Tom Chivers discusses his book 'Everything is Predictable: How Bayesian Statistics Explain Our World' and the applications of Bayesian statistics in various fields. He explains how Bayesian reasoning can be used to make predictions and evaluate the likelihood of hypotheses. Chivers also touches on the intersection of AI and ethics, particularly in relation to AI-generated art. The conversation explores the history of Bayes' theorem and its role in science, law, and medicine. Overall, the discussion highlights the power and implications of Bayesian statistics in understanding and navigating the world.

The conversation explores the role of AI in prediction and the importance of Bayesian thinking. It discusses the progress of AI in image classification and the challenges it still faces, such as accurately depicting fine details like hands. The conversation also delves into the topic of predictions going wrong, particularly in the context of conspiracy theories. It highlights the Bayesian nature of human beliefs and the influence of prior probabilities on updating beliefs with new evidence. The conversation concludes with a discussion on the relevance of Bayesian statistics in various fields and the need for beliefs to have probabilities and predictions attached to them.

Takeaways

• Bayesian statistics can be used to make predictions and evaluate the likelihood of hypotheses.
• Bayes' theorem has applications in various fields, including science, law, and medicine.
• The intersection of AI and ethics raises complex questions about AI-generated art and the predictability of human behavior.
• Understanding Bayesian reasoning can enhance decision-making and critical thinking skills. AI has made significant progress in image classification, but still faces challenges in accurately depicting fine details.
• Predictions can go wrong due to the influence of prior beliefs and the interpretation of new evidence.
• Beliefs should have probabilities and predictions attached to them, allowing for updates with new information.
• Bayesian thinking is crucial in various fields, including AI, pharmaceuticals, and decision-making.
• The importance of defining predictions and probabilities when engaging in debates and discussions.

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Summary

**Tensor Poster - If you are interested in the Breaking Math Tensor Poster on the mathematics of General Relativity, email us at BreakingMathPodcast@gmail.com

In this episode, Gabriel Hesch and Autumn Phaneuf interview Steve Nadis, the author of the book 'The Gravity of Math.' They discuss the mathematics of gravity, including the work of Isaac Newton and Albert Einstein, gravitational waves, black holes, and recent developments in the field. Nadis shares his collaboration with Shing-Tung Yau and their journey in writing the book. They also talk about their shared experience at Hampshire College and the importance of independent thinking in education. In this conversation, Steve Nadis discusses the mathematical foundations of general relativity and the contributions of mathematicians to the theory. He explains how Einstein was introduced to the concept of gravity by Bernhard Riemann and learned about tensor calculus from Gregorio Ricci and Tullio Levi-Civita. Nadis also explores Einstein's discovery of the equivalence principle and his realization that a theory of gravity would require accelerated motion. He describes the development of the equations of general relativity and their significance in understanding the curvature of spacetime. Nadis highlights the ongoing research in general relativity, including the detection of gravitational waves and the exploration of higher dimensions and black holes. He also discusses the contributions of mathematician Emmy Noether to the conservation laws in physics. Finally, Nadis explains Einstein's cosmological constant and its connection to dark energy.

Chapters

00:00 Introduction and Book Overview

08:09 Collaboration and Writing Process

25:48 Interest in Black Holes and Recent Developments

35:30 The Mathematical Foundations of General Relativity

44:55 The Curvature of Spacetime and the Equations of General Relativity

56:06 Recent Discoveries in General Relativity

01:06:46 Emmy Noether's Contributions to Conservation Laws

01:13:48 Einstein's Cosmological Constant and Dark Energy

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In the United States, the fourth of July is celebrated as a national holiday, where the focus of that holiday is the war that had the end effect of ending England’s colonial influence over the American colonies. To that end, we are here to talk about war, and how it has been influenced by mathematics and mathematicians. The brutality of war and the ingenuity of war seem to stand at stark odds to one another, as one begets temporary chaos and the other represents lasting accomplishment in the sciences. Leonardo da Vinci, one of the greatest western minds, thought war was an illness, but worked on war machines. Feynman and Von Neumann held similar views, as have many over time; part of being human is being intrigued and disgusted by war, which is something we have to be aware of as a species. So what is warfare? What have we learned from refining its practice? And why do we find it necessary?

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]

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]

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]

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

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[Featuring: Sofía Baca, Meryl Flaherty]

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

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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]

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

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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.

Episode used in the ad was Buffering by Quiet Music for Tiny Robots.

[Featuring: Sofía Baca; Meryl Flaherty]

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.

This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.

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The theme for this episode was written by Elliot Smith.

Music in the ad was Tiny Robot Armies by Quiet Music for Tiny Robots.

[Featuring: Sofía Baca, Gabriel Hesch]

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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Music used in the The Great Courses ad was Portal by Evan Shaeffer

[Featuring: Sofía Baca, Gabriel Hesch]

The fabric of the natural world is an issue of no small contention: philosophers and truth-seekers universally debate about and study the nature of reality, and exist as long as there are observers in that reality. One topic that has grown from a curiosity to a branch of mathematics within the last century is the topic of cellular automata. Cellular automata are named as such for the simple reason that they involve discrete cells (which hold a (usually finite and countable) range of values) and the cells, over some field we designate as "time", propagate to simple automatic rules. So what can cellular automata do? What have we learned from them? And how could they be involved in the future of the way we view the world?

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"

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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Neuroscience is a topic that, in many ways, is in its infancy. The tools that are being used in this field are constantly being honed and reevaluated as our understanding of the brain and mind increase. And it's no surprise: the brain is responsible for the way we interact with the world, and the idea that ideas hone one another is not new to anyone who possesses a mind. But how can the tools that we use to study the brain and the mind be linked? How do the mind and the brain encode one another? And what does Bayes have to do with this? All of this and more on this episode of Breaking Math.

[Featuring: Sofía Baca, Gabriel Hesch; Peter Zeidman]

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This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.

Spheres and circles are simple objects. They are objects that are uniformly curved throughout in some way or another. They can also be defined as objects which have a boundary that is uniformly distant from some point, using some definition of distance. Circles and spheres were integral to the study of mathematics at least from the days of Euclid, being the objects generated by tracing the ends of idealized compasses. However, these objects have many wonderful and often surprising mathematical properties. To this point, a circle's circumference divided by its diameter is the mathematical constant pi, which has been a topic of fascination for mathematicians for as long as circles have been considered.

[Featuring Sofía Baca; Meryl Flaherty]

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Join Sofia and Gabriel on this problem episode where we explore "base 3-to-2" — a base system we explored on the last podcast — and how it relates to "base 3/2" from last episode.

[Featuring: Sofía Baca; Gabriel Hesch]

A numerical base is a system of representing numbers using a sequence of symbols. However, like any mathematical concept, it can be extended and re-imagined in many different forms. A term used occasionally in mathematics is the term 'exotic', which just means 'different than usual in an odd or quirky way'. In this episode we are covering exotic bases. We will start with something very familiar (viz., decimal points) as a continuation of our previous episode, and then progress to the more odd, such as non-integer and complex bases. So how can the base systems we covered last time be extended to represent fractional numbers? How can fractional numbers be used as a base for integers? And what is pi plus e times i in base i + 1?

This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.

[Featuring: Sofía Baca; Merryl Flaherty]

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Numbering was originally done with tally marks: the number of tally marks indicated the number of items being counted, and they were grouped together by fives. A little later, people wrote numbers down by chunking the number in a similar way into larger numbers: there were symbols for ten, ten times that, and so forth, for example, in ancient Egypt; and we are all familiar with the Is, Vs, Xs, Ls, Cs, and Ds, at least, of Roman numerals. However, over time, several peoples, including the Inuit, Indians, Sumerians, and Mayans, had figured out how to chunk numbers indefinitely, and make numbers to count seemingly uncountable quantities using the mind, and write them down in a few easily mastered motions. These are known as place-value systems, and the study of bases has its root in them: talking about bases helps us talk about what is happening when we use these magical symbols.

Machines, during the lifetime of anyone who is listening to this, have advanced and revolutionized the way that we live our lives. Many listening to this, for example, have lived through the rise of smart phones, 3d printing, massive advancements in lithium ion batteries, the Internet, robotics, and some have even lived through the introduction of cable TV, color television, and computers as an appliance. All advances in machinery, however, since the beginning of time have one thing in common: they make what we want to do easier. One of the great tragedies of being imperfect entities, however, is that we make mistakes. Sometimes those mistakes can lead to war, famine, blood feuds, miscalculation, the punishment of the innocent, and other terrible things. It has, thus, been the goal of many, for a very long time, to come up with a system for not making these mistakes in the first place: a thinking machine, which would help eliminate bias in situations. Such a fantastic machine is looking like it's becoming closer and closer to reality, especially with the advancements in artificial intelligence. But what are the origins of this fantasy? What attempts have people made over time to encapsulate reason? And what is ultimately possible with the automated manipulation of meaning? All of this and more on this episode of Breaking Math. Episode 48: Thinking Machines References: * https://publicdomainreview.org/essay/let-us-calculate-leibniz-llull-and-the-computational-imagination * https://spectrum.ieee.org/tag/history+of+natural+language+processing https://en.wikipedia.org/wiki/Characteristica_universalis https://ourworldindata.org/coronavirus-source-data This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org. [Featuring: Sofía Baca, Gabriel Hesch]

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Join Gabriel and Sofía as they delve into some introductory calculus concepts.

[Featuring: Sofía Baca, Gabriel Hesch]

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The spectre of disease causes untold mayhem, anguish, and desolation. The extent to which this spectre has yielded its power, however, has been massively curtailed in the past century. To understand how this has been accomplished, we must understand the science and mathematics of epidemiology. Epidemiology is the field of study related to how disease unfolds in a population. So how has epidemiology improved our lives? What have we learned from it? And what can we do to learn more from it?

Time is something that everyone has an idea of, but is hard to describe. Roughly, the arrow of time is the same as the arrow of causality. However, what happens when that is not the case? It is so often the case in our experience that this possibility brings not only scientific and mathematic, but ontological difficulties. So what is retrocausality? What are closed timelike curves? And how does this all relate to entanglement?

This episode is distributed under a CC BY-SA 4.0 license. For more information, visit CreativeCommons.org.

[Featuring: Sofía Baca, Gabriel Hesch]

Learn more about radiative forcing, the environment, and how global temperature changes with atmospheric absorption with this Problem Episode about you walking your (perhaps fictional?) dog around a park.  This episode is distributed under a CC BY-SA license. For more information, visit CreativeCommons.org.

[Featuring: Sofía Baca, Gabriel Hesch]

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Since time immemorial, blacksmiths have known that the hotter metal gets, the more it glows: it starts out red, then gets yellower, and then eventually white. In 1900, Max Planck discovered the relationship between an ideal object's radiation of light and its temperature. A hundred and twenty years later, we're using the consequences of this discovery for many things, including (indirectly) LED TVs, but perhaps one of the most dangerously neglected (or at least ignored) applications of this theory is in climate science. So what is the greenhouse effect? How does blackbody radiation help us design factories? And what are the problems with this model?

This episode is distributed under a CC BY-SA license. For more information, visit CreativeCommons.org.

[Featuring: Sofía Baca, Gabriel Hesch]

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Climate change is an issue that has become frighteningly more relevant in recent years, and because of special interests, the field has become muddied with climate change deniers who use dishonest tactics to try to get their message across. The website SkepticalScience.com is one line of defense against these messengers, and it was created and maintained by a research assistant professor at the Center for Climate Change Communication at George Mason University, and both authored and co-authored two books about climate science with an emphasis on climate change. He also lead-authored a 2013 award-winning paper on the scientific consensus on climate change, and in 2015, he developed an open online course on climate change denial with the Global Change Institute at the University of Queensland. This person is John Cook.

This episode is distributed under a CC BY-SA license. For more information, visit CreativeCommons.org.

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

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Mathematics, like any intellectual pursuit, is a constantly-evolving field; and, like any evolving field, there are both new beginnings and sudden unexpected twists, and things take on both new forms and new responsibilities. Today on the show, we're going to cover a few mathematical topics whose nature has changed over the centuries. So what does it mean for math to be extinct? How does this happen? And will it continue forever?

This episode is distributed under a CC BY-SA license. For more information, visit CreativeCommons.org.

[Featuring: Sofía Baca, Gabriel Hesch]

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Learn more about calculus, derivatives, and the chain rule with this Problem Episode about you walking your (perhaps fictional?) dog around a park.

This episode is distributed under a CC BY-SA license. For more information, visit CreativeCommons.org.

[Featuring: Sofía Baca, Gabriel Hesch]

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Ben Orlin has been a guest on the show before. He got famous with a blog called 'Math With Bad Drawings", which is what it says on the tin: he teaches mathematics using his humble drawing skills. His last book was a smorgasbord of different mathematical topics, but he recently came out with a new book 'Change is the Only Constant: the Wisdom of Calculus in a Madcap World', which focuses more on calculus itself.

This episode is distributed under a CC BY-SA license. For more info, visit creativecommons.org

Statistics is a field that is considered boring by a lot of people, including a huge amount of mathematicians. This may be because the history of statistics starts in a sort of humdrum way: collecting information on the population for use by the state. However, it has blossomed into a beautiful field with its fundamental roots in measure theory, and with some very interesting properties. So what is statistics? What is Bayes' theorem? And what are the differences between the frequentist and Bayesian approaches to a problem?

Distributed under a Creative Commons Attribution-ShareAlike 4.0 International License (creativecommons.org)

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Children who are being taught mathematics often balk at the idea of negative numbers, thinking them to be fictional entities, and often only learn later that they are useful for expressing opposite extremes of things, such as considering a debt an amount of money with a negative sum. Similarly, students of mathematics often are puzzled by the idea of complex numbers, saying that it makes no sense to be able to take the square root of something negative, and only realizing later that these can have the meaning of two-dimensional direction and magnitude, or that they are essential to our modern understanding of electrical engineering. Our discussion today will be much more abstract than that. Much like in our discussion in episode five, "Language of the Universe", we will be discussing how math and physics draw inspiration from one another; we're going to talk about what different fields (such as the real, complex, and quaternion fields) seem to predict about our universe. So how are real numbers related to classical mechanics? What does this mean complex numbers and quaternions are related to? And what possible physicses exist?

Update:  Dr. Alex Alaniz and the Breaking Math Podcast have teamed up to create a new youtube show called the "Turing Rabbit Holes Podcast."  We discuss science, math, and society with spectacular visuals.    Available at youtube.com/TuringRabbitHolesPodcast and on all other podcast platforms.  

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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.

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.

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We communicate every day through languages; not only human languages, but other things that could be classified as languages such as internet protocols, or even the structure of business transactions. The structure of words or sentences, or their metaphorical equivalents, in that language is known as their syntax. There is a way to describe certain syntaxes mathematically through what are known as formal grammars. So how is a grammar defined mathematically? What model of language is often used in math? And what are the fundamental limits of grammar?

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?

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Hello listeners. You don't know me, but I know you. I want to play a game. In your ears are two earbuds. Connected to the earbuds are a podcast playing an episode about game theory. Hosting that podcast are two knuckleheads. And you're locked into this episode. The key is at the end of the episode. What is game theory? Why did we parody the Saw franchise? And what twisted lessons will you learn?

-See our New Youtube Show "Turing Rabbit Holes Podcast" at youtube.com/TuringRabbitHolesPodcast.   Also available on all podcast players.  

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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.

Centuries ago, there began something of a curiosity between mathematicians that didn't really amount to much but some interesting thoughts and cool mathematical theorems. This form of math had to do with strictly integer quantities; theorems about whole numbers. Things started to change in the 19th century with some breakthroughs in decrypting intelligence through examining the frequency of letters. In the fervor that followed to increase the security of existing avenues of communication, and to speed up the newfound media of telegraphy, came a field of mathematics called discrete math. It is now an essential part of our world today, with technologies such as online banking being essentially impossible without it. So what have we learned from discrete math? What are some essential methods used within it? And how is it applied today?

An interview with Ben Orlin, author of the book 'Math with Bad Drawings,' as well as the blog of the same name.  The blog can be found at www.mathwithbaddrawings.com.

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The hosts of Breaking Math had too much time on their hands.

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A lot of the information in this episode of Breaking Math depends on episodes 30 and 31 entitled "The Abyss" and "Into the Abyss" respectively. If you have not listened to those episodes, then we'd highly recommend going back and listening to those. We're choosing to present this information this way because otherwise we'd waste most of your time re-explaining concepts we've already covered.

Black holes are so bizarre when we measured against the yardstick of the mundanity of our day to day lives that they inspire fear, awe, and controversy. In this last episode of the Abyss series, we will look at some more cutting-edge problems and paradoxes surrounding black holes. So how are black holes and entanglement related? What is the holographic principle? And what is the future of black holes?

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The history of physics as a natural science is filled with examples of when an experiment will demonstrate something or another, but what is often forgotten is the fact that the experiment had to be thought up in the first place by someone who was aware of more than one plausible value for a property of the universe, and realized that there was a way to word a question in such a way that the universe could understand. Such a property was debated during the quantum revolution, and involved Einstein, Polodsky, Rosen, and Schrödinger. The question was 'do particles which are entangled "know" the state of one another from far away, or do they have a sort of "DNA" which infuses them with their properties?' The question was thought for a while to be purely philosophical one until John Stewart Bell found the right way to word a question, and proved it in a laboratory of thought. It was demonstrated to be valid in a laboratory of the universe. So how do particles speak to each other from far away? What do we mean when we say we observe something? And how is a pair of gloves like and unlike a pair of walkie talkies?

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)

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

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

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Summary:

A paradox is characterized either by a logical problem that does not have a single dominant expert solution, or by a set of logical steps that seem to lead somehow from sanity to insanity. This happens when a problem is either ill-defined, or challenges the status quo. The thing that all paradoxes, however, have in common is that they increase our understanding of the phenomena which bore them. So what are some examples of paradox? How does one go about resolving it? And what have we learned from paradox?

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Information theory was founded in 1948 by Claude Shannon, and is a way of both qualitatively and quantitatively describing the limits and processes involved in communication. Roughly speaking, when two entities communicate, they have a message, a medium, confusion, encoding, and decoding; and when two entities communicate, they transfer information between them. The amount of information that is possible to be transmitted can be increased or decreased by manipulating any of the aforementioned variables. One of the practical, and original, applications of information theory is to models of language. So what is entropy? How can we say language has it? And what structures within language with respect to information theory reveal deep insights about the nature of language itself?

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This episode is sponsored by 

· Anchor: The easiest way to make a podcast.  https://anchor.fm/app

Support this podcast: https://anchor.fm/breakingmathpodcast/support

In the study of mathematics, there are many abstractions that we deal with. For example, we deal with the notion of a real number with infinitesimal granularity and infinite range, even though we have no evidence for this existing in nature besides the generally noted demi-rules 'smaller things keep getting discovered' and 'larger things keep getting discovered'. In a similar fashion, we define things like circles, squares, lines, planes, and so on. Many of the concepts that were just mentioned have to do with geometry; and perhaps it is because our brains developed to deal with geometric information, or perhaps it is because geometry is the language of nature, but there's no doubt denying that geometry is one of the original forms of mathematics. So what defines geometry? Can we make progress indefinitely with it? And where is the line between geometry and analysis?

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This episode is sponsored by 

· Anchor: The easiest way to make a podcast.  https://anchor.fm/app

Support this podcast: https://anchor.fm/breakingmathpodcast/support