160 Matching Annotations
  1. Jul 2021
    1. John Sweller’s cognitive load theory argues that problem solving is often inefficient.2 His studies showed that students learned to solve algebra problems faster when they were shown lots of examples of solved problems, rather than trying to solve them on their own.3

      Problem solving is often inefficient, seeing lots of solved problems may be better than solving them on one's own.

      (This was the sort of model I used in learning most of my math over the years, though solving a few problems along the way also helped to reinforce things for me.)

      Sweller, John. “Cognitive load during problem solving: Effects on learning.” Cognitive science 12, no. 2 (1988): 257-285. Sweller, John, and Graham A. Cooper. “The use of worked examples as a substitute for problem solving in learning algebra.” Cognition and instruction 2, no. 1 (1985): 59-89.

  2. Jun 2021
    1. Page 1

      These advantages alone claim for it a place in the education of all, not excepting that of women.

      Interesting to see a male mathematician advocating for the education of women in 1860 England.

    1. If you have a vector space, any vector space, you can define linear functions on that space. The set of all those functions is the dual space of the vector space. The important point here is that it doesn't matter what this original vector space is. You have a vector space V

      One of the better "simple" discussions of dual spaces I've seen:

      If you have a vector space, any vector space, you can define linear functions on that space. The set of all those functions is the dual space of the vector space. The important point here is that it doesn't matter what this original vector space is. You have a vector space V, you have a corresponding dual V∗.

      OK, now you have linear functions. Now if you add two linear functions, you get again a linear function. Also if you multiply a linear function with a factor, you get again a linear function. Indeed, you can check that linear functions fulfill all the vector space axioms this way. Or in short, the dual space is a vector space in its own right.

      But if V∗ is a vector space, then it comes with everything a vector space comes with. But as we have seen in the beginning, one thing every vector space comes with is a dual space, the space of all linear functions on it. Therefore also the dual space V∗ has a corresponding dual space, V∗∗, which is called double dual space (because "dual space of the dual space" is a bit long).

      So we have the dual space, but we also want to know what sort of functions are in that double dual space. Well, such a function takes a vector from V∗, that is, a linear function on V, and maps that to a scalar (that is, to a member of the field the vector space is based on). Now, if you have a linear function on V, you already know a way to get a scalar from that: Just apply it to a vector from V. Indeed, it is not hard to show that if you just choose an arbitrary fixed element v∈V, then the function Fv:ϕ↦ϕ(v) indeed is a linear function on V∗, and thus a member of the double dual V∗∗. That way we have not only identified certain members of V∗∗ but in addition a natural mapping from V to V∗∗, namely F:v↦Fv. It is not hard to prove that this mapping is linear and injective, so that the functions in V∗∗ corresponding to vectors in V form a subspace of V∗∗. Indeed, if V is finite dimensional, it's even all of V∗∗. That's easy to see if you know that dim(V∗)=dimV and therefore dim(V∗∗)=dimV∗=dimV. On the other hand, since F is injective, dim(F(V))=dim(V). However for finite dimensional vector spaces, the only subspace of the same dimension as the full space is the full space itself. However if V is infinite dimensional, V∗∗ is larger than V. In other words, there are functions in V∗∗ which are not of the form Fv with v∈V.

      Note that since V∗∗again is a vector space, it also has a dual space, which again has a dual space, and so on. So in principle you have an infinite series of duals (although only for infinite vector spaces they are all different).

    1. There are some very beautiful and easily accessible applications of duality, adjointness, etc. in Rota's modern reformulation of the Umbral Calculus. You'll quickly gain an appreciation for the power of such duality once you see how easily this approach unifies hundreds of diverse special-function identities, and makes their derivation essentially trivial. For a nice introduction see Steven Roman's book "The Umbral Calculus".

      Note to self: Look at [[Steven Roman]]'s book [[The Umbral Calculus]] to follow up on having a more intuitive idea of what a dual space is and how it's useful

    2. Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R

      Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R where M is a smooth manifold is an object that produces, for any point p∈M and tangent vector v∈TpM, a number, the directional derivative, in a linear way. In other words, ==a differentiable function defines an element of the dual to the tangent space (the cotangent space) at each point of the manifold.==

    3. This also happens to explain intuitively some facts. For instance, the fact that there is no canonical isomorphism between a vector space and its dual can then be seen as a consequence of the fact that rulers need scaling, and there is no canonical way to provide one scaling for space. However, if we were to measure the measure-instruments, how could we proceed? Is there a canonical way to do so? Well, if we want to measure our measures, why not measure them by how they act on what they are supposed to measure? We need no bases for that. This justifies intuitively why there is a natural embedding of the space on its bidual.
    4. The dual is intuitively the space of "rulers" (or measurement-instruments) of our vector space. Its elements measure vectors. This is what makes the dual space and its relatives so important in Differential Geometry, for instance.

      A more intuitive description of why dual spaces are useful or interesting.

    1. The Internet, an immeasurably powerful computing system, is subsuming most of our other intellectual technologies. It’s becoming our map and our clock, our printing press and our typewriter, our calculator and our telephone, and our radio and TV.

      An example of technological progress subsuming broader things and abstracting them into something larger.

      Most good mathematical and physical theories exhibit this sort of behaviour. Cross reference Simon Singh's The Big Bang.

    1. To put it succinctly, differential topology studies structures on manifolds that, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds that do have an interesting local (or sometimes even infinitesimal) structure.

      Differential topology take a more global view and studies structures on manifolds that have no interesting local structure while differential geometry studies structures on manifolds that have interesting local structures.

  3. May 2021
    1. In an individual model of privacy, we are only as private as our least private friend.

      So don't have any friends?

      Obviously this isn't a thing, but the implications of this within privacy models can be important.

      Are there ways to create this as a ceiling instead of as a floor? How might we use topology to flip this script?

    1. Standard economic theory uses mathematics as its main means of understanding, and this brings clarity of reasoning and logical power. But there is a drawback: algebraic mathematics restricts economic modeling to what can be expressed only in quantitative nouns, and this forces theory to leave out matters to do with process, formation, adjustment, creation and nonequilibrium. For these we need a different means of understanding, one that allows verbs as well as nouns. Algorithmic expression is such a means. It allows verbs (processes) as well as nouns (objects and quantities). It allows fuller description in economics, and can include heterogeneity of agents, actions as well as objects, and realistic models of behavior in ill-defined situations. The world that algorithms reveal is action-based as well as object-based, organic, possibly ever-changing, and not fully knowable. But it is strangely and wonderfully alive.

      Read abstract.

      The analogy of adding a "verb" to mathematics is intriguing here.

  4. Apr 2021
    1. The form of the obelus as a horizontal line with a dot above and a dot below, ÷, was first used as a symbol for division by the Swiss mathematician Johann Rahn in his book Teutsche Algebra in 1659.
  5. 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.
  6. 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

  7. 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.

  8. 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.”
  9. 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.

  10. 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.

  11. 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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  12. Sep 2020
  13. 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.
  14. 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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  16. 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.
  17. 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.

  18. 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,

  19. 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.

  20. 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.