CapitalistImperialistPig

Describes the evolution of a fluid. Quanta magazine has a nice article on mathematicians' attempts to probe the limits of the equation, and more importantly, for me, a lovely video of a Kelvin-Helmholtz instability evolving under the equation. Mathematicians have long suspected that there might be something dodgy about the equation, and there is big money, a Millenium Prize, riding on the conjecture that the equation doesn't always have consistent solutions. Quanta notes that these potential problems don't bother physicists, but doesn't bother to say why. The more fundamental reason is that physicists know that the N-S is not a faithful description of nature. If you look at a fluid in close detail, it becomes a seething mass of individual particles, not infinitely divisible fluid elements. There is also another fact: because the NS has chaotic solutions, their predictivity is always limited in practice, whether it is in theory or not.

Prime Numbers and the Riemann Hypothesis by Barry Mazur and William Stein is a slender (142 pg.) book aimed at a varied audience of the mathematically curious. It is profusely illustrated, mainly with pictures of what the authors call the staircase of primes, a function that starts at zero and goes up by one each time a prime is encountered, though several recarpentried versions of the staircase also make the scene. The book is divided into 38 very short chapters, organized into four sections, with the first and longest section (chapters 1-24) aimed at readers without a calculus background. The second section demands a bit of calculus (not much!) and the third some Fourier analysis, while the fourth gets to the nitty-gritty of the zeta function. The figures and many of the calculations were done with Sage, a free mathware package developed by the second author, and made available to the eager experimenter. The first section has a lot of the lore primes that is accessible at the e...

The Lumonator has a recent post on the importance of mathematical thinking, and of teaching it . He summarizes some of this in ten points, which I endorse, but I would like to add one more point which he doesn't quite state explicitly: mathematical thinking teaches disciplined methods of thought. I am reminded of the fact that Lincoln taught himself to prove all the theorems of Euclid's first six elements at sight not because he thought they would come up in his legal practice, but because he thought that would sharpen his logical and analytical skills. It also teaches a language for expressing analysis in a disciplined manner.

And I'm not talking about Trump's inauguration yet. Well, maybe that's part of it. My actual subject is my reaction when I saw the first homework assignment in my Astronomy class in Dynamics and Hydrodynamics. Of course it wasn't based on the class material, since there wasn't any yet. Instead it was more of a basic math pretest: Differential equations, analyzing the behavior of integrals and deriving vector identities - stuff I hadn't done, for the most part, in fifty years. I panicked when I couldn't see how to get the inhomogeneous solution to the very elementary first differential equation. It reminded me of the feeling I had when I first saw the problem set on my PhD comprehensive and realized that there was not a single problem on it that I knew how to solve. However, just as on that long ago comprehensive, once I pondered the problems a bit I gradually realized that I did have the tools, in this case rusted, dull, and buried deep, for solving ...

The expression "going to the mattresses" should be familiar to fans of The Godfather or of Nora Ephron's You've Got Mail. It's what Mafioso, or presumably, book store owners, do when they go to war. In linear algebra and geometry, the sophisticated prefer to speak of the advantages of coordinate free representations, but, when the rubber meets the road, they often "shut the doors and compute with matrices," as one wag put it.* I was reminded of that by my current interest in tensor networks, where the action is precisely in matrices (and their higher rank analogs. * Actual quote, from Irving Kaplansky, speaking of himself and Paul Halmos: We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury. And a different opinion from Dieudonne: There is hardly any theory which is more elementary [than linear algebra], in spite of the fac...

It seems to be time to return to that old time theme: why does American education suck? The current items in evidence are the 2012 scores in the Program for International Student Assessment (PISA), in particular, the mathematical literacy assessment. Tyler Cowen notes that even our rich kids are below average : The data was provided to The WorldPost by Pablo Zoido, an analyst at the Organisation for Economic Co-operation and Development, the group behind PISA. It shows that students’ wealth does not necessarily make them more competitive on an international scale. In the United States, for example, the poorest kids scored around a 433 out of 700 on the math portion of PISA, while the wealthiest ones netted about a 547. The lower score comes in just below the OECD average for the bottom decile (436), but the higher score also comes in below the OECD average for the top decile (554). “At the top of the distribution, our performance is surprisingly bad given our top decile is among th...

One of the scientists I most admire, Edward O. Wilson, recently wrote an article in the WSJ arguing that you don't need much math to be a scientist - even a great scientist. He doesn't admit knowing much himself, but he did learn some calculus as a tenured Professor. Fortunately, exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory. Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition. Everyone sometimes daydreams like a scientist. Ramped up and disciplined, fantasies are the fountainhead of all creative thinking. Newton dreamed, Darwin dreamed, you dream. The images evoked are at first vague. They may shift in form and fade in and out. They grow a bit firmer when sketched as diagrams on pads of paper, and they take on life as real examples are sought and found. Pioneers in science only rarely ma...

Economics and climatology have become too depressing to talk about, so I picked a quarrel with my commenters over autism spectrum disorder. Arun was moved to write: "...the difference between Americans and other cultures is that Americans seem to believe that there is a math gene, if you have it, you are good at math;..." There is quite a bit of literature indicating that mathematical talent (like talent in music, chess and a wide variety of other areas) is strongly influenced by genetics. See, e.g., Behav Genet. 2009 Jul;39(4):380-92. Epub 2009 Mar 15. The heritability of aptitude and exceptional talent across different domains in adolescents and young adults. Vinkhuyzen AA, van der Sluis S, Posthuma D, Boomsma DI. SourceDepartment of Biological Psychology, VU University Amsterdam, Amsterdam, The Netherlands. aae.vinkhuyzen@psy.vu.nl I'm not sure this is a uniquely American prejudice.

We hear that students are abandoning science and engineering studies in droves, not least of all because it turns out that those subjects are hard and graded rather strictly. Perhaps I can help alleviate the pain with the current version of my twenty minute calculus. {Best presented with the help of a blackboard} If you have opened your calculus book, you may have noticed that it consists of about 1700 pages of closely spaced text, diagrams, formulas, and equations. Perhaps that experience has already convinced some of you to change your major to psychology or art history. For those of you who plan to leave, then, as well as those of you who plan to stay, I would like to start this lecture by mentioning that there are only a few key ideas in calculus, and those are handy to know even if you do plan to major in psychology or art history. Depending on your point of view, those ideas are one, two, or three in number. I should add that none of those ideas will be exactly new to you...

Math is hard.................Barbie The next thing I probably ought to do in my current project is figure out how to do multi-fractal analysis and modeling of my data. But like Barbie said, and I don't know how to do that, and learning gets hard when you get old and dumb.

About big numbers , by Scott Aaronson, via Steve Landsburg . Beyond ordinary notions of bigness, like exponentials and factorials, says Scott, lie more exotic numbers defined by recursive functions and Turing machines. These big numbers make my head hurt, for reasons also explained by Scott. Whence the cowering before big numbers, then? Does it have a biological origin? In 1999, a group led by neuropsychologist Stanislas Dehaene reported evidence in Science that two separate brain systems contribute to mathematical thinking. The group trained Russian-English bilinguals to solve a set of problems, including two-digit addition, base-eight addition, cube roots, and logarithms. Some subjects were trained in Russian, others in English. When the subjects were then asked to solve problems approximately—to choose the closer of two estimates—they performed equally well in both languages. But when asked to solve problems exactly, they performed better in the language of their training. What’s m...

Calculus is the gateway to math, engineering, and the sciences. It's hard for me to consider anyone educated who doesn't know at least a little. It's a forbiddingly fortified gateway though. Typical calculus texts these days have 1300 pages or so and weigh and cost roughly as much as a Mercedes S600. Could that be overkill? I mean there are only one or two real ideas in calculus - the rest is technology. Right?

Once more into the breach: another statistics problem from The Burg: Suppose you’ve somehow found yourself in a game of Russian Roulette. Russian roulette is not, perhaps, the most rational of games to be playing in the first place, so let’s suppose you’ve been forced to play. Question 1: At the moment, there are two bullets in the six-shooter pointed at your head. How much would you pay to remove both bullets and play with an empty chamber? Question 2: At the moment, there are four bullets in the six-shooter. How much would you pay to remove one of them and play with a half-full chamber? The hardest part of this kind of problem is figuring exactly how to frame it. Suppose, for example, that objective here is to maximize your lifetime, and that your expected lifetime, should you survive the game, is a function of your remaining wealth W, say f(W). For question 1, then, without the payoff, your expected future lifetime becomes: L = (1/3)*0 + (2/3)*f(W) = (2/3)*f(W), and L = f(W-P) with ...

I asked The Statistical Mechanic (AKA Wolfgang) for his opinion on the subject of the Landsburg-Motl Gotterdammerung . I hope he won't mind my quoting most of his post on the subject here: If E() denotes expectation values, then E(x/y) is in general not E(x)/E(y). If x is the number of girls and y enumerates the boys then we have pretty much described the whole debate about this puzzle already. Furthermore, notice that E(x/y) is in many cases not well defined and the sum or (in general) the integral Int[ dx dy (x/y) p(x) p(y) ] will not necessarily equal 1 even if p is normalized and does vanish around zero values of x, y.

Captain James Kirk certainly was a silly sentimentalist. Imagine him keeping that old fraud Spock on the payroll despite repeated demonstrations of incompetence in his supposed expertise. Whenever a dangerous mission loomed, Spock could be depended on to pull one of his patented fake probability predictions out of his ... - usually something like a 99.99973% chance of failure. In one way I couldn't blame him - Kirk was a total klutz, always going off half-cocked and without a clue. Still, I couldn't resist my own predictions, namely that Spock was off by about 99.99973%. The poor dolt had no head for figures. Of course this post is actually about Steve Landsburg - I think he might miss me - who has a series of posts on the question of what should constitute a "reasonable doubt" in a murder trial. There is a certain amount of amusement to be obtained by doing his arithmetic, but fundamentally he is just making Spock's mistake - assigning arbitrary numbers wh...

What is curvature? We have an intuitive notion that some curves are curvier than others, so how have mathematicians sorted this out? I have been reading The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions by Shing-Tung Yau and Steve Nadis. It turns out that the notion of curvature, and in particular, Ricci curvature, is fundamental to all the considerations therein. From my deeply shallow and mostly forgotten studies of general relativity I recalled that the Ricci tensor was (a) an index contracted Riemann curvature tensor and (b) an essential component of the Einstein tensor . Neither bit of intellectual flotsam gave me any significant insight into what Ricci curvature really was. For me to understand something, I need to have a mental picture that can be expressed in familiar notions. The simplest notion of curvature is that we associate with a circle. We have an intuitive notion that a smaller circle is “more curved” than a larger one. We...

Steve Landsburg (OK, I can’t help myself - I can’t quit you Steve. He’s frequently wrong but often writes about interesting stuff) is arguing that the universe, and its contents, are mathematical objects. 1. A “mathematical object” consists of abstract entities (that is, “things” with no intrinsic properties) together with some relations among them. For example, the euclidean plane that you studied in high school geometry consists of points, together with certain relations among them (such as “points A, B and C are collinear”). Mathematical objects can be very complicated. Mathematical objects can have “substructures”, which is a fancy name for “parts”. A line in the plane, for example, is a substructure of the plane. 2. Every modern theory of physics says that our universe is a mathematical object, and that we are substructures of that object. Theories differ only with regard to which mathematical object we happen to be a part of. Particles, forces and energy are not just described by...

What’s in a name? Would not a turdorch* by any other name smell as sweet? Today's question is inspired (or perhaps provoked) by this Steve Landsburg claim: "well formed statements about arithmetic are either true or false, regardless of whether they have proofs or disproofs." For example, the following: "Every even number is the sum of two primes." Steve Landsburg has a series of posts on the foundations of arithmetic. I won’t try to summarize them in any detail, so it’s not likely that you will be able to follow my argument below without reading them at least in part. There is a central point I want to dispute, that “every well-formed arithmetical statement is either true or false,” whether provable in some axiomatic description or other. This claim assumes that numbers, and their arithmetic, have an existence independent of that axiomatic description. As a philosophical point, I tend to agree, but the point is hardly self-evident. The strange thing to me is t...

One of the unhappiest marriages of modern technology is that of economists and multiple regression. With the aid of a computer, multiple regression is simple enough for anyone to do it, and far too many of them do. Another egregious example seems to be Dr Dave Berri (with details here ). His technique produces a number of counterintuitive results - for example, he rates Derek Fisher of LA as the least productive player on a winning team. This is sufficient evidence for me to rate Berri as an idiot among economist multi-regressors. The are several kinds of pitfall for multiple regression, prominent among them neglect of correlations and of variables not in the regression. This sort of thing is especially evident in the kinds of statistics Berri uses. Key stats are what he calls "defensive efficiency" which is essentially just points scoring of the guy you are guarding and "offensive efficiency," scoring per unit time. These stats are very easily contaminated. If yo...

There are two sorts of truth, Bohr claimed. Ordinary truths, whose opposites are falsities, and great truths, whose opposites are also great truths. Now it should go without saying that one shouldn't take Bohr too literally, but these latter are the ones that interest me. Libertarians have some of these truths, I think, mixed together with a gross misinterpretation of the nature of the human condition. One of those confused libertarians links to this interesting great truth about the teaching of mathematics . Briefly, Paul Lockhart's Lament, as he styles it, is that mathematics is not being taught as one of the arts, as it ought. An emphasis on technique and memorization, at the expense of ideas, makes students dislike mathematics and resist learning it. I'm not sure the comparison with music that he chooses really makes his point: musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are hel...