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Mathematics under the Microscope
Personal blog · micromath.wordpress.commath
Mathematics under the Microscope is a personal blog at micromath.wordpress.com about math. Since 2008, 19 different Hacker News readers have linked it 27 times, most often “Donald Knuth's letter about teaching O notation in calculus classes”. It is still writing: the last post was on 2025-06-01, and you can follow it by feed.
Too few sampled comments (20) to say how often HN recommends it.?
Why HN links it
HN readers link this blog for posts on mathematics teaching and history, especially Donald Knuth's letter proposing that calculus be taught with big-O and little-o notation, and on the time lag in learning maths.
Summarised by Claude from a sample of 20 HN comments that link here. Each comment is marked recommending, critical or neutral (citing without judging); how this works.
Atomic objects, structures and concepts of mathematics
Most-linked posts?
- Donald Knuth's letter about teaching O notation in calculus classes8 people · 8 mentions · first linked 2008-04-14
- /2008/04/14/donald-knuth-calculus-via-o-notation/3 people · 5 mentions · first linked 2015-03-04
- /2011/11/06/andrew-wiles-on-doing-mathematics/2 people · 2 mentions · first linked 2020-06-30
- Women and Mathematics2 people · 2 mentions · first linked 2008-12-27
What Hacker News says?
Your history of Calculus is a little..muddled. Newton's approach was based on the idea that rate of change was a fundamental quantity. This lead to fluxions. His notation for Calculus was x with a dot over it. He used his method as a rough notation to figure out what the answers should be, before coming up with pure geometric constructions to use in the Principia Mathematica. Inspired by the Principia, Leibniz reinv…
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Calculus Learning Guide”Submitted Donald Knuth's letter about teaching O notation in calculus classes as a story · 50 points
HN discussionThe 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 fina…
Comment linking /2011/11/06/andrew-wiles-on-doing-mathematics/
on “Software estimation is hard – do it anyway”Submitted Women and Mathematics as a story · 17 points
HN discussionWhen 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-calc... 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.
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Calculus for Mathematicians, Computer Scientists, and Physicists [pdf]”- ot · 2012-02-11
Submitted Donald Knuth: Calculus via O notation (2008) as a story · 12 points
HN discussion 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-learn...
Comment linking /2011/05/17/time-lag-in-learning-mathematics/
on “The Math Myth”Submitted Donald Knuth: Calculus via O notation (1998) as a story · 3 points
HN discussionBig 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-calc...
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Stuff I've Messed Up While Interviewing”Submitted Some pre-school proto-mathematics as a story · 2 points
HN discussionOne 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…
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Big-O notation explained by a self-taught programmer”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…
Comment linking /2009/02/16/achilles-tortoise-and-yessenin-volpin/
on “The Essence of Mathematics Through Elementary Problems”- gz · 2010-07-16
Knuth's take: http://micromath.wordpress.com/2008/04/14/donald-knuth-calcu...
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Plain english explanation of Big O” Dimensional analysis, one of the topics in the book, is a favorite of mine. Unfortunately the more useful parts of the subject are not that well known. Fortunately, this book seems to have a fairly standard introduction to the subject. Most people seem to think dimensional analysis is limited to checking that the dimensions are consistent ("dimensional homogeneity"). But dimensional homogeneity is a constraint which…
Comment linking /2008/04/04/kolmogorovs-53-law/
on “The Art of Approximation in Science and Engineering [pdf]”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…
Comment linking /2011/11/06/andrew-wiles-on-doing-mathematics/
on “Does scrum ruin great engineers or are you doing it wrong?”I've thought before about Knuth's proposal (see http://micromath.wordpress.com/2008/04/14/donald-knuth-calcu... for his description) that Calculus be introduced using Big-O, little-o notation. I once drew up a curriculum based on it, and it seems to avoid the issue that you describe. In particular the idea of approximation seems inherently useful. All approximations take the form something = approximation + error, s…
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “You don't understand something until you think it's obvious.”Wouldn't that be backwards? The purpose of notation is to aid human thought and communication. Technology should adapt to whatever notation would be most useful to mathematicians, rather than mathematicians adapting their notation for the sake of technology (like “more easily typesettable”). The blackboard-bold fonts are an interesting example of this, with computer typefaces being created for what mathematicians us…
Comment linking /2010/09/24/psychophysiology-of-blackboard-teaching-2/
on “Boiling lead and black art: a history of mathematical typography”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 pa…
Comment linking /2012/02/25/alan-turing-and-linear-algebra-2/
on “Konrad Zuse: Nearly the German Turing”> I would of course explain that the equality sign is not symmetric with respect to such notations; we have 3=A(5) and 4=A(5) but not 3=4, nor can we say that A(5)=4. We can, however, say that A(0)=0. As de Bruijn points out in [1, 1.2], mathematicians customarily use the = sign as they use the word “is” in English: Aristotle is a man, but a man isn’t necessarily Aristotle. — Donald Knuth, Teach Calculus via O Notat…
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Cure Dolly's Japanese Grammar Lessons”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…
Comment linking /
on “Who listens to scientists? Mostly just other scientists.”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-mathemat...
Comment linking /2008/12/26/women-and-mathematics/
on “Ask HN: Do you criticize?”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-calcu... An example of alternative uses for big-O is the description of the "order of error" introduced by taking the Ta…
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Plain English Explanation of Big O Notation”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 t…
Comment linking /2008/04/04/kolmogorovs-53-law/
on “Visualizing turbulence with a home demo [video]”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 cl…
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “What is f(x) ≤ g(x) + O(1)? Inequalities With Asymptotics”I think that Big O / little o are both more approachable and provide a richer understanding than limits. See, for example, https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... to see Donald Knuth agreeing. As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation an…
Comment linking /2008/04/14/donald-knuth-calculus-via-o-notation/
on “Calculus Made Easy by Silvanus P. Thompson (1910)”
