 Last 7 days


Gemignani, Michael C. Elementary Topology. 2nd ed. AddisonWesley Series in Mathematics. Reading, MA: AddisonWesley Publishing Company, 1971.

 Aug 2024

www.youtube.com www.youtube.com

It Only Takes Two Weeks by [[The Math Sorcerer]]
Within a particular class versus their peers, a dedicated student can usually catch up to the best students in 2 weeks.


www.youtube.com www.youtube.com

Undergraduate Topology by Kasriel by [[The Internet Sorcerer]]
Undergraduate Topology by Robert Herman Kasriel
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www.youtube.com www.youtube.com

Mary E. Rudin: "Set theory and General Topology" by [[UMMilwaukee Department of Mathematical Sciences]]


www.youtube.com www.youtube.com

Fancy Topology Book by [[The Math Sorcerer]]
1974 text on topology by Murray Eisenberg


www.topologywithouttears.net www.topologywithouttears.net

Topology Without Tears by [[Sidney A. Morris]]


www.youtube.com www.youtube.com

The Best Topology Book For Beginners is Free by [[The Math Sorcerer]]
Topology Without Tears by Sidney A. Morris<br /> https://www.topologywithouttears.net/


www.youtube.com www.youtube.com

Topology: A Short Introduction by [[badbettybooks]]
only the most surface level review here from the perspective of a relatively unsophisticated undergraduate


www.youtube.com www.youtube.com

Best Books for Learning Topology by [[The Math Sorcerer]]
Some heavy math shaming at the start of this video... ugh.
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Maybe a democracy can work, if it's voting system works well enough.


www.reddit.com www.reddit.com

Monopoly is not played on a cartesian plane. It's played on a directed circular graph. Therefore, it is inappropriate to use the Euclidean distance metric to compare the distances between places on the board. We must instead use minimum path lengths. Example: If we used Euclidean distance, then you would have to agree that the distance between, say, Go and Jail is equal to the distance between the Short Line and the Pennsylvania Railroad. Clearly, this is not the intention. In your example, the "nearest railroad" would be the railroad square having the shortest path from wherever you stand. With the game board representing a directed graph, there are no "backwards" paths. Thus, the distance from the pink Chance square to the Reading railroad is not 2. It's 38.

 Jul 2024

arxiv.org arxiv.org

Tao, Terence. “What Is Good Mathematics?,” February 13, 2007. http://arxiv.org/abs/math/0702396.
Variations of this can also be applied to other fields, like history. What makes good history, good historians, good history teachers, etc.?


www.jonmsterling.com www.jonmsterling.com

https://www.jonmsterling.com/index.xml
ᔥRichard in A forest of evergreen notes at 20240602<br /> (accessed:: 20240607 11:14:53)

 Jun 2024

languagelog.ldc.upenn.edu languagelog.ldc.upenn.edu

It's an interesting position and had me rethinking things a bit, but the way I look at it, the actions themselves are negative; it's their boundary conditions which are different. Take for instance embark/disembark. In pseudomathematical terms, I would tend to think they increment or decrement one's embarkedness, with an upper boundary of 1 (aboard), and a lower boundary of 0 (ashore). The nonexistence of values >1 (superaboard) or <0 (antiaboard) shouldn't affect the relative polarity of the actions themselves. I think. Looking through the rest of the list, there's a variety of different boundary conditions. Prove/disprove would range from 1 to 1 (1=proven, 0=asserted but untested, 1=proven false), entangle/disentangle seems to range from 0 to infinity (because you can always be a little more entangled, can't you?), and please/displease is perhaps wholly unbounded (if we imagine that humanity has an infinite capacity for both suffering and joy).


coevolving.com coevolving.com

The formal systems of mathematics are systems in this sense. The parts numbers, variables, and signs like + and =. The rules specify ways of combining three parts to form expressions, and ways of forming expressions from other expressions, and ways of forming true sentences from expressions, and ways of forming true sentences from other true sentences. The combinations of parts, generated by such a system, are the true sentences, hence theorems, of mathematics. Any combination of parts which is not formed according to the rules is either meaningless or false


writingslowly.com writingslowly.com

A forest of evergreen notes by Richard
See also comments at crossposting at https://www.reddit.com/r/Zettelkasten/comments/1d69inw/a_forest_of_evergreen_notes/

 Apr 2024

www.reddit.com www.reddit.com

scihuy 0 points1 point2 points 2 hours ago (1 child)Hi, Can you point out any articles on notetaking in the sciences as opposed to history or social sciences? Any pointers would be very helpful
reply to u/scihuy at https://www.reddit.com/r/Zettelkasten/comments/1c2b2d6/note_taking_in_the_past/kzcg3qa/
I posed your question to my own card index:
Generally scientists haven't spent the time to talk about their methods the way those in the social sciences and humanities are apt to do. This being said, their methods are unsurprisingly all the same.
If you want to look up examples, you can delve into the nachlass (digitized or not) of most of the famous scientists and mathematicians out there to verify this. Ramon Llull certainly wrote, but broadly memorized all of his work; Newton had his wastebooks; Leibnitz used Thomas Harrison's Ark of Studies cabinet; Carl Linnaeus "invented" index cards for his work (search for the work of Staffan MüllerWille and Isabelle Charmantier); Erasmus Darwin and Charles Darwin both used commonplace books; physicist Mario Bunge had a significant zettelkasten practice; Richard Feynman used notebooks; engineer Ross Ashby used a combination of notebooks which he indexed using a card index.
For historical reasons, most used a commonplace book method in which they indexed against keywords rather than Luhmann's variation, but broadly the results are the same either way.
Computer scientist Gerald Weinberg is one of the few I'm aware of within the sciences who's written a note taking manual, but again, his method is broadly the same as that described by other writers for centuries:
Weinberg, Gerald M. Weinberg on Writing: The Fieldstone Method. New York, N.Y: Dorset House, 2005.
I identify as both a mathematician and an engineer, and I have a paperbased zettelkasten for these areas, primarily as I prefer writing out equations versus attempting to write everything out as LaTeX. I'm sure others here could add their experiences as well. I've previously written about zettelkasten from the framing of set theory, topology, dense sets, and have even touched on it with respect to the ideas of equivalence classes and category theory, though I haven't published much in depth here as most don't have the mathematical sophistication to appreciate the structures and analogies.


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classify
Ideally, when classifying, one should draw up classes which are mutually exclusive so as to better differentiate the items under consideration. This is not always possible and there are some cases where having overlap my be beneficial in one's work.

 Feb 2024

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He had helped Murray with the very firstentry in the Dictionary – A: not only the sound A, ‘the lowbackwide vowelformed with the widest opening of the jaws, pharynx, and lips’, but also themusical sense of A, ‘the 6th note of the diatonic scale of C major’, and finallythe algebraic sense of A, ‘as in a, b, c, early letters of the alphabet used toexpress known quantities, as x, y, z are to express the unknown’. Ellis washappy to see these and other results of his work on the printed page, includingthe words air, alert, algebra.
He here is A. J. Ellis

 Jan 2024

www.mccoyskecatalogs.com www.mccoyskecatalogs.com

www.mccoyskecatalogs.com www.mccoyskecatalogs.com


Kells, Lyman M., Willis F. Kern, and James R. Bland. K+E Slide Rules: A Self Instruction Manual. New York: Keuffel & Esser Co., 1955.


mathstodon.xyz mathstodon.xyzMastodon1

Christian LawsonPerfect @christianp@mathstodon.xyz@liseo there are lots of ways of representing colours numerically. The most basic way that computers use is to use a number between 0 and 255 for each of the red, green and blue components, called RGB encoding. The problem with that is that colours that look close to each other don't necessarily have close RGB values. There are other colour spaces which try to get closer to the ideal of having similar colours close together. Oklab, which I use in this tool, is currently the best for that.
https://mastodon.social/@christianp@mathstodon.xyz/111759984202211741
Is there a way to mathematically encode colors, similar to RGB perhaps, such that the colors in nearby neighborhoods all have values close to each other?


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Silverman, Joseph H., and John T. Tate. Rational Points on Elliptic Curves. 2nd ed. Undergraduate Texts in Mathematics. 1992. Reprint, New York: Springer, 2015.
urn:xpdf:e59b1cd55350541e359fbb11896edd0c
annotations at: https://jonudell.info/h/facet/?user=chrisaldrich&max=100&exactTagSearch=true&expanded=true&url=urn%3Axpdf%3Ae59b1cd55350541e359fbb11896edd0c

 Dec 2023

www.youtube.com www.youtube.com

How to fold and cut a Christmas star<br /> Christian LawsonPerfect 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.

 Nov 2023

www.browndailyherald.com www.browndailyherald.com


www.reddit.com www.reddit.com

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
reply to u/Apprehensive_Net5630 and u/marco89lcdm at https://www.reddit.com/r/antinet/comments/17m7ggz/comment/k83bou9/?utm_source=reddit&utm_medium=web2x&context=3
Don't sweat the difference as there is a onetoone 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 makework exercise, you'd end up with the same exact thing with the same ordering in the end.

 Oct 2023


Applying zettelkasten for math heavy subjects. .t3_17bqztm._2FCtqQzlfuNSwVMUZMM3 { postTitleVisitedLinkColor: #9b9b9b; postTitleLinkVisitedLinkColor: #9b9b9b; postBodyLinkVisitedLinkColor: #989898; }
reply to u/acosmicjoke at https://www.reddit.com/r/Zettelkasten/comments/17bqztm/applying_zettelkasten_for_math_heavy_subjects/
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 allin 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 afterthefact. 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.

 Sep 2023

delong.typepad.com delong.typepad.com

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.

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.



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



www.uclaextension.edu www.uclaextension.edu

 Aug 2023


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....


hub.jhu.edu hub.jhu.edu

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 60foot 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!


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

I think mathematicians do math in part because we think it's beautiful."

"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

"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.

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?"

But then, so are numbers, for all their illusion of concrete specificity and precision.
Too many nonmathematicians 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 nonmathematical 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?
Tags
 writing
 quotes
 analogies
 category theory
 beauty
 ontologies
 Emily Riehl
 read
 exploration
 mathematics why
 John Conway
 metaphors
 Dale Keiger
 mathematics pedagogy
 mathematics
 creative mathematics
 mathematical definitions
 patterns
 Emily Rieh
 abstract mathematics
 Jordan curve theorem
 topology
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 Jun 2023

opentextbooks.uregina.ca opentextbooks.uregina.ca

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

 May 2023

kleinbottle.com kleinbottle.com

www.amazon.com www.amazon.com

https://www.amazon.com/VaughnCubeMultiplicationPetersons/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.


oeis.org oeis.org

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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 Apr 2023

www.popularmechanics.com www.popularmechanics.com

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

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 1620 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.

 Mar 2023

www.ams.org www.ams.org

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 InquiryBased 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.


mathbabe.org mathbabe.org

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

 Feb 2023

texnicalstuff.blogspot.com texnicalstuff.blogspot.com

Zettelkasten workflow<br /> by pqnelson


en.wikipedia.org en.wikipedia.org

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)?



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


math.stackexchange.com math.stackexchange.com
 Jan 2023


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.


ncase.me ncase.me

An interesting interactive model for segregation here. See also https://www.bloomberg.com/news/articles/20141210/animmersivegameshowshoweasilysegregationarisesandhowwemightfixit for press coverage.


www.reddit.com www.reddit.com

Equations and Formulas in Mathematics, Physics and Chemistry Using a Digital Zettelkasten .t3_10dbza7._2FCtqQzlfuNSwVMUZMM3 { postTitleVisitedLinkColor: #9b9b9b; postTitleLinkVisitedLinkColor: #9b9b9b; postBodyLinkVisitedLinkColor: #989898; } zkstructureHow do you handle relations between mathematical, physical or chemical formulas in a digital Zettelkasten? Since I would like to use a futureproof 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?
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 prenumbered 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 zettelkastenlike 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 GayLussac'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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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.


www.cambridge.org www.cambridge.org

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.

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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 history of mathematics
 ordinals
 cardinals
 anthropology
 abstraction
 place values
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McCoy, Neal Henry. The Theory of Rings. 1964. Reprint, The Bronx, New York: Chelsea Publishing Company, 1973.

 Dec 2022

andreiwonge.github.io andreiwonge.github.io

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!

 Nov 2022

learnapsoutheast2prodfleet01xythos.content.blackboardcdn.com learnapsoutheast2prodfleet01xythos.content.blackboardcdn.comview1

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?


watermark.silverchair.com watermark.silverchair.com

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

 Oct 2022

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).

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



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
<script async src="https://platform.twitter.com/widgets.js" charset="utf8"></script>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, 2022Bianca 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


www.sub.unihamburg.de www.sub.unihamburg.de

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.


physicstoday.scitation.org physicstoday.scitation.org

The nature of physics problemsolvingBelow are 29 sets of questions that students and physicists need to ask themselves during the research process. The answers at each step allow them to make the 29 decisions needed to solve a physics problem. (Adapted from reference 33. A. M. Price et al., CBE—Life Sci. Edu. 20, ar43 (2021). https://doi.org/10.1187/cbe.20120276.)A. Selection and planning1. What is important in the field? Where is the field heading? Are there advances in the field that open new possibilities?2. Are there opportunities that fit the physicist’s expertise? Are there gaps in the field that need solving or opportunities to challenge the status quo and question assumptions in the field? Given experts’ capabilities, are there opportunities particularly accessible to them?3. What are the goals, design criteria, or requirements of the problem solution? What is the scope of the problem? What will be the criteria on which the solution is evaluated?4. What are the important underlying features or concepts that apply? Which available information is relevant to solving the problem and why? To better identify the important information, create a suitable representation of core ideas.5. Which predictive frameworks should be used? Decide on the appropriate level of mechanism and structure that the framework needs to be most useful for the problem at hand.6. How can the problem be narrowed? Formulate specific questions and hypotheses to make the problem more tractable.7. What are related problems or work that have been seen before? What aspects of their problemsolving process and solutions might be useful?8. What are some potential solutions? (This decision is based on experience and the results of decisions 3 and 4.)9. Is the problem plausibly solvable? Is the solution worth pursuing given the difficulties, constraints, risks, and uncertainties?Decisions 10–15 establish the specifics needed to solve the problem.10. What approximations or simplifications are appropriate?11. How can the research problem be decomposed into subproblems? Subproblems are independently solvable pieces with their own subgoals.12. Which areas of a problem are particularly difficult or uncertain in the solving process? What are acceptable levels of uncertainty with which to proceed at various stages?13. What information is needed to solve the problem? What approach will be sufficient to test and distinguish between potential solutions?14. Which among the many competing considerations should be prioritized? Considerations could include the following: What are the most important or most difficult? What are the time, materials, and cost constraints?15. How can necessary information be obtained? Options include designing and conducting experiments, making observations, talking to experts, consulting the literature, performing calculations, building models, and using simulations. Plans also involve setting milestones and metrics for evaluating progress and considering possible alternative outcomes and paths that may arise during the problemsolving process.B. Analysis and conclusions16. Which calculations and data analysis should be done? How should they be carried out?17. What is the best way to represent and organize available information to provide clarity and insights?18. Is information valid, reliable, and believable? Is the interpretation unbiased?19. How does information compare with predictions? As new information is collected, how does it compare with expected results based on the predictive framework?20. If a result is different from expected, how should one follow up? Does a potential anomaly fit within the acceptable range of predictive frameworks, given their limitations and underlying assumptions and approximations?21. What are appropriate, justifiable conclusions based on the data?22. What is the best solution from the candidate solutions? To narrow down the list, decide which of those solutions are consistent with all available information, and which can be rejected. Determine what refinements need to be made to the candidate solutions. For this decision, which should be made repeatedly throughout the problemsolving process, the candidate list need not be narrowed down to a single solution.23. Are previous decisions about simplifications and predictive frameworks still appropriate in light of new information? Does the chosen predictive framework need to be modified?24. Is the physicist’s relevant knowledge and the current information they have sufficient? Is more information needed, and if so, what is it? Does some information need to be verified?25. How well is the problemsolving approach working? Does it need to be modified? A physicist should reflect on their strategy by evaluating progress toward the solution and possibly revising their goals.26. How good is the chosen solution? After selecting one from the candidate solutions and reflecting on it, does it make sense and pass disciplinespecific tests for solutions to the problem? How might it fail?Decisions 27–29 are about the significance of the work and how to communicate the results.27. What are the broader implications of the results? Over what range of contexts does the solution apply? What outstanding problems in the field might it solve? What novel predictions can it enable? How and why might the solution be seen as interesting to a broader community?28. Who is the audience for the work? What are the audience’s important characteristics?29. What is the best way to present the work to have it understood and to have its correctness and importance appreciated? How can a compelling story be made of the work?

 Sep 2022

www.uclaextension.edu www.uclaextension.edu


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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 20220927 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.


www.science.org www.science.org

twitter.com twitter.com

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.

 Jul 2022

docdrop.org docdrop.org

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/timestamped 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 preexisting 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/YMZhofbEeyvXyf1gjXZCg (Vienna library catalogue system)  https://hypothes.is/a/Jlnn3IfSEey_3uboxHsOA (Vienna conscription numbers)

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.


www.intheknow.com www.intheknow.com

https://news.yahoo.com/mathematiciantiktokgivesexampleinsane192823997.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.


www.tricki.org www.tricki.org

archive copy: https://web.archive.org/web/20220326200254/http://www.tricki.org/
a Wikistyle site with a large store of useful mathematical problemsolving techniques.

 Jun 2022

www.youtube.com www.youtube.com

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."
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www.aeaweb.org www.aeaweb.org

algebra.org algebra.org

www.nytimes.com www.nytimes.com


The Algebra Project was born.At its core, the project is a fivestep 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?



www.theescalanteprogram.org www.theescalanteprogram.org

thecalculusproject.org thecalculusproject.org

windowsontheory.org windowsontheory.org

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


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.


sites.google.com sites.google.com

<small><cite class='hcite via'>ᔥ <span class='pauthor hcard'>Boaz Barak</span> in Brian Conrad takes down the CMF – Windows On Theory (<time class='dtpublished'>06/01/2022 21:35:06</time>)</cite></small>


scottaaronson.blog scottaaronson.blog

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

 May 2022

slate.com slate.com

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.


minimalistquotes.com minimalistquotes.com

Curve: The loveliest distance between two points. Mae West

 Apr 2022

en.wikipedia.org en.wikipedia.org
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In the course of teaching hundredsof firstyear 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?
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twitter.com twitter.com

Adam Kucharski. (2020, December 13). I’ve turned down a lot of COVIDrelated 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


www.sfchronicle.com www.sfchronicle.com

 Mar 2022

www.linkedin.com www.linkedin.com

https://www.linkedin.com/pulse/incorrectuseinformationtheoryrafaelgarc%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?


lithub.com lithub.com

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"



To signify that an angle is acute, Jeffreys taught them, “make PacMan 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 PacMan 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.

Washington State mathteacher Brendan Jeffreys turned to gesture as a way of easing the mental loadcarried by his students, many of whom come from lowincome 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.

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.

 Feb 2022

www.dynamicmath.xyz www.dynamicmath.xyz

thinkzone.wlonk.com thinkzone.wlonk.com

https://thinkzone.wlonk.com/Numbers/NumberSets.htm
A relatively clear explanation of the types of numbers and their properties.
I particularly like this diagram:


linear.axler.net linear.axler.net

https://linear.axler.net/LADRvideos.html
I've seen Gilbert Strang's excellent lecture series. I'm curious how these rank in relation.


www.theguardian.com www.theguardian.com

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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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 handwaving 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.

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 counterexamples. 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?
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www.latimes.com www.latimes.com

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


archive.org archive.org

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,

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.

man sich zu der gegebenen Reihe von Ob jecten eine inverse hinzudenkt
one adds an inverse in thought to the given series of objects

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 nonreal, imaginary number, as a "false" one.

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.

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, thoughtthings, 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

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;

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.

 Jan 2022

www.nytimes.com www.nytimes.com

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.


www.nytimes.com www.nytimes.com