229 Matching Annotations
  1. Jul 2022
    1. The numbers themselves have also been a source ofdebate. Some digital users identify a new notechronologically. One I made right now, for example,might be numbered “202207201003”, which would beunique in my system, provided I don’t make another thisminute. The advantage of this system is that I could keeptrack of when I had particular ideas, which might comein handy sometime in the future. The disadvantage is thatthe number doesn’t convey any additional information,and it doesn’t allow me to choose where to insert a newnote “behind” the existing note it is most closely relatedto.

      Allosso points out some useful critiques of numbering systems, but doesn't seem to get to the two core ideas that underpin them (and let's be honest, most other sources don't either). As a result most of the controversies are based on a variety of opinions from users, many of whom don't have long enough term practices to see the potential value.

      The important things about numbers (or even titles) within zettelkasten or even commonplace book systems is that they be unique to immediately and irrevocably identify ideas within a system.

      The other important piece is that ideas be linked to at least one other idea, so they're less likely to get lost.

      Once these are dealt with there's little other controversy to be had.

      The issue with date/time-stamped numbering systems in digital contexts is that users make notes using them, but wholly fail to link them to anything much less one other idea within their system, thus creating orphaned ideas. (This is fine in the early days, but ultimately one should strive to have nothing orphaned).

      The benefit of Luhmann's analog method was that by putting one idea behind its most closely related idea was that it immediately created that minimum of one link (to the thing it sits behind). It's only at this point once it's situated that it can be given it's unique number (and not before).


      Luhmann's numbering system, similar to those seen in Viennese contexts for conscription numbers/house numbers and early library call numbers, allows one to infinitely add new ideas to a pre-existing set no matter how packed the collection may become. This idea is very similar to the idea of dense sets in mathematics settings in which one can get arbitrarily close to any member of a set.

      link to: - https://hypothes.is/a/YMZ-hofbEeyvXyf1gjXZCg (Vienna library catalogue system) - https://hypothes.is/a/Jlnn3IfSEey_-3uboxHsOA (Vienna conscription numbers)

    2. Even physicists,when they leave equations behind and try to describetheir discoveries to the rest of us in plain English, findthemselves employing analogies, metaphors, and theother language tools we all use

      Within mathematical contexts one of the major factors often at play is the idea of abstraction: how can one use a basic idea and then abstract it to other situations to see what results.

      The idea of abstraction in mathematics is analogous to analogy and metaphor in literature.

  2. Jun 2022
    1. Two mathematicians at a chalk board looking at line two that reads "Then a miracle occurs".

      S. Harris cartoon "I think you should be more explicit here in step two."

    1. The Algebra Project was born.At its core, the project is a five-step philosophy of teaching that can be applied to any concept: Physical experience. Pictorial representation. People talk (explain it in your own words). Feature talk (put it into proper English). Symbolic representation.

      The five step philosophy of the Algebra Project: - physical experience - pictorial representation - people talk (explain it in your own words) - feature talk (put it into proper English) - symbolic representation


      "people talk" within the Algebra project is an example of the Feynman technique at work

      Link this to Sonke Ahrens' method for improving understanding. Are there research links to this within their work?

    1. There are efforts that actually do work to decrease educational gaps: these include Bob Moses’ Algebra Project, Adrian Mims’ (contact person for one of the letters) Calculus Project,  Jaime Escalante  (from “stand and deliver”) math program, and the Harlem Children’s Zone.

      Mathematical education programs that are attempting to decrease educational gaps: - Bob Moses' Algebra Project - Adrian Mims' Calculus Project - Jaime Escalante math program - Harlem Children's Zone

    2. Shouldn’t CS and STEM faculty stay out of this debate, and leave it to the math education faculty that are the true subject matter experts?

      In querying math professors at many universities, I've discovered that many feel as if they're spending all their time and energy preparing students in the sciences and engineering and very little of their time supporting students in the math department. If one left things up to them, then it's likely that STEM and CS would die on the vine.

  3. May 2022
    1. The recipe details, moreover, assume that these “unmarry’d Women” had the kind of knowledge of arithmetic that the book’s earlier instructional sections had taught. The recipe insists on careful attention to measurement and counting. And it asks the preparer to work with repeated multiples of three. Franklin had a track record of promoting female education, and of arithmetic for them in particular. He advocates for it in his early, anonymous “Silence Dogood” articles, and in his Autobiography singles out a Dutch printer’s widow who saved the family business thanks to her education. There, Franklin makes an explicit call “recommending that branch of education for our young females.”

      Evidence for Benjamin Franklin encouraging the education of women in mathematics.

  4. Apr 2022
    1. In the course of teaching hundredsof first-year law students, Monte Smith, a professor and dean at Ohio StateUniversity’s law school, grew increasingly puzzled by the seeming inability ofhis bright, hardworking students to absorb basic tenets of legal thinking and toapply them in writing. He came to believe that the manner of his instruction wasdemanding more from them than their mental bandwidth would allow. Studentswere being asked to employ a whole new vocabulary and a whole new suite ofconcepts, even as they were attempting to write in an unaccustomed style and anunaccustomed form. It was too much, and they had too few mental resources leftover to actually learn.

      This same analogy also works in advanced mathematics courses where students are often learning dense and technical vocabulary and then moments later applying it directly to even more technical ideas and proofs.

      How might this sort of solution from law school be applied to abstract mathematics?

    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

  5. Mar 2022
    1. https://www.linkedin.com/pulse/incorrect-use-information-theory-rafael-garc%C3%ADa/

      A fascinating little problem. The bigger question is how can one abstract this problem into a more general theory?

      How many questions can one ask? How many groups could things be broken up into? What is the effect on the number of objects?

    1. Semasiography is a system of conventional symbols— iconic, abstract—that carry information, though not in any specific language. The bond between sign and sound is variable, loose, unbound by precise rules. It’s a nonphonetic system (in the most technical, glottographic sense). Think about mathematical formulas, or music notes, or the buttons on your washing machine: these are all semasiographic systems. We understand them thanks to the conventions that regulate the way we interpret their meaning, but we can read them in any language. They are metalinguistic systems, in sum, not phonetic systems.

      Semasiography are iconic and abstract symbols and languages not based on spoken words, but which carry information.

      Mathematical formulas, musical notation, computer icons, emoji, buttons on washing machines, and quipu are considered semasiographic systems which communicate information without speech as an intermediary.

      semasiography from - Greek: σημασία (semasia) "signification, meaning" - Greek: γραφία (graphia) "writing") is "writing with signs"

    1. To signify that an angle is acute, Jeffreys taught them, “make Pac-Man withyour arms.” To signify that it is obtuse, “spread out your arms like you’re goingto hug someone.” And to signify a right angle, “flex an arm like you’re showingoff your muscle.” For addition, bring two hands together; for division, make akarate chop; to find the area of a shape, “motion as if you’re using your hand asa knife to butter bread.”

      Math teacher Brendan Jeffreys from the Auburn school district in Auburn, WA created simple hand gestures to accompany or replace mathematical terms. Examples included making a Pac-Man shape with one's arms to describe an acute angle, spreading one's arms wide as if to hug someone to indicate an obtuse angle, or flexing your arm to show your muscles to indicate a right triangle. Other examples included a karate chop to indicate division or a motion imitating using a knife to butter bread to indicate finding the area of a shape.

    2. Washington State mathteacher Brendan Jeffreys turned to gesture as a way of easing the mental loadcarried by his students, many of whom come from low-income households,speak English as a second language, or both. “Academic language—vocabularyterms like ‘congruent’ and ‘equivalent’ and ‘quotient’—is not something mystudents hear in their homes, by and large,” says Jeffreys, who works for theAuburn School District in Auburn, a small city south of Seattle. “I could see thatmy kids were stumbling over those words even as they were trying to keep trackof the numbers and perform the mathematical operations.” So Jeffreys devised aset of simple hand gestures to accompany, or even temporarily replace, theunfamiliar terms that taxed his students’ ability to carry out mental math.

      Mathematics can often be more difficult compared to other subjects as students learning new concepts are forced not only to understand entirely new concepts, but simultaneously are required to know new vocabulary to describe those concepts. Utilizing gestures to help lighten the cognitive load of the new vocabulary to allow students to focus on the concepts and operations can be invaluable.

    3. gesture is often scorned as hapless“hand waving,” or disparaged as showy or gauche.

      The value of gesture is sometimes disparaged with the phrase "handy waving".

      Some of this statement is misleading as a hand waving argument relies solely on the movement of the hands as "proof" of something which is neither communicated well with the use of either words or the physical hand movements. The communication fails on both axes, but the blame is placed on the gestural portion of the communication, perhaps because it may have been the more important of the two?


      Link this to the example of the Riverside teacher who used both verbal and visual gestures and acting to cement the trigonometry ideas of soh, cah, toa to her students and got fired for it. In her example, the gauche behaviour was overamplified by extreme exaggeration as well as racist expression.

  6. Feb 2022
    1. Also, we shouldn’t underestimate the advantages of writing. In oralpresentations, we easily get away with unfounded claims. We candistract from argumentative gaps with confident gestures or drop acasual “you know what I mean” irrespective of whether we knowwhat we meant. In writing, these manoeuvres are a little too obvious.It is easy to check a statement like: “But that is what I said!” Themost important advantage of writing is that it helps us to confrontourselves when we do not understand something as well as wewould like to believe.

      In modern literate contexts, it is easier to establish doubletalk in oral contexts than it is in written contexts as the written is more easily reviewed for clarity and concreteness. Verbal ticks like "you know what I mean", "it's easy to see/show", and other versions of similar hand-waving arguments that indicate gaps in thinking and arguments are far easier to identify in writing than they are in speech where social pressure may cause the audience to agree without actually following the thread of the argument. Writing certainly allows for timeshiting, but it explicitly also expands time frames for grasping and understanding a full argument in a way not commonly seen in oral settings.

      Note that this may not be the case in primarily oral cultures which may take specific steps to mitigate these patterns.

      Link this to the anthropology example from Scott M. Lacy of the (Malian?) tribe that made group decisions by repeating a statement from the lowest to the highest and back again to ensure understanding and agreement.


      This difference in communication between oral and literate is one which leaders can take advantage of in leading their followers astray. An example is Donald Trump who actively eschewed written communication or even reading in general in favor of oral and highly emotional speech. This generally freed him from the need to make coherent and useful arguments.

    2. his suggests that successful problem solvingmay be a function of flexible strategy application in relation to taskdemands.” (Vartanian 2009, 57)

      Successful problem solving requires having the ability to adaptively and flexibly focus one's attention with respect to the demands of the work. Having a toolbelt of potential methods and combinatorially working through them can be incredibly helpful and we too often forget to explicitly think about doing or how to do that.

      This is particularly important in mathematics where students forget to look over at their toolbox of methods. What are the different means of proof? Some mathematicians will use direct proof during the day and indirect forms of proof at night. Look for examples and counter-examples. Why not look at a problem from disparate areas of mathematical thought? If topology isn't revealing any results, why not look at an algebraic or combinatoric approach?

      How can you put a problem into a different context and leverage that to your benefit?

    1. https://www.latimes.com/california/story/2022-02-09/riverside-sohcahtoa-teacher-viral-video-mocked-native-americans-fired

      Riverside teacher who dressed up and mocked Native Americans for a trigonometry lesson involving a mnemonic using SOH CAH TOA in Riverside, CA is fired.

      There is a right way to teach mnemonic techniques and a wrong way. This one took the advice to be big and provocative went way overboard. The children are unlikely to forget the many lessons (particularly the social one) contained here.

      It's unfortunate that this could have potentially been a chance to bring indigenous memory methods into a classroom for a far better pedagogical and cultural outcome. Sad that the methods are so widely unknown that media missed a good teaching moment here.

      referenced video:

      https://www.youtube.com/watch?v=Bu4fulKVv2c

      A snippet at the end of the video has the teacher talking to rocks and a "rock god", but it's extremely unlikely that she was doing so using indigenous methods or for indigenous reasons.

      read: 7:00 AM

    1. Diesen gebrochenen Zahlen, welche zunächst als reine Zeichen auftreten, kann in vielen Fällen eine actuelle Bedeutung beigelegt werden.

      A presented meaning can in many cases be attributed to these rational numbers, which at first appear as pure signs,

    2. Wie wir die Regeln der rein formalen Verknüpfungen, d. h. der mit den mentalen Objecten vorzunehmenden Operationen definiren, steht in unserer Willkühr, nur muss eine Bedingung als wesentlich festgehalten werden: nämlich dass irgend welche logische Widersprüche in den- selben nicht implicirt sein dürfen.

      How we define the rules of purely formal operations (Verknüpfungen), i.e., of carrying out operations (Operationen) with mental objects, is our arbitrary choice, except that one essential condition must be adhered to: namely that no logical contradiction may be implied in these same rules.

    3. man sich zu der gegebenen Reihe von Ob- jecten eine inverse hinzudenkt

      one adds an inverse in thought to the given series of objects

    4. Man sieht aber nicht, wie unter — 3 eine reale Substanz verstanden werden kann, wenn das ursprünglich gesetzte Object eine solche ist, und würde im Rechte sein, wenn man — 3 als eine nicht reelle, imaginäre Zahl als eine „falsche" bezeichnete.

      one cannot see how a real substance can be understood by -3... and would be within his rights if he refers to -3 as a non-real, imaginary number, as a "false" one.

    5. Eine andere Definition des Begriffes der formalen Zahlen kann nicht gegeben werden; jede andere muss aus der Anschauung oder Erfahrung Vorstellungen zu Hilfe nehmen, welche zu dem Begriffe in einer nur zufälligen Beziehung stehen, und deren Beschränktheit einer allgemeinen Untersuchung der Rechnungsoperationen unüber- steigliche Hindemisse in den Weg legt..

      A different definition of the concept of the formal numbers cannot be given; every other definition must rely on ideas from intuition or experience, which stand in only an accidental relation to the concept, and the limitations of which place insurmountable obstacles in the way of a general investigation of the arithmetic operations.

    6. Die Bedingung zur Aufstellung einer allgemeinen Arithmetik ist daher eine von aller Anschauung losgelöste, rein intellectuelle Mathem&tik, eine reine Formenlehre, in welcher nicht Quanta oder ihre Bilder, die Zahlen verknüpft werden, sondern intellectuelle Objecte, Gedankendinge, denen actuelle Objecte oder Relationen solcher entsprechen kön- nen, aber nicht müssen.

      The condition for the establishment of a general arithmetic is therefore a purely intellectual mathematics detached from all intuition, a pure theory of form, in which quanta or their images, the numbers, are not combined, but rather intellectual objects, thought-things, to which presented objects or relations of such objects can, but need not, correspond.

    7. Wie überhaupt die Entwicklung mathematischer Begriffe und Vorstellungen historisch zwei entgegengesetzte Phasen zu durchlaufen pflegt, so auch die des Imaginären. Zunächst erschien dieser Begriff' als paradox, streng genommen unzulässig, unmög- lich;

      As the development of mathematical concepts and ideas generally goes historically through two opposed phases, so goes also that of the imaginary numbers. At first this concept appeared as a paradox, strictly inadmissible, impossible;

    8. Wissenschaft leistete, im Laufe der Zeit alle Zweifel an seiner Legitimität nieder und es bildete sich die Ueberzeugung seiner inneren Wahrheit und Nothwendigkeit in solcher Entschiedenheit aus, dass die Schwierigkeiten und Widersprüche, welche man anfangs in ihm bemerkte, kaum noch gefühlt wurden. In diesem zweiten Stadium befindet sich die Frage des Imaginären heut zu Tage ; — indessen bedarf es keines Beweises, dass die eigentliche Natur von Begriffen und Vorstellungen erst dann hinreichend auf- geklärt ist, wenn man unterscheiden kann, was an ihnen noth- wendig ist, und was arbiträr, d. h. zu einem gewissen Zwecke in sie hineingelegt ist.

      however, in the course of time, the essential services which it affords to science subdue all doubts of its legitimacy, and one is convinced in such decisiveness of its inner truth and necessity, that the difficulties and contradictions which one noticed in it at the beginning are hardly felt. Today, the question of imaginary numbers is in this second stage; --- however it needs no proof that the actual nature of concepts and ideas is only sufficiently clarified when one can distinguish what is necessary in them, and what is arbitrary, i.e., is put to a certain purpose in them.

  7. Jan 2022
    1. And there are, again, ethical questions that must be asked and answered when dealing with the quantitative study of human atrocity, which is what we’re ultimately doing when we bring statistical and mathematical methods to the study of slavery.
    1. https://www.youtube.com/watch?v=z3Tvjf0buc8

      graph thinking

      • intuitive
      • speed, agility
      • adaptability

      ; graph thinking : focuses on relationships to turn data into information and uses patterns to find meaning

      property graph data model

      • relationships (connectors with verbs which can have properties)
      • nodes (have names and can have properties)

      Examples:

      • Purchase recommendations for products in real time
      • Fraud detection

      Use for dependency analysis

  8. Dec 2021
    1. Order RelationsA relation C on a set A is called an order relation (or a simple order, or a linear order)if it has the following properties:(1) (Comparability) For every x and y in A for which x = y, either xCy or yCx.(2) (Nonreflexivity) For no x in A does the relation xCx hold.(3) (Transitivity) If xCy and yCz, then xCz.Note that property (1) does not by itself exclude the possibility that for some pair ofelements x and y of A, both the relations xCy and yCx hold (since “or” means “oneor the other, or both”). But properties (2) and (3) combined do exclude this possibil-ity; for if both xCy and yCx held, transitivity would imply that xCx, contradictingnonreflexivity.EXAMPLE 7. Consider the relation on the real line consisting of all pairs (x, y) of real

      Link to idea from The Dawn of Everything about comparative anthropology.

    1. ‘Noble’ savages are, ultimately, just as boring as savageones; more to the point, neither actually exist. Helena Valero washerself adamant on this point. The Yanomami were not devils, sheinsisted, neither were they angels. They were human, like the rest ofus.

      This is an interesting starting point for discussing the ills of comparative anthropology which will tend to put one culture or society over another in some sort of linear way and an expectation of equivalence relations (in a mathematical sense).

      Humans and their societies and cultures aren't always reflexive, symmetric, or transitive. There may not be an order relation (aka simple order or linear order) on humanity. We may not have comparability, nonreflexivity, or transitivity.

      (See page 24 on Set Theory and Logic in Topology by James R. Munkres for definition of "order relation")

    1. “That’s one of the exciting things about math,” said Jack Morava, a mathematician at Johns Hopkins University and the inventor of Morava K-theory. “You can go through a door and you wind up in a completely different universe. It’s very much like Alice in Wonderland.”
    2. Morava K-theory is an invariant

      Morava K-theory is an invariant.

    3. “[Floer] homology theory depends only on the topology of your manifold. [This] is Floer’s incredible insight,” said Agustin Moreno of the Institute for Advanced Study.

      Floer homology theory depends only on the topology of the manifold.

    4. Mathematicians already had a method, known as Morse theory, for studying these critical points.

      Morse theory can be used to study critical points.

    5. In the field of topology, homology is the formal way to count holes. Homology associates to each shape an algebraic object, which can be used to extract information like the number of holes in each dimension.

      A relatively simple definition of homology and what it is.

  9. Nov 2021
  10. Oct 2021
    1. "Vielmehr", so Schmidt et al., "notiert Luhmann in der Regel nur maximal drei Systemstellen, an denen der jeweilige Begriff zu finden ist, da er annimmt, dass man dann über das interne Verweisungsnetz schnell die anderen relevanten Stellen findet."

      machine translation:

      "Rather," says Schmidt et al., "Luhmann usually only notes a maximum of three system points at which the respective term can be found, since he assumes that the other relevant points can then be found quickly via the internal network of references."

      I wonder how many tags one might use in practice to maximize this? Can we determine such a thing mathematically?

    1. trailblazing physicist David Bohm and Indian spiritual philosopher Jiddu Krishnamurti sat down for a mind-bending, soul-stretching series of conversations about some of the most abiding human concerns: time, transcendence, compassion, death, the nature of reality, and the meaning of existence.

      What came up for me in exploring the parallels between writing and mathematics.

    1. General relativity implies that information gets destroyed; quantum theory says it’s preserved. Hence the paradox.

      Isn't this an example of the law of the excluded middle? If LoEM doesn't exist (in Gisin's theory), then could there be information that isn't either created or destroyed?

  11. Sep 2021
    1. I knew that Sol Golomb had been collaborating on a textbook going back almost fifteen years. It's great to see it not only finally come out, but to see it published with his name in the title!

      I had the pleasure of taking Sol's combinatorics class at USC several years before he passed away, so I also got an early look at much of the material as he was using it in class. It was scheduled at my lunchtime, so I took the time to drive over to USC at lunch twice a week to sit in. My favorite part was seeing proofs for various things I'd seen in other branches of mathematics, but done in a combinatorial way.

      Somewhere knocking around I think I've got audio recordings and notes of the class that I'll have to do something with one day.

      Many talk about Sol's ability to do calculations in his head, but like most mathematicians he knew the standard tricks and shortcuts. To me this was underlined by the fact that he always did long division on the board when there wasn't a simple short cut.

    1. https://fs.blog/2021/07/mathematicians-lament/

      What if we taught art and music the way we do mathematics? All theory and drudgery without any excitement or exploration?

      What textbooks out there take math from the perspective of exploration?

      • Inventional geometry does

      Certainly Gauss, Euler, and other "greats" explored mathematics this way? Why shouldn't we?

      This same problem of teaching math is also one we ignore when it comes to things like note taking, commonplacing, and even memory, but even there we don't even delve into the theory at all.

      How can we better reframe mathematics education?

      I can see creating an analogy that equates math with art and music. Perhaps something like Arthur Eddington's quote:

      Suppose that we were asked to arrange the following in two categories–

      distance, mass, electric force, entropy, beauty, melody.

      I think there are the strongest grounds for placing entropy alongside beauty and melody and not with the first three. —Sir Arthur Stanley Eddington, OM, FRS (1882-1944), a British astronomer, physicist, and mathematician in The Nature of the Physical World, 1927

    2. “We don’t need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience.”

      Paul Lockhart in Lockhart's Lament

    3. “What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources—beautiful works of art by some of the most creative minds in history—in favor of third-rate textbook bastardizations?”

      ---Paul Lockhart

    4. We don’t teach the process of creating math. We teach only the steps to repeat someone else’s creation, without exploring how they got there—or why.

      This is the primary problem with mathematics education!

  12. Aug 2021
  13. 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.

  14. 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.

  15. 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.

  16. 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.
  17. 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.
  18. 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.
  19. 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.

  20. 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.”
  21. 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.

  22. 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.

  23. 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