278 Matching Annotations
1. Dec 2023
1. How to fold and cut a Christmas star<br /> Christian Lawson-Perfect https://www.youtube.com/watch?v=S90WPkgxvas

What a great simple example with some interesting complexity.

For teachers trying this with students, when one is done making some five pointed stars, the next questions a curious mathematician might ask are: how might I generalize this new knowledge to make a 6 pointed star? A 7 pointed star? a 1,729 pointed star? Is there a maximum number of points possible? Is there a minimum? Can any star be made without a cut? What happens if we make more than one cut? Are there certain numbers for which a star can't be made? Is there a relationship between the number of folds made and the number of points? What does all this have to do with our basic definition of what a paper star might look like? What other questions might we ask to extend this little idea of cutting paper stars?

Recalling some results from my third grade origami days, based on the thickness of most standard office paper, a typical sheet of paper can only be folded in half at most 7 times. This number can go up a bit if the thickness of the paper is reduced, but having a maximum number of potential folds suggests there is an upper bound for how many points a star might have using this method of construction.

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3. Nov 2023
4. www.browndailyherald.com www.browndailyherald.com

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5. www.reddit.com www.reddit.com
1. level 2Apprehensive_Net5630Op · 3 hr. agoI've come to think the thousands category is kind of superfluous. Instead of starting 2000, just start a card "2 management" and then create a card "25 leadership" and add below, e.g. "251 blah blah", "252 blah blah"2ReplyShareReportSaveFollowlevel 3marco89lcdm · 2 hr. agoMmm.. interesting .. Although I think is too late as I’m already well into the “2000” category and the problem presented once I’ve started to do some leadership cards. Those are 3 or 4 and I can still amend the ID number, but the management one are almost 30 already, I don’t feel like changing everything while so well advanced.. this could put me off from keeping doing it completely. Maybe worth knowing that I didn’t have an exhaustive index yet.. because of the fact that IDing the card is not clear for me

Don't sweat the difference as there is a one-to-one and onto (or bijective) relationship between what you're doing and what u/Apprehensive_Net5630 suggests. Mathematicians would call the relationship homomorphic (ie: of the same shape), so other than the make-work exercise, you'd end up with the same exact thing with the same ordering in the end.

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6. Oct 2023
7. www.reddit.com www.reddit.com

Most of my math section in my ZK is primarily very basic definitions and theorems. I have very few proofs of basic things outside of my own personal work. There is an occasional useful example or two or lists of various lists of things that fit certain structures (lists of groups, rings, fields, categories, etc.)

Digital notes just don't work for me at all with respect to math, so I'm all-in on index cards and simply typeset all the necessary parts when I'm done with something if I intend to publish or share with others. Paper also makes it much easier to shuffle things around and reshape pieces as necessary to try out different structural approaches.

I also have a short stack of method cards (not dissimilar to Eno & Schmidt's "Oblique Strategies", but with a mathematical bent) to remind me of different approaches to try out when I get stuck which has been pretty beneficial.

I also take a fairly segmented approach between the writing I do for understanding a math text as I'm reading it and the permanent sort of notes I specifically make after-the-fact. My goal is never to recreate entire textbooks within the main section of my own zettelkasten, but create material for new works I might be writing for others to read.

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8. Sep 2023
9. docdrop.org docdrop.org
1. e hard scientist doesis to say that he "stipulates his usage"-that is, he informs youwhat terms are essential to his argument and how he is goingto use them. Such stipulations usually occur at the beginningof the book, in the form of definitions, postulates, axioms, andso forth. Since stipulation of usage is characteristic of thesefields, it has been said that they are like games or have a"game structure."

Depending on what level a writer stipulates their usage, they may come to some drastically bad conclusions. One should watch out for these sorts of biases.

Compare with the results of accepting certain axioms within mathematics and how that changes/shifts one's framework of truth.

2. Many people are frightened of mathematics and thinkthey cannot read it at all. No one is quite sure why this is so.Some psychologists think there is such a thing as "symbolblindness"-the inability to set aside one's dependence on theconcrete and to follow the controlled shifting of symbols.

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10. docdrop.org docdrop.org
1. this is just a hypothesis there's a thing called the unreasonable effectiveness of mathematics like why is it the case that you go off and you invent complex numbers and quaternions and do abstract algebra and somehow that has something 00:33:26 to say about the physical world still a mystery but there's an unreasonable effectiveness of deep learning there is no a priori reason why DNA and images and video and speech synthesis and fmri 00:33:40 should share a kind of universal shape but they do and I think that's telling us something very deep about the structure of the universe
• for: unreasonable effectiveness of mathematics, emptiness, emptiness - unreasonable effectiveness of mathematics

• comment

• everything is an expression of emptiness

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11. math.ucr.edu math.ucr.edu

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12. www.uclaextension.edu www.uclaextension.edu

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13. Aug 2023
14. zettelkasten.de zettelkasten.de
1. category theory + zettelkasten... hmmm... feels a bit like Leibniz chasing a universal language

2. Might we coin "zettelkastenly" as a function of (box) composed with (manually+cerebrally)?

hand:manually::brain:cerebrally::box:zettelkastenly

I would use the LaTeX \boxtimes symbol for this composition I think....

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15. hub.jhu.edu hub.jhu.edu
1. In the ensuing decades, mathematicians began working with this new thing, the category, and this new idea, equivalence. In so doing they created a revolutionary new approach to mathematics, category theory, that many see as supplanting set theory. Imagine if writers had spent 150 years representing the world only through basic description: This is a red ball. That is a 60-foot tree. This is a dog. Then one day someone discovers metaphor. Suddenly, our ability to find new ways to represent the world explodes, as does our knowledge of writing as a discipline.

I love the idea here of analogizing the abstract nature of category theory in math with the abstract nature of metaphor in writing!

Good job Dale Keiger!

2. "If mathematics is the science of analogy, the study of patterns, then category theory is the study of patterns of mathematical thought."

original source?

3. I think mathematicians do math in part because we think it's beautiful."
4. "My teachers really gave me a glimpse of what math was like at the college level, the creative side of mathematics as opposed to the calculational side of mathematics.

creative mathematics versus computational or calculational mathematics...

we need more of the creative in early education

partial quote from Emily Riehl

5. "But there's a very famous theorem in topology called the Jordan curve theorem. You have a plane and on it a simple curve that doesn't intersect and closes—in other words, a loop. There's an inside and an outside to the loop." As Riehl draws this, it seems obvious enough, but here's the problem: No matter how much your intuition tells you that there must be an inside and an outside, it's very hard to prove mathematically that this holds true for any loop that can be drawn.

How does one concretely define "inside" and "outside"? This definition is part of the missing space between the intuition and the mathematical proof.

6. In an article she wrote recently for Scientific American, Riehl quoted John Horton Conway, an esteemed English mathematician: "What's the ontology of mathematical things? There's no doubt that they do exist, but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, despite having been a mathematician all my life. How can things be there without actually being there?"
7. But then, so are numbers, for all their illusion of concrete specificity and precision.

Too many non-mathematicians view numbers as solid, concrete things which are meant to make definite sense and quite often their only experience with it is just that. Add two numbers up and always get the same thing. Calculate something in physics with an equation and get an exact, "true" answer. But somehow to be an actual mathematician, one must not see it as a "solid area" (using these words in their non-mathematical senses), but a wholly abstract field of abstraction built upon abstraction. While each abstraction has a sense of "trueness", it will need to be abstracted over and over while still maintaining that sense of "trueness". For many, this is close to being impossible because of the sense of solidity and gravity given to early mathematics.

How can we add more exploration for younger students?

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16. Jun 2023
17. opentextbooks.uregina.ca opentextbooks.uregina.ca
1. Sardarli, Arzu, and Ida Swan. Cree Dictionary of Mathematical Terms with Visual Examples. University of Regina, 2022. https://opentextbooks.uregina.ca/creemathdictionary/.

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18. May 2023
19. kleinbottle.com kleinbottle.com

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20. www.amazon.com www.amazon.com
1. https://www.amazon.com/Vaughn-Cube-Multiplication-Petersons/dp/0768941776

Dean Vaughn has apparently renamed the method of loci as a commodity to be able to market a method for memorizing the multiplication tables.

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21. oeis.org oeis.org
1. Trying to follow an argument given here: https://youtu.be/mFZs7uGwNBo?t=3413

The sequence A002267 is claimed by Haris Neophytou to be the 1st 15 "super singular prime numbers" (ie, the primes that divides the order of the Monster Group). The order is the number of elements in the group.

Note that the last 3 elements [47, 59, 71] multiply to give the number of dimensions in which the Monster group exists: 196,883.

Neophytou believes A002267 gives a different way of looking at the monster group $$M$$.

Around 1:02:45, Neophytou says he'll start from A002822...

(a list of numbers, $$m\text{,}$$ such that $$6m - 1$$ and $$6m + 1$$ are twin primes)

... and construct "the minimal order of the monster" (what?)

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22. Apr 2023
23. www.popularmechanics.com www.popularmechanics.com
1. To Solve the Rubik’s Cube, You Have to Understand the Amazing Math Inside<br /> by Dave Linkletter

suggested by Matt Maldre's annotations

2. Only small tidbits of math remain unresolved for Rubik’s Cube. While God’s number is 20, it’s unknown exactly how many of the 43,252,003,274,489,856,000 combinations require a whole 20 moves to be solved.

We've got solutions for the number of configurations there are to solve a Rubic's cube with from 1 move up to 15, but we don't know how many cube configurations there are that can be solved with 16-20 moves.

• Example: the number of positions that require exactly one move solve them is 18, which is counted by multiplying the six faces and each of the three ways they can be twisted.

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24. Mar 2023
25. www.ams.org www.ams.org
1. Resources and inspiration for math instruction and learning

Welcome to AMS Open Math Notes, a repository of freely downloadable mathematical works hosted by the American Mathematical Society as a service to researchers, faculty and students. Open Math Notes includes: - Draft works including course notes, textbooks, and research expositions. These have not been published elsewhere and are subject to revision. - Items previously published in the Journal of Inquiry-Based Learning in Mathematics, a refereed journal - Refereed publications at the AMS

Visitors are encouraged to download and use any of these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

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26. mathbabe.org mathbabe.org
1. What tensor products taught me about living my life<br /> by Cathy O'Neil aka mathbabe

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27. Feb 2023
28. texnicalstuff.blogspot.com texnicalstuff.blogspot.com
1. Zettelkasten workflow<br /> by pqnelson

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29. en.wikipedia.org en.wikipedia.org
1. Whewell was prominent not only in scientific research and philosophy but also in university and college administration. His first work, An Elementary Treatise on Mechanics (1819), cooperated with those of George Peacock and John Herschel in reforming the Cambridge method of mathematical teaching.

What was the specific change in mathematical teaching instituted by Whewell, Peacock, and Herschel in An Elementary Treatise on Mechanics (1819)?

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30. docdrop.org docdrop.org
1. Eugene P. Wigner, The unreasonable effectiveness of mathematics in thenatural sciences, Comm. Pure Appl. Math. 13 (1960), 1–14.

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31. math.stackexchange.com math.stackexchange.com

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32. Jan 2023
33. docdrop.org docdrop.org
1. Woit, Peter. Quantum Theory, Groups and Representations: An Introduction. Revised and Expanded version [2022]. Springer, 2017. https://www.math.columbia.edu/~woit/QM/qmbook.pdf.

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34. ncase.me ncase.me

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35. www.reddit.com www.reddit.com
1. Equations and Formulas in Mathematics, Physics and Chemistry Using a Digital Zettelkasten .t3_10dbza7._2FCtq-QzlfuN-SwVMUZMM3 { --postTitle-VisitedLinkColor: #9b9b9b; --postTitleLink-VisitedLinkColor: #9b9b9b; --postBodyLink-VisitedLinkColor: #989898; } zk-structureHow do you handle relations between mathematical, physical or chemical formulas in a digital Zettelkasten? Since I would like to use a future-proof system, my files are written in markdown.Is it possible to write down those formulas on a tablet and save the pdfs inside the digital Zettelkasten (along with a new ID and a descriptive title) and then just reference on it from the markdown Zettel? Or should I create attachments for markdown Zettel that require some formulas or images providing the *same* ID to them?Or, do I completely overthinking this?

Perhaps you've not gotten far enough in your studies to see the pattern yet, but most advanced mathematics texts (Hungerford's Algebra or Rudin's Real Analysis for example) and many physics texts are written as if they were pre-numbered zettelkasten. Generally every definition, theorem, corollary, proposition, and lemma in the text will be written out succinctly with its own unique number and arranged in some sort of branching order. Most of these texts you could generally cut up and paste onto cards and have something zettelkasten-like without any additional work.

I generally follow this same pattern and usually separate proofs on cards behind their associated theorems (typically only for those I've written out myself). Individual equations can be numbered, but I rarely give an equation its own card and instead reference them by card number with a particular equation line in parenthesis when necessary. I can then cross reference definitions, theorems, etc. easily as I continue building things up. Over time you can eventually cross reference various branches of math, physics, chemistry, biology, etc. For example I've got a dozen different proofs and uses of Schwartz' inequality in six different branches of mathematics which goes toward showing how closely knit various disparate branches of math can be.

In most cases I might also suggest against too heavy a focus on the equations, but on what they may say/mean, and what you can use it for. Alternately drawing diagrams or pictures of the relationships can be valuable. Understand it first then write it down. As an example you can look at Boyle's Law, Charles' law, Avogadro's Law and Gay-Lussac's law, but if you know and understand them properly then you should be able to write down and understand the more generic Ideal Gas Law, which shows how they're interlinked. Later in your studies you might also then be able to derive it from microscopic kinetic theory with statistical mechanics as well, then you'll be able to link up those concepts at that time.

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36. Local file Local file
1. It is necessary that the student be alert to reason as the speakerreasons. It is very dangerous to jot down the results of reasoning if youhave not followed it in your own mind.

Dramatically important in mathematics, but also in every other area.

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37. www.cambridge.org www.cambridge.org
1. The value of <Y>, the position of which varies in the sequences, may be the precursor of place value, in which, for example, 5, 50 and 500 represent different values according to their position, thought to have been a Sumerian invention (d'Errico et al. Reference d'Errico, Doyon and Colage2017).

The idea of place value is thought to have been a Sumerian invention (d'Errico et al., 2017), but the example of <Y> in the work of B. Bacon, et al (2023) may push the date of the idea of place value back significantly.

2. The sequences of dots/lines associated with animal images certainly meet the criteria for representing numbers: they are usually organized in registers that are horizontal relative to the image with which they appear to be associated, and are of regular (rather than random) size and spacing, akin to the notion of a Mental Number Line being central to the development of mathematical abilities (Brannon Reference Brannon2006; Dehaene et al. Reference Dehaene, Bossini and Giraux1993; Pinel et al. Reference Pinel, Piazza, Le Bihan and Dehaene2004; Previtali et al. Reference Previtali, Rinaldi and Girelli2011; Tang et al. Reference Tang, Ward and Butterworth2008).

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38. Local file Local file
1. McCoy, Neal Henry. The Theory of Rings. 1964. Reprint, The Bronx, New York: Chelsea Publishing Company, 1973.

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39. Dec 2022
40. andrei-wonge.github.io andrei-wonge.github.io
1. formula

Consider that: 1. Sine and cosine are orthogonal to each other 2. Hence, you can rewrite -sin(Θ) = cos(Θ + π/2) cos(Θ) = sin(Θ + π/2) 3. Therefore, the angle between the standard basis vectors, and their orientation, are preserved!

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41. Nov 2022
42. desales.brightspace.com desales.brightspace.com
1. Social Annotation and Mathematics Education

My internal mathematician wishes there was more substance in this particular portion.

• Use of $$LaTeX$$
• annotating the breakdown of logic in problems
• providing missing context
• filling in details of problems left as an exercise for the student
• others?

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43. watermark.silverchair.com watermark.silverchair.com
1. Introduction to Daniel Rosiak's spectacular "Sheaf Theory through Examples" available open access from MIT Direct Press: https://doi.org/10.7551/mitpress/12581.003.0003

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44. Oct 2022
45. www.reddit.com www.reddit.com

reply (unsent)<br /> I appreciate where you're coming from, and it's an excellent thought experiment. However, knowing that there was a clear older prior zettelkasten tradition for several hundred years prior to Luhmann which also included a number of mathematician practitioners including not only Leibnitz but also Newton, who incidentally invented his version of calculus in his waste book (also a part of that tradition). (See also: https://www.newtonproject.ox.ac.uk/texts/notebooks?sort=date&order=desc).

2. His social theory, developed over thirty years, owes a massive intellectual debt to the work of the English philosopher and mathematician George Spencer-Brown. Spencer-Brown's work of algebraic locic, Laws of Form (1969), was a minor cult hit in the 1970s

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1. If you give a title to your notes, "claim notes" are simply notes with a verb. They invite you to say: "Prove it!" - "The positive impact of PKM" (not a claim) - "PKM has a positive impact in improving writer's block" (claim) A small change with positive mindset consequences

If you give a title to your notes, "claim notes" are simply notes with a verb.<br><br>They invite you to say: "Prove it!"<br><br>- "The positive impact of PKM" (not a claim)<br>- "PKM has a positive impact in improving writer's block" (claim)<br><br>A small change with positive mindset consequences

— Bianca Pereira | PKM Coach and Researcher (@bianca_oli_per) October 6, 2022

Bianca Pereira coins the ideas of "concept notes" versus "claim notes". Claim notes are framings similar to the theorem or claim portion of the mathematical framing of definition/theorem(claim)/proof. This set up provides the driving impetus of most of mathematics. One defines objects about which one then advances claims for which proofs are provided to make them theorems.

Framing one's notes as claims invites one to provide supporting proof for them to determine how strong they may or may not be. Otherwise, ideas may just state concepts which are far less interesting or active. What is one to do with them? They require more active work to advance or improve upon in more passive framings.

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47. www.sub.uni-hamburg.de www.sub.uni-hamburg.de
1. Jungius beschäftigte sich lange mit verschiedenen Kalendersystemen. In seinen Briefen kann man sehen, welche Probleme es damals bereitete, dass einige Gebiete dem Gregorianischen Kalender folgten und andere, darunter auch Hamburg, noch dem Julianischen.

Jungius worked for a long time on various calendar systems. In his letters one can see the problems it caused at the time that some areas followed the Gregorian calendar and others, including Hamburg, still followed the Julian.

Joachim Jungius worked on various calendar systems at a time when the Gregorian calendar was coming into use while his home city of Hamburg still followed the Julian calendar. Jungius used the Julian calendar as well, but generally only dated his notes with the month and the last two digits of the year and excluded the century.

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48. physicstoday.scitation.org physicstoday.scitation.org

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49. Sep 2022
50. www.uclaextension.edu www.uclaextension.edu

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51. Local file Local file
1. formerly was a partof the mathematical program at the intermediate level, has now beendropped from that program, and hence is no longer included in thenew textbooks on elementary algebra. On the other hand, the curriculaat the more advanced levels (even in the mathematics divisions ofuniversities) also omit this theory.

Mike Miller relayed to me on 2022-09-27 on the first meeting of his class Theory and Applications of Continued Fractions that he met a professor colleague walking on campus who mistakenly thought that he was teaching an 8th grade class on basic fractions for the UCLA math department.

2. Khinchin, Aleksandr Yakovlevich. Continued Fractions. 3rd ed. Chicago: University of Chicago Press, 1964.

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52. www.science.org www.science.org

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1. Originality has three dimensions: new sources, new arguments, and/or new audiences.

Originality has many facets including:<br /> - new arguments - new sources - new audiences

What other facets may exist? Think about communication theory to explore this.

I particularly like the new audiences aspect as there are dramatically different audiences for different pieces. (Eg: academic articles, newspapers, magazines, books, blog posts, social media, etc.) A plethora of audiences may be needed to reach the right audiences.

Compare this with Terry Tao's list of mathematical talents which includes communication.

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54. Jul 2022
55. docdrop.org docdrop.org
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.

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56. www.intheknow.com www.intheknow.com
1. https://news.yahoo.com/mathematician-tiktok-gives-example-insane-192823997.html

A sad, but subtle bit of math shaming going on here. Worse it's indicating that math is hard for even the elite without providing proper context.

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57. www.tricki.org www.tricki.org
1. http://www.tricki.org/

a Wiki-style site with a large store of useful mathematical problem-solving techniques.

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58. Jun 2022

It's often said that, 'The hand is the visible part of our brain.'

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

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61. www.aeaweb.org www.aeaweb.org

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62. algebra.org algebra.org

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63. www.nytimes.com www.nytimes.com
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?

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64. www.theescalanteprogram.org www.theescalanteprogram.org

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65. thecalculusproject.org thecalculusproject.org

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66. windowsontheory.org windowsontheory.org
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.

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1. <small><cite class='h-cite via'> <span class='p-author h-card'>Boaz Barak</span> in Brian Conrad takes down the CMF – Windows On Theory (<time class='dt-published'>06/01/2022 21:35:06</time>)</cite></small>

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68. scottaaronson.blog scottaaronson.blog
1. https://scottaaronson.blog/?p=6389

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69. May 2022
70. slate.com slate.com
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.

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71. minimalistquotes.com minimalistquotes.com
1. Curve: The loveliest distance between two points. Mae West

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72. Apr 2022
73. en.wikipedia.org en.wikipedia.org

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74. Local file Local file
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?

#### Annotators

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

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76. www.sfchronicle.com www.sfchronicle.com

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77. Mar 2022

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?

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79. lithub.com lithub.com
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"

#### URL

80. Local file Local file
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.

#### Annotators

81. Feb 2022
82. www.dynamicmath.xyz www.dynamicmath.xyz

#### URL

83. thinkzone.wlonk.com thinkzone.wlonk.com
1. https://thinkzone.wlonk.com/Numbers/NumberSets.htm

A relatively clear explanation of the types of numbers and their properties.

I particularly like this diagram:

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84. linear.axler.net linear.axler.net

I've seen Gilbert Strang's excellent lecture series. I'm curious how these rank in relation.

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85. www.theguardian.com www.theguardian.com
1. https://www.theguardian.com/us-news/2015/oct/17/berkeley-math-professor-alexander-coward-campus-battle

A mathematics lecturer at UC Berkeley went against the grain and got fired for it. I'm curious what his teaching secrets were.

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86. Local file Local file
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?

#### Annotators

87. www.latimes.com www.latimes.com
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:

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.

#### URL

88. archive.org archive.org
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.

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89. Jan 2022
90. www.nytimes.com www.nytimes.com
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.

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91. www.nytimes.com www.nytimes.com
1. https://www.nytimes.com/2022/01/28/nyregion/steven-strogatz-sundays.html

Nice to see the normalization of math and a mathematician without any math shaming in sight.

I'm more curious to hear about his Mondays or Tuesdays...

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92. xkcd.com xkcd.com

## graph thinking

• intuitive
• speed, agility

; 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

#### URL

94. Dec 2021
95. Local file Local file
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

#### Annotators

96. docdrop.org docdrop.org
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")

#### URL

97. www.quantamagazine.org www.quantamagazine.org
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.

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98. Nov 2021
99. www.animatedknots.com www.animatedknots.com
1. https://www.animatedknots.com/

This looks awesomely cool.

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100. www.theatlantic.com www.theatlantic.com
1. <small><cite class='h-cite via'> <span class='p-author h-card'>Yusufa</span> in Improving how universities teach science (<time class='dt-published'>11/04/2021 13:26:34</time>)</cite></small>

#### URL

101. Oct 2021
102. www.heise.de www.heise.de
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?

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103. fermatslibrary.com fermatslibrary.com
1. https://fermatslibrary.com/s/perfect-numbers

Apparently Terry Tao's first paper

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104. www.brainpickings.org www.brainpickings.org
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.

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105. www.quantamagazine.org www.quantamagazine.org
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?

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106. Sep 2021
107. www.springer.com www.springer.com
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.

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108. fs.blog fs.blog
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!

#### URL

109. www.maa.org www.maa.org
1. https://www.maa.org/external_archive/devlin/devlin_03_08.html

<small><cite class='h-cite via'> <span class='p-author h-card'>FARNAM STREET </span> in Why Math Class Is Boring—and What to Do About It (<time class='dt-published'>09/01/2021 09:22:50</time>)</cite></small>

#### URL

110. Aug 2021
111. www.nature.com www.nature.com
1. Murphy, C., Laurence, E., & Allard, A. (2021). Deep learning of contagion dynamics on complex networks. Nature Communications, 12(1), 4720. https://doi.org/10.1038/s41467-021-24732-2

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112. Jul 2021
113. www.skolverket.se www.skolverket.se
1. Flertusenårig historia

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114. arxiv.org arxiv.org

#### URL

115. www.scotthyoung.com www.scotthyoung.com
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.

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116. Jun 2021
117. newclasses.nyu.edu newclasses.nyu.edu
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.

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118. math.stackexchange.com math.stackexchange.com
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).

#### URL

119. math.stackexchange.com math.stackexchange.com
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.