Logical literacy is an awareness and understanding of the metalanguage in which propositions, conjectures, lemmas and theorems are written.
Futurama has always been a haven for geek humor, but last week's episode "The Prisoner of Benda" pushed things to the next level. First hinted at in an American Physical Society interview with showrunner David X. Cohen (previously), staff writer and mathematics Ph.D. Ken Keeler devised a novel mathematical proof based on group theory to resolve the logic puzzle spawned by the episode's brain-swapping (but no backsies!) conceit. Curious how it works? Read the proof (in the show or in plain text), then see it in action using this handy chart. Too much math for a lazy Sunday? Then entertain your brain with lengthy clips from the episode -- including two of the funniest moments in the series in the span of two minutes.
The 300th issue of This Week's Finds in Mathematical Physics will be the last. It is not an exaggeration to say that when John Baez started publishing TWF in 1993, he invented the science blog, and an (academic) generation has now grown up reading his thoughts on higher category theory, zeta functions, quantum gravity, crazy pictures of roots of polynomials, science fiction, and everything else that can loosely be called either "mathematical" or "physics." Baez continues to blog actively at n-category cafe and the associated nLab (an intriguingly fermented commune of mathematicians, physicists, and philosophers.) He is now starting a new blog, Azimuth, "centered around the theme of what scientists can do to help save the planet."
I am pleased to announce a proof that P is not equal to NP. In this paper, Vinay Deolalikar (HP Labs) proposes a proof to answer the most important problem in its field of mathematics. [more inside]
Editors of the pop-culture magazine Wired provided the title "iPhone 4’s ‘Retina’ Display Claims Are False Marketing" to a highly critical article about the new iPhone's high-resolution "Retina" display, so-called as the human eye cannot resolve individual pixels when viewing it. A technician who worked on the Hubble telescope disagreed with the Wired editors' choice of rhetoric in very strong technical terms and issued less stringent disagreement with Raymond Soneira, the writer of the piece. Neuroscientist and photographer Bryan Jones published his own highly readable technical analysis of the display's pixel arrangement, that helped him decide whether Apple's claims were truthful or not.
The great[pdf] Russian mathematican Vladimir Igorevich Arnol'd, foremost modern practitioner of classical mechanics, influential teacher, namesake of a minor planet, and semi-nude cross-country skier has died.
"Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"" [more inside]
Plus magazine has compiled all their articles on mathematics and the arts into one handy-dandy page full of highly enjoyable articles ranging from limericks and screeching violins to the restoration of frescoes.
Dan Meyer is a high school math teacher with a clever idea: make math about the real world. On his blog, he writes about classroom management, the real skills of teaching, labels, information design, and assessment.
Since its first printing in 1964, Abramowitz and Stegun's Handbook of Mathematical Functions has been a standard (and public domain) reference manual for special functions and applied mathematics. This week, NIST released its successor, the Digital Library of Mathematical Functions, online to the public.
If politicians were mathematicians. "I would like to suggest two systems for parliamentary votes, one that would weaken the party system but without killing it off entirely, and one that would protect large minorities. Neither has the slightest chance of being adopted, because they are both too complicated to be taken seriously. But mathematicians wouldn’t find them complicated at all — hence the title of this post." Fields medalist Tim Gowers messes around with political axioms.
Mathematician Barbara Shipman speculates that a honey bee's sense of the quantum world could be as important to their perception of the world as sight, sound or smell: "the mathematics implies that bees are doing something with quarks."
Every number from 1 to 9,999 has a special meaning. (much mathematical terminology, scrolling)
Mathematics Illuminated is a set of thirteen surveys in varied topics in mathematics, nicely produced with video, text, and interactive Flash gadgets for each of the topics.
"Crazy as it sounds, over the next several weeks I’m going to try to do something close to that. I’ll be writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it." Mathematics in the pages of the New York Times! [more inside]
"Take a little bad psychology, add a dash of bad philosophy and ethics, and liberal quantities of bad logic, and any economist can prove that the demand curve for a commodity is negatively inclined." MIT economist Andrew Lo and string theorist turned asset manager Mark Mueller on the "physics envy" that plagues economics, and how to stop worrying and love uncertainty.
Does a group of indigenous South Americans hold the key to our relationship with maths? Still, I thought it odd that numbers larger than five did not crop up at all in Amazonian daily life. What if you ask a Munduruku with six children how many kids they have? "He will say, 'I don't know,'" Pica said. "It is impossible to express."
Grigori Perelman has refused one million dollars from the Clay Mathematics Institute for his solution to the Poincaré conjecture. Despite some pressure to take the money and give it to one party or another, Perelman insists "I am not a hero of mathematics. I am not successful at all, and I do not want to be observed by everyone." Perelman previously refused the Fields Medal, mathematics' highest honor. (Previously.)
Nature by Numbers is a new animated short film by Cristóbal Vila (previously) inspired by some mathematical constructs found in nature. (via)
The Shannon number? Skewes' number? Graham's number? Please. When you're ready to get serious, here are some truly large numbers. (previously, but with dead links)
Found Functions. An elegant demonstration of beauty in mathematics (and landscape). Nikki Graziano is a math and photography student at Rochester Institute of Technology; some of her photographs were recently featured in Wired. Graziano "overlays graphs and their corresponding equations onto her carefully composed photos. ... Graziano doesn’t go out looking for a specific function but lets one find her instead. Once she’s got an image she likes, Graziano whips up the numbers and tweaks the function until the graph it describes aligns perfectly with the photograph."
"Back in 1993 I was tutoring my sister in algebra. Her quizzes and tests were always made of word problems with a running storyline involving many recurring places and characters. I tied the fate of the main characters to how well she did on the previous quiz, so a good performance brought them good fortune. Unfortunately, one test she completely bombed, and, well, this is a transcription of the quiz she got next." [more inside]
The Unreasonable Effectiveness of Mathematics in the Natural Sciences is a 1960 essay by Eugene Wigner. Via Steve Strogatz.
Early elementary school teachers in the United States are almost exclusively female (>90%), and we provide evidence that these female teachers’ anxieties relate to girls’ math achievement via girls’ beliefs about who is good at math. A study (abstract and full-text [pdf]) by the University of Chicago Department of Psychology and Committee on Education found a link between math anxiety in elementary school teachers and their female students' math abilities. [more inside]
The Calculus Lifesaver lectures -- videos available here in streaming (Real Player), mp4 and wmv formats -- were originally given as "review sessions for the Princeton introductory calculus courses MAT103 and MAT104 during the 2006/7 academic year". Each lecture is about 2 hours.
The beauty of roots. From Dan Christensen and Sam Derbyshire via John Baez. If you like algebra: these are plots of the density in the complex plane of roots of polynomials with small integral coefficients. If you don't: these are extravagantly beautiful images produced from the simplest of mathematical procedures. Explore the image interactively here.
Alice's adventures in algebra: Wonderland solved "Outgunned in the specialist press, Dodgson took his mathematics to his fiction. Using a technique familiar from Euclid's proofs, reductio ad absurdum, he picked apart the "semi-logic" of the new abstract mathematics, mocking its weakness by taking these premises to their logical conclusions, with mad results. The outcome is Alice's Adventures in Wonderland."
Math Overflow is the first attempt to use the Stack Exchange platform, already popular with programmers, as a scientific research tool. Founded this month by a group of young mathematicians, including Scott Morrison and Ben Webster of the Secret Blogging Seminar, the site is already wrestling with hundreds of questions, ranging from the technical ("When is a map given by a word surjective?") to the historical ("Most interesting mathematics mistake?")
The Anatomy of Spiral Arms, shows how galaxies naturally evolve to form grand-design two-arm spirals. The Hubble Ultra Deep Field in 3D. [more inside]
Figure 3. Basic model outbreak scenario. Susceptibles are quickly eradicated and zombies take over, infecting everyone.
Durango Bill's Home Page. With topics that include: 3D end-to-end tour of the Grand Canyon, the origin and formation of the Colorado River, and examples of river systems that cut through mountain ranges instead of taking easier routes around them in Ancestral Rivers of the World. [more inside]
"I can see the audience tonight, so I can see also from the size of it that there must many of you here who are not thoroughly familiar with physics, and also a number that are not too versed in mathematics- and I don't doubt that there are some who know neither physics nor mathematics very well. That puts a considerable challenge on a speaker who is going to speak on the relation of physics and mathematics- a challenge which I, however, will not accept: I published the title of the talk in clear and precise language, and didn't make it sound like it was something it wasn't- it's the relation of physics and mathematics - and if you find that in some spots it assumes some minor knowledge of physics or mathematics, I cannot help it. It was named." The Feynman Messenger series at Cornell has been made available online for the first time thanks to Bill Gates.
The principles of Harmonics were discovered by Pythagoras c.587-c.507 B.C. during travels to Egypt and throughout the ancient world. Hans Kayser made a profound philosophic study of harmonics in the 20th century. Algorithmic composition is the technique of using harmonic algorithms to create music. Drew Lesso has been creating algorithmic music since 1975. Samples like Crystal, Constellations, or Planet Earth demonstrate the math behind the music. Over the years, Lesso has collaborated with many other musicians and poets to create an airy, evolutionary legacy.
The Cornell Historical Math Monographs archive has a great many famous papers, including works by De Morgan, Hamilton, Descartes (warning: French) and of course Lewis Carroll. [more inside]
Parts 1, 2, 3 of a 1959 interview with philosopher, mathematician and peace campaigner Bertrand Russell (1872-1970). Works and pictures online include Anti-suffragist Anxieties, Why I am not a Christian, the Russell-Einstein Manifesto against nuclear weapons and the book The Conquest of Happiness. Russell is also known for his pithy quotes, his teapot and was the subject of poem Mr Apollinax by T.S. Eliot.
Stephen Wolfram discusses Wolfram|Alpha: Computational Knowledge Engine - at the same time Google Adds Search to Public Data, viz: "Nobody really paid attention to the two hour snorecast" -- like a cross between designing for big data and a glossary of game theory terms -- on Wolfram|Alpha (previously), yet the veil is being lifted nonetheless: "[on] a platonic search engine, unearthing eternal truths that may never have been written down before," cf. hunch & cyc (and in other startup news...) [via] [more inside]
Logicomix: An Epic Search for Truth by Apostolos Doxiadis, Christos H. Papadimitriou, Alecos Papadatos, and Annie Di Donna. Covering a span of sixty years, the graphic novel Logicomix was inspired by the epic story of the quest for the Foundations of Mathematics. This is another kind of epic battle that does not quite lead where the characters thought it would take them. Featuring Bertrand Russell as narrator and the awesome might of the villain (of sorts) Incompleteness. Themes include the high personal price paid for knowledge.
"the scale-free network modeing paradigm is largely inconsistent with the engineered nature of the Internet..." For a decade it's been conventional wisdom that the Internet has a scale-free topology, in which the number of links emanating from a site obeys a power law. In other words, the Internet has a long tail; compared with a completely random network, its structure is dominated by a few very highly connected nodes, while the rest of the web consists of a gigantic list of sites attached to hardly anything. Among its other effects, this makes the web highly vulnerable to epidemics. The power law on the internet has inspired a vast array of research by computer scientists, mathematicians, and engineers. According to an article in this month's Notices of the American Math Society, it's all wrong. How could so many scientists make this kind of mistake? Statistician Cosma Shalizi explains how people see power laws when they aren't there: "Abusing linear regression makes the baby Gauss cry."
Joe Palca, a science correspondent for NPR's Morning Edition, was meditating on the best way to convey the magnitude of the world's largest known prime number, 243112609-1. He contacted H&FJ at Typography.com to discuss the implications of typesetting a number with more than twelve million digits. Crunching of numbers and fonts ensued.
Topology and Geometry Software by Jeff Weeks.
NYT Guesstimation Quiz. Enrico Fermi estimated the yield of the Trinity A-bomb test by dropping some shredded paper. He also asked his students to estimate unusual quantities like the number of piano tuners in Chicago - to show that just about anything can be estimated without detailed knowledge.
Oxford Professor & Fellow of the Royal Society James Murray uses mathematical modelling to predict whether a marriage will survive or end in divorce, with 94% accuracy. His lecture to the Royal Society will be available for view on demand within two days.
Ever wondered what comes next, and why? The On-Line Encyclopedia of Integer Sequences has the answers. (Previously.)
The MacTutor History of Mathematics archive is an astounding collection of historical material on mathematics, especially biographies. (Previously: 1 2 3 4.)
"Do you like fiction and mathematics? Are you interested in what our society thinks about mathematicians?" [more inside]
World of Science contains budding encyclopedias of astronomy, scientific biography, chemistry, and physics. This resource has been assembled over more than a decade by internet encyclopedist Eric Weisstein with assistance from the internet community. MeFi visited Weisstein's Mathworld a couple years ago.