32 Matching Annotations
  1. Aug 2023
    1. "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.

  2. Jul 2023
    1. I'm using LaTeX to create my Zettel notes. .t3_158gy35._2FCtq-QzlfuN-SwVMUZMM3 { --postTitle-VisitedLinkColor: #9b9b9b; --postTitleLink-VisitedLinkColor: #9b9b9b; --postBodyLink-VisitedLinkColor: #989898; }

      reply to u/AndreSanch at https://www.reddit.com/r/Zettelkasten/comments/158gy35/im_using_latex_to_create_my_zettel_notes/

      This sort of thing has certainly been done before by many. Be careful of going overboard.

      If you don't already have a list of most of the common LaTeX math symbols, here's a good starter list, but make sure that your assigned meaning to them from an argumentation perspective is either "standard" or you've written it down for later use/memory. (There's nothing worse than a 10 year old note whose symbols you no longer remember.)

      If you haven't done a course in philosophy or logic (something along the lines of Elements of Logic), then that may also help you in terms of many of the common uses/meanings, though there are a variety of meanings to various symbols through time, so take care.

      Scribes and scholars over time have used a variety of symbols and annotations to mean various things, some of which were standardized in various contexts. For more on this take a look at some of Evina Stein's work and research on historic texts. Some of this might include:

      Steinová, Evina. “Nota and Require. The Oldest Western Annotation Symbols and Their Dissemination in the Early Middle Ages.” Scribes and the Presentation of Texts (from Antiquity to c. 1550). Proceedings of the 20th Colloquium of the Comité International de Paléographie Latine, 2021, 473–89. https://doi.org/10.1484/M.BIB-EB.5.124987.<br /> ———. Notam Superponere Studui: The Use of Annotation Symbols in the Early Middle Ages. Brepols, 2019.<br /> Steinova, Evina. “Technical Signs in Early Medieval Manuscripts Copied in Irish Minuscule.” In The Annotated Book in the Early Middle Ages: Practices of Reading and Writing, edited by M. J. Teeuwen and I. Van Renswoude, 37–85. Brepols, 2017.

      For those interested in scratching the surface of some possibilities and history, I might recommend:

      Scheinerman, Edward R. Mathematical Notation: A Guide for Engineers and Scientists. CreateSpace, 2011.

      Your note about Forte, while cute and clever doesn't necessarily mean that he's an old man, however, so take care about your propositions and what you draw from them or else your system won't hold up for long.

  3. Apr 2023
  4. Mar 2023
    1. 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.

  5. Feb 2023
    1. One of the problems in approaching quantum gravity is the choice for how to best represent it mathematically. Most of quantum mechanics is algebraic in nature but gravity has a geometry component which is important. (restatement)

      This is similar to the early 20th century problem of how to best represent quantum mechanics: as differential equations or using group theory/Lie algebras?

      This prompts the question: what other potential representations might also work?

      Could it be better understood/represented using Algebraic geometry or algebraic topology as perspectives?

      [handwritten notes from 2023-02-02]

  6. Jan 2023
    1. Woit does provide problems, but they are all at the back of the book. It would have been better to see them between chapters. That provides a natural break in the material and gives the student a quick check on his understanding.

      Homework problems are pedagogical devices and many (most) authors place them in the text near where they would be profitably be done. They also provide a useful break in the text to prompt more novice students to actually perform them at the end of a section.

      More advanced students, however, should have caught on eventually at the need to work out examples for themselves which are presented in a textbook, but they should also be seeking out additional problems where ever they appear in the text, not to mention seeing out any outside additional problems, making up their own, and exploring any additional questions these pose.

      In mathematics textbooks this working of problems, expanding on them and seeking out new ones is often a large part of what is lurking behind the sometimes nebulous sounding idea of "mathematical sophistication". The rest of that equation typically includes experience with the various methods and means of proofs and some basic background in logic.

    2. Woit does not, for the most part, follow the death march of proposition, lemma, proof. He writes in the style of a theoretical physicist.

      "death march of proposition, lemma, proof"

      This is a bit harsh n'cest pas?

  7. Dec 2022
    1. My freely downloadable Beginning Mathematical Logic is a Study Guide, suggesting introductory readings beginning at sub-Masters level. Take a look at the main introductory suggestions on First-Order Logic, Computability, Set Theory as useful preparation. Tackling mid-level books will help develop your appreciation of mathematical approaches to logic.

      This is a reference to a great book "Beginning Mathematical Logic: A Study Guide [18 Feb 2022]" by Peter Smith on "Teach Yourself Logic A Study Guide (and other Book Notes)". The document itself is called "LogicStudyGuide.pdf".

      It focuses on mathematical logic and can be a gateway into understanding Gödel's incompleteness theorems.

      I found this some time ago when looking for a way to grasp the difference between first-order and second-order logics. I recall enjoying his style of writing and his commentary on the books he refers to. Both recollections still remain true after rereading some of it.

      It both serves as an intro to and recommended reading list for the following: - classical logics - first- & second-order - modal logics - model theory<br /> - non-classical logics - intuitionistic - relevant - free - plural - arithmetic, computability, and incompleteness - set theory (naïve and less naïve) - proof theory - algebras for logic - Boolean - Heyting/pseudo-Boolean - higher-order logics - type theory - homotopy type theory

  8. Nov 2022
  9. Aug 2022
  10. Oct 2021
  11. Sep 2021
  12. Jul 2021
  13. May 2021
  14. Mar 2021
    1. To the consternation of some users, 3.x employed Unicode variable names such as λ, φ, τ and π for a concise representation of mathematical operations. A downside of this approach was that a SyntaxError would occur if you loaded the non-minified D3 using ISO-8859-1 instead of UTF-8. 3.x also used Unicode string literals, such as the SI-prefix µ for 1e-6. 4.0 uses only ASCII variable names and ASCII string literals (see rollup-plugin-ascii), avoiding encoding problems.
  15. Feb 2021
  16. Jul 2020
    1. Most of Algol's "special" characters (⊂, ≡, ␣, ×, ÷, ≤, ≥, ≠, ¬, ⊃, ≡, ∨, ∧, →, ↓, ↑, ⌊, ⌈, ⎩, ⎧, ⊥, ⏨, ¢, ○ and □) can be found on the IBM 2741 keyboard with the APL "golf-ball" print head inserted; these became available in the mid-1960s while ALGOL 68 was being drafted. These characters are also part of the Unicode standard and most of them are available in several popular fonts.
  17. Jun 2020
    1. Kucharski, A. J., Klepac, P., Conlan, A. J. K., Kissler, S. M., Tang, M. L., Fry, H., Gog, J. R., Edmunds, W. J., Emery, J. C., Medley, G., Munday, J. D., Russell, T. W., Leclerc, Q. J., Diamond, C., Procter, S. R., Gimma, A., Sun, F. Y., Gibbs, H. P., Rosello, A., … Simons, D. (2020). Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: A mathematical modelling study. The Lancet Infectious Diseases, 0(0). https://doi.org/10.1016/S1473-3099(20)30457-6

  18. Jan 2014
    1. Once a searchable atlas has been constructed there are fundamentally two approaches that can be used to analyze the data: one visual, the other mathematical.