Podcasts about banach tarski

Subscribe: Apple Podcasts | Spotify | Stitcher | Overcast Support the Show. Get the AudioBook! AudioBook: Audible| Kobo| Authors Direct | Google Play | Apple Introduction On this episode of The Entrepreneur Ethos, the host, Jarie Bolander, talks with Nick Spoors and Mike Mills about their mathematical investing aggressive growth fund and the importance of knowing your target audience. The discussion covers topics such as creating machine learning algorithms from scratch, expanding networks, and navigating heavily regulated industries. Nick and Mike share their history and discuss their interest in applying mathematical theories to investing. The speaker also delves into the importance of pure mathematics and explores concepts such as the Axiom of Choice and hypercomputing. Overall, this episode provides insight into the intersection of mathematics and investing, as well as the challenges and innovations in the industry. Timestamps [00:02:40] College friendship led to successful fund. [00:04:35] Nick found investing applied abstract mathematics [00:10:36] Researching pure math for novel applications. [00:12:43] "Axiom of choice: Math's fundamental principle" [00:17:16] Inventing investment strategy into a business product. [00:19:06] Learning from Nick; sometimes needs help. [00:22:44] Know your audience to target effectively. [00:28:24] Launching investment firm through regulated industry. Communication hurdle. [00:31:14] Mathematical investing fund seeks aggressive long-term growth. [00:36:30] Hypercomputing: A new frontier beyond Turing machines. [00:38:42] Accelerating advancements in computing and imagination's limits. [00:42:34] Small steps add up to exponential change. [00:45:49] Podcast outro: Learn, share, follow, read, get better. Key Topics - Explanation of the mathematical investing aggressive growth fund - Analyzing the short term past to predict the long term future - Use of own machine learning algorithms rather than overcrowded public ones - Plan to hire 100 mathematicians to replicate old think tank - Differential advantage in using math and investing together - Importance of knowing target audience and dialogue responses to potential investors - Expansion of network over the years - Specific dialogue responses and ability to discern conversations not worth pursuing - Encounter with potential investors who seem like a good fit but do not like what they do - Challenges faced by the speaker's company - Navigating a heavily regulated industry - Communicating message to appropriate audience - Fine-tuning message to potential clients' interests - Background and shared interest of Nick Spoors and Mike Mills - Nick's interest in applying mathematical theories to investing - Mike's introduction of the Banach-Tarski paradox - Appreciation for entrepreneurs who think differently and take risks to create new categories - Speaker's learning from interactions with Nick Spoors - Nick's ability to explain things in a way that fits the listener's learning style - Learning to communicate without Nick's help but knowing when to bring him in for technical and deep conversations - Interest in researching pure mathematics for new applications in investing - Discussion of mathematics principles and concepts - Importance of mathematics in understanding the physical world and foundational principles of AI and machine learning - Axiom of choice and its connection to complex adaptive systems - Hypercomputing and its potential for the future with advancements in AI and quantum computing Links https://www.infinitaryfund.com/ Keep In Touch Book or Blog or Twitter or LinkedIn or Get Story-Driven Learn more about your ad choices. Visit podcastchoices.com/adchoices

Well, you're in a real pickle. You see, a few years ago, the king decided your life would be forfeit unless you tripled the gold coins in his treasury. Fortunately for you, a strange little man appeared and magically performed the feat. He placed handfuls of coins in and out of a magical bag, and sang a strange rhyme:“The more gold goes in and more comes out, as sure as I am me. And in again, and out again, and now it's it times three!”好吧，你真的遇到麻烦了。你看，几年前，国王决定你的生命将被没收，除非你将他国库中的金币增加三倍。对你来说幸运的是，一个奇怪的小男人出现了，他神奇地完成了这一壮举。他把一把硬币从一个神奇的袋子里取出来，并唱着一首奇怪的韵律：“金子进得越多，出得越多，我就是我。又进又出，现在是三倍了！”Incredibly, that tripled the coins and saved your life. Were you grateful? Yes. Were you desperate? Yes. Did you promise him your first-born child in exchange for his help? Yes.Fast forward to today. No sooner have you given birth to a beautiful baby boy than the little man shows up to claim his prize. You cry and beg him not to take the baby. Softening, he begins, “If you can guess my name—” “Banach-Tarski?” you say. “It's on the front of your shirt.”令人难以置信的是，这使硬币增加了三倍并挽救了您的生命。你感激吗？是的。你绝望了吗？是的。你答应他你的第一个孩子以换取他的帮助吗？是的。快进到今天。你刚生下一个漂亮的男婴，小个子就来领奖了。你哭着求他不要带孩子。软化，他开始说，“如果你能猜出我的名字——” “Banach-Tarski？”你说。 “它在你衬衫的前面。”“What! That won't do. Aha. My bag,” he explains, “increases the number of gold coins placed inside it in a very special way. If I take any number of coins and place them in, more will come out. And if I place those in the bag again, the total that comes out will be three times whatever I began with.”He takes 13 coins and places them in the bag, then removes the contents. “I've used the magic once, not twice,” he says. “Tell me how many coins are in my hand and I'll have mercy.”How many coins is he holding?“什么！那不行。啊哈。我的包，”他解释道，“以一种非常特殊的方式增加了里面放置的金币数量。如果我拿出任意数量的硬币并将它们放入，就会有更多硬币出来。如果我再把它们放进袋子里，出来的总数将是我开始时的三倍。”他拿出 13 枚硬币放在袋子里，然后取出里面的东西。 “我用过一次魔法，不是两次，”他说。 “告诉我我手上有多少硬币，我手下留情。”他手上有多少硬币？The bag's magic works just like what in mathematics is called a “function,” and it's convenient in both cases to use an arrow to denote the transformation. We can write what we know like this.We want to know what goes in this particular blank.Maybe the bag just multiplies the number of coins by some number. In that case, multiplying by that number twice would be the same as multiplying by 3, which means the multiplier would be the square root of 3. That's not a whole number, though. And we don't have bits of gold coins coming out of the bag. Something else is going on.这个袋子的魔法就像数学中所谓的“函数”一样，在这两种情况下都可以方便地使用箭头来表示转换。我们可以像这样写下我们所知道的。我们想知道这个特定的空白中有什么。也许袋子只是将硬币的数量乘以某个数字。在那种情况下，乘以该数字两次与乘以 3 相同，这意味着乘数将是 3 的平方根。不过，这不是整数。而且我们没有从袋子里出来的金币。其他事情正在发生。Well, if filling in the blank between 13 and 39 is too hard, maybe we can start with something easier. Can we figure out what'll happen to 1 coin?If you use the bag on a single coin twice, you end up with triple; that's three gold pieces.Because the bag always increases the number of gold coins, the blank must be between 1 and 3, so 2. It's a start. What's next?Let's think about a few other possible starting places.好吧，如果填写 13 到 39 之间的空白太难了，也许我们可以从更简单的开始。我们能算出 1 个硬币会发生什么吗？如果你在一个硬币上使用袋子两次，你最终会得到三倍；那就是三个金币。因为袋子总是增加金币的数量，所以空白必须在1到3之间，所以2。这是一个开始。下一步是什么？让我们考虑一些其他可能的起点。We already know 2 becomes 3 and that lets us fill in the next blank as well. Now we're getting somewhere!We just need to extend this out to 13. Remember the other rule, though: when you put more coins in, you get more coins out. That means the numbers in every column must go in increasing order as well. In other words, because 6 coins become 9, it's not possible for 4 coins to become 10. Nor could 4 become 5, since 3 becomes 6.So 7 and 8 fill those blanks on the right of 4 and 5, which in turn gives the answer for two more blanks.我们已经知道 2 变成 3，这也让我们可以填补下一个空白。现在我们有所进展！我们只需要将其扩展到 13。不过请记住另一条规则：当你投入更多的硬币时，你会得到更多的硬币。这意味着每一列中的数字也必须按递增顺序排列。换句话说，因为 6 个硬币变成 9，4 个硬币不可能变成 10。4 也不可能变成 5，因为 3 变成 6。所以 7 和 8 填补了 4 和 5 右边的空白，这反过来给出两个空白的答案。Knowing that the numbers go in increasing order in every column, the only choices for the remaining blanks are 19, 20, 22, and 23.And look! We have our answer! There must be 22 gold coins in his hand.“I'll give you three guesses,” the little man begins to say.“22 coins,” you respond.“What?! How did you know?”“I enjoy a good riddle,” you say. “Also, it's on the back of your shirt.”知道每一列中的数字都是按递增顺序排列的，剩余空白的唯一选择是 19、20、22 和 23。看！我们有答案！他手里一定有 22 个金币。“我猜你猜 3 次，”小个子开始说。“22 个金币，”你回答。“什么？！你怎么知道的？” “我喜欢猜谜语，”你说。 “还有，它在你衬衫的背面。”

We can't believe it took 75 episodes to get to the Banach-Tarski paradox, but finally Dave Kung chose it as his favorite theorem. Also, Enigma Variations.

Everything is shareholder derivative claimsThe basic rule is that when there is a devastating plane crash, the shareholders of the company that makes ... I call ittwo 737 Max planes crashedbrought securities fraud lawsuitsDelaware Vice Chancellor Morgan Zurn ruled sue you diversity policyButinvestor presentation Bloomberg Businessweek story did not always work this wayis Facebook Inc.merchant cash advancenegative $1.4 billionrevenueSilverbacktalks about Bitcoingives away popcorn recently talked aboutstuff gets weird talked on Thursday talked on Friday sold each of them separately back in 1986 Million Dollar Homepage the Banach-Tarski paradoxthis projectSHL0MSsecondary marketthis infamous tweetsold an NFT about itbreak up Ant’s AlipayCanadian Pacific Bidtightens his gripwealth business booms Stock SellersStokes Sales 75% Haircut26.5% Corporate Tax Rate Advance Refundings Rush to CryptoNed Flanders Bridgesee your friends againsubscribe at this linkherethe most recent 10-Kthe 2017 10-KherehereFTSE Russell itself

Dr. Robert Simon shares his views on the Banach-Tarski Paradox, Myopic Equilibria and Computational applications of Game Theory.

This 86th episode of Learning Machines 101 discusses the problem of assigning probabilities to a possibly infinite set of outcomes in a space-time continuum which characterizes our physical world. Such a set is called an “environmental event”. The machine learning algorithm uses information about the frequency of environmental events to support learning. If we want to study statistical machine learning, then we must be able to discuss how to represent and compute the probability of an environmental event. It is essential that we have methods for communicating probability concepts to other researchers, methods for calculating probabilities, and methods for calculating the expectation of specific environmental events. This episode discusses the challenges of assigning probabilities to events when we allow for the case of events comprised of an infinite number of outcomes. Along the way we introduce essential concepts for representing and computing probabilities using measure theory mathematical tools such as sigma fields, and the Radon-Nikodym probability density function. Near the end we also briefly discuss the intriguing Banach-Tarski paradox and how it motivates the development of some of these special mathematical tools. Check out: www.learningmachines101.com and www.statisticalmachinelearning.com for more information!!!

Doç. Dr. Serhan Yarkan ve Halil Said Cankurtaran'ın yer aldığı, Bilim Tarihi Serisi'nin Kümeler Kuramı odaklı dördüncü kısmında: başta Kümeler Kuramı'nın kurucusu Georg Cantor olmak üzere, kuraltanımaz bilim insanlarına, açmazlara ve paradokslara değinilmiştir. Bu başlıklar, Vitali kümesi, Hilbert'in Oteli, Banach-Tarski ve Russell paradoksu örnekleri üzerinden ele alınmıştır. Sonrasında ise bilim insanlarının bu sorunlar ile nasıl başa çıktığı ve bilimsel ilerlemenin nasıl sağlandığı üzerinde konuşulmuştur. Bölüm, gelecek tahminleri ve planları üzerine konuşularak sonlandırılmıştır. Keyifli dinlemeler. #73. Kümeler Kuramı'nın Önemi ve Tarihsel Gelişimi (Bilim Tarihi Serisi B5: I. Kısım): https://youtu.be/pSksJkWK6wU #76. Kümeler Kuramı'nın Etkileri (Bilim Tarihi Serisi B6: II. Kısım): https://youtu.be/gtpdAUaCgzw #77. Kümeler Kuramı ve Hesaplama (Bilim Tarihi Serisi B7: III. Kısım): https://youtu.be/TMt_rUbE4M4 Tapir Lab. GitHub: @TapirLab, https://www.github.com/tapirlab Tapir Lab. Instagram: @tapirlab, https://www.instagram.com/tapirlab/ Tapir Lab. Twitter: @tapirlab, https://www.twitter.com/tapirlab Tapir Lab.: http://www.tapirlab.com

i just watched a video of a guy who took a chocolate bar and cut it a certain way. it was 4 squares by 8 squares, and after he cut it he rearranged the pieces in a way where the resulting bar has one square extra that doesn’t fit into the finished bar. Fuck me i was stunned, so weird these things. if you’re looking for the dodging bullets series it has its own location now click the link to find it! - anchor.fm/dodgingbullets Check out my new podcast too! Encephalon click this link anchor.fm/mrtyler --- This episode is sponsored by · Anchor: The easiest way to make a podcast. https://anchor.fm/app --- Send in a voice message: https://anchor.fm/tystrust/message Support this podcast: https://anchor.fm/tystrust/support

Neste podcast: Conheça mundos horríveis da ficção que não seriam legais viver, qual o nome do Jaeger brasileiro e prepare-se para a filosofia de Azaghal! ARTE DA VITRINE:  Felipe Santos O HOBBIT ESTÁ NA NERDSTORE! Reserve agora! https://bit.ly/2JDNPFX NerdCash! NerdCast extra toda segunda sexta-feira do mês para você aprender a investir o seu dinheiro! Link para abertura de conta com a Jovem Nerd: http://bit.ly/2YFQsxb Link para Futura Academy: http://bit.ly/2YHGf3g Ouça o NerdCash 13 - Sonhos de consumo: https://bit.ly/2SciDl6 Site da nova Futura: http://bit.ly/2Lor4oL Youtube: http://bit.ly/2LsKUzk Instagram: http://bit.ly/2sBKbod Facebook: http://bit.ly/2JehoQV Twitter: http://bit.ly/2qBKDlj CITADO NA LEITURA DE E-MAILS Modular Elliptic Curves and Fermat's Last Theorem (em inglês): https://bit.ly/2xHvKkL Vídeo explicando o paradoxo de Banach-Tarski (em inglês): https://youtu.be/s86-Z-CbaHA  E-MAILS Mande suas críticas, elogios, sugestões e caneladas para nerdcast@jovemnerd.com.br EDIÇÃO COMPLETA POR RADIOFOBIA PODCAST E MULTIMÍDIA http://radiofobia.com.br

Neste podcast: Conheça mundos horríveis da ficção que não seriam legais viver, qual o nome do Jaeger brasileiro e prepare-se para a filosofia de Azaghal! ARTE DA VITRINE:  Felipe Santos O HOBBIT ESTÁ NA NERDSTORE! Reserve agora! https://bit.ly/2JDNPFX NerdCash! NerdCast extra toda segunda sexta-feira do mês para você aprender a investir o seu dinheiro! Link para abertura de conta com a Jovem Nerd: http://bit.ly/2YFQsxb Link para Futura Academy: http://bit.ly/2YHGf3g Ouça o NerdCash 13 - Sonhos de consumo: https://bit.ly/2SciDl6 Site da nova Futura: http://bit.ly/2Lor4oL Youtube: http://bit.ly/2LsKUzk Instagram: http://bit.ly/2sBKbod Facebook: http://bit.ly/2JehoQV Twitter: http://bit.ly/2qBKDlj CITADO NA LEITURA DE E-MAILS Modular Elliptic Curves and Fermat's Last Theorem (em inglês): https://bit.ly/2xHvKkL Vídeo explicando o paradoxo de Banach-Tarski (em inglês): https://youtu.be/s86-Z-CbaHA  E-MAILS Mande suas críticas, elogios, sugestões e caneladas para nerdcast@jovemnerd.com.br EDIÇÃO COMPLETA POR RADIOFOBIA PODCAST E MULTIMÍDIA http://radiofobia.com.br

Neste podcast: Conheça mundos horríveis da ficção que não seriam legais viver, qual o nome do Jaeger brasileiro e prepare-se para a filosofia de Azaghal! ARTE DA VITRINE:  Felipe Santos O HOBBIT ESTÁ NA NERDSTORE! Reserve agora! https://bit.ly/2JDNPFX NerdCash! NerdCast extra toda segunda sexta-feira do mês para você aprender a investir o seu dinheiro! Link para abertura de conta com a Jovem Nerd: http://bit.ly/2YFQsxb Link para Futura Academy: http://bit.ly/2YHGf3g Ouça o NerdCash 13 - Sonhos de consumo: https://bit.ly/2SciDl6 Site da nova Futura: http://bit.ly/2Lor4oL Youtube: http://bit.ly/2LsKUzk Instagram: http://bit.ly/2sBKbod Facebook: http://bit.ly/2JehoQV Twitter: http://bit.ly/2qBKDlj CITADO NA LEITURA DE E-MAILS Modular Elliptic Curves and Fermat's Last Theorem (em inglês): https://bit.ly/2xHvKkL Vídeo explicando o paradoxo de Banach-Tarski (em inglês): https://youtu.be/s86-Z-CbaHA  E-MAILS Mande suas críticas, elogios, sugestões e caneladas para nerdcast@jovemnerd.com.br EDIÇÃO COMPLETA POR RADIOFOBIA PODCAST E MULTIMÍDIA http://radiofobia.com.br

Neste podcast: Conheça mundos horríveis da ficção que não seriam legais viver, qual o nome do Jaeger brasileiro e prepare-se para a filosofia de Azaghal! ARTE DA VITRINE:  Felipe Santos O HOBBIT ESTÁ NA NERDSTORE! Reserve agora! https://bit.ly/2JDNPFX NerdCash! NerdCast extra toda segunda sexta-feira do mês para você aprender a investir o seu dinheiro! Link para abertura de conta com a Jovem Nerd: http://bit.ly/2YFQsxb Link para Futura Academy: http://bit.ly/2YHGf3g Ouça o NerdCash 13 - Sonhos de consumo: https://bit.ly/2SciDl6 Site da nova Futura: http://bit.ly/2Lor4oL Youtube: http://bit.ly/2LsKUzk Instagram: http://bit.ly/2sBKbod Facebook: http://bit.ly/2JehoQV Twitter: http://bit.ly/2qBKDlj CITADO NA LEITURA DE E-MAILS Modular Elliptic Curves and Fermat's Last Theorem (em inglês): https://bit.ly/2xHvKkL Vídeo explicando o paradoxo de Banach-Tarski (em inglês): https://youtu.be/s86-Z-CbaHA  E-MAILS Mande suas críticas, elogios, sugestões e caneladas para nerdcast@jovemnerd.com.br EDIÇÃO COMPLETA POR RADIOFOBIA PODCAST E MULTIMÍDIA http://radiofobia.com.br

Neste podcast: Conheça mundos horríveis da ficção que não seriam legais viver, qual o nome do Jaeger brasileiro e prepare-se para a filosofia de Azaghal! ARTE DA VITRINE:  Felipe Santos O HOBBIT ESTÁ NA NERDSTORE! Reserve agora! https://bit.ly/2JDNPFX NerdCash! NerdCast extra toda segunda sexta-feira do mês para você aprender a investir o seu dinheiro! Link para abertura de conta com a Jovem Nerd: http://bit.ly/2YFQsxb Link para Futura Academy: http://bit.ly/2YHGf3g Ouça o NerdCash 13 - Sonhos de consumo: https://bit.ly/2SciDl6 Site da nova Futura: http://bit.ly/2Lor4oL Youtube: http://bit.ly/2LsKUzk Instagram: http://bit.ly/2sBKbod Facebook: http://bit.ly/2JehoQV Twitter: http://bit.ly/2qBKDlj CITADO NA LEITURA DE E-MAILS Modular Elliptic Curves and Fermat's Last Theorem (em inglês): https://bit.ly/2xHvKkL Vídeo explicando o paradoxo de Banach-Tarski (em inglês): https://youtu.be/s86-Z-CbaHA  E-MAILS Mande suas críticas, elogios, sugestões e caneladas para nerdcast@jovemnerd.com.br EDIÇÃO COMPLETA POR RADIOFOBIA PODCAST E MULTIMÍDIA http://radiofobia.com.br

Neste podcast: Conheça mundos horríveis da ficção que não seriam legais viver, qual o nome do Jaeger brasileiro e prepare-se para a filosofia de Azaghal! ARTE DA VITRINE:  Felipe Santos O HOBBIT ESTÁ NA NERDSTORE! Reserve agora! https://bit.ly/2JDNPFX NerdCash! NerdCast extra toda segunda sexta-feira do mês para você aprender a investir o seu dinheiro! Link para abertura de conta com a Jovem Nerd: http://bit.ly/2YFQsxb Link para Futura Academy: http://bit.ly/2YHGf3g Ouça o NerdCash 13 - Sonhos de consumo: https://bit.ly/2SciDl6 Site da nova Futura: http://bit.ly/2Lor4oL Youtube: http://bit.ly/2LsKUzk Instagram: http://bit.ly/2sBKbod Facebook: http://bit.ly/2JehoQV Twitter: http://bit.ly/2qBKDlj CITADO NA LEITURA DE E-MAILS Modular Elliptic Curves and Fermat's Last Theorem (em inglês): https://bit.ly/2xHvKkL Vídeo explicando o paradoxo de Banach-Tarski (em inglês): https://youtu.be/s86-Z-CbaHA  E-MAILS Mande suas críticas, elogios, sugestões e caneladas para nerdcast@jovemnerd.com.br EDIÇÃO COMPLETA POR RADIOFOBIA PODCAST E MULTIMÍDIA http://radiofobia.com.br

Math is way more than formulas learned in algebra class in high school. This episode's guest, Yen Duong, earned her PhD studying graph theory and topology! While she's now a science reporter in North Carolina, she was happy to tell me all about her PhD research! Suggested Reading: Yen's blog on geometric group theory: https://bakingandmath.com/2015/03/02/what-is-geometric-group-theory/ and several other math posts Example of a funky manifold: https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-three-torus/ Background on the field: https://www.quantamagazine.org/from-hyperbolic-geometry-to-cube-complexes-and-back-20121002/ Video on Banach Tarski paradox: https://www.youtube.com/watch?v=s86-Z-CbaHA&t=689s Follow Yen Duong: @yenergy, www.yenduong.com Follow me: PhDrinking@gmail.com, @PhDrinking, @SadieWit, www.facebook.com/PhDrinking/ Thanks to www.bensound.com/ for the intro/outro Thanks to @TylerDamme for audio editing

"I'm a good person! It was an accident!"Tune in as we – and this week’s guest, the immaculate Henry Potts-Rubin – take on rap beef, inequality, offensive comedy, accidental shoplifting, personality flaws, and the Banach-Tarski paradox.

Nicolas Monod teaches at the École polytechnique fédérale in Lausanne and leads the Ergodic and Geometric Group Theory group there. In May 2016 he was invited to give the Gauß lecture of the German Mathematical Society (DMV) at the Technical University in Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski Paradox. The morning after his lecture we met to talk about paradoxes and hidden assumptions our mind makes in struggling with geometrical representations and measures. A very well-known game is Tangram. Here a square is divided into seven pieces (which all are polygons). These pieces can be rearranged by moving them around on the table, e.g.. The task for the player is to form given shapes using the seven pieces – like a cat etc.. Of course the Tangram cat looks more like a flat Origami-cat. But we could take the Tangram idea and use thousands or millions of little pieces to build a much more realistic cat with them – as with pixels on a screen. In three dimensions one can play a similar game with pieces of a cube. This could lead to a LEGO-like three-dimensional cat for example. In this traditional Tangram game, there is no fundamental difference between the versions in dimension two and three. But in 1914 it was shown that given a three-dimensional ball, there exists a decomposition of this ball into a finite number of subsets, which can then be rearranged to yield two identical copies of the original ball. This sounds like a magical trick – or more scientifically said – like a paradoxical situation. It is now known under the name Banach-Tarski paradox. In his lecture, Nicolas Monod dealt with the question: Why are we so surprised about this result and think of it as paradoxical? One reason is the fact that we think to know deeply what we understand as volume and expect it to be preserved under rearrangements (like in the Tangram game, e.g.).Then the impact of the Banach-Tarski paradox is similar for our understanding of volume to the shift in understanding the relation between time and space through Einstein's relativity theory (which is from about the same time). In short the answer is: In our every day concept of volume we trust in too many good properties of it. It was Felix Hausdorff who looked at the axioms which should be valid for any measure (such as volume). It should be independent of the point in space where we measure (or the coordinate system) and if we divide objects, it should add up properly. In our understanding there is a third hidden property: The concept "volume" must make sense for every subset of space we choose to measure. Unfortunately, it is a big problem to assign a volume to any given object and Hausdorff showed that all three properties cannot all be true at the same time in three space dimensions. Couriously, they can be satisfied in two dimensions but not in three. Of course, we would like to understand why there is such a big difference between two and three space dimensions, that the naive concept of volume breaks down by going over to the third dimension. To see that let us consider motions. Any motion can be decomposed into translations (i.e. gliding) and rotations around an arbitrarily chosen common center. In two dimensions the order in which one performs several rotations around the same center does not matter since one can freely interchange all rotations and obtains the same result. In three dimensions this is not possible – in general the outcomes after interchanging the order of several rotations will be different. This break of the symmetry ruins the good properties of the naive concept of volume. Serious consequences of the Banach-Tarski paradox are not that obvious. Noone really duplicated a ball in real life. But measure theory is the basis of the whole probability theory and its countless applications. There, we have to understand several counter-intuitive concepts to have the right understanding of probabilities and risk. More anecdotally, an idea of Bruno Augenstein is that in particle physics certain transformations are reminiscent of the Banach-Tarski phenomenon. Nicolas Monod really enjoys the beauty and the liberty of mathematics. One does not have to believe anything without a proof. In his opinion, mathematics is the language of natural sciences and he considers himself as a linguist of this language. This means in particular to have a closer look at our thought processes in order to investigate both the richness and the limitations of our models of the universe. References: F. Hausdorff: Bemerkung über den Inhalt von Punktmengen. Math. Ann. 75 (3), 428–433, 1914. S. Banach and A.Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae 6, 244–277, 1924. J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta Mathematicae 13, 73–116, 1929. S. Wagon: The Banach–Tarski Paradox. Cambridge University Press, 1994. B.W. Augenstein: Links Between Physics and Set Theory, Chaos, Solitons and Fractals, 7 (11), 1761–1798, 1996. N. Monod: Groups of piecewise projective homeomorphisms, PNAS 110 (12), 4524-4527, 2013. Vsauce-Video on the Banach-Tarksi Paradox

Nicolas Monod teaches at the École polytechnique fédérale in Lausanne and leads the Ergodic and Geometric Group Theory group there. In May 2016 he was invited to give the Gauß lecture of the German Mathematical Society (DMV) at the Technical University in Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski Paradox. The morning after his lecture we met to talk about paradoxes and hidden assumptions our mind makes in struggling with geometrical representations and measures. A very well-known game is Tangram. Here a square is divided into seven pieces (which all are polygons). These pieces can be rearranged by moving them around on the table, e.g.. The task for the player is to form given shapes using the seven pieces – like a cat etc.. Of course the Tangram cat looks more like a flat Origami-cat. But we could take the Tangram idea and use thousands or millions of little pieces to build a much more realistic cat with them – as with pixels on a screen. In three dimensions one can play a similar game with pieces of a cube. This could lead to a LEGO-like three-dimensional cat for example. In this traditional Tangram game, there is no fundamental difference between the versions in dimension two and three. But in 1914 it was shown that given a three-dimensional ball, there exists a decomposition of this ball into a finite number of subsets, which can then be rearranged to yield two identical copies of the original ball. This sounds like a magical trick – or more scientifically said – like a paradoxical situation. It is now known under the name Banach-Tarski paradox. In his lecture, Nicolas Monod dealt with the question: Why are we so surprised about this result and think of it as paradoxical? One reason is the fact that we think to know deeply what we understand as volume and expect it to be preserved under rearrangements (like in the Tangram game, e.g.).Then the impact of the Banach-Tarski paradox is similar for our understanding of volume to the shift in understanding the relation between time and space through Einstein's relativity theory (which is from about the same time). In short the answer is: In our every day concept of volume we trust in too many good properties of it. It was Felix Hausdorff who looked at the axioms which should be valid for any measure (such as volume). It should be independent of the point in space where we measure (or the coordinate system) and if we divide objects, it should add up properly. In our understanding there is a third hidden property: The concept "volume" must make sense for every subset of space we choose to measure. Unfortunately, it is a big problem to assign a volume to any given object and Hausdorff showed that all three properties cannot all be true at the same time in three space dimensions. Couriously, they can be satisfied in two dimensions but not in three. Of course, we would like to understand why there is such a big difference between two and three space dimensions, that the naive concept of volume breaks down by going over to the third dimension. To see that let us consider motions. Any motion can be decomposed into translations (i.e. gliding) and rotations around an arbitrarily chosen common center. In two dimensions the order in which one performs several rotations around the same center does not matter since one can freely interchange all rotations and obtains the same result. In three dimensions this is not possible – in general the outcomes after interchanging the order of several rotations will be different. This break of the symmetry ruins the good properties of the naive concept of volume. Serious consequences of the Banach-Tarski paradox are not that obvious. Noone really duplicated a ball in real life. But measure theory is the basis of the whole probability theory and its countless applications. There, we have to understand several counter-intuitive concepts to have the right understanding of probabilities and risk. More anecdotally, an idea of Bruno Augenstein is that in particle physics certain transformations are reminiscent of the Banach-Tarski phenomenon. Nicolas Monod really enjoys the beauty and the liberty of mathematics. One does not have to believe anything without a proof. In his opinion, mathematics is the language of natural sciences and he considers himself as a linguist of this language. This means in particular to have a closer look at our thought processes in order to investigate both the richness and the limitations of our models of the universe. References: F. Hausdorff: Bemerkung über den Inhalt von Punktmengen. Math. Ann. 75 (3), 428–433, 1914. S. Banach and A.Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae 6, 244–277, 1924. J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta Mathematicae 13, 73–116, 1929. S. Wagon: The Banach–Tarski Paradox. Cambridge University Press, 1994. B.W. Augenstein: Links Between Physics and Set Theory, Chaos, Solitons and Fractals, 7 (11), 1761–1798, 1996. N. Monod: Groups of piecewise projective homeomorphisms, PNAS 110 (12), 4524-4527, 2013. Vsauce-Video on the Banach-Tarksi Paradox

The Banach-Tarski ParadoxA mouthful no doubt! This week we go over just how a mathematical universe differs from a physical one on the basis of the Axiom of Choice. More interestingly, if we exploit this paradox, we can clone anything we want with the exact same properties as the original. As per Andrew's note, you can see an example of a fractal here. Really amazing stuff to say the least. Barbarian Horde is by Hans Zimmer