How math predicts life on Earth and the universe beyond
Math doesn't suck. It is one of humanity's greatest and most mysterious journeys.
MICHELLE THALLER: Mathematics is in some ways kind of scary in how useful it is at really describing how the universe works around us. Now, I mean to give you an idea, the origin of mathematics seems very straightforward. We can count on our fingers up to ten and maybe it was useful to understand how many sheep you had so you could start counting sheep. And then you either added or subtracted sheep as you got more or as you lost some. It was a simple thing. We learned how to count. We learned how to add and subtract. The idea of multiplying and dividing is a little more abstract but that also makes sense. That's something that we can kind of visualize. But then what amazes me is that this led us on a tremendously complicated journey that's still going on to this day and we had no idea where this would lead us.
EDWARD FRENKEL: It's a very unfortunate situation when you can't even begin a conversation about mathematics without people saying 'Oh, my gosh. I don't want to talk about it.' And it's kind of strange because no one would ever say 'I hate literature' or 'I hate art' or 'I hate music.' At least, intelligent people would never say that. It's kind of shameful to say that. But it's perfectly okay in our society to say 'I hate mathematics.' And so what I dream of is a society in which it's not that everyone has a Ph.D. in mathematics, but rather I would like to live in a society in which if mathematics is brought up someone would say 'Oh, mathematics. Interesting. How do I find out more? Can you give me a gist of the idea?' I'm not scared of it. I'm curious about it, the way I'm curious about the solar system, about the atoms and the DNA. All these things which are in our collective consciousness, in our public discourse, which are no simpler than mathematical concepts. Mathematical concepts are no more complicated than these concepts in physics and biology that are in the air, that are what we are aware of. So I would like people to be aware of this, of these mathematical ideas. I'd like them to be more and more aware of how mathematics invades our lives. How it's controlling our lives.
MICHIO KAKU: In the 1600s, Isaac Newton asked a simple question: If an apple falls, then does the Moon also fall? That is perhaps one of the greatest questions ever asked by a member of homo sapiens since the six million years since we parted ways with the apes. If an apple falls, does the Moon also fall? Isaac Newton said yes, the Moon falls because of the inverse square law, so does an apple. He had a unified theory of the heavens, but he didn't have the mathematics to solve the falling Moon problem. So what did he do? He invented calculus. So calculus is a direct consequence of solving the falling moon problem.
PO-SHEN LOH: I do think that everyone in America could benefit from having that mathematical background in reasoning just to help everyone make very good decisions. And here I'm distinguishing already between math as people usually conceive of it and decision making and analysis which is actually what I think math is. So, for example, I don't think that being a math person means that you can recite the formulas between sins, cosines, tangents and to use logarithms and exponentials interchangeably. That's not necessarily what I think everyone should try to concentrate to understand. The main things to concentrate to understand are the mathematical principles of reasoning.
FRENKEL: And I understand why people are scared and frightened. It's not their fault. It's because of how mathematics is taught in our schools.
LOH: I think that actually one reason mathematics is difficult to understand is actually because of that network of prerequisites. You see, math is one of these strange subjects for which the concepts are chained in sequences of dependencies. When you have long chains there are very few starting points. In mathematics you have very few that you memorize and the rest you deduce as you go through. And this chain of deductions is actually what's critical. Now, let me contrast that with other subjects like say history. History doesn't have this long chain. In fact, if you fully understand the War of 1812 that's great and it is true that that will influence perhaps your understanding later of the Women's Movement, but it won't be as absolutely prerequisite. In the sense that if you think about the concepts, I actually think that history has more concepts than mathematics, it's just that they're spread out broader and they don't depend on each other as strongly. Math has fewer concepts but they're chained deeper. And because of the way that we usually learn, when you have deep chains it's very fragile because you lose any one link meaning if you miss a few concepts along the chain you can actually be completely lost. Now, I think that the way to help to address this is to provide a way for everyone to learn at their own pace and, in fact, to fill in the holes whenever they are sensed. I actually feel like if everyone was able to pick up every one of those prerequisites as necessary filling in any gap they have, mathematics would change from being the hardest subject to the easiest subject. So after thinking for some time I actually came to an idea which was based on using these core mathematical areas that I had been working with to actually build a solution for education that could be delivered for free on every smartphone. This is actually the project that I'm working on right now called Expii. Our principle is that actually you could turn that smartphone into a virtual tutor which automates what a person would get if they hired a tutor. It's not that the class comes first and then the homework and then the exam. The first thing that comes is the exam essentially followed by these practice problems which adapt to you, followed by the class for anything that you don't know. The idea is that this should cure boredom at the high end and also cure confusion at the struggling end.
KAKU: Some people ask the question of what good is math? What is the relationship between math and physics? Well, sometimes math leads, sometimes physics leads, sometimes they come together because, of course, there's a use for the mathematics.
JANNA LEVIN: As a theoretical physicist I rely very much on calculating to understand results. So let's say I want to know what happens when a big black hole swallows a little tiny black hole. My starting point will be to think what's my first mathematical sentence that I know if I crack it open will answer this question? Now even that step can be very hard. And then once I do that I'll be moving in a very structured way through the steps to unlock that. When I work with much younger students I like to kind of sit back in the room when we're calculating now and ask them to try to find the solution in the way that we do know how to go one step after another, but I'll be trying at the same time to try to find that cute way of unpacking it that's highly non-obvious. And sometimes I'm sitting there for two days while they're generating 12 pages of very tough, beautifully done calculations where I'm trying to find a way to try to do it in a half a page. And sometimes I succeed and sometimes I don't. Sometimes I can't do that. It's just not possible or I haven't figured out how to do it. And sometimes I do. And when you do it's not just that it's shorter, it's that you're working with such bigger structures that you can see so much more at a time.
KAKU: Then here comes Einstein asking a different question and that is: What is the nature in origin of gravity? Einstein said that gravity is nothing but the byproduct of curved space. So, why am I sitting in this chair? A normal person would say I'm sitting in this chair because gravity pulls me to the ground. But Einstein said no, no, no, no. There's no such thing as gravitational pull. The earth has curved this space over my head and around my body so space is pushing me into my chair. So to summarize Einstein's theory: Gravity does not pull. Space pushes. But, you see, the pushing of the fabric of space and time requires differential calculus. That is the language of curved surfaces, differential calculus, which you learn in fourth-year calculus. So again, here's a situation where math and physics were very closely combined but this time math came first. The theory of curved surfaces came first. Einstein took that theory of curved surfaces and then imported it into physics.
THALLER: If you can do multiplication and subtraction it's not too long before you begin to develop the basic building blocks of calculus. And calculus describes how moving objects can change, how things can accelerate. If you want to describe an apple falling from a tree to the ground or a ball rolling down a hill, that's calculus. It's the mathematics of how things can change over time. That's really interesting and the amazing thing is it works so well. If you use these equations to predict how a ball will roll down a hill, reality matches that. It really does tell you how something is going to behave. So now we've gone from counting on our fingers how many sheep we have to being able to predict what the universe around us is going to do. That's incredibly powerful.
GEOFFREY WEST: Just taking mammals, that the largest mammal, the whale, is in terms of measurable quantities, that is of its physiology and its life history, is actually a scaled-up version of the smallest mammal which is actually the shrew but a mouse is very close to that and everything in between, but they are scaled versions of one another. The most well-known of these is the scaling of metabolic rate and metabolic rate is maybe the most fundamental quantity of life because metabolic rate simply means how much energy or maybe just how much food does an animal need to eat each day in order to stay alive. And everybody's used to that and is familiar with that, it's sort of roughly 2,000 food calories a day for a human being. So you can ask what is that for different mammals? And what you find is that they're related to one another in a very simple way despite the fact that metabolism may be the most complex physical chemical process in the universe, for all we know. It's phenomenal, because metabolism is taking essentially something that's inorganic and making it into life. So here's this extraordinary, complex process and yet it scales in a very simple way and you can express it in English—you can express it quite precisely in a very simple mathematical equation—but in English, it's roughly speaking that every time you double the size of an organism from say two grams to four grams or from 20 grams to 40 grams or 20 kilograms to 40 kilograms or whatever, just doubling anywhere, instead of what you might naively expect, double the size, double the number of cells, roughly speaking, therefore you would expect to double the amount of energy, the amount of metabolic energy you need to keep that organism alive because you have twice as many cells. Quite the contrary, you don't need twice as much. Systematically, you only need, roughly speaking, 75 percent as much. So there's this kind of systematic 25 percent, one-quarter, savings. And it turns out that anything else you measure, as I mentioned a moment ago, scales in a similar way with this sort of 25 percent rule occurring in some interesting way. The same mathematical—and this is extremely important—the same mathematical and physical principles apply to a mammal, which has a beating heart, as applies to a tree. And a mammal, our circuitry system is a bunch of tubes like in your house, the plumbing in this building we're sitting in, that's our circulatory system. But a tree and a plant, they're not like that. They're a bunch of fiber bundles kind of joined together like electrical cables that spray out and that's what you see when you see a tree. But even though they—and they don't have beating hearts, as we well know. And yet they satisfy the same mathematical principles and those mathematical principles give rise to this quarter-power scaling in mammals, but also in plants and trees, but also in fish and birds and crustacea, in principle, and insects and so on, and that's the idea.
THALLER: Now we look around us and we see things like planets orbiting the stars or the galaxy turning around and we realize those equations of motion apply to everything else in the universe. It's not just here. It's not just on the surface of the earth, but we can look at things literally billions of lightyears out in space and they're following those same rules of mathematics. So we keep getting led farther and farther down this rabbit hole. Where does math lead us? Now we realize that you can describe physics incredibly well if you allow the universe to exist in many different dimensions, more than the three dimensions that we're familiar with. In fact, specifically if you want to do particle physics it requires 11 dimensions. That's not something our minds comprehend but we can do the math. We can do the math of how things would behave if they could move in 11 different directions. And it turns out to predict exactly the results we get from particle physics. That's kind of scary. Does that mean that's real? Are there really 11 dimensions? The math works so well and the predictions are so strong that it can't just be nonsense. But now we've gone to the limit of what I can tell you: Is it real or not? Our math has given us something incredibly useful, but it's taken us completely out of our realm of common sense, of human scale, of how our minds work and even our sense of space and time. I don't think that journey's over yet. Where is math going to lead us? It may lead us to understand things like the universe is a type of a hologram. That was a mathematical solution to how things work around a black hole and it works really, really well. So, I think it's wonderful and a little bit scary that you start counting on your fingers, you get to 11 dimensions of space and time and where else?
- There is a pervasive cultural attitude against mathematics, but it is actually a mind-blowing tool for analyzing and predicting the world around us—and far beyond. We asked mathematicians Edward Frenkel and Po-Shen Loh, and physicists Michio Kaku, Michelle Thaller, Janna Levin and Geoffrey West to explain the wonders of math.
- West explains the rule of 'quarter-power scaling' in biology—there is a mathematical equation that predicts how much food an organism needs to eat to survive and it's remarkably consistent, whether you're looking at ladybugs, cats, elephants, and even trees and flowers. Math underpins our lives in incredible ways.
- Infinitesimal calculus—the math that describes how moving bodies change over time—turns out to predict not just phenomena on Earth but far out in the universe. The 11-dimensional math used by physicists turns out to predict the exact results of particle physics experiments. Humanity is on an incredible journey with mathematics and every day it opens up the world and universe in eye-opening ways.
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Innovators don't ignore risk; they are just better able to analyze it in uncertain situations.
The Labour Economics study suggests two potential reasons for the increase: corruption and increased capacity.
Cool hand rebuke<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDQyMTIyNy9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTY0NjY1NTYyOH0.0MCPKN3If94mYCNf3mMNrnTvJXjXN_bKLhgk9203EXk/img.jpg?width=917&coordinates=0%2C0%2C0%2C0&height=453" id="1627b" class="rm-shortcode" data-rm-shortcode-id="6d76421ba1ea0de4b09956b97e80c384" data-rm-shortcode-name="rebelmouse-image" />
A chart showing prison population rates (per 100,000 people) in 2018. The United States has the highest rate of incarceration in the world.
Who profits with for-profit prisons?<span style="display:block;position:relative;padding-top:56.25%;" class="rm-shortcode" data-rm-shortcode-id="97ac37e6c7f6f22ec130ea2d56871701"><iframe type="lazy-iframe" data-runner-src="https://www.youtube.com/embed/dB78NV2WpWc?rel=0" width="100%" height="auto" frameborder="0" scrolling="no" style="position:absolute;top:0;left:0;width:100%;height:100%;"></iframe></span><p>The Labour Economics study suggests that privately-run prisons do convicts a few favors at the moment of sentencing. However, proponents of private prisons often point to other benefits when making their case. Specifically, they argue that private prisons reduce operating costs, stimulate innovation in the correctional system, and reduce recidivism—the rate at which released prisoners are rearrested and return to prison.</p><p>In regard to recidivism, the research is mixed. <a href="https://journals.sagepub.com/doi/abs/10.1177/0011128799045001002" target="_blank">One study</a> compared roughly 400 former prisoners from Florida, 200 released from private prisons and 200 from state-run facilities. It found the private-prison cohort maintained lower rates of recidivism. However, <a href="https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1745-9133.2005.00006.x" target="_blank" rel="noopener noreferrer">another Florida study</a> found no significant rate differences. And two other studies—one from <a href="https://journals.sagepub.com/doi/abs/10.1177/0011128799045001002" target="_blank" rel="noopener noreferrer">Oklahoma</a> and another out of <a href="https://journals.sagepub.com/doi/abs/10.1177/0734016813478823" target="_blank" rel="noopener noreferrer">Minnesota</a>, both comparing much larger cohorts than the first Florida study— found that prisoners leaving private prisons had a greater risk of recidivism.</p><p>The research is also inconclusive regarding cost savings. <a href="https://www.hamiltonproject.org/assets/files/economics_of_private_prisons.pdf" target="_blank" rel="noopener noreferrer">A Hamilton Project analysis</a> noted that such comparisons are difficult because private prisons, like all private companies, are not required to release operational details. In comparing what studies were available, the authors estimate the costs to be comparable and that "in practice the primary mechanism for cost saving in private prisons is lower salaries for correctional officers"—about $7,000 less than their public peers. They add that competition-driven innovation is lacking as the three largest firms control nearly the entire market.</p><p>"We aren't saying private prisons are bad," Galinato said. "But states need to be careful with them. If your state has previous and regular issues with corruption, I wouldn't be surprised to see laws being more skewed to give longer sentences, for example. If the goal is to reduce the number of incarcerated individuals, increasing the number of private prisons may not be the way to go."</p>
A vertical map might better represent a world dominated by China and determined by shipping routes across the iceless Arctic.
- Europe has dominated cartography for so long that its central place on the world map seems normal.
- However, as the economic centre of gravity shifts east and the climate warms up, tomorrow's map may be very different.
- Focusing on both China and Arctic shipping lanes, this vertical representation could be the world map of the future.
The world, but not as we know it<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yMDU1Nzg1NS9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYzNTkwMjIyNn0.qmQfwUdjQka8JX6q4KGANagleiuucpWay5ytMenZxUU/img.jpg?width=980" id="b95e4" class="rm-shortcode" data-rm-shortcode-id="ac088ec55c0585a93a9a310faab9a4c7" data-rm-shortcode-name="rebelmouse-image" />
A Chinese 'vertical world map,' showing the world in a different perspective from the one we're used to.
Image: Prior Probability<p>Europe is tucked away in a corner, an appendage of Asia dwarfed by neighboring Africa. North America is stood on its head, facing the rest of the world from the top of the map — cut off from South America, which cuts a solitary figure at the bottom. Africa is justifiably huge, but equally eccentric. </p><p>The eye scouts elsewhere for a place to land: not the Indian Ocean, which dominates the middle of the map, but some terra firma. Antarctica and Australia are too small, mere stepping stones for the land mass of Asia. Ultimately our gaze is drawn toward China, the lynchpin of this unfamiliar world. </p><p>Managing to leave both poles intact, this "vertical" world map is about as far away as you can get from the classic Mercator projection – which slices up both, giving center stage to a puffed-up Europe. Perhaps this new map will become more familiar soon: It may do more justice to the world of the near future, dominated by China and determined by shipping routes across the iceless Arctic. <br></p>
China's 'ten-dash line'<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yMDU1Nzg1Ni9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTY1NTI4MzQyNn0.sBe0oFTif4Jef1vWh1kAnUylU_QMPXT5xQjm-5aA3sA/img.jpg?width=980" id="a3b81" class="rm-shortcode" data-rm-shortcode-id="80fc6e4f5c9c1c978f698be2c8de5484" data-rm-shortcode-name="rebelmouse-image" />
'China without any part left out': includes Taiwan and the islands and atolls in the South China Sea, surrounded by a ten-dash line
Image: Global Times<p>While there's no indication that this map represents the Chinese government's "official" worldview, it is no secret that China has a thing with maps – and more specifically, the country's representation on them. </p><p>In China, the country's current economic success is seen as a redress of the unequal treatment meted out by western superpowers in the 19th century. China's world dominance is a return to a more natural state of world affairs, many feel. Cartographic rectifications are a symbolically significant corollary of that sentiment.</p><p><a href="https://www.citylab.com/equity/2015/12/china-cracks-down-on-politcally-incorrect-maps/421032/" target="_blank">Fines are regularly imposed</a> on companies – domestic and foreign – that fail to represent China to the fullest extent of its external borders, disputed though they may be by others (e.g. India, Taiwan and any of the countries with claims overlapping China's in the South China Sea). But the People's Republic's cartographic obsession doesn't end at China's territory itself. It also includes the country's position on the world map. <br></p>
The Kingdom at the Middle of the World<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yMDU1Nzg2MS9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYyOTkwODEzMX0.SGrAZBH6iJVggFYSaIahzv9GvfEh17y1SwUNINbVicQ/img.jpg?width=980" id="1774c" class="rm-shortcode" data-rm-shortcode-id="99790d80a909d17a948f7c5d463d7d98" data-rm-shortcode-name="rebelmouse-image" />
Early Japanese color copy of Ricci's world map
Image: public domain<p>China's name for itself is <em>Zhōngguó</em>, which means 'Central State' or 'Middle Kingdom', reflecting its ancient self-image as the civilized center (<em>Huá</em>) of the world, with wild tribes (<em>Yí</em>) at the edge. That view is not unique to China. Vietnam, for example, at certain times also styled itself as the "central state" (<em>Trung Quóc</em>) – considering the Chinese in turn as the uncouth outsiders.</p><p>It may be surprising to recall, but Europeans themselves once considered their own continent a relative backwater, viewing Jerusalem as the true center of the world. That changed with the Age of Discovery, which placed Europe at the center of an ever-expanding world. Maps reflected that worldview, and largely continue to do so. That's why today's standard world map still has Europe at its center – with China off toward the periphery on the map's right-hand side. </p><p>The most notable feature of the very first major modern world map produced in China, the <em>Kunyu Wanguo Quantu</em> (1602), is that it places China firmly at the center of the world. Produced for the Chinese emperor by Jesuit missionary Matteo Ricci, it was the first map ever to combine that perspective with modern western knowledge: it was the first Chinese map to show the Americas, for instance. </p><p>That representation may not have taken off elsewhere, but it will be instantly recognizable to Chinese students, as it's the standard format for world maps in China's schools today.<br></p>
America on its head<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yMDU1Nzg2My9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYwMzQ5NTc0MH0.EqadI2Yp-2dPwi3VccFZelIDK4V9t0ZOfTfHjdB6wVw/img.jpg?width=980" id="97104" class="rm-shortcode" data-rm-shortcode-id="2b66e8de389b3d736bc28e019e445cd0" data-rm-shortcode-name="rebelmouse-image" />
Upside down you turn me: North America on its head, in Chinese characters
Image: Prior Probability<p>For those used to "classic" Eurocentric world maps, Europe's marginalization may come across as a bit of an upset. America's new position on the horizontal Chinese world map is less jarring: It merely moves from the left- to the right-hand side of the picture. But then there's this vertical world map, which deals a similar blow to the American land mass: divided in two and pushed to the upper and lower edges of the map.</p><p>Unfamiliar? Sure. Shocking? Perhaps. Wrong? Not really. First off, no world map is totally right, since it's mathematically impossible to transfer the surface of a three-dimensional object onto a flat surface without some distortion. And since the world is a globe, where you center that map is a matter of purely subjective choice.<br></p><p>Those choices have historical reasons. Mercator's map was not specifically designed to put an inflated Europe at the center of the world. That was just a side effect; its main purpose was to aid shipping: Straight lines on the map correspond to straight lines sailed on the seas.</p>
By 2050, a completely melted Arctic could enable the Transpolar Passage, shortening trade routes between Asia and Europe and boosting business for Alaskan ports like Nome and Dutch Harbor.
Image: The Maritime Executive<p>The vertical world map, showing the relative proximity of China (and the rest of Asia) to Europe and (even the East Coast of) North America, has a similarly maritime <em>raison d'être</em>, or it will have by mid-century. <a href="https://www.maritime-executive.com/editorials/the-arctic-shipping-route-no-one-s-talking-about" target="_blank" rel="noopener noreferrer">Experts project</a> that by 2050 (if not sooner), the Arctic will be sufficiently ice-free to enable the so-called Transpolar Passage, i.e. shipping straight across the North Pole. </p><p>That would shave more than three weeks off a traditional sea voyage between Europe and Asia, via the Suez Canal – and even be significantly faster than other northern alternatives like the Northwest Passage (via Canada) or the Northern Sea Route (hugging the Siberian coast). Since ships would not need to go through locks or pass over shallow waters, it would also remove current restrictions on tonnage per ship. <br></p><p>The only country seriously preparing for such a future: China. None of the other Arctic powers is giving the Transpolar route any strategic thought. On the other hand, China's Arctic Policy document, released in January 2018, already matter-of-factly refers to the Transpolar route as the 'Central Passage' – one of several 'Polar Silk Roads' that China seems to want to develop. And they already have the world map to go with it.</p>
What exactly does "questions are the new answers" mean?
- Traditionally, intelligence has been viewed as having all the answers. When it comes to being innovative and forward-thinking, it turns out that being able to ask the right questions is an equally valuable skill.
- The difference between the right and wrong questions is not simply in the level of difficulty. In this video, geobiologist Hope Jahren, journalist Warren Berger, experimental philosopher Jonathon Keats, and investor Tim Ferriss discuss the power of creativity and the merit in asking naive and even "dumb" questions.
- "Very often the dumb question that is sitting right there that no one seems to be asking is the smartest question you can ask," Ferriss says, adding that "not only is it the smartest, most incisive, but if you want to ask it and you're reasonably smart, I guarantee you there are other people who want to ask it but are just embarrassed to do so."