understanding Genetic Mutations20:51 Brain Organoids in Rat Bodies22:14 Dendrite Growth in Brain Organoids23:11 Duration of Dendrite Growth24:26 Genetic Memory Transfer in Brain Organoids25:19 Social Media Presence of Brain Organoid Companies26:15 Brain Organoids Controlling Robot Spiders27:14 Conclusion and Invitation to Part 2



References:

Muotri Labs (Brain Organelle piloting Spider Robot)

Cortical Labs (Brain Organelle's trained to play Pong)

*For a copy of the episode transcript, email us at [email protected]


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This is a follow up on our previous episode on OpenAi's SORA. We attempt to answer the question, "Can OpenAi's SORA model real-world physics?"

We go over the details of the technical report, we discuss some controversial opinoins by experts in the field at Nvdia and Google's Deep Mind.

The transcript for episode is avialable below upon request.

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Summary

OpenAI's Sora, a text-to-video model, has the ability to generate realistic and imaginative scenes based on text prompts. This conversation explores the capabilities, limitations, and safety concerns of Sora. It showcases various examples of videos generated by Sora, including pirate ships battling in a cup of coffee, woolly mammoths in a snowy meadow, and golden retriever puppies playing in the snow. The conversation also discusses the technical details of Sora, such as its use of diffusion and transformer models. Additionally, it highlights the potential risks of AI fraud and impersonation. The episode concludes with a look at the future of physics-informed modeling and a call to action for listeners to engage with Breaking Math content.


Takeaways

OpenAI's Sora is a groundbreaking text-to-video model that can generate realistic and imaginative scenes based on text prompts.Sora has the potential to revolutionize various industries, including entertainment, advertising, and education.While Sora's capabilities are impressive, there are limitations and safety concerns, such as the potential for misuse and the need for robust verification methods.The conversation highlights the importance of understanding the ethical implications of AI and the need for ongoing research and development in the field.
Chapters

00:00 Introduction to OpenAI's Sora
04:22 Overview of Sora's Capabilities
07:08 Exploring Prompts and Generated Videos
12:20 Technical Details of Sora
16:33 Limitations and Safety Concerns
23:10 Examples of Glitches in Generated Videos
26:04 Impressive Videos Generated by Sora
29:09 AI Fraud and Impersonation
35:41 Future of Physics-Informed Modeling
36:25 Conclusion and Call to Action

#OpenAiSora #

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Transcripts are available upon request. Email us at [email protected]
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Visit our guest Levi McClain's Pages:
youtube.com/@LeviMcClain
levimcclain.com/

Summary
Levi McClean discusses various topics related to music, sound, and artificial intelligence. He explores what makes a sound scary, the intersection of art and technology, sonifying data, microtonal tuning, and the impact of using 31 notes per octave. Levi also talks about creating instruments for microtonal music and using unconventional techniques to make music. The conversation concludes with a discussion on understanding consonance and dissonance and the challenges of programming artificial intelligence to perceive sound like humans do.


Takeaways:

The perception of scary sounds can be analyzed from different perspectives, including composition techniques, acoustic properties, neuroscience, and psychology.Approaching art and music with a technical mind can lead to unique and innovative creations.Sonifying data allows for the exploration of different ways to express information through sound.Microtonal tuning expands the possibilities of harmony and offers new avenues for musical expression.Creating instruments and using unconventional techniques can push the boundaries of traditional music-making.Understanding consonance and dissonance is a complex topic that varies across cultures and musical traditions.Programming artificial intelligence to understand consonance and dissonance requires a deeper understanding of human perception and cultural context.

Chapters
00:00 What Makes a Sound Scary
03:00 Approaching Art and Music with a Technical Mind
05:19 Sonifying Data and Turning it into Sound
08:39 Exploring Music with Microtonal Tuning
15:44 The Impact of Using 31 Notes per Octave
17:37 Why 31 Notes Instead of Any Other Arbitrary Number
19:53 Creating Instruments for Microtonal Music
21:25 Using Unconventional Techniques to Make Music
23:06 Closing Remarks and Questions
24:03 Understanding Consonance and Dissonance
25:25 Programming Artificial Intelligence to Understand Consonance and Dissonance

Listen to episodes commercial Free on Patreon at patreon.com/breakingmath

We are joined today by content creator Levi McClain to discuss the mathematics behind music theory, neuroscience, and human experiences such as fear as they relate to audio processing.

For a copy of the episode transcript, email us at [email protected].

For more in depth discussions on these topics and more, check out Levi's channels at:

Patreon.com/LeviMcClain

youtube.com/@LeviMcClain

Tiktok.com/@levimcclain

Instagram.com/levimcclainmusic

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Part 2/2 of the interview with Brit Cruise, creator of the YouTube channel "Art of the Problem," about interesting mathematics,, electrical and computer engineering problems.

In Part 1, we explored what 'intelligence' may be defined as by looking for examples of brains and proto-brains found in nature (including mold, bacteria, fungus, insects, fish, reptiles, and mammals).

In Part 2, we discuss aritifical neural nets and how they are both similar different from human brains, as well as the ever decreasing gap between the two.

Brit's YoutTube Channel can be found here: Art of the Problem - Brit Cruise

Transcript will be made available soon! Stay tuned. You may receive a transcript by emailing us at [email protected].

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In this episode (part 1 of 2), I interview Brit Cruise, creator of the YouTube channel 'Art of the Problem.' On his channel, he recently released the video "ChatGPT: 30 Year History | How AI learned to talk." We discuss examples of intelligence in nature and what is required in order for a brain to evolve at the most basic level. We use these concepts to discuss what artificial intelligence - such as Chat GPT - both is and is not.

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How is Machine Learning being used to further original scientific discoveries?

Transcripts of this episode are avialable upon request. Email us at [email protected].

A link to the paper discussed in this episode can be found here-->

Digital Discovery - Generative adversarial networks and diffusion models in material discovery

In this episode Gabriel Hesch interviews Taylor Sparks, a professor of material science and engineering, about his recent paper on the use of generative modeling a.i. for material disovery. The paper is published in the journal Digital Discovery and is titled 'Generative Adversarial Networks and Diffusion MOdels in Material Discovery. They discuss the purpose of the call, the process of generative modeling, creating a representation for materials, using image-based generative models, and a comparison with Google's approach. They also touch on the concept of conditional generation of materials, the importance of open-source resources and collaboration, and the exciting developments in materials and AI. The conversation concludes with a discussion on future collaboration opportunities.
Takeaways
Generative modeling is an exciting approach in materials science that allows for the prediction and creation of new materials.Creating a representation for materials, such as using the crystallographic information file, enables the application of image-based generative models.Google's approach to generative modeling received attention but also criticism for its lack of novelty and unconditioned generation of materials.Open-source resources and collaboration are crucial in advancing materials informatics and machine learning in the field of materials science.

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In October of 2023, Sofia Baca passed away unexpectedly from natural causes. Sofia was one of the founders and cohosts of the Breaking Math Podcast. In this episode, host Gabriel Hesch interviews Diane Baca, mother of Sofia Baca as we talk about her passions for creativity, mathematics, science, and discovering what it means to be human.


Sofia lived an exceptional life with explosive creativity, a voracious passion for mathematics, physics, computer science, and creativity. Sofia also struggled immensely with mental health issues which included substance abuse as well as struggling for a very long time understand the source of their discontent. Sofia found great happiness in connecting with other people through teaching, tutoring, and creative expression. The podcast will continue in honor of Sofia. There are many folders of ideas that Sofia left with ideas for the show or for other projects. We will continue this show with sharing some of these ideas, but also with sharing stories of Sofia - including her ideas and her struggles in hopes that others may find solace in that they are not alone in their struggles. But also in hopes that others may find inspiration in what Sofia had to offer.

We miss you dearly, Sofia.

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Join Sofía Baca and her guests, the host and co-host of the Nerd Forensics podcast, Millicent Oriana and Jacob Urban, as they explore what it means to be able to solve one problem in multiple ways.

This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For full text, visit: https://creativecommons.org/licenses/by-sa/4.0/

[Featuring: Sofía Baca; Millicent Oriana, Jacob Urban[

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The history of mathematics, in many ways, begins with counting. Things that needed, initially, to be counted were, and often still are, just that; things. We can say we have twelve tomatoes, or five friends, or that eleven days have passed. As society got more complex, tools that had been used since time immemorial, such as string and scales, became essential tools for counting not only concrete things, like sheep and bison, but more abstract things, such as distance and weight based on agreed-upon multiples of physical artifacts that were copied. This development could not have taken place without the idea of a unit: a standard of measuring something that defines what it means to have one of something. These units can be treated not only as counting numbers, but can be manipulated using fractions, and divided into arbitrarily small divisions. They can even be multiplied and divided together to form new units. So where does the idea of a unit come from? What's the difference between a unit, a dimension, and a physical variable? And how does the idea of physical dimension allow us to simplify complex problems? All of this and more on this episode of Breaking Math.

Distributed under a CC BY-SA 4.0 International License. For full text, visit: https://creativecommons.org/licenses/by-sa/4.0/

[Featuring: Sofía Baca; Millicent Oriana, Jacob Urban]

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Join Sofia Baca and Nerd Forensics co-host Jacob Urban as they discuss all things counting!

Counting is the first arithmetic concept we learn, and we typically learn to do so during early childhood. Counting is the basis of arithmetic. Before people could manipulate numbers, numbers had to exist. Counting was first done on the body, before it was done on apparatuses outside the body such as clay tablets and hard drives. However, counting has become an invaluable tool in mathematics itself, as became apparent when counting started to be examined analytically. How did counting begin? What is the study of combinatorics? And what can be counted? All of this and more, on this episode of Breaking Math.

This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License (full text: https://creativecommons.org/licenses/by-sa/4.0/)

[Featuring: Sofia Baca; Jacob Urban]

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As you listen to this episode, you'll be exerting mental effort, as well as maybe exerting effort doing other things. The energy allowing your neurons to continually charge and discharge, as well as exert mechanical energy in your muscles and chemical energy in places like your liver and kidneys, came from the food you ate. Specifically, it came from food you chewed, and then digested with acid and with the help of symbiotic bacteria. And even if that food you're eating is meat, you can trace its energy back to the sun and the formation of the earth. Much of this was established in the previous episode, but this time we're going to explore a fundamental property of all systems in which heat can be defined. All of these structures had a certain order to them; the cow that might have made your hamburger had all the same parts that you do: stomach, lips, teeth, and brain. The plants, such as the tomatoes and wheat, were also complex structures, complete with signaling mechanisms. As you chewed that food, you mixed it, and later, as the food digested, it became more and more disordered; that is to say, it became more and more "shuffled", so to speak, and at a certain point, it became so shuffled that you'd need all the original information to reconstruct it: reversing the flow of entropy would mean converting vomit back into the original food; you'd need all the pieces. The electrical energy bonding molecules were thus broken apart and made available to you. And, if you're cleaning your room while listening to this, you are creating order only at the cost of destroying order elsewhere, since you are using energy from the food you ate. Even in industrial agriculture where from 350 megajoules of human and machine energy, often 140 gigajoules of corn can be derived per acre, a ratio of more than 400:1, the order that the seeds seem to produce from nowhere is constructed from the energy of the chaotic explosion from a nearby star. So why are the concepts of heat, energy, and disorder so closely linked? Is there a general law of disorder? And why does the second law mean you can't freeze eggs in a hot pan? All of this and more on this episode of Breaking Math.

Distributed under a CC BY-SA 4.0 License (https://creativecommons.org/licenses/by-sa/4.0/)

[Featuring: Sofia Baca; Millicent Oriana, Jacob Urban]

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Christopher Roblesz is a math educator who, until the pandemic, worked as a teacher. It was his experiences during the pandemic, and his unwavering passion for preparing disadvantaged youth for STEM careers, that eventually led him to developing mathnmore, a company focused on providing an enriched educational experience for sstudents who are preparing for these careers.More on energy and entropy next time!All of this and more on this interview episode of Breaking Math!

[Featuring: Sofia Baca; Christopher Roblesz]

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Join Sofia Baca and her guests Millicent Oriana from Nerd Forensics and Arianna Lunarosa as they discuss energy.

The sound that you're listening to, the device that you're listening on, and the cells in both the ear you're using to listen and the brain that understands these words have at least one thing in common: they represent the consumption or transference of energy. The same goes for your eyes if you're reading a transcript of this. The waves in the ears are pressure waves, while eyes receive information in the form of radiant energy, but they both are still called "energy". But what is energy? Energy is a scalar quantity measured in dimensions of force times distance, and the role that energy plays depends on the dynamics of the system. So what is the difference between potential and kinetic energy? How can understanding energy simplify problems? And how do we design a roller coaster in frictionless physics land?[Featuring: Sofia Baca; Millicent Oriana, Arianna Lunarosa]

This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. Full text here: https://creativecommons.org/licenses/by-sa/4.0/

Become a supporter of this podcast: https://www.spreaker.com/podcast/breaking-math-podcast--5545277/support.

The world around us is a four-dimensional world; there are three spatial dimensions, and one temporal dimension. Many of these objects emit an almost unfathomable number of photons. As we developed as creatures on this planet, we gathered the ability to sense the world around us; and given the amount of information represented as photons, it is no surprise that we developed an organ for sensing photons. But because of the amount of photons that are involved, and our relatively limited computational resources, it is necessary to develop shortcuts if we want to simulate an environment in silico. So what is raytracing? How is that different from what happens in games? And what does Ptolemy have to do with 3D graphics? All of this and more on this episode of Breaking Math.

Theme was Breaking Math Theme and outro was Breaking Math Outro by Elliot Smith of Albuquerque.

This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International license. License information can be found here: https://creativecommons.org/licenses/by-sa/4.0/

[Featuring: Sofía Baca, Gabriel Hesch]

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Physical objects are everywhere, and they're all made out of molecules, and atoms. However, the arrangement and refinement of these atoms can be the difference between a computer and sand, or between a tree and paper. For a species as reliant on tool use, the ability to conceieve of, design, create, and produce these materials is an ongoing concern. Since we've been around as humans, and even before, we have been material scientists in some regard, searching for new materials to make things out of, including the tools we use to make things. So what is the difference between iron and steel? How do we think up new things to make things out of? And what are time crystals? All of this and more on this episode of Breaking Math.

This episode is released under a Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) license. More information here: https://creativecommons.org/licenses/by-nc/4.0/

[Featuring: Sofía Baca, Gabriel Hesch; Taylor Sparks]

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Robert Black is an author who has written a six-book series about seven influential mathematicians, their lives, and their work. We interview him and his books, and take a peek into the lives of these influential mathematicians.

Addendum: Hey Breaking Math fans, I just wanted to let y'all know that the second material science podcast is delayed.

[Featuring: Sofía Baca; Robert Black]

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Matter is that which takes up space, and has mass. It is what we interact with, and what we are. Imagining a world without matter is to imagine light particles drifting aimlessly in space. Gasses, liquids, solids, and plasmas are all states of matter. Material science studies all of these, and their combinations and intricacies, found in examining foams, gels, meshes, and other materials and metamaterials. Chris Cogswell is a material scientist, and host of The Mad Scientist Podcast, a podcast that takes a critical look at things ranging from technological fads, to pseudoscience, and topics that deserve a critical eye. On the first of a pair of two episodes about material science, we interview Chris about his experience with studying material science, and ask questions about the subject in general.
Links referenced by Chris Cogswell:
- https://www.youtube.com/watch?v=bUvi5eQhPTc is about nanomagnetism and cool demonstration of ferrofluid
- https://www.youtube.com/watch?v=4Dlt63N-Uuk goes over nanomagnetic applications in medicine
- http://yaghi.berkeley.edu/pdfPublications/04MOFs.pdf Great review paper on new class of materials known as MOFs which are going to be very important in coming years
- https://www.youtube.com/watch?v=IkYimZBzguw Crash course engineering on nanomaterials, really good introduction to the field
- https://www.youtube.com/watch?v=t7EYQLOlwDM Oak Ridge national lab paper on using nano materials for carbon dioxide conversion to other carbon molecules
- https://www.youtube.com/watch?v=cxVFopLpIQY Really good paper on carbon capture technology challenges and economics
[Featuring: Sofía Baca, Gabriel Hesch, Meryl Flaherty; Chris Cogswell]

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Seldom do we think about self-reference, but it is a huge part of the world we live in. Every time that we say 'myself', for instance, we are engaging in self-reference. Long ago, the Liar Paradox and the Golden Ratio were among the first formal examples of self-reference. Freedom to refer to the self has given us fruitful results in mathematics and technology. Recursion, for example, is used in algorithms such as PageRank, which is one of the primary algorithms in Google's search engine. Elements of self-reference can also be found in foundational shifts in the way we understand mathematics, and has propelled our understanding of mathematics forward. Forming modern set theory was only possible due to a paradox called Russel's paradox, for example. Even humor uses self-reference. Realizing this, can we find harmony in self-reference? Even in a podcast intro, are there elements of self-reference? Nobody knows, but I'd check if I were you. Catch all of this, and more, on this episode of Breaking Math. Episode 70.1: Episode Seventy Point One of Breaking Math Podcast

[Featuring: Sofía Baca, Gabriel Hesch; Millicent Oriana]

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Hello, listeners! This is Sofia with an announcement.
Season 4 is about to start, and we have some great episodes planned. The last few weeks have been busy for us in our personal lives, and we apologize for our spotty release schedule lately. We're excited to bring you more of the content you've grown to love.
Today, we're going to have a rerun of our first episode on. This episode is a little rough at points, but we're choosing to rerun it because it captures the spirit of the podcast so elegantly. So, without further ado, here is Breaking Math episode 1: Forbidden Formulas.

[Featuring: Sofía Baca, Gabriel Hesch; Amy Lynn]

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On this problem episode, join Sofía and guest Diane Baca to learn about what an early attempt to formalize the natural numbers has to say about whether or not m+n equals n+m.

[Featuring: Sofía Baca; Diane Baca]

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Michael Brooks is a science writer who specializes in making difficult concepts easier to grasp. In his latest book, Brooks goes through several mathematical concepts and discusses their motivation, history, and discovery. So how do stories make it easier to learn? What are some of the challenges associated with conveying difficult concepts to the general public? And who, historically, has been a mathematician? All of this and more on this episode of Breaking Math.  Songs were Breaking Math Intro and Outro by Elliot Smith of Albuquerque.  This episode is published under a Creative Commons 4.0 Attribute-ShareAlike-NonCommercial license. For more information, visit CreativeCommons.org  [Featuring: Sofía Baca, Gabriel Hesch, Meryl Flaherty; Michael Brooks]

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There are times in mathematics when we are generalizing the behavior of many different, but similar, entities. One such time that this happens is the use cases of Big O notation, which include describing the long-term behavior of functions, and talking about how accurate numerical calculations are. On this problem episode, we are going to discuss Big O notation and how to use it.

This episode is licensed by Sofia Baca under a Creative Commons Attribution-ShareAlike-NonCommercial 4.0 International License. For more information, visit CreativeCommons.org.
[Featuring: Sofía Baca]

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The world is often uncertain, but it has only been in the last half millennium that we've found ways to interact mathematically with that concept. From its roots in death statistics, insurance, and gambling to modern Bayesian networks and machine learning, we've seen immense productivity in this field. Every way of looking at probability has something in common: the use of random variables. Random variables let us talk about events with uncertain outcomes in a concrete way. So what are random variables? How are they defined? And how do they interact? All of this, and more, on this episode of Breaking Math.

Interact with the hosts:
@SciPodSofia
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Featuring music by Elliot Smith. For info about music used in ads, which are inserted dynamically, contact us at [email protected]

[Featuring: Sofía Baca, Gabriel Hesch; Millicent Oriana]

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Mathematics is a subject that has been used for great things over time: it has helped people grow food, design shelter, and in every part of life. It should be, then, no surprise that sometimes mathematics is used for evil; that is to say, there are times where mathematics is used to either implement or justify regressive things like greed, racism, classism, and even genocide. So when has math been used for destructive purposes? What makes us mis-apply mathematics? And why can oversimplification lead to devastation? All of this, and more, on this episode of Breaking Math.

Theme song is Breaking Math Theme by Elliot Smith of Albuquerque.

This episode is distributed under a Creative Commons Attribution-ShareAlike-NonCommercial 4.0 International License. For more information, go to CreativeCommons.org

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Join Sofía Baca with her guest Millicent Oriana from the newly launched Nerd Forensics podcast as they discuss some apparent paradoxes in probability and Russian roulette.

Intro is "Breaking Math Theme" by Elliot Smith. Ads feature "Ding Dong" by Simon Panrucker

[Featuring: Sofía Baca; Millicent Oriana]

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Katharine Hayhoe was the lead author on the 2018 US Climate Assessment report, and has spent her time since then spreading the word about climate change. She was always faced with the difficult task of convincing people who had stakes in things that would be affected by acknowledging the information in her report. In her newest book, ?Saving Us: A Climate Scientist?s Case for Hope and Healing in a Divided World?, she discusses the challenges associated with these conversations, at both the micro and macro level. So who is Katherine Heyhoe? How has she learned to get people to acknowledge the reality of climate science? And is she the best, or worst, person to strike up a discussion about how the weather?s been? All of this, and more, on this episode of Breaking Math. Papers Cited: -?99.94 percent of papers agree with the scientific consensus.?

More info: https://journals.sagepub.com/doi/10.1177/0270467617707079

This episode is distributed under a CC BY-NC 4.0 International License. For more information, visit creativecommons.org.
Intro is "Breaking Math Theme" by Elliot Smith. Ads feature "Ding Dong" by Simon Panrucker

[Featuring: Sofía Baca, Gabriel Hesch, Meryl Flaherty; Katherine Heyhoe, Elliot Smith]

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One tells a lie, the other the truth! Have fun with Sofía and Meryl as they investigate knight, knave, and spy problems!

Intro is "Breaking Math Theme" by Elliot Smith. Music in the ads were Plug Me In by Steve Combs and "Ding Dong" by Simon Panrucker. You can access their work at freemusicarchive.org.

[Featuring: Sofia Baca; Meryl Flaherty]

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Ian is an author who has written many math and science books, and collaborated with Terry Pratchett.

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The world is a big place with a lot of wonderful things in it. The world also happens to be spherical, which can make getting to those things a challenge if you don't have many landmarks. This is the case when people are navigating by sea. For this reason, map projections, which take a sphere and attempt to flatten it onto a sheet, were born. So what is a map projection? Why are there so many? And why is Gall-Peters the worst? All of this, and more, on this episode of Breaking Math.
Theme was written by Elliot Smith.
This episode is distributed under a Creative Commons 4.0 Attribution-ShareAlike-NonCommercial International License. For more information, visit CreativeCommons.org.

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This is a rerun of one of our favorite episodes! We hope that you enjoy it if you haven't listened to it yet. We'll be back next week with new content! Thank you so much for listening to Breaking Math!

Math is a gravely serious topic which has been traditionally been done by stodgy people behind closed doors, and it cannot ever be taken lightly. Those who have fun with mathematics mock science, medicine, and the foundation of engineering. That is why on today's podcast, we're going to have absolutely no fun with mathematics. There will not be a single point at which you consider yourself charmed, there will not be a single thing you will want to tell anyone for the sake of enjoyment, and there will be no tolerance for your specific brand of foolishness, and that means you too, Kevin.

Theme by Elliot Smith.

Distributed under a CC BY-SA-NC 4.0 license. For more information visit CreativeCommons.org

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Voting systems are, in modern times, essential to the way that large-scale decisions are made. The concept of voicing an opinion to be, hopefully, considered fairly is as ancient and well-established as the human concept of society in general. But, as time goes on, the recent massive influx of voting systems in the last 150 years have shown us that there are as many ways to vote as there are flaws in the way that the vote is tallied. So what problems exist with voting? Are there any intrinsic weaknesses in group decision-making systems? And what can we learn by examining these systems? All of this, and more, on this episode of Breaking Math.

Licensed under Creative Commons Attribution-ShareAlike-NonCommercial 4.0 International License. For more information, visit CreativeCommons.org.

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Forecasting is a constantly evolving science, and has been applied to complex systems; everything from the weather, to determining what customers might like to buy, and even what governments might rise and fall. John Fuisz is someone who works with this science, and has experience improving the accuracy of forecasting. So how can forecasting be analyzed? What type of events are predictable? And why might Russia think a Missouri senator's race hinges upon North Korea? All of this and more on this episode of Breaking Math.

The theme for this episode was written by Elliot Smith.

[Featuring: Sofía Baca, Gabriel Hesch; John Fuisz]

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This is a rerun of one of our favorite episodes while we change our studio around.
Game theory is all about decision-making and how it is impacted by choice of strategy, and a strategy is a decision that is influenced not only by the choice of the decision-maker, but one or more similar decision makers. This episode will give an idea of the type of problem-solving that is used in game theory. So what is strict dominance? How can it help us solve some games? And why are The Obnoxious Seven wanted by the police?
Distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For more information, visit CreativeCommons.or
[Featuring: Sofía Baca; Diane Baca]

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In mathematics, nature is a constant driving inspiration; mathematicians are part of nature, so this is natural. A huge part of nature is the idea of things like networks. These are represented by mathematical objects called 'graphs'. Graphs allow us to describe a huge variety of things, such as: the food chain, lineage, plumbing networks, electrical grids, and even friendships. So where did this concept come from? What tools can we use to analyze graphs? And how can you use graph theory to minimize highway tolls? All of this and more on this episode of Breaking Math.
Episode distributed under an Creative Commons Attribution-ShareAlike-NonCommercial 4.0 International License. For more information, visit CreativeCommons.org
[Featuring: Sofía Baca, Meryl Flaherty]

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How many piano tuners are there in New York City? How much cheese is there in Delaware? And how can you find out? All of this and more on this problem-episode of Breaking Math.
This episode distributed under a Creative Commons Attribution-ShareAlike-Noncommercial 4.0 International License. For more information, visit creativecommons.org
Featuring theme song and outro by Elliot Smith of Albuquerque.

[Featuring: Sofía Baca, Meryl Flaherty]

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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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Mathematics is full of all sorts of objects that can be difficult to comprehend. For example, if we take a slip of paper and glue it to itself, we can get a ring. If we turn it a half turn before gluing it to itself, we get what's called a Möbius strip, which has only one side twice the length of the paper. If we glue the edges of the Möbius strip to each other, and make a tube, you'll run into trouble in three dimensions, because the object that this would make is called a Klein flask, and can only exist in four dimensions. So what is a fiber? What can fiber bundles teach us about higher dimensional objects?

All of this, and more, on this episode of Breaking Math.

[Featuring: Sofía Baca, Meryl Flaherty]

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In introductory geometry classes, many of the objects dealt with can be considered 'elementary' in nature; things like tetrahedrons, spheres, cylinders, planes, triangles, lines, and other such concepts are common in these classes. However, we often have the need to describe more complex objects. These objects can often be quite organic, or even abstract in shape, and include things like spirals, flowery shapes, and other curved surfaces. These are often described better by differential geometry as opposed to the more elementary classical geometry. One helpful metric in describing these objects is how they are curved around a certain point. So how is curvature defined mathematically? What is the difference between negative and positive curvature? And what can Gauss' Theorema Egregium teach us about eating pizza?

This episode distributed under a Creative Commons Attribution ShareAlike 4.0 International License. For more information, go to creativecommons.org

Visit our sponsor today at Brilliant.org/BreakingMath for 20% off their annual membership! Learn hands-on with Brilliant.

[Featuring: Sofía Baca, Meryl Flaherty]

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Join Sofía and Gabriel as they talk about Morikawa's recently solved problem, first proposed in 1821 and not solved until last year!

Also, if you haven't yet, check out our sponsor The Great Courses at thegreatcoursesplus.com/breakingmath for a free month! Learn basically anything there.

The paper featured in this episode can be found at https://arxiv.org/abs/2008.00922

This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For more information, visit CreativeCommons.org!

[Featuring: Sofía Baca, Gabriel Hesch]

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Join Sofía and Gabriel as they discuss an old but great proof of the irrationality of the square root of two.

[Featuring: Sofía Baca, Gabriel Hesch]

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Ad contained music track "Buffering" from Quiet Music for Tiny Robots.
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If you are there, and I am here, we can measure the distance between us. If we are standing in a room, we can calculate the area of where we're standing; and, if we want, the volume. These are all examples of measures; which, essentially, tell us how much 'stuff' we have. So what is a measure? How are distance, area, and volume related? And how big is the Sierpinski triangle? All of this and more on this episode of Breaking Math.
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The theme for this episode was written by Elliot Smith.
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[Featuring: Sofía Baca; Meryl Flaherty]

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Sofía and Gabriel discuss the question of "how many angles are there in a circle", and visit theorems from Euclid, as well as differential calculus.
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The theme for this episode was written by Elliot Smith.
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[Featuring: Sofía Baca, Gabriel Hesch]

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Look at all you phonies out there.
You poseurs.
All of you sheep. Counting 'til infinity. Counting sheep.
*pff*
What if I told you there were more there? Like, ... more than you can count?
But what would a sheeple like you know about more than infinity that you can count?
heh. *pff*
So, like, what does it mean to count til infinity? What does it mean to count more? And, like, where do dimensions fall in all of this?

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(Correction: at 12:00, the paradox is actually due to Galileo Galilei)
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[Featuring: Sofía Baca, Gabriel Hesch]

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As a child, did you ever have a conversation that went as follows:
"When I grow up, I want to have a million cats"
"Well I'm gonna have a billion billion cats"
"Oh yeah? I'm gonna have infinity cats"
"Then I'm gonna have infinity plus one cats"
"That's nothing. I'm gonna have infinity infinity cats"
"I'm gonna have infinity infinity infinity infinity *gasp* infinity so many infinities that there are infinity infinities plus one cats"
What if I told you that you were dabbling in the transfinite ordinal numbers? So what are ordinal numbers? What does "transfinite" mean? And what does it mean to have a number one larger than another infinite number?

[Featuring: Sofía Baca; Diane Baca]

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This episode is released under a Creative Commons attribution sharealike 4.0 international license. For more information, go to CreativeCommoms.org
This episode features the song "Buffering" by "Quiet Music for Tiny Robots"

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There are a lot of things in the universe, but no matter how you break them down, you will still have far fewer particles than even some of the smaller of what we're calling the 'very large numbers'. Many people have a fascination with these numbers, and perhaps it is because their sheer scale can boggle the mind. So what numbers can be called 'large'? When are they useful? And what is the Ackermann function? All of this and more on this episode of Breaking Math

[Featuring: Sofía Baca; Diane Baca]

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