Episodes
 00:52:40
23: Don't Touch My Circles!
Breaking Math Podcast starstarstarstarstar addIn 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 demirules '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?
 00:56:31
22: Incomplet
Breaking Math Podcast starstarstarstarstar addGödel, Escher, Bach is a book about everything from formal logic to the intricacies underlying the mechanisms of reasoning. For that reason, we've decided to make a tribute episode; specifically, about episode IV. There is a Sanskrit word "maya" which describes the difference between a symbol and that which it symbolizes. This episode is going to be all about the math of maya. So what is a string? How are formal systems useful? And why do we study them with such vigor?

Missing episodes?
 01:02:38
21: Einstein's Biggest Idea
Breaking Math Podcast starstarstarstarstar addSome see the world of thought divided into two types of ideas: evolutionary and revolutionary ideas. However, the truth can be more nuanced than that; evolutionary ideas can spur revolutions, and revolutionary ideas may be necessary to create incremental advancements. General relativity is an idea that was evolutionary mathematically, revolutionary physically, and necessary for our modern understanding of the cosmos. Devised in its full form first by Einstein, and later proven correct by experiment, general relativity gives us a framework for understanding not only the relationship between mass and energy and space and time, but topology and destiny. So why is relativity such an important concept? How do special and general relativity differ? And what is meant by the equation G=8πT?
 00:40:22
20: Rational
Breaking Math Podcast starstarstarstarstar addFrom MC²’s statement of mass energy equivalence and Newton’s theory of gravitation to the sex ratio of bees and the golden ratio, our world is characterized by the ratios which can be found within it. In nature as well as in mathematics, there are some quantities which equal one another: every action has its equal and opposite reaction, buoyancy is characterized by the displaced water being equal to the weight of that which has displaced it, and so on. These are characterized by a qualitative difference in what is on each side of the equality operator; that is to say: the action is equal but opposite, and the weight of water is being measured versus the weight of the buoyant object. However, there are some formulas in which the equality between two quantities is related by a constant. This is the essence of the ratio. So what can be measured with ratios? Why is this topic of importance in science? And what can we learn from the mathematics of ratios?
 00:39:51
19: Tune of the Hickory Stick
Breaking Math Podcast starstarstarstarstar addFrom MC2’s statement of mass energy equivalence and Newton’s theory of gravitation to the sex ratio of bees and the golden ratio, our world is characterized by the ratios which can be found within it. In nature as well as in mathematics, there are some quantities which equal one another: every action has its equal and opposite reaction, buoyancy is characterized by the displaced water being equal to the weight of that which has displaced it, and so on. These are characterized by a qualitative difference in what is on each side of the equality operator; that is to say: the action is equal but opposite, and the weight of water is being measured versus the weight of the buoyant object. However, there are some formulas in which the equality between two quantities is related by a constant. This is the essence of the ratio. So what can be measured with ratios? Why is this topic of importance in science? And what can we learn from the mathematics of ratios?
 00:44:05
18: Frequency
Breaking Math Podcast starstarstarstarstar addDuration and proximity are, as demonstrated by Fourier and later Einstein and Heisenberg, very closely related properties. These properties are related by a fundamental concept: frequency. A high frequency describes something which changes many times in a short amount of space or time, and a lower frequency describes something which changes few times in the same time. It is even true that, in a sense, you can ‘rotate’ space into time. So what have we learned from frequencies? How have they been studied? And how do they relate to the rest of mathematics?
 01:00:24
17: Navier Stoked
Breaking Math Podcast starstarstarstarstar addFrom our first breath of the day to brushing our teeth to washing our faces to our first sip of coffee, and even in the waters of the rivers we have built cities upon since antiquity, we find ourselves surrounded by fluids. Fluids, in this context, mean anything that can take the shape of its container. Physically, that means anything that has molecules that can move past one another, but mathematics has, as always, a slightly different view. This view is seen by some as more nuanced, others as more statistical, but by all as a challenge. This definition cannot fit into an introduction, and I’ll be picking away at it for the remainder of this episode. So what is a fluid? What can we learn from it? And how could learning from it be worth a million dollars?
 01:09:41
BFNB2: Thought for Food
Breaking Math Podcast starstarstarstarstar addSponsored by www.brilliant.org/breakingmath, where you can take courses in calculus, computer science, chemistry, and other STEM subjects. All online; all at your own pace; and accessible anywhere with an internet connection, including your smartphone or tablet! Start learning today!
Check out: https://blankfornonblank.podiant.co/e/357f09da787bac/
What you're about to hear is part two of an episode recorded by the podcasting network ___forNon___ (Blank for NonBlank), of which Breaking Math, along with several other podcasts, is a part. To check out more ___forNon___ content, you can click on the link in this description. And of course, for more info and interactive widgets you can go to breakingmathpodcast.com, you can support us at patreon.com/breakingmathpodcast, and you can contact us directly at breakingmathpodcast@gmail.com. We hope you enjoy the second part of the first ___forNon___ group episode. You can also support ___forNon___ by donating at patreon.com/blankfornonblank.  00:33:38
BFNB1: Food for Thought
Breaking Math Podcast starstarstarstarstar addThis is the first group podcast for the podcasting network ___forNon___ (pronounced "Blank for NonBlank"), a podcasting network which strives to present expertlevel subject matter to nonexperts in a way which is simultaneously engaging, interesting, and simple. The episode today delves into the problem of learning. We hope you enjoy this episode.
 00:01:01
A Special Message
Breaking Math Podcast starstarstarstarstar addHello. This is Jonathan from Breaking Math to bring you a special
message. Gabriel, my cohost, has recently had a child. The child
is healthy, but both children and Breaking Math take time,
and we're still figuring out how to make use of said time most
efficiently. So I'm here to tell you what you can expect in the
mean time.
In the mean time, you can expect some minisodes from us. These will
be covering a variety of topics, hopefully including the millennium
problems.
You can also expect us to release new episodes again in a very short
amount of time. The hosts and their families have discussed how time
is going to be spent, and all that remains to be seen is if this plan
is realistic, and to tweak it to make sure that you all get the same
content you've grown to know and love.
So thank you all for your patience, and if you have anything to say
to us in the mean time, you can write to us at breakingmathpodcast@gmail.com
or write to us on our facebook page, which is at facebook.com/breakingmathpodcast.
Thank you, and until we see you again, don't forget to check
periodically for updates of Breaking Math. Bye!  00:24:13
Minisode 0.6: Four Problems
Breaking Math Podcast starstarstarstarstar addJonathan and Gabriel discuss four challenging problems.
 01:00:42
15: Consciousness I
Breaking Math Podcast starstarstarstarstar addWhat does it mean to be a good person? What does it mean to make a mistake? These are questions which we are not going to attempt to answer, but they are essential to the topic of study of today’s episode: consciousness. Conscious is the nebulous thing that lends a certain air of importance to experience, but as we’ve seen from 500 centuries of fascination with this topic, it is difficult to describe in languages which we’re used to. But with the advent of neuroscience and psychology, we seem to be closer than ever to revealing aspects of consciousness that we’ve never beheld. So what does it mean to feel? What are qualia? And how do we know that we ourselves are conscious?
 00:12:17
Minisode 0.5: ___forNon___
Breaking Math Podcast starstarstarstarstar addJonathan and Gabriel discuss ___forNon___ (blank for nonblank); a podcasting collective they've recently joined. Check out more at blankfornonblank.com.
 01:05:38
14: Artificial Thought
Breaking Math Podcast starstarstarstarstar addGo to www.brilliant.org/breakingmathpodcast to learn neural networks, everyday physics, computer science fundamentals, the joy of problem solving, and many related topics in science, technology, engineering, and math.
Mathematics takes inspiration from all forms with which life interacts. Perhaps that is why, recently, mathematics has taken inspiration from that which itself perceives the world around it; the brain itself. What we’re talking about are neural networks. Neural networks have their origins around the time of automated computing, and with advances in hardware, have advanced in turn. So what is a neuron? How do multitudes of them contribute to structured thought? And what is in their future?  00:49:01
13: Math and Prison Riots
Breaking Math Podcast starstarstarstarstar addFrank Salas is an statistical exception, but far from an irreplicable result. Busted on the streets of Albuquerque for selling crack cocaine at 17, an age where many of us are busy honing the skills that we've chosen to master, and promply incarcerated in one of the myriad concrete boxes that comprise the United States penal system. There, he struggled, as most would in his position, to better himself spiritually or ethically, once even participating in a prison riot. After two stints in solitary confinement, he did the unthinkable: he imagined a better world for himself. One where it was not all him versus the world. With newfound vigor, he discovered what was there all along: a passion for mathematics and the sciences. After nine years of hard time he graduated to a halfway house. From there, we attended classes at community college, honing his skills using his second lease on life. That took him on a trajectory which developed into him working on a PhD in electrical engineering from the University of Michegan. We're talking, of course, about Frank Salas; a man who is living proof that condition and destiny are not forced to correlate, and who uses this proof as inspiration for many in the halway house that he once roamed. So who is he? What is his mission? And who is part of that mission? And what does this have to do with Maxwell's equations of electromagnetism?
 00:50:44
12: Math Factory
Breaking Math Podcast starstarstarstarstar addIn a universe where everything is representable by information, what does it mean to interact with that world? When you follow a series of steps to accomplish a goal, what you're doing is taking part in a mathematical tradition as old as math itself: algorithms. From time immemorial, we've accelerated the growth of this means of transformation, and whether we're modeling neurons, recognizing faces, designing trusses on a bridge, or coloring a map, we're involving ourselves heavily in a fantastic world, where everything is connected to everything else through a massive network of mathematical factories. So does it mean to do something? What does it mean for something to end? And what is time relative to these questions?
 00:59:59
11: A Culture of Hacking
Breaking Math Podcast starstarstarstarstar addThe culture of mathematics is a strange topic. It is almost as important to the history of mathematics as the theorems that have come from it, yet it is rarely commented upon, and it is almost never taught in schools. One form of mathematical inquiry that has cropped up in the last two centuries has been the algorithm. While not exclusive to this time period, it has achieved a renaissance, and with the algorithm has come what has come to be known as "hacker culture". From Lord Byron to Richard Stallman, from scratches on paper to masses of wire, hacker culture has influenced the way in which we interact with conveniences that algorithms have endowed upon our society. So what are these advances? How have they been affected by the culture which birthed them? And what can we learn from this fragile yet pervasive relationship?
 01:14:19
10: Cryptomath
Breaking Math Podcast starstarstarstarstar addLanguage and communication is a huge part of what it means to be a person, and a large part of this importance is the ability to direct the flow of that information; this is a practice known as cryptography. There are as many ways to encrypt data as there are ways to use them, ranging from cryptoquips solvable by children in an afternoon to four kilobit RSA taking eons of time. So why are there so many forms of encryption? What can they be used for? And what are the differences in their methodology, if not philosophy?
 00:52:08
9: Humanity 2.0
Breaking Math Podcast starstarstarstarstar addHumanity, since its inception, has been nebulously defined. Every technological advancement has changed what it means to be a person, and every person has changed what it means to advance. In this same vein, there is a concept called “transhumanism”, which refers to what it will mean to be a person. This can range from everything from genetic engineering, to artificial intelligence, to technology which is beyond our current physical understanding. So what does it mean to be a person? And is transhumanism compatible with our natural understanding, if it exists, of being?
 00:13:56
Minisode 0.4: Comin' Up Next
Breaking Math Podcast starstarstarstarstar addJonathan and Gabriel talk about the next four episodes coming down the pike, including Humanity 2.0, which debuts Tuesday, April 2nd 2017.
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