144 Matching Annotations
  1. Apr 2021
  2. mathshistory.st-andrews.ac.uk mathshistory.st-andrews.ac.uk
    1. Hérigone’s only published work of any consequence is the Cursus mathematicus, a six-volume compendium of elementary and intermediate mathematics in French and Latin. Although there is little substantive originality in the Cursus, it shows an extensive knowledge and understanding of contemporary mathematics. Its striking feature is the introduction of a complete system of mathematical and logical notation, very much in line with the seventeenth-century preoccupation with universal languages.

      Interesting that this links the idea of universal languages to his mathematical notation and NOT to the idea of translating numbers into words using and early form of the major system.

    1. <small><cite class='h-cite via'> <span class='p-author h-card'>Martin Gardner </span> in Hexaflexagons, Probability Paradoxes & the Tower of Hanoi in Chapter 11 Memorizing Numbers (<time class='dt-published'>04/02/2021 14:31:10</time>)</cite></small>

    1. In Germany the great Gottfried Wil-helm von Leibniz was sufficiently intrigued by the notion to incor-porate it into his scheme for a universal language;

      I wish he'd written more here about this. Now I'll have to dig up the reference and the set up as I've long had a similar thought for doing this myself.

      I'll also want to check into the primacy of the idea as others have certainly thought about the same thing. My initial research indicates that both François Fauvel Gouraud and Isaac Pitman both wrote about or developed this possibility. In Pitman's case he used it to develop his version of shorthand which was likely informed by earlier versions of shorthand.

    2. Reading just chapter eleven about "Memorizing Numbers"

      Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi by Martin Gardner (Cambridge University Press, 2008) (The New Martin Gardner Mathematical Library, Series Number 1)

    3. I know of no similar aids in English to recalle, the other commontranscendental number. However, if you memorizeeto five deci-mal places (2.71828), you automatically know it to nine, becausethe last four digits obligingly repeat themselves (2.718281828). InFranceeis memorized to 10 places by the traditional memory aid:Tu aideras rappeler ta quantit beaucoup de docteurs amis.Perhapssome reader can construct an amusing English sentence that willcarryeto at least 20 decimals.
  3. Mar 2021
    1. In a broader sense, taxonomy also applies to relationship schemes other than parent-child hierarchies, such as network structures. Taxonomies may then include a single child with multi-parents, for example, "Car" might appear with both parents "Vehicle" and "Steel Mechanisms"
    1. This article demonstrates how Monte Carlo simulation can be used to solve a real-world, every day problem: Of these three games, which one will provide entertainment for my four-year-old yet let me retain my sanity? If your child is inflexible regarding changing the rules, choose Cootie or Chutes and Ladders which have similar average game lengths. Of the two, Chutes and Ladders is probably the more interesting because of the possibility of moving both forward and backward. If your child insists on Candyland, consider changing the rules as suggested above. An alternative strategy, of course, is simply to let your child cheat. This not only shortens the games, but has the additional incentive that it usually causes the child to win and puts them in a better mood (though it certainly doesn't teach much about ethics). On the bright side, in the few weeks it has taken to complete this study, we have progressed from board games to card games, specifically Uno, which are much more interesting for adults and children. Perhaps there is a God after all.

      The most awesome set of conclusions I've read in a paper in years!

    2. In each of the games moves are entirely determined by chance; there is no opportunity to make decisions regarding play. (This, of course, is one reason why most adults with any intellectual capacity have little interest in playing the games for extended times, especially since no money or alcohol is involved.)

      The entire motivation for this study.

    3. Monte Carlo simulation is used to determine the distribution of game lengths in number of moves for three popular children's games: Cootie, Candyland and Chutes and Ladders. The effect of modifications to the existing rules are investigated. Recommendations are made for preserving the sanity of parents who must participate in the games.

      And people say math isn't important...

    4. <small><cite class='h-cite via'> <span class='p-author h-card'>Lou Scheffer</span> in Mathematical Analysis of Candyland (<time class='dt-published'>10/26/2007 21:55:07</time>)</cite></small>

    1. <small><cite class='h-cite via'> <span class='p-author h-card'>hyperlink.academy</span> in The Future of Textbooks (<time class='dt-published'>03/18/2021 23:54:19</time>)</cite></small>

    1. The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.
    2. An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs. A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a list of ingredients). You can argue that these examples are not equations, in the sense that hydrogen and oxygen are not "equal" to water, yet you can use them to make water.
    1. Adam Kucharski. (2020, December 13). I’ve turned down a lot of COVID-related interviews/events this year because topic was outside my main expertise and/or I thought there were others who were better placed to comment. Science communication isn’t just about what you take part in – it’s also about what you decline. [Tweet]. @AdamJKucharski. https://twitter.com/AdamJKucharski/status/1338079300097077250

  4. Feb 2021
    1. In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology)
    1. Benford’s Law is a theory which states that small digits (1, 2, 3) appear at the beginning of numbers much more frequently than large digits (7, 8, 9). In theory Benford’s Law can be used to detect anomalies in accounting practices or election results, though in practice it can easily be misapplied. If you suspect a dataset has been created or modified to deceive, Benford’s Law is an excellent first test, but you should always verify your results with an expert before concluding your data has been manipulated.

      This is a relatively good explanation of Benford's law.

      I've come across the theory in advanced math, but I'm forgetting where I saw the proof. p-adic analysis perhaps? Look this up.

  5. Jan 2021
    1. In a more recent paper, Michelle Feng and Mason Porter used a new technique called persistent homology to detect political islands — geographical holes in one candidate’s support that serve as spots of support for the other candidate — in California during the 2016 presidential election.
    1. As Goldston wrote years later on accepting the prestigious Cole Prize for these ideas: “While mathematicians often do not have much humility, we all have lots of experience with humiliation.”
  6. Dec 2020
    1. In trying to capture the essence of a system through a minimum of unambiguous symbols, scientists and artists are driven by a similar concern for beauty and symmetry, a similar thirst for light. What makes mathematics special is its promise of prophecy, the promise that it will help us understand all mysteries and all knowledge.

      We all want desperately to know what the future holds.

    2. But something about the comforting rigidity of the process, its seductive notation, but perhaps mostly its connotations of intellectual privilege, has drawn a diverse selection of disciplines to the altar of mathematical reasoning. Indeed, the widespread misappropriation of the language of mathematics in the social and biological sciences has to be one of the great tragedies of our time.

      The deliberate misappropriation of the language of mathematics.

    3. the language of mathematics- which has proved to be an indispensable tool in scientific inquiry- distinguishes itself by the lack of ambiguity in its terms.

      The associations of words can obfuscate instead of clarify.

  7. Nov 2020
    1. the adjective strong or the adverb strongly may be added to a mathematical notion to indicate a related stronger notion; for example, a strong antichain is an antichain satisfying certain additional conditions, and likewise a strongly regular graph is a regular graph meeting stronger conditions. When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as "antichain")
    1. Jan Zoń - A New Revolutionary Cards Method

      This highlights a question I've had for a while: What is the best encoding method for very quickly memorizing a deck of cards while still keeping a relatively small ceiling on the amount of space to memorize and work out in advance.

      I want to revisit it and do the actual math to maximize the difference between the methods.

  8. Oct 2020
    1. it encourages a “growth” mindset: the belief that your abilities can improve with your efforts.

      I'll be this also helps with their feeling of "flow" too.

    2. “Many thought, okay to get from A to B there are these three steps, but it turns out there are really five or six,”

      Sounds a lot like the mathematicians who came after Perelman to show that his proof of Poincare was correct--they needed help in getting from A to B too.

    1. A statistician is the exact same thing as a data scientist or machine learning researcher with the differences that there are qualifications needed to be a statistician, and that we are snarkier.
    1. If you think mathematics is difficult, tough, or you're scared of it, this article will indicate why and potentially show you a way forward for yourself and your children.

    2. The low achievers did not know less, they just did not use numbers flexibly—probably because they had been set on the wrong pathway, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly.4
    3. Unfortunately for low achievers, they are often identified as struggling with math and therefore given more drill and practice—cementing their beliefs that math success means memorizing methods, not understanding and making sense of situations. They are sent down a damaging pathway that makes them cling to formal procedures, and as a result, they often face a lifetime of difficulty with mathematics.
    4. Unfortunately, many classrooms focus on math facts in isolation, giving students the impression that math facts are the essence of mathematics, and, even worse, that mastering the fast recall of math facts is what it means to be a strong mathematics student. Both of these ideas are wrong, and it is critical that we remove them from classrooms, as they play a key role in creating math-anxious and disaffected students.

      This article uses the word "unfortunately quite a lot.

    5. Notably, the brain can only compress concepts; it cannot compress rules and methods.
    6. The hippocampus, like other brain regions, is not fixed and can grow at any time,15 but it will always be the case that some students are faster or slower when memorizing, and this has nothing to do with mathematics potential.
    1. Moreover, the teacher repeatedly asks, “Did anyone get a different answer?” or “Did anyone use a different method?” to elicit multiple solutions strategies. This highlights the connections between different problems, concepts, and areas of mathematics and helps develop students’ mathematical creativity. Creativity is further fostered through acknowledging “good mistakes.” Students who make an error are often commended for the progress they made and how their work contributed to the discussion and to the collective understanding of the class.
    2. Andrews, P., & Hatch, G. (2001). Hungary and its characteristic pedagogical flow. Proceedings of the British Congress of Mathematics Education, 21(2). 26-40. Stockton, J. C. (2010). Education of Mathematically Talented Students in Hungary. Journal of Mathematics Education at Teachers College, 1(2), 1-6.

      references to read

    1. This result of Erd ̋os [E] is famous not because it has large numbers of applications,nor because it is difficult, nor because it solved a long-standing open problem. Its famerests on the fact that it opened the floodgates to probabilistic arguments in combinatorics.If you understand Erd ̋os’s simple argument (or one of many other similar arguments) then,lodged in your mind will be a general principle along the following lines:if one is trying to maximize the size of some structure under certain constraints, andif the constraints seem to force the extremal examples to be spread about in a uniformsort of way, then choosing an example randomly is likely to give a good answer.Once you become aware of this principle, your mathematical power immediately increases.
    2. Once again, Atiyah writes very clearlyand sensibly on this matter (while acknowledging his debt to earlier great mathematicianssuch as Poincar ́e and Weyl). He makes the point (see for example [A2]) that so muchmathematics is produced that it is not possible for all of it to be remembered. The processesof abstraction and generalization are therefore very important as a means of making senseof the huge mass of raw data (that is, proofs of individual theorems) and enabling at leastsome of it to be passed on. The results that will last are the ones that can be organizedcoherently and explained economically to future generations of mathematicians. Of course,some results will be remembered because they solve very famous problems, but even these,if they do not fit into an organizing framework, are unlikely to be studied in detail by morethan a handful of mathematicians.

      bandwidth in mathematics is an important concept

      We definitely need ways of simplifying and encoding smaller cases into bigger cases to make the abstractions easier to encapsulate and pass on so that new ground can be broken

    3. The important ideasof combinatorics do not usually appear in the form of precisely stated theorems, but moreoften as general principles of wide applicability.
    4. I mean the distinction between mathematicians who regard their centralaim as being to solve problems, and those who are more concerned with building andunderstanding theories.
    5. many clever techniquesinvented. Some of these can again be encapsulated in the form of useful principles. Oneof them is the following, known to its friends as Concentration of Measure:if a functionfdepends in a reasonably continuous way on a large number of smallvariables, thenf(x) is almost always close to its expected value.
    1. their name gives no mnemonic boost whatsoever. Whatever faint associations it might once have held fade away, especially when the discover was neither famous nor narrow, and the reader is several generations removed.

      This might be debatable as many of the names in the example are relatively famous names. Any associations they provide might also extend to the dates of the mathematician which also then places the ideas historically as well.

      More often I see the problem with some of the bigger greats like Euler and Cauchy who discovered so many things and everything is named after them.

      The other problem is mis-attribution of the discovery, which happens all-too-frequently, and the thing is named after the wrong person.

    2. The average number of coauthors on math papers has gone up since 1900. So has the number of working mathematicians in the world, which raises the odds of independent rediscoveries, separated in time or space. These two trends have opened the door to triple and even quadruple hyphen situations, as in the Albert-Brauer-Hasse-Noether Theorem and the Grothendieck-Hirzebruch-Riemann-Roch Theorem.

      But this also gets rid of naming quirks for multiple people like the Cox-Zucker Machine.

    3. The worst answer I can imagine is the one Pope Gregory VII gave for refusing to let the Holy Scripture be translated out of Latin: “... [I]f it were plainly apparent to all men, perchance it would be little esteemed and be subject to disrespect; or it might be falsely understood by those of mediocre learning, and lead to error.”

      I'd push back on this a bit by saying that there are huge swaths of people looking at English translations, of Latin translations, of Greek, Hebrew, and Aramaic translations. Not only is there some detail lost in the multiple levels of translation, but many modern Christians are actively mis-applying the stories in the Bible to apply to their modern lives in radically different ways than was intended.

    1. It’s easier to “bridge” science and art when you don’t really think there’s a gap between them in the first place, as I don’t. The boundaries between subjects are really artificial constructs by humans, like the boundaries between colors in a rainbow.
    1. In 1945 Jacques S. Hadamard surveyed mathematicians to determine their mental processes at work by posing a series of questions to them and later published his results in An Essay on the Psychology of Invention in the Mathematical Field.

      I suspect this might be an interesting read.

    1. The parameterization is said to be identifiable if distinct parameter valuesgive rise to distinct distributions; that is,Pθ=Pθ′impliesθ=θ′.

      Definition of identifiable parameterization

    1. In fact, if you do the math, if failure equals knowledge and knowledge equals power, then failure equals power by the transitive property.

      But first we have to prove that this system has the transitive property to begin with!

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  9. Sep 2020
  10. Jul 2020
    1. You should see me doing a write up for a Ruby feature suggestion only to discover that Array#- is hijacked for non-mathematical reasons
    2. Oh. Oh no. That's... not a thing that should be. People wonder why we can't get be more popular with science / math academics.
  11. Jun 2020
    1. In many ways, though, mathematicians treat the problems they are attempting to solve—problems that require highly specialized background and sophisticated thinking and technique—in much the way that non-mathematicians treat puzzles.
    1. Research tells us that for skills like the ones students need for mathematics short practices that recur frequently are far more effective than the same amount of time packed into one session.
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  12. May 2020
  13. Apr 2020
    1. The programming language is augmented with natural language description details, where convenient, or with compact mathematical notation.
    1. In mathematical jargon, one says that two objects are the same up to an isomorphism.
  14. Jan 2020
    1. Norbert Wiener was a mathematician with extraordinarily broad interests. The son of a Harvard professor of Slavic languages, Wiener was reading Dante and Darwin at seven, graduated from Tufts at fourteen, and received a PhD from Harvard at eighteen. He joined MIT's Department of Mathematics in 1919, where he remained until his death in 1964 at sixty-nine. In Ex-Prodigy, Wiener offers an emotionally raw account of being raised as a child prodigy by an overbearing father. In I Am a Mathematician, Wiener describes his research at MIT and how he established the foundations for the multidisciplinary field of cybernetics and the theory of feedback systems. This volume makes available the essence of Wiener's life and thought to a new generation of readers.

  15. Dec 2019
    1. “The pupil is thereby ‘schooled’ to confuse teaching with learning, grade advancement with education, a diploma with competence, and fluency with the ability to say something new. His imagination is ‘schooled’ to accept service in place of value.” (1)

      I think this issue is particularly important in mathematics. One of the seminal researchers in my field, Les Steffe, distinguishes "school mathematics" from the mathematics of students as a modeling construct; others have conceptualized situated cognition; informal mathematics,

  16. Oct 2019
    1. Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists.
    1. Harvard Science of Psychedelics Club: The Hyperbolic Geometry of DMT Experiences

      Craft 5 mathematics & psychedelics tokens for this video.

  17. Feb 2019
    1. And so it makes most sense to regard epoch 280 as the point beyond which overfitting is dominating learning in our neural network.

      I do not get this. Epoch 15 indicates that we are already over-fitting to the training data set, on? Assuming both training and test set come from the same population that we are trying to learn from.

    2. If we see that the accuracy on the test data is no longer improving, then we should stop training

      This contradicts the earlier statement about epoch 280 being the point where there is over-training.

    3. It might be that accuracy on the test data and the training data both stop improving at the same time

      Can this happen? Can the accuracy on the training data set ever increase with the training epoch?

    4. What is the limiting value for the output activations aLj

      When c is large, small differences in z_j^L are magnified and the function jumps between 0 and 1, depending on the sign of the differences. On the other hand, when c is very small, all activation values will be close to 1/N; where N is the number of neurons in layer L.

  18. Nov 2018
    1. Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks

      这个哥们的文章和这个月内好几篇的立意基本一致(1811.03804/1811.03962/1811.04918) [抓狂] ,估计作者正写的时候,内心是崩溃的~[笑cry] 赶快强调自己有着不同的 assumption~

    2. Accelerating Natural Gradient with Higher-Order Invariance

      每次看到研究梯度优化理论的 paper,都感觉到无比的神奇!厉害到爆表。。。。

    3. Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers

      全篇的数学理论推导,意在回答2/3层过参的网络可以足够充分地学习和有良好的泛化表现,即使在简单的优化策略(类SGD)等假定下。(FYI: 文章可谓行云流水,直截了当,标准规范,阅读有种赏心悦目的感觉~)

    4. Gradient Descent Finds Global Minima of Deep Neural Networks

      全篇的数学理论证明:深度过参网络可以训练到0。(仅 train loss,非 test loss)+(GD,非 SGD)

      烧脑!CMU、北大等合著论文真的找到了神经网络的全局最优解

    5. A Convergence Theory for Deep Learning via Over-Parameterization

      又一个全篇的数学理论证明,但是没找到 conclusion 到底是啥,唯一接近的是 remark 的信息,但内容也都并不惊奇。不过倒是一个不错的材料,若作为熟悉DNN背后的数学描述的话。

  19. Apr 2018
    1. ConvexHull

      In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X. -Wikipedia

  20. Nov 2017
    1. As Uri Treisman said, “The most common use of algebra in the adult world is helping their kids with algebra.”

      CUNY found that more students went on to graduate when they were allowed to take statistics instead of remedial algebra. And the results were the same regardless of race or ethnicity.

  21. Jul 2017
    1. there are two different measurements for the length of a foot in the United States: the International Foot (also commonly called the foot) and the U.S. Survey Foot. The International Foot (which we were all taught in school) is defined as 0.3048 meters, whereas the U.S. Survey Foot is defined as 0.3048006096 meters. The difference of the two equates to 2 parts per million.

      For example, in a measurement of 10,000 feet, the difference would be 0.02 feet (just less than one-quarter of an inch). In a measurement of 1 million feet, the difference is 2 feet.

  22. Apr 2017
    1. Two people can have one conversation. Three people have four unique conversation groups (three different two-person conversations and a fourth conversation between all three as a group). Five people have 26. Twenty people have 1,048,554.

      what's the equation for that?

  23. Jan 2017
    1. Whether you're a student, parent, or teacher, this book is your key to unlocking the aha! moments that make math click -- and learning enjoyable.

      You had me already at the Coffee Cup picture over the equations! :)

  24. Nov 2016
    1. If you accept this, thenit seems fair to say that untilPversusNPis solved, the story of Hilbert's Entscheidungsproblem|itsrise, its fall, and the consequences for philosophy|is not yet over.

      If you accept this bizarre interpretation, then you can suspend the belief in the fact, known to everybody, that \(P \ne NP\), because it hasn't been mathematically proved, and say the question isn't solved yet. Wow, how interesting!

  25. Oct 2016
    1. Sunil Singh asks us to stop promoting mathematics based on its current applications in business and science. Math is an art that should be enjoyed for its own sake.

      This reminded me of A Mathematician's Lament by Paul Lockhart. This is a 25-page essay which was later worked into a 140-page book. (And Sunil Singh has read at least one of them. He credits Lockhart in one of the replies.)

      It also reminds me of this article on the history of Gaussian elimination and the birth of matrix algebra. Newton's algebra text included instructions for solving systems of equations -- but it didn't have much practical use until later. (Silly word problems are as old as mathematics.)

    1. We should let people learn at their own pace. We should neither rush them, nor hold them back. If they show a talent, then encouraging them to push themselves is fine.

    1. Math isn't for everyone, and that's fine. The same is true of any other subject. We should help people learn what they are interested in learning.

  26. Jul 2016
    1. I always found it incredible. He would start with some problem, and fill up pages with calculations. And at the end of it, he would actually get the right answer! But he usually wasn’t satisfied with that. Once he’d gotten the answer, he’d go back and try to figure out why it was obvious. And often he’d come up with one of those classic Feynman straightforward-sounding explanations. And he’d never tell people about all the calculations behind it. Sometimes it was kind of a game for him: having people be flabbergasted by his seemingly instant physical intuition, not knowing that really it was based on some long, hard calculation he’d done.

      Straightforward intuition isn't just intuition.

  27. Jun 2016
    1. The Hardy-Littlewood Rule

      "The rule states that anyone that joins collaboration in good faith will be listed equally as an author, regardless of the relative contributions they end up making.

  28. Apr 2016
  29. Mar 2016