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enProbing Wikipedia editors’ hive mind for rules on cooperative behavior
https://www.sciencenews.org/article/probing-wikipedia-editors%E2%80%99-hive-mind-rules-cooperative-behavior
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<h2>Wikipedia, Encyclopedia, cooperation</h2>
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<span class="field-content">3:12pm, August 30, 2013</span> </div>
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<p>Wikipedia acts a bit like one big brain. Similar to how independently firing neurons somehow in aggregate produce thought, independently operating editors together produce the vast online encyclopedia, which, it has been <a target="_blank" href="http://www.nature.com/nature/journal/v438/n7070/full/438900a.html">argued</a>, has an accuracy approaching that of the carefully curated <em>Encyclopedia Brittanica</em>. Taken as a whole, the system performs a sort of enormous computation, taking in the collective knowledge of the Internet and spitting out encyclopedia articles.</p>
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So just as neuroscientists look for rules that explain how the individual firing of neurons comes together to create the large-scale phenomenon of thought, Simon DeDeo of the Santa Fe Institute is looking for rules that explain how zillions of individual edits come together to produce encyclopedia articles. Those articles can be remarkably different from what any individual editor would produce. The entry for <a target="_blank" href="http://en.wikipedia.org/wiki/George_W._Bush">George W. Bush</a>, for example, is the most heavily edited on Wikipedia, with 45,000 changes. Though each editor likely had strong opinions about him, the entry reads as a dispassionate exposition. Somehow the system was able to channel partisan discord into a single, generally coherent article. How? What rules underlie the group’s behavior?</p>
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These, of course, are such broad questions that scientists will be chewing on them for a long time to come — especially because DeDeo really wants to divine these types of driving rules for a wide variety of social systems. But Wikipedia made an especially good target for an initial stab at the questions because it makes so much data available, recording every last edit. DeDeo’s challenge was to figure out how to get a handle on the emergent dynamics, not just the individual behavior of editors. </p>
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The degree of cooperation between editors was one key behavior to analyze. A notable noncooperative behavior on Wikipedia is reversion, when an editor cancels a previous editor’s changes, returning the article to its earlier incarnation. This can lead to “revert wars” with individual editors insisting on their preferred version. </p>
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DeDeo analyzed data on the 10 most edited pages on Wikipedia, and he found that the longer the editors had worked together cooperatively, without reverts, the lower the likelihood the next edit would be a revert. It seems logical enough: Cooperation begets cooperation. But in addition, he found that the likelihood of a revert fell predictably according to a consistent mathematical law: On each of the pages, the probability of a revert declined as the square root of the length of time since the last revert. So the longer it’s been since the last revert, the lower the chance that the next edit will be a revert — but the probability falls quickly at first and more slowly later on. DeDeo described his results in July at the Santa Fe Institute and online at <a target="_blank" href="http://arxiv.org/abs/1212.0018">arXiv.org</a>. </p>
<p><br />
From one perspective, this is a surprisingly pessimistic result. Many standard models of reasoning would imply that people believe that there will be about as much cooperation in the future as they’ve seen in the past. DeDeo, together with Seth Lloyd of MIT, have realized that the square root law implies that although cooperation begets cooperation, it doesn’t beget quite that much: The square root law means that people believe that they will see less cooperation in the future than they have so far.</p>
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DeDeo has an intriguing hint that this square-root law might describe something fundamental to social systems. Together with Drew Cabaniss of the University of North Carolina, he has performed a similar analysis of revolutions in ancient Greece, figuring out the probability of revolution given how long a ruler has been in power. He has found preliminary evidence of the same thing: A revolution was less likely the longer a ruler had been in power, and the probability fell in proportion to the square root of the current ruler’s longevity.</p>
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DeDeo points out that his work has another implication: “Somehow the group itself has a memory.” The amount of time since the last revert influences the behavior of current editors, even though editors change over time and it’s possible that none have been actively editing the page the entire time since the last revert. DeDeo wants to understand how this institutional memory functions. One possibility is that editors assiduously read through the entire history of edits, but that is sufficiently time-consuming — particularly for the very heavily edited entries DeDeo studied — to make it unlikely. Another possibility is that the information is stored in the entry itself: A long period of cooperation may smooth out the issues that might attract a revert. A final possibility is that the community of editors might be affected by extended cooperation, forming a culture that makes them disinclined to revert.</p>
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DeDeo’s results give just a glimpse into how Wikipedia works at a deeper, hidden level, but it’s a tantalizing view. Our lives are increasingly influenced by social systems whose properties emerge from the loosely constrained behavior of many individuals, and we’re just beginning to develop the tools necessary to understand them. DeDeo’s work gives a glimmer of how mathematicians and scientists might proceed in untangling the workings of the hive mind. </p>
<span property="rnews:name schema:name" content="Probing Wikipedia editors’ hive mind for rules on cooperative behavior" class="rdf-meta element-hidden"></span>Fri, 30 Aug 2013 19:21:34 +0000Erin20579 at https://www.sciencenews.orgNew system offers way to defeat decryption by quantum computers
https://www.sciencenews.org/article/new-system-offers-way-defeat-decryption-quantum-computers
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<span class="field-content">11:20am, July 25, 2011</span> </div>
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<p>Quantum computers elicit dreams of great computational feats to come. But they also promise a nightmare: They could break today’s security codes, rendering them no more secure than a TSA-approved luggage lock.</p>
<p>Now, for the first time, researchers have shown a security method to be immune to the type of attack that could bring down RSA, the cryptosystem in almost universal current use. </p>
<p>Modern cryptography methods are generally based on some mathematical problem that is hard to solve without special information. Breaking RSA, for example, would require factoring very large numbers, a task thought to be insurmountably difficult even for supercomputers. But in 1994, mathematician Peter Shor, now of MIT, found an algorithm that a quantum computer (once one exists) could use to factor big numbers in seconds. </p>
<p>There’s no way of proving that a cryptosystem is impervious to all possible attacks. Still, Hang Dinh of Indiana University South Bend, Cristopher Moore of the Santa Fe Institute and Alexander Russell of the University of Connecticut in Storrs have now shown that one approach, known as the McEliece cryptosystem, is at least immune to the type of attack Shor devised to bring down RSA. “There may be an algorithm out there that can break it,” Moore says, “but it would have to use ideas that are completely different from any now known.” The team’s findings will appear in the Proceedings of CRYPTO 2011.</p>
<p>McEliece is based on the methods used for correcting errors in codes. If Alice sends Bob the binary message 011 directly and a bit gets corrupted along the way — say, the first bit flips to a 1 — then Bob won’t get her message accurately. To avoid this problem, she can encode the message using these strings:</p>
<p>String 1: 0001111</p>
<p>String 2: 0110011</p>
<p>String 3: 1010101 </p>
<p>Not only do these strings differ from one another in at least four places, but so do all eight possible strings produced by adding them together. Thus any combination of these strings can be distinguished from any other combination, even if a bit or two gets corrupted.</p>
<p>Because the second and third bits in Alice’s message are both 1, she would encode it by adding together the second and third strings (getting 1100110). Bob can use an algebraic method to decode this and retrieve her original message. </p>
<p>Even if a bit or two of Alice’s message gets corrupted along the way, Bob can figure out what she probably meant, because he knows that she must have sent one of the eight uncorrupted combinations of the original three coding strings. He just has to figure out which of the eight is closest to the message he received. Compact disc players use this method to play accurately even when the disc is somewhat scratched. </p>
<p>This decoding step isn’t always easy, though. The eight uncorrupted strings can be visualized as grid points in a seven-dimensional space (one dimension for each bit in the message). Bob’s problem then is to find the grid point nearest to a general point in the space. This is known as the “closest vector problem,” and for most lattices in very high-dimensional spaces it’s extraordinarily difficult; mathematicians since Carl Friedrich Gauss have tried to solve it for 200 years. But if the code for correcting errors is chosen carefully, Bob’s problem is easy.</p>
<p>Very hard problems like the closest vector problem can, of course, form the core of a cryptographic system. In the McEliece system, Alice comes up with an error-correcting code that is a scrambled, twisted version of one that’s simple to decode, and anyone can send her a message by using this code and adding a bit of noise. She can decode it easily because she knows how the code has been scrambled and twisted, but no one else can.</p>
<p>The researchers started out trying to use Shor’s algorithm to break McEliece but changed tactics when they couldn’t. “One of the fun things about computer science is that you can switch hats,” Moore says. “If a problem is hard to solve, you can try to turn it into a code that’s hard to break and make cryptographic lemonade from algorithmic lemons.”</p>
<p>The McEliece system isn’t commonly used because to be secure, the secret key has to be inconveniently long. As computational power and bandwidth increases, though, this may become less of an obstacle.</p>
<p>Other encryption systems are also thought to be immune to attack by quantum computers. The leading contender is a lattice-based cryptosystem that, like McEliece, has the closest vector problem at its heart. That system may be more secure and flexible but hasn’t yet been proven to be immune to Shor’s algorithm. Dinh is now interested in seeing if her team can do that.</p>
<span property="rnews:name schema:name" content="New system offers way to defeat decryption by quantum computers" class="rdf-meta element-hidden"></span>Wed, 31 Jul 2013 15:28:48 +0000Allison20570 at https://www.sciencenews.orgGrocers stacking oranges demonstrate intuitive grasp of sphere-packing math
https://www.sciencenews.org/article/grocers-stacking-oranges-demonstrate-intuitive-grasp-sphere-packing-math
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<span class="field-content">11:02am, November 14, 2011</span> </div>
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<p>They may not know it, but grocers face some of the most difficult questions in mathematics when stacking produce each day. </p>
<p>Four centuries ago, the astronomer and mathematician Johannes Kepler guessed that the standard grocers’ method of piling oranges packs the most fruit into the least space. Confirming he was right had to wait until 1998, when mathematician Thomas Hales of the University of Pittsburgh, working with his student Samuel Ferguson, proved Kepler’s conjecture with the aid of 180,000 lines of computer code. </p>
<p>But even that achievement didn’t settle all the grocers’ dilemmas. Researchers have known that other sphere-stacking methods can be equally dense. So how does one recognize when a stack is the densest possible? A partial answer appears in a paper posted October 3 at arXiv.org.</p>
<p>Back in 1969, Hungarian mathematician László Fejes Tóth thought he’d figured out one simple method: If each sphere is in contact with as many others as possible, the packing would be the densest achievable. But he couldn’t prove it, and even the techniques that Hales and Ferguson used to settle the formidable Kepler conjecture weren’t powerful enough to crack it. But now Hales himself has shown that Tóth’s guess was accurate. </p>
<p>“This is really a major advance,” says Henry Cohn of Microsoft Research in Cambridge, Mass. “As with the Kepler conjecture, if you mess around with ping-pong balls, you can convince yourself that there isn’t another way to do it. But proving it is another matter entirely.”</p>
<p>Tóth started with what mathematicians call the “kissing number.” Imagine placing a penny on a table and putting as many pennies as you can touching it — or, in mathematical parlance, “kissing” it. You’ll find that exactly six pennies fit, forming a hexagonal pattern. As a result, the kissing number in two dimensions is six. This pattern can be extended to cover the entire table, with each penny surrounded by six others.</p>
<p>Now do the same thing in three dimensions: Take an orange and arrange as many oranges as possible kissing it. It turns out that you can’t fit more than 12, so the kissing number in three dimensions is 12. Unlike with pennies, though, the 12 oranges won’t fit snugly around the central one; there’s a bit of room to spare, so the oranges can jiggle into different positions. As a result, if you extend this pattern out to fill space, with each ball surrounded by 12 others, the jiggliness will give you a lot of choices about how to arrange the balls. </p>
<p>Tóth’s idea was that no matter what choices you make, the balls will have the densest possible packing — that is, their arrangement will fit the most oranges possible into a given space. Furthermore, he figured that he knew every possible arrangement the oranges could have: They’d all be variations on the way grocers do it. In particular, they would be arranged in layers, and each layer would have a hexagonal pattern just like the pennies. </p>
<p>Grocers stack fruit by laying the bottom row of the bottom layer first. They then put the next row in the crevices between the oranges of the first row, the following row in the crevices of the second row, and so on. This process ends up creating a hexagonal pattern, with each orange touching six others in its layer. The next layer goes in the pockets created between the oranges in the first layer. But only half the pockets are filled, because placing an orange in every pocket would force the oranges to overlap. So the grocer has a choice of which of the two sets of pockets to fill with each layer. This means that there are infinitely many possible ways to stack the oranges (assuming, of course, that you start with an infinite orange supply). The upshot of Tóth’s guess is that if each orange touches 12 others, then the overall packing must be laid according to this pattern. </p>
<p>Hales’ confirmation, available at<a href="http://arxiv.org/abs/1110.0402" target="_blank"> http://arxiv.org/abs/1110.0402</a>, grew out of his effort to settle a lingering uncertainty about his proof of the Kepler conjecture. Because that proof relied on such extensive computer code, mathematicians couldn’t guarantee that no hidden bug had led to an error. Inspired by the way that Georges Gonthier of Microsoft Research verified his proof of a different theorem, Hales has been formally checking his own work by spelling out each step explicitly and using a computer to check the logical deductions. “The constraints of doing everything rigorously force you to understand the structure of a problem and simplify it,” Gonthier says. “And in this case, that led to a very nice advance.”</p>
<span property="rnews:name schema:name" content="Grocers stacking oranges demonstrate intuitive grasp of sphere-packing math" class="rdf-meta element-hidden"></span>Wed, 31 Jul 2013 15:10:56 +0000Allison20569 at https://www.sciencenews.orgTurning numbers into shapes offers potential medical benefits
https://www.sciencenews.org/article/turning-numbers-shapes-offers-potential-medical-benefits
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<span class="field-content">10:39am, October 3, 2011</span> </div>
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<p>Until recently, topology was seen as being among the most abstract fields of mathematics, one that bore out Henry John Stephen Smith’s 19th century toast: “Pure mathematics — may it never be of use to anyone!” But now the field, which deals with the shape of many-dimensional objects, has unexpectedly proved its usefulness in, of all places, medicine. Researchers have used topology to discover a new subgroup of breast cancer patients with a 100 percent survival rate. More generally, the method may prove powerful for making sense of the massive, high-dimensional, noisy datasets modern science is producing.</p>
<p>Genetics experiments can produce vast quantities of data — determining the activity of each of the approximately 20,000 genes in a sample of breast cancer tissue, for example. Each sample can be seen as a point in 20,000-dimensional space. But these readings aren’t absolutely accurate, so each point may not be in exactly the right place. That makes plucking information out of that sea of data particularly challenging.</p>
<p>One key is to recognize that “data has shape, and that shape matters,” says mathematician Gunnar Carlsson of Stanford University.</p>
<p>Topology turns out to be especially useful for identifying the shape of noisy data, because it characterizes shapes in a flexible, qualitative way. Squish, twist or enlarge an object and topology will consider it unchanged, as long as you don’t punch holes or glue bits together. So from the perspective of topology, a coffee cup and a doughnut have the same shape: By squishing the cup down, the handle turns into a doughnutlike ring. This qualitative understanding turns out to deal perfectly with the noisiness of data sets, since the precise location of data points doesn’t matter.</p>
<p>To show the power of topological methods for data analysis, Carlsson and his colleagues Monica Nicolau of Stanford and Arnold Levine of the Institute for Advanced Study in Princeton, N.J., analyzed gene activity data from about 300 Dutch breast cancer patients. </p>
<p>To turn the discrete data points into a surface that topology could analyze, the researchers calculated how different each breast cancer sample was from normal tissue and decreed two data points to be close to one another if they had a similar degree of difference from the normal tissue. The scientists then “fattened up” the data points to form a surface by essentially considering all the points within a certain distance from the existing data points to be within the surface.</p>
<p>The next step was to understand the shape of this 20,000-dimensional surface. Carlsson notes that the crucial details probably fit in many fewer than 20,000 dimensions. A cylinder, for example, lives in three dimensions, but since it can be squished flat it’s topologically equivalent to a circle, which lives in only two. Carlsson’s team created a version of the data in two dimensions that captured essential aspects of the data’s shape, if not every detail. </p>
<p>The resulting shape looked like a Y, with normal patients at the bottom. The right-hand flare consisted of known subgroups of patients with mostly poor prognoses, but the left-hand flare consisted of patients who had not been previously identified as a coherent subgroup. </p>
<p>To see what these patients had in common biologically, Nicolau checked their survival rates. She was shocked: “I saw the best survival curve I’ve seen in my entire life.” Eight percent of the patients fell into the newly identified group, and not one of them had died from their cancer in the 10 years they’d been followed. Further analysis showed that gene activity patterns among these patients were extremely similar, suggesting that the same gene had been mutated in each case. Applying the same analysis to two more groups comprising 134 women yielded the same result.</p>
<p>The impact on breast cancer treatment is not yet clear. Most of the patients in the new subgroup were already known to have good prognoses, so they were unlikely to receive aggressive treatment. Further studies would be needed to know if these patients would do just as well with no treatment. A private company that Carlsson cofounded, Ayasdi Inc., is working to bring the result to clinical practice. The scientists are also working to identify subgroups of leukemia patients with the hope of understanding which treatments are appropriate for which patients.</p>
<p>Since publishing the breast cancer work in the <em>Proceedings of the National Academy of Sciences </em>in April, the researchers have applied their method to many other datasets. Among other things, they tracked the fate of the oil plume after the Gulf of Mexico spill, the disappearance of moderate votes in Congress in 2009–2010 and the number of functional positions that basketball players assume on the court. </p>
<span property="rnews:name schema:name" content="Turning numbers into shapes offers potential medical benefits" class="rdf-meta element-hidden"></span>Wed, 31 Jul 2013 14:47:04 +0000Allison20564 at https://www.sciencenews.orgApp for analyzing leafy curves lets amateur botanists identify trees
https://www.sciencenews.org/article/app-analyzing-leafy-curves-lets-amateur-botanists-identify-trees
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<span class="field-content">4:46pm, August 29, 2011</span> </div>
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<p>Peter Belhumeur just wanted to know what type of tree was outside his apartment. He scanned field guides, looking for willowlike leaves with a wavy, spiked edge, but he couldn’t find the right one. Neither could his neighbors.</p>
<p>As it happened, though, he was building a tool designed to address just this problem. Belhumeur is a computer scientist at Columbia University who has worked on face recognition, and he and David Jacobs of the University of Maryland in College Park had dreamed up an iPhone app that would use similar mathematical methods to recognize trees. Leafsnap (<a href="http://leafsnap.com/">www.leafsnap.com</a>) allows a person to pick a leaf from the U.S. Northeast, snap a picture of it against a white background, and find out which tree species it is most likely to have come from. </p>
<p>“We thought we could build an electronic field guide for the 21st century,” Belhumeur says, “so that the search would be initiated by an image, much like a Google search is initiated by a text query.”</p>
<p>To teach a smartphone to identify trees, Belhumeur and Jacobs first needed to know how a botanist does the job, so they turned to John Kress of the Smithsonian Institution in Washington, D.C., and his colleagues. Although botanists consider many aspects of the tree — the bark, the flowers, the overall shape of the tree, its height — the key feature is the shape of the leaves. </p>
<p>So the scientists had to define “shape” in a way that a computer could understand. They ended up focusing on curvature at varying scales: At the finest scale, a sharply curving leaf would be serrated, like an elm leaf. At the middle level, a leaf with a lot of curvature would have lobes, like an oak leaf. And at the largest scale, a highly curved leaf would be round like an aspen rather than elongated like a willow. </p>
<p>To calculate curvature, the researchers place a disk under the edge of the leaf and then compute the percentage of the disk that is covered by the leaf. If the edge is perfectly straight, exactly half the disk will be covered; if the edge is concave, less than half will be; and if it is convex, more than half will be. The app analyzes a leaf by comparing the distribution of its curvature at each scale with a set of 8,000 leaf photographs. </p>
<p>Leafsnap then returns a ranked list of the species that are the closest matches. Clicking on a species gives more information, including high-quality photos of the entire tree, its bark, leaves, flowers and fruit. </p>
<p>About 70 percent of the time, the team found, the app returns the correct species as its top choice. It gets within the top three about 90 percent of the time, as long as the picture is taken against a bright background and is free of shadows.</p>
<p>Face-recognition algorithms often use a similar approach with different criteria. One common criterion involves drawing an arrow at each point of the image toward the lightest nearby area. For example, a point at the top of an eyebrow would have an arrow pointing upward, since the skin above the eyebrow is lighter than the eyebrow hair below. Crucially, that will be true regardless of how the face is lit. The face-recognition algorithm can then compare the distributions of the arrows to match up different images of the same face.</p>
<p>In the first two months after its May release, Leafsnap was downloaded more than 300,000 times, swamping the servers in Belhumeur’s lab. The team now wants to expand to include trees across the entire United States. </p>
<p>Shape analysis has the potential for much broader application. Google, for example, would like to develop its Google Goggles app to recognize a pair of shoes from their shape in a photograph and take you to a website to buy them. Doctors would like to be able to take a CAT scan, identify the organs from their shapes, and use a computer to precisely guide radiation treatment. Other researchers are working to identify Alzheimer’s patients from the shape of their hippocampi. Leafsnap’s basic mathematical approach could be applied to all these problems.</p>
<p>For now, the team is pretty happy with just leaves. Belhumeur tried the app prototype on the puzzling tree outside his apartment. The answer made his heart sink: a sawtooth oak. “I thought, that’s discouraging, because this is clearly not an oak,” he says. “I know what an oak looks like, and this isn’t it.” But even though he knew that oaks have round, lobed leaves, not the pointed ones on his tree, he checked out the sawtooth oak anyway. And sure enough, the app had it right.</p>
<span property="rnews:name schema:name" content="App for analyzing leafy curves lets amateur botanists identify trees" class="rdf-meta element-hidden"></span>Tue, 30 Jul 2013 20:54:46 +0000Allison20560 at https://www.sciencenews.orgFlatland and its sequel bring the math of higher dimensions to the silver screen
https://www.sciencenews.org/article/flatland-and-its-sequel-bring-math-higher-dimensions-silver-screen
<div class="field field-name-field-article-type field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even">Math Trek</div></div></div><div class="field field-name-field-op-section-term field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/search?tt=97" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numbers</a></div></div></div><div class="field field-name-field-sn-subtitle">
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<span class="field-content">10:00am, July 29, 2013</span> </div>
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<p>In 1884, Edwin Abbott wrote a strange and enchanting novella called <em>Flatland</em>, in which a square who lives in a two-dimensional world comes to comprehend the existence of a third dimension but is unable to persuade his compatriots of his discovery. Through the book, Abbott skewered hierarchical Victorian values while simultaneously giving a glimpse of the mathematics of higher dimensions. </p>
<p>In 2007, <em><a href="http://www.flatlandthemovie.com/">Flatland</a></em> was made into an animated movie with the voices of Martin Sheen, Kristen Bell and Michael York. Now there’s a sequel called <em><a href="http://www.spherelandthemovie.com/">Flatland 2: Sphereland</a></em>, which expands the story into non-Euclidean geometry. It’s a gem: an engaging, beautiful and mathematically rich film that children and adults alike can enjoy.</p>
<p>The first film, <em>Flatland: The Movie</em>, takes immediate advantage of its medium by opening with a dazzling flyover of Flatland. The inhabited parts of Flatland look like ornate Islamic tilings, and the uninhabited parts are filled with exotic fractal patterns that could come from the surface of Mercury. </p>
<p>We then zoom into the home of Arthur Square, who is late to take his granddaughter Hex to school at St. Euclid. On the way, Arthur drills Hex — a darling little hexagon, complete with a bow and a big expressive eye — on Flatland’s uncompromising hierarchy. Each subsequent generation, Hex dutifully reports, acquires an additional side, so that Arthur Square’s children are pentagons and Hex is a hexagon. In Flatland, having more sides supposedly means that you’re smarter. Lowly triangles are good only for manual labor; squares are part of the professional class; and creatures with so many sides that they look circular are priests, who, frowns Hex, “just make rules that everyone else has to obey.”</p>
<p>The circles pronounce a frightening decree: Anyone espousing the nonsensical and heretical notion that the third dimension exists will be executed. What, Hex asks, is a dimension? So Arthur explains: A point is zero-dimensional; a point moving straight traces out a one-dimensional line; and a line moving perpendicular to itself traces out a two-dimensional square. Hex immediately makes the forbidden leap: A square that somehow moves perpendicularly to itself, she reasons, would trace out a “super square” in three dimensions. She even calculates how big such an object would be. But her mathematical insights only earn her a scolding from Arthur.</p>
<p>By the end, Hex wins happy vindication. Both she and Arthur get a mind-blowing tour of the full three-dimensional universe from a sphere, Spherius — and they even manage to proclaim their discovery and save their own skins (er, perimeters). </p>
<p>But when they ask Spherius about a fourth or fifth dimension, following their mathematical logic, he’s as skeptical as their compatriots had been about the third.</p>
<p>The sequel joins Hex 20 years later, with her bow long lost and a disillusioned cast to her eye. Although Hex and Arthur’s discoveries have knocked the circle priests from power and brought equal rights to all shapes, Flatlanders still deny the reality of the third dimension. The sphere never returned, and Arthur has died heartbroken and disgraced. Hex is now living in isolation, pursuing her mathematics. </p>
<p>Then a fellow hexagon, Puncto, seeks her out for help with a mathematical problem he can’t get anyone else to take seriously. He’s an engineer for the Flatland space program, and his data haven’t made any sense. By his calculations, the angles on some very big triangular paths that Flatland’s rockets will follow to other planets add up to more than 180 degrees. Everyone has been telling him that he must just be making a mistake, but he’s convinced there’s a deeper issue. Space itself must be warped, he says. And if space is warped, the rocket they’re about to send out could hit an asteroid in the Sierpinski belt!</p>
<p>Hex and Puncto end up on an otherworldly adventure through multiple dimensions and worlds. Hex stumbles on a key mathematical insight — the key to Puncto’s dilemma — when they visit one-dimensional Lineland. The world appears to be a straight line, but when they travel high above it they discover that it’s a circle. Hex realizes that similarly, Flatland itself might not be flat, even though it seems so — it could be curved into the third dimension. Perhaps Flatland is on the surface of a sphere: Sphereland! If so, Hex realizes, the edges of a triangle in Flatland would actually curve outward in three dimensions, making the angles a bit more than 180 degrees, just as Puncto had found. </p>
<p>But if Hex is right, the rocket’s path is off, and unless she and Puncto convince the Flatlanders of their discovery, it could crash. Thus begins a madcap race back to Flatland, complete with other mathematical revelations along the way.</p>
<p>Calling the films educational somehow seems an insult. They manage to accomplish that so-rare feat of giving viewers a taste of the delight of mathematical discovery while carrying them along through a quirky, multi-dimensional story.</p>
<span property="rnews:name schema:name" content="Flatland and its sequel bring the math of higher dimensions to the silver screen" class="rdf-meta element-hidden"></span>Mon, 29 Jul 2013 14:08:34 +0000Allison20524 at https://www.sciencenews.orgBeer bubble math helps to unravel some mysteries in materials science
https://www.sciencenews.org/article/beer-bubble-math-helps-unravel-some-mysteries-materials-science
<div class="field field-name-field-article-type field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even">Math Trek</div></div></div><div class="field field-name-field-op-section-term field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/search?tt=97" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numbers</a></div></div></div><div class="field field-name-field-sn-subtitle">
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<span class="field-content">12:00am, June 10, 2011</span> </div>
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<p>Before downing your next beer, pause to contemplate the bubbles. You’ll find that they grow and shrink in odd, hard-to-predict ways. A mathematician and an engineer have found a simple and surprising equation to describe this process, using a field of mathematics no one expected to be relevant. </p>
<p>Now, new simulations are building on that result to illuminate more than just foams. Metals and ceramics are made of crystals that grow and shrink the same way that beer bubbles do, affecting the properties of the materials. The new work may thus lead to more resilient airplane wings, more reliable computer chips and stronger steel beams.</p>
<p>Over time, the bubbles in foam tend to consolidate, becoming fewer and larger. The physics driving this process has long been understood: When two bubbles adjoin one another, gas tends to pass from the bubble with higher pressure to the one with lower pressure. The higher-pressure bubble bulges into the lower-pressure one, so the shape of a bubble reflects the pressure in all the bubbles around it. As a result, the shape of the bubble should be enough to determine how fast it’s going to grow or shrink. </p>
<p>In 1952, the mathematician John von Neumann used this principle to find an elegant equation that predicts the growth rate of a bubble in a two-dimensional spread of bubbles. But the three-dimensional case — which is the one scientists and beer drinkers most care about — proved far more difficult. Some researchers even believed the problem unsolvable. </p>
<p>What was needed, it turned out, was to introduce another field of mathematics: topology. Topology is essentially the study of connections. For a topologist, two objects are the same if one can be shrunk or stretched into the same shape as the other, but without punching holes or gluing anything together, since that would change the way the parts of the object connect to one another. So, for example, a doughnut and a coffee cup are the same shape to a topologist: Squish down the cup part of the coffee cup, and you’ll end up with a doughnut-shaped ring. A doughnut is topologically different, however, from a beach ball.</p>
<p>Topology would seem to offer little help in determining how quickly bubbles in foam grow or shrink, because squishing a bubble into a different shape will change its growth rate. But several years ago researchers found an equation involving the Euler characteristic, a property of a shape that stays the same no matter how the shape is stretched or smushed. “It’s really beautiful,” says Robert MacPherson of the Institute for Advanced Study in Princeton, N.J., the mathematician on the project, “because the Euler characteristic shouldn’t have anything to do with it.” </p>
<p>To calculate the Euler characteristic, first imagine slicing a bubble in different directions. Usually, the resulting shape would be roughly circular. But suppose that the bubble had a couple of little bumps on its surface. If your slice went through these bumps, you could end up with two disconnected circles. Or, if the bubble had a small divot in it and your slice went through the divot, you could end up with a little hole inside your slice. The Euler characteristic of the slice is the number of disconnected pieces it contains, minus the number of holes within it.</p>
<p>The equation that MacPherson and David Srolovitz of the Agency for Science, Technology and Research in Singapore developed shows that bubbles grow quickly when the beer is warm and when the bubbles have divots in them rather than bumps, or when they are connected to lots of other bubbles.</p>
<p>Now, the team is taking the equation one step further, using it to create computer simulations of foams, metals and ceramics. The work is a way to expose a material’s inner structures that are not normally visible, but that influence the material’s overall properties. “If you beat an egg white enough, it becomes almost a paste and you can make peaks out of it,” MacPherson says. “If you looked at an individual egg white bubble, you wouldn’t expect that. We want to understand these collective properties.”</p>
<p>“Topology is going to be a very powerful tool for understanding these structures,” says Jeremy Mason, a materials scientist at the Institute for Advanced Study who has joined the research team. He points out that the behavior of foam as a whole is unlikely to change dramatically if you stretch or squish the bubbles a bit without changing the way they’re connected to one another, even though changing the shape of an individual bubble will affect its growth rate. Focusing on the connections — that is, the foam’s topology — may then allow researchers to home in on its most crucial aspects. </p>
<p>“We’re awash in data, and the challenge is to identify what is meaningful and compare it between two different structures,” Mason says. “Measuring topological characteristics may give us a language for that.”</p>
<span property="rnews:name schema:name" content="Beer bubble math helps to unravel some mysteries in materials science" class="rdf-meta element-hidden"></span>Tue, 09 Jul 2013 20:27:46 +0000Kate20507 at https://www.sciencenews.orgA field where breakthroughs are hard to come by produces two big advances on a single day
https://www.sciencenews.org/article/field-where-breakthroughs-are-hard-come-produces-two-big-advances-single-day
<div class="field field-name-field-article-type field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even">Math Trek</div></div></div><div class="field field-name-field-op-section-term field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/search?tt=97" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numbers</a></div></div></div><div class="field field-name-field-sn-subtitle">
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<span class="field-content">9:52am, June 24, 2013</span> </div>
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<div class="field-item even"><br />H.A. Helfgott. Major arcs for Goldbach's theorem. arXiv.org. Posted May 13, 2013. <a href="http://arxiv.org/abs/1305.2897">[Go to]</a></div>
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<p>Problems in number theory often have a certain exasperating charm: They are extraordinarily simple to state, but so difficult to prove that centuries of effort haven’t sufficed to crack them. So it’s pretty remarkable that on one day this May, mathematicians announced results on two of these mathematical conundrums. Both proofs address one of the most fundamental questions in all of mathematics, the relationship between multiplication and addition.</p>
<p>On May 13, a virtually unknown lecturer at the University of New Hampshire, Yitang Zhang, shocked experts when he announced in a talk at Harvard a proof that takes steps toward solving one of the oldest problems in all of mathematics: the twin prime conjecture.</p>
<p>“Zhang’s result came completely out of the blue,” says Andrew Granville of the University of Montreal. “It’s huge.”</p>
<p>Prime numbers — those divisible only by 1 and themselves — are like the fundamental particles of mathematics, the indivisible building blocks out of which all other numbers are formed. Mathematicians have long noticed that primes often occur in pairs that differ by 2, like 5 and 7 or 137 and 139. They suspect that there are infinitely many pairs of primes that differ by 2, as well as infinitely many that differ by any even number.</p>
<p>Since primes get rarer as they get larger, it’s easy to imagine the opposite — that the gaps between them also grow. </p>
<p> Zhang has shown that that’s not true. He has shown that there’s at least one number <em>N</em> such that there are infinitely many pairs of primes that differ by <em>N</em>. Zhang can’t tell you what <em>N</em> is — but he does know it’s smaller than 70 million. </p>
<p>True, 70 million is a lot more than 2. But it’s a start.</p>
<p>“All of the experts, including me, had thought about how to develop the idea,” says Granville. “We <em>thought</em> we thought through the idea that Zhang ultimately used, but we decided there was no way. Zhang fortunately didn’t know the experts, so he didn’t get put off. He found the way that we all missed, not by a little bit but by a lot.”</p>
<p>Perhaps Zhang was able to find a different way because after earning his doctorate in 1991, he chose an unorthodox path. He apparently didn’t seek an academic job and worked for a while in a sandwich shop before finding work as a lecturer. Still, he kept doing mathematics. His thesis advisor, T.T. Moh of Purdue University, describes him in extraordinary terms: “When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer.” </p>
<p>The same day that Zhang gave his talk, Harald Helfgott of the École Normale Supérieure in Paris <a href="http://arxiv.org/abs/1305.2897">posted a paper online</a> with a proof closing in on another major open problem in number theory. The Goldbach conjecture says that any even number greater than 2 is the sum of two primes. The number 8, for example, is 3+5. Helfgott has proven something a bit easier, a proposition known as the odd Goldbach conjecture: that any odd number greater than 5 is the sum of three primes. The number 7, for example, is 2+2+3.</p>
<p>The Goldbach conjecture implies the odd Goldbach conjecture: subtract 3 from any odd number greater than 5, express the result as a sum of two primes, add back the 3, and you’ve expressed your original number as a sum of three primes. Unfortunately, the trick doesn’t work the other way. </p>
<p>Helfgott’s finding didn’t shock mathematicians the way Zhang’s did because he was following a well-understood approach. In 1919, mathematicians G.H. Hardy and John Littlewood produced a function that counts how many different ways one can represent a number as the sum of three primes. As long as the result of that function is always greater than 0, the odd Goldbach conjecture is true. Because this function is created using the same methods that decompose a radio signal into its basic frequencies, Helfgott describes this as “listening to the primes.” </p>
<p>Generally, big numbers can be represented in many different ways as the sum of three primes. In 1937, Ivan Vinogradov used Hardy and Littlewood’s function to prove the odd Goldbach conjecture for big numbers. Really, really big, that is: at least 10<sup>6,846,168</sup> or so. </p>
<p>That number is boggling, far beyond the number of atoms in the universe and the magnitude computers can handle. So the challenge has been to bring that limit down to something manageable and then to allow computers to do the rest, which Helfgott has finally managed to do. </p>
<p>“Helfgott’s treatment must have 60 or 70 clever new ideas in it, and each of them produces a small improvement, and eventually he got it down to something he could deal with in practice,” says Roger Heath-Brown of the University of Oxford. “It’s a great achievement.”</p>
<p>Unfortunately, proving either Goldbach or the twin primes in full is likely to require an approach fundamentally different from the ones Helfgott and Zhang used. Mathematicians hope to hone down Zhang’s 70 million number — the most wildly optimistic guess is that they might reduce it to 16 — but very significant new ideas will be needed to get all the way to 2. And Helfgott’s method is known not to extend beyond sums of three primes.</p>
<span property="rnews:name schema:name" content="A field where breakthroughs are hard to come by produces two big advances on a single day" class="rdf-meta element-hidden"></span>Mon, 24 Jun 2013 14:02:31 +0000Allison20505 at https://www.sciencenews.orgOne of the most abstract fields in math finds application in the 'real' world
https://www.sciencenews.org/article/one-most-abstract-fields-math-finds-application-real-world
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<span class="field-content">9:53am, May 20, 2013</span> </div>
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<div class="field-item even"><br />D.I. Spivak. Category theory for scientists. arXiv.org. Published online February 27, 2013. arXiv:1302.6946v2 <a href="http://arxiv.org/abs/1302.6946">[Go to]</a> </div>
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<p>Every pure mathematician has experienced that awkward moment when asked, “So what’s your research good for?” There are standard responses: a proud “Nothing!”; an explanation that mathematical research is an art form like, say, Olympic gymnastics (with a much smaller audience); or a stammered response that so much of pure math has ended up finding application that maybe, perhaps, someday, it will turn out to be useful.</p>
<p>That last possibility is now proving itself to be dramatically true in the case of category theory, perhaps the most abstract area in all of mathematics. Where math is an abstraction of the real world, category theory is an abstraction of mathematics: It describes the architectural structure of any mathematical field, independent of the specific kind of mathematical object being considered. Yet somehow, what is in a sense the purest of all pure math is now being used to describe areas throughout the sciences and beyond, in computer science, quantum physics, biology, music, linguistics and philosophy. </p>
<p>Samuel Eilenberg of Columbia University and Saunders Mac Lane of the University of Chicago developed category theory in the 1940s to build a bridge between abstract algebra (the generalization of high school algebra) and topology (the qualitative study of shapes, including those in very high dimensions). Very similar arguments repeatedly cropped up in the two fields in different contexts, so the mathematicians reasoned that some deeper structure must unite these situations. </p>
<p>They created an organizing framework that any field of mathematics could be put in. A “category” is a collection of mathematical objects together with arrows connecting them. So, for example, the natural numbers are the objects of a category, and one particular arrow in that category would connect each number to its double. Eilenberg and Mac Lane could then analyze maps between entire categories, and maps between those maps. This allowed the connections between different fields of mathematics to be formulated precisely. </p>
<p>Mathematicians sardonically dubbed the field “abstract nonsense.” Its extreme level of abstraction drains all the content out of the theory, since the objects can represent nearly anything. Draining the content, many expected, would also drain its power: What can anyone possibly say that will apply to essentially all mathematical objects? </p>
<p>Surprisingly, a lot. The recurrent arguments that had spurred the theory were ones that applied to all categories. Eilenberg and Mac Lane’s framework revealed an entire world of theorems that could be applied throughout mathematics.</p>
<p>Logicians started using category theory, viewing a deduction of one theorem from another as an arrow connecting the two. Then computer scientists carried category theory further still, viewing programs as maps connecting input of one category to output of another. A program that multiplies two numbers, for example, would go from the category of pairs of numbers (the numbers being multiplied) to the category of numbers (their product). </p>
<p>These connections turned out to be extraordinarily deep — indeed, the theory of programming languages and the field of logic can be seen as essentially identical to category theory. Computer scientist Robert Harper of Carnegie Mellon University jokingly calls this “computational trinitarianism,” imitating the Christian notion that God is a trinity of Father, Son and Holy Ghost. </p>
<p>“The central dogma of computational trinitarianism,” he wrote on his <a href="http://existentialtype.wordpress.com/2011/03/">blog</a>, “holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation. There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives. Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two.” Porting ideas between the fields has led to profound insights for all three.</p>
<p>Category theory’s spread has continued. Many results in quantum information theory turn out to follow directly from category theory. Category theory’s hierarchical structure has made it useful for modeling complex biological systems. Category theoretic models of language have outperformed conventional ones in distinguishing, for example, the meaning of “saw” in sentences like “I saw a man with a saw.” It’s even proving valuable in developing rigorous models of music theory.</p>
<p>David Spivak of MIT has perhaps the boldest vision for category theory’s potential. In a paper posted February 27 on <a href="http://arxiv.org/abs/1302.6946">arXiv.org</a>, he argues that all scientific thought can be expressed in a structured way using category theory. Both ideas and the data supporting them can be encoded in the universal language of category theory, allowing scientists to present a database with their full work. Spivak even imagines a Facebook-like interface with people’s full thoughts and experiences presented in a category theoretic database that would connect people whose databases overlap.</p>
<p>“If people adopt the level of rigor of category theory,” he says, “it will provide a precise language for science as a whole, and it will help individual scientists to clarify their thinking. My ultimate dream is that communication problems would only happen because someone is trying to lie.” </p>
<span property="rnews:name schema:name" content="One of the most abstract fields in math finds application in the 'real' world" class="rdf-meta element-hidden"></span>Mon, 20 May 2013 14:02:52 +0000Allison20501 at https://www.sciencenews.orgA theorem in limbo shows that QED is not the last word in a mathematical proof
https://www.sciencenews.org/article/theorem-limbo-shows-qed-not-last-word-mathematical-proof
<div class="field field-name-field-article-type field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even">Math Trek</div></div></div><div class="field field-name-field-sn-subtitle">
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<span itemprop="name"><a href="/author/julie-rehmeyer">Julie Rehmeyer</a></span> </div>
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<span class="field-content">11:56am, March 25, 2013</span> </div>
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<p>When a top-tier mathematician announced in August that he had proved one of the greatest problems in mathematics, the claim was trumpeted in the <em>New York Times</em>, <em>Nature</em>, <em>Science</em> and the<em> Boston Globe</em>.</p>
<p>
But that didn’t make the abc conjecture proven. People often think of mathematics as a solitary pursuit, with a written proof as final product. In fact, it’s an unavoidably social activity, even for mathematicians who prefer to work alone. A theorem isn’t proven until the mathematical community is persuaded that it’s proven. And proofs today are often so complex that that persuasion must happen in person.</p>
<p>
Six months after Shinichi Mochizuki of Kyoto University in Japan released his 500-page proof of the abc conjecture, that vetting process has yet to occur. No one has been able to explain the central ideas of the proof. And few people are trying to understand it anymore, with the possible exception of a mathematician or two in Japan. </p>
<p>
Whenever major proofs are announced, mathematicians caution that the work might not hold up. Ordinarily it’s a matter of checking for hidden errors, as mathematicians in the field quickly understand the strategy of the proof. </p>
<p>
But this time, no one except Mochizuki seems to have any glimmering of how his proof works. It is so peculiar that mathematicians might have dismissed it as the work of a crank, except that Mochizuki is known as a deep thinker with a record of strong results.</p>
<p>
Also, they really hope he is right. Though the abcconjecture is only 30 years old, it has become one of the greatest prizes in mathematics, subsuming Fermat’s Last Theorem along with four other major theorems in number theory. </p>
<p>
To understand what it says, start with two whole numbers, <em>a</em> and <em>b</em>, that are divisible by small primes raised to large powers. Say <em>a</em> = 2<sup>10</sup> = 1,024 and <em>b</em> = 3<sup>4</sup> = 81. Add them together to get <em>c</em>: In this case, <em>c</em> =1,024 + 81 = 1,105. That number happens to be the result when you multiply three other prime numbers raised to small powers (in this case the power is 1): 5 x 13 x 17 = 1,105.</p>
<p>
This pattern of two numbers <em>a</em> and <em>b</em> that are divisible by small primes raised to large powers adding up to a number <em>c</em> that is divisible by large primes raised to small powers turns out to be quite common. In the 1980s, mathematicians formulated this observation precisely into the abc conjecture, encapsulating a deep connection between the two most basic mathematical operations, addition and multiplication.</p>
<p>
In August, when Mochizuki released the four papers explaining his proof, a number of mathematicians dove into them eagerly. But they couldn’t even understand his vocabulary. Mochizuki had built an entirely new mathematical field, one he named “inter-universal Teichmüller geometry,” and populated it with objects no one had ever heard of: “anabelioids,” “Frobenoids,” “NF-Hodge theaters.” Without any sense of the overarching logic of the proof, his readers bogged down in minutiae. </p>
<p>
This isn’t all Mochizuki’s fault: Mathematicians generally don’t necessarily like to read mathematics. “It’s so painful to read someone else’s paper, even if it is a short paper,” says Minhyong Kim of the University of Oxford. The formality and precision necessary for accuracy can interfere with developing intuition, particularly with such unfamiliar mathematics.</p>
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Mathematicians began clamoring for Mochizuki to explain the kernel of his ideas, but he refused. Kim explains the refusal this way: “Imagine asking a poet what a poem means: They’d probably say no. What they meant by the poem is what they wrote. I suspect that this is the psychology of the situation for Mochizuki. He said what he wanted to say in the paper.”</p>
<p>
Mochizuki has, however, been happy to answer specific questions by e-mail. And recently he released a “panoramic overview,” though many mathematicians find it nearly as impenetrable as the full proof.</p>
<p>
Rumors circulated that some of the leading figures in the field had become skeptical of the proof. Out of respect for Mochizuki — and hope that his proof will, in the end, turn out to be right — an effort was made to keep these rumors from circulating on the Internet, and no mathematician would go on the record expressing them. Still, the rumors have further dampened enthusiasm for the hard work of slogging through the four papers. </p>
<p>
Hope rests on a couple of Japanese mathematicians believed to be talking through the proof with Mochizuki, but they are unwilling to talk to the press.</p>
<p>
“If something is very, very familiar to you, it’s kind of hard to put yourself in the position of someone who’s never seen it,” says Jordan Ellenberg of the University of Wisconsin–Madison. “What we need is for this stuff to exist in someone else’s brain besides [Mochizuki’s]. We need someone who has just understood it to help us understand it.” </p>
<span property="rnews:name schema:name" content="A theorem in limbo shows that QED is not the last word in a mathematical proof" class="rdf-meta element-hidden"></span>Mon, 25 Mar 2013 16:04:29 +0000Erin20493 at https://www.sciencenews.org