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.

• for: unreasonable effectiveness of mathematics, emptiness, emptiness - unreasonable effectiveness of mathematics

• comment

  • everything is an expression of emptiness

https://math.ucr.edu/home/baez/act_course/

https://www.uclaextension.edu/sciences-math/math-statistics/course/topics-ring-theory-and-modules-math-900

category theory + zettelkasten... hmmm... feels a bit like Leibniz chasing a universal language

what is a zettelkasten monad?

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....

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!

https://hub.jhu.edu/magazine/2021/winter/emily-riehl-category-theory/

original source?

creative mathematics versus computational or calculational mathematics...

we need more of the creative in early education

partial quote from Emily Riehl

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

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?

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

via Tagged Cooltech: Cree Dictionary of Mathematical Terms with Visual Examples by Adam Levine

https://kleinbottle.com/

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.

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?)

To Solve the Rubik’s Cube, You Have to Understand the Amazing Math Inside<br /> by Dave Linkletter

suggested by Matt Maldre's annotations

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.

AMS Open Math Notes

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.

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

Zettelkasten workflow<br /> by pqnelson

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

https://math.stackexchange.com/questions/160039/contradiction-any-symbol-for

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.

https://ncase.me/polygons/

An interesting interactive model for segregation here. See also https://www.bloomberg.com/news/articles/2014-12-10/an-immersive-game-shows-how-easily-segregation-arises-and-how-we-might-fix-it for press coverage.

reply to u/phil98f at https://www.reddit.com/r/Zettelkasten/comments/10dbza7/equations_and_formulas_in_mathematics_physics_and/

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.

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

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.

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

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!

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?

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

https://www.reddit.com/r/Zettelkasten/comments/yg2g8l/a_thought_experiment_what_if_luhmann_had_been_a/

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).

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.

link to: - Maggie Delano's reading framing: https://hypothes.is/a/4xBvpE2TEe2ZmWfoCX_HyQ

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.

https://www.uclaextension.edu/sciences-math/math-statistics/course/theory-and-applications-continued-fractions-math-x-45150

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.

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

https://www.science.org/doi/10.1126/science.145.3631.478.b

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.

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)

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.

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.

http://www.tricki.org/

archive copy: https://web.archive.org/web/20220326200254/http://www.tricki.org/

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

https://www.youtube.com/watch?v=s6OmqXCsYt8

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

see also: https://en.wikipedia.org/wiki/Soroban

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

https://www.aeaweb.org/articles?id=10.1257/app.3.3.158

https://algebra.org/wp/

https://www.nytimes.com/2001/01/07/education/algebra-project-bob-moses-empowers-students.html

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?

https://www.nytimes.com/2001/01/07/education/algebra-project-bob-moses-empowers-students.html

https://www.theescalanteprogram.org/

https://thecalculusproject.org/

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

https://windowsontheory.org/2022/04/27/a-personal-faq-on-the-math-education-controversies/

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.

<small><cite class='h-cite via'>ᔥ <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>

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

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

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?

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

https://www.sfchronicle.com/bayarea/article/California-math-wars-get-ugly-Accusations-of-17060072.php

clickbait mostly

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?

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"

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.

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.

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.

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

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

I particularly like this diagram:

https://linear.axler.net/LADRvideos.html

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

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.

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.

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?

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

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

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.

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

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.

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.

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.

Frege, Hankel, and Formalism in the Foundations discusses this text in relation to Frege's Grundlagen

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;

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.

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...

https://xkcd.com/2566/

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

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

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")

Morava K-theory is an invariant.

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

Morse theory can be used to study critical points.

A relatively simple definition of homology and what it is.

https://www.animatedknots.com/

This looks awesomely cool.

<small><cite class='h-cite via'>ᔥ <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>

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?

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

Apparently Terry Tao's first paper

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

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?

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.

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

Paul Lockhart in Lockhart's Lament

---Paul Lockhart

This is the primary problem with mathematics education!

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

<small><cite class='h-cite via'>ᔥ <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>

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

Flertusenårig historia

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.

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.

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).

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

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.==

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