Micromath

When I saw it was for computer scientists, I briefly hoped that it would take the Big-O, little-o approach as Knuth recommended in 1998. See https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o… for a repost of Knuth's letter on the topic. Sadly, no. It just seems to start with a gentle version of real analysis, leading into basic Calculus.

The business might not know enough for that estimate to be possible either. It reminds me of this quote from Andrew Wiles about mathematics: "Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were." [1] [1] Source: https://micromath.wordpress.com/2011/11/06/andrew-wiles-on-doing-mathe…

Note that the first problem is not actually to learn the multiplication tables! Problem 1 is to calculate a few things like 0.004×0.02 and 1.08÷1.2 in one's head; it's just that these real questions are placed in 1(b), with a precursor 1(a), purely to indicate prerequisites, that states "Compute for yourself, and learn by heart, the times tables up to 9×9." (Incidentally, even here the "compute for yourself" part is the real heart of the matter, though of course knowing the times table up to 9×9 is something very useful. You'll see a lot more connections once there's some "fuel", as mentioned in the preface and the introduction to the chapter.) In any case, don't let a cursory look at Problem 1 decide the nature of the book for you. There's more than meets the eye. I don't know much about Tony Gardiner but the other author, Alexandre Borovik, has a wonderful blog called "Mathematics under the Microscope" (https://micromath.wordpress.com/) (and also a book of that name); to everything he brings a profound perspective with a light touch, a unique combination. A couple of memorable posts I still remember from over a decade ago: • “Psychophysiology of blackboard teaching” https://micromath.wordpress.com/2010/09/24/psychophysiology-of-blackbo… • “Achilles, Tortoise and Yessenin-Volpin” https://micromath.wordpress.com/2009/02/16/achilles-tortoise-and-yesse… • All the posts about "a childhood story". Anyway, I haven't gone through this book so far, but…

We use Scrum at work and I have to say, I am pretty annoyed by it. I think it is fundamentally rooted in the idea that ideal software development is just a continuous production of small improvements to the code. And from this come all its micromanagement failure modes (which there are plenty). I think in reality, SW development done right is nothing like that. It is highly discrete when it comes to output, because it takes time and experimentation to come up with the correct way to approach a problem, but if you do it right, you save yourself an incredible amount of time. It's more like mathematics, where it takes time to discover that things are actually rather easy (this quote comes to mind https://micromath.wordpress.com/2011/11/06/andrew-wiles-on-doing-mathe… Of course, if you look at the project from the longer time perspective, it can appear as continuous production of improvements. But it has a kind of quantum nature, if you decrease the scale of time, you will see less continuity and more discrete bursts of code production. There is something like uncertainty principle that on certain scale, it's pointless to try to predict things, because making the correct prediction (which is needed at least to define thing DONE) will take as long as doing the thing in the first place. I don't consider myself special, but it is possible that Scrum ruins great minds because they seek conceptual clarity (we sometimes humbly call it "lack of technical debt"). I am not…

Fluid dynamicist here. This is a fairly nice video, though there are a few flaws. I was waiting for a derivation of the 5/3rds law, but it's not in the video. This is a shame as it can be derived very simply: http://micromath.wordpress.com/2008/04/04/kolmogorovs-53-law/ I don't buy the estimate of the limits of the inertial subrange given. That might be true for a particular velocity, but I imagine the limits of the inertial subrange vary greatly. Unfortunately 3blue1brown falls into a common trap when defining turbulence. Turbulent flows are chaotic... but laminar flows can be chaotic too. They are both solutions to the same non-linear equations. I think it would be better to describe turbulence as seemingly random or more precisely, probabilistic. The governing equations are not stochastic, but practically speaking you can treat the velocity field as a random variable. The uncertainty for laminar flows is much smaller. This is the basis for a lot of turbulence theory. This is not to say that there is not order in turbulence; there is both order and disorder. The Navier-Stokes millennium prize problem is also not as valuable as it seems to outsiders. The solutions most likely exist and are unique, etc. Your prior on the solutions existing should be high as Newtonian physics problems typically have solutions. If the solutions are not unique, add an extra condition to obtain unique solutions. That's already often needed for the compressible Navier-Stokes equations. I'd…

One of the nice things about big-O notation from a mathematical point of view is that it allows you to avoid dealing with limits. As everyone knows, calculus is based on limits, but when handled rigorously limits have a lot of subtle pitfalls that take a lot of work to deal with. For example, given a function f(x,y), if we want to take one limit after another, the result might depend on the order in which the limits are taken. In other words, taking limits is not commutative unless certain conditions are satisfied. None other than the great Donald Knuth went so far as to claim that all of calculus could be taught without limits! Quoting from [1]: "Students will be motivated to use O notation for two important reasons. First, it significantly simplifies calculations because it allows us to be sloppy — but in a satisfactorily controlled way. Second, it appears in the power series calculations of symbolic algebra systems like Maple and Mathematica, which today’s students will surely be using. For more than 20 years I have dreamed of writing a calculus text entitled O Calculus, in which the subject would be taught along the lines sketched above." [1] http://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-…

I have heard it complained that we scientists don't "talk to the general public" enough. I would very much welcome specific suggestions for how to do this. That's very commendable. I am a nonmathematician who has become a mathematics teacher as part of figuring out why so few learners in the United States succeed in learning mathematics well. Here are some examples for you. Good mathematicians (some Fields medalists) with readable blogs: http://terrytao.wordpress.com/ http://gowers.wordpress.com/ http://www.arsmathematica.net/ http://www.scottaaronson.com/blog/ http://scientopia.org/blogs/goodmath/ http://wildaboutmath.com/ http://micromath.wordpress.com/ A list found by a Google search http://www.onlinedegree.net/50-best-blogs-for-math-majors/ suggests more. A few popular authors who are mathematicians write very approachable books about mathematics. http://www.amazon.com/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&sort=r… http://www.amazon.com/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&sort=r… http://www.amazon.com/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&sort=r… Following links from links to the books by those authors will reveal other books by mathematicians who write for popular audiences. Best wishes in gaining readers other than your mother (like me, for instance) as you communicate your…

The reason it's preferred to use "=" instead of "\in" is because the way that Landau notation is generally used in practice is as a kind of ellipsis or placeholder. For example, the Taylor expansion e^x = 1 + x + O(x^2). I could just as well write e^x = 1 + x + ..., but the former conveys more meaning about what is hidden behind the ellipsis. It's an abuse of notation, but in the contexts that it's used, it's not clear what additional clarity using "\in" would bring over "=". Maybe also that big O is mainly used as a notation to facilitate doing calculations, less describing what family a function belongs to. Here are Knuth's thoughts, which I agree with: https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o…

One of the commenters to the original article talks about this phenomenon specifically. Each math course you took taught you some ideas or techniques, but those techniques weren't really learned until you used them in the next course. For example, you didn't learn algebra well until you used it in calc 1; calc 1 skills are really cemented in difeq. https://micromath.wordpress.com/2011/05/17/time-lag-in-learning-mathem…

Big O is, in some senses, about giving a natural intuition for reasoning with equations. As Knuth says, "it significantly simplifies calculations because it allows us to be sloppy."[1] So, having a natural intuition for it could be a way of saying that one has a natural intuition for equations, in general. [1] https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o…

Knuth's take: http://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-…

Did anybody know that big o notation is more general than either of these articles lets on? These two articles are explaining the use of big-O notation in algorithms, which is just one possible use. The article by Knuth gives a better insight: http://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-… An example of alternative uses for big-O is the description of the "order of error" introduced by taking the Taylor-series truncation of a math function. This is useful in evaluating the accuracy of numerical techniques.

Personal criticism is harsh and a bit rude. However criticism of code or architecture should be encouraged and seen as a constructive thing as it lets you discuss different approaches and why the current approach was taken and how it can be changed or improved. Criticism and debate in Math, Engineering and Science should be encouraged. Related: http://micromath.wordpress.com/2008/12/26/women-and-mathematics/

I agree that in terms of practical impact, Turing has had limited impact. And, probably he had political connections related to the war effort. But I don't feel that he is overrated. He was a first class theoretician. Some non-trivial results that I know of are, 1. A version of the central limit theorem [1] 2. A fixed-point combinator in his proof of equivalence of lambda calculus and computable functions, in his paper. Any one who has tried to derive the Y-combinator (appropriate for this website) knows how tricky it is to get one. Turing seemed to have done this all this in an appendix. [2] 3. LU decomposition. [3] 4. I don't understand his work on morphogenesis, but it seems to be his most cited paper. [4] I largely agree with the list of pioneers that you have pointed out, without berating Turing. [1] http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511614293&cid=C… [2] http://en.wikipedia.org/wiki/Fixed-point_combinator#Other_fixed-point_… [3] https://micromath.wordpress.com/2012/02/25/alan-turing-and-linear-alge… [4] http://en.wikipedia.org/wiki/The_Chemical_Basis_of_Morphogenesis