10 ms·

Using spaced repetition systems to see through a piece of mathematics

256 points by uijl 8y ago

ohum 8y ago
There are defined ways of simplifying mathematics, computationally, such as via the CSP-NT computational laws
- yshklarov 8y ago
  I'm not sure what this has to do with the content of the article. Could you elaborate?
PacifyFish 8y ago
I love Michael Nielsen. He's so earnest and thoughtful. His recent forays into spaced repetition have been an interesting glimpse into his thought process. I am a bit of a spaced rep fanatic, but use it to internalize more general "Mental Models" (a la Farnam Street), useful tidbits e.g. how to horizontally align using flexbox, or surface-level concepts like product management frameworks. I love the effort Michael puts into grappling with ideas to strengthen his intuition. This is something I'd like to do more of, but feel that few of the ideas in my day-to-day are sufficiently complex (I don't have any use for linear algebra proofs, for example). Anyways, that's my word salad on Michael Nielsen and spaced repetition.
- mxstbr 8y ago
  How do you form flashcards out of mental models you encounter and want to remember and use? Could you post an example?
  - PacifyFish 8y ago
    Sure thing. I’ll typically make a few cards at first trying to test different aspects of the MM (general concept, example applications, given definition identify the MM, quiz any internal associations I have with it like if I learned it in a specific context, etc.) then I can delete ones that aren’t useful and/or add others later. Let’s use the mental model of inversion as an example. I found this one in a Farnam Street blog post about Charlie Munger. Here are some cards I might create: Front - what is the principle of inversion? Back - when you want to try to maximize something, instead try minimizing its converse. Or vice-versa. Front - what mental model might help if you’ve unsuccessfully tried implementing programs to increase innovation in your company? Back - inversion. Rather than thinking of ways to increase innovation, can you instead think of things that are decreasing innovation and eliminate those? Front - what’s another way to think about reducing time spent on work tasks? Back - invert the problem. Try to increase time saved on nonessentials (e.g. laundry service, meal prep, outsourcing) Front - what’s it called when you work backwards through a problem you’ve already tried to work through forwards? Back - inversion Front - what would Charlie Munger ask you if you came to him with a tough optimization problem? Back - “have you tried inverting it?” This is a bit contrived, but I hope this gives you a sense of how I think about creating cards for mental models.
tmyklebu 8y ago
> The converse is also true (and is much easier to prove, so we won’t be concerned with it): a diagonalizable matrix is always normal. It is actually more difficult to prove. [2, 1; 0, 1] is diagonalisable (via [1, -sqrt(2)/2; 0, sqrt(2)/2]) but not normal.
- michael_nielsen 8y ago
  I mean diagonalizable by a unitary matrix. Fixed.
  - Escapado 8y ago
    I never knew you were on HN but I'll use this opportunity hoping that you'll read this. Recently I finished my masters thesis in quantum machine learning (I'm a physicist) and I think without the amazing book you wrote with Isaac Chuang I would not have been able to do it. So this is meant as a huge thank you! I tried other QC books but I found yours the most insightful and approachable. On another note I also enjoy your essays on SRS and it encouraged me to start learning Japanese using it.
  - fhars 8y ago
    You might also want to mention early that when you talk about the length of a row or column, you do not mean its (thanks, Apple, for autocorrecting that to it’s) length (i.e. number of elements), but its L2 norm. That just caused me to lose half an hour...
branweb 8y ago
As a long-time SRS user, I've really enjoyed Michael's recent essays on the subject though I haven't had much luck using SRS to learn structured, abstract things, despite several attempts. These days I use it mostly for things that are naturally atomic--vocabulary words, quotations, etc. It seems he and I approach SRS very differently. I see it as a place to drop bits of information I want to remember, and once the info is in there, I no longer have to think about it, outsourcing the effort of remembering to the algorithm. He seems to use it as a tool to mentally digest a subject, constantly revising his understanding and readjusting his cards. That process isn't foreign to me, but I prefer to use something besides SRS for it. If I had to constantly revise my cards, that would seem to defeat the purpose of using SRS anyway. Very interesting to get a different perspective on this though, so I'll have to think more about his approach. I'm also interesting in the creation details of his math-cards. Does he use LaTex? Maybe he goes into it in one of his earlier posts...
- michael_nielsen 8y ago
  Glad you enjoyed the essays. On LaTeX - I sometimes cut-and-paste screenshots, but usually just write raw LaTeX or pseudo-LaTeX (i.e., something LaTeX-like that wouldn't compile, but is easier to read). It'd be much, much better to use some kind of plugin that turned it into images, but I've been too lazy to try. This has, incidentally, made many of my matrix-based Anki cards rather ugly. I'll probably update a bunch of answers with cut-and-paste from this essay. Not to put too fine a point on it, but it's pretty bad I'm using raw LaTeX in this way - I'm sure it hurts my understanding! On the other hand, I've written so much LaTeX over the years that reading it seems really natural.
  - yorwba 8y ago
    You wouldn't even need la plugin. Anki supports [latex][/latex] out of the box, provided you install LaTeX in a way that Anki can find it. https://apps.ankiweb.net/docs/manual.html#latex-support https://apps.ankiweb.net/docs/manual.html#latex-support
    - sbmassey 8y ago
deleted 8y ago
[deleted]
glaberficken 8y ago
>"You might suppose a great mathematician such as Kolmogorov would be writing about some very complicated piece of mathematics, but his subject was the humble equals sign: what made it a good piece of notation, and what its deficiencies were. Kolmogorov discussed this in loving detail, and made many beautiful points along the way, e.g., that the invention of the equals sign helped make possible notions such as equations (and algebraic manipulations of equations). (...) (I found Kolmogorov’s essay in my University library as a teenager. I’ve unsuccessfully tried to track it down several times in the intervening years. If anyone can identify the essay, I’d appreciate it. I’ve put enough effort into tracking it down that I must admit I’ve sometimes wondered if I imagined the essay. If so, I have no idea where the above story comes from.)" Can anyone find the essay mentioned above? Maybe a Russian speaker could track down the original (as presumably what Michael Nielsen is referencing here is an English translation).
- e12e 8y ago
  Judging from http://kolmogorov.com/ http://kolmogorov.com/ there seems to be a fair chance the original is in French. But I can't find anything that looks like the essay in question... Might be a bit easier if the name of the essay was known...
- hopler 8y ago
  A great article on a relatively similar topic, equality and isomorphism: http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
- drsopp 8y ago
  Perhaps it is the article On Sense and Reference by Gottlob Frege? http://www.scu.edu.tw/philos/98class/Peng/05.pdf http://www.scu.edu.tw/philos/98class/Peng/05.pdf
- jacobolus 8y ago
  There are a bunch of Kolmogorov essays that might be relevant, judging by their titles. https://www.turpion.org/php/reference.phtml?journal_id=rm&paper_id=1991&volume=43&issue=6&type=xrf https://www.turpion.org/php/reference.phtml?journal_id=rm&pa... Perhaps "On the school definition of identity", Matematika v shkole 1966 * * * Maybe a Russian speaker wants to go search? There doesn’t seem to be an English translation of most of Kolmogorov’s essays online. https://www.google.com/search?tbm=bks&q=Математика+в+школе+Колмого́ров+1966 https://www.google.com/search?tbm=bks&q=Математика+в+школе+К...
tankenmate 8y ago
This article seems the describe what it means to grok something; and that maybe one (easy?) way to reach that is via spaced repetition systems.
tomxor 8y ago
> You might suppose a great mathematician such as Kolmogorov would be writing about some very complicated piece of mathematics, but his subject was the humble equals sign: I've only just discovered Kolmogorov.. not being particularly mathematically talented myself I'm happy to find his work conceptually fascinating from an amateur's perspective, I hope the essay is found.
burtonator 8y ago
You guys might like this: https://getpolarized.io/ https://getpolarized.io/ Polar is an SRS based document management platform and personal knowledge repository I've been working on. It syncs with anki and allows you to create and read flashcards directly in the reader itself. Supports text annotations, highlights, comments, keeps track of where you're reading in documents, supports tags, rich formatting of notes, etc. It's also Open Source and supports cloud sync across your desktop machines.
- verletx64 8y ago
  Looks pretty good for building a first pass of more atomic Anki notes. I've found that over time, I need to make multiple editing passes on Anki Cards, to try to boil them down into something actually recallable. How has your experience been?
- harperlee 8y ago
  Wow, thanks a lot for this. I was considering building something similar but barebones, and now I can use this and hopefully contribute! Particularly happy about offline first!
  - burtonator 8y ago
    Thanks.. I think one of the cool things is it supports HTML capture so you can take pages you see on Hacker News and store them for posterity!
    - harperlee 8y ago
      Sure! I was using evernote and later pocket for this but they ended up as a bin for past read content. With polar that can further go to anki and then to my memory :)
- IanCal 8y ago
  I've not really used this beyond just installing this now, but I'd encourage anyone that sounds a bit interested from this comment to go checkout the website. If the features it talks about are smooth, it looks absolutely wonderful.
Schiphol 8y ago
I wonder if that Kolmogorov essay on the equal sign might not rather be Frege's Begriffsschrift (the Wikipedia page links to a facsimile version: https://en.m.wikipedia.org/wiki/Begriffsschrift https://en.m.wikipedia.org/wiki/Begriffsschrift).
philip-b 8y ago
Hello Michael, I've read your 2 articles about anki. I am an avid user of anki myself and I use it for math as well. In your 2 articles you tell about using it to understand and retain memory of an Alphago paper and to do the same for the theorem about orthogonal diagonalizability of normal matrices. I wonder, how do you organize anki cards and in which order do you study them? Personally I put all cards in one deck and anki shows them in kinda random order, so I might get a question about convex optimization, then a question on numerical linear algebra, then a question on some dance moves. I take it you do it in a different way? Because you create many small cards for a topic I think you would spend too much time context switching if you did random order like me. For this reason I make larger cards, e.g. "Prove that for any linear operator on a finite dimensional complex vector space there is a basis such that the operator has upper triangular matrix in it"; and when preparing for an exam in institute I might even create a card "Definition, existence, uniqueness, and computational complexity of SVD". Also I think creating small cards might be bad for chunking, i.e. you won't get large chunks of all the related knowledge about a theorem and instead you will have small chunks - a chunk per card. Another question is how do you add all this information to anki? Anki obviously sucks as an exploratory medium. I often find that even clearing up my paper notes, taking photos of them, cutting them, and putting them into anki takes a lot of time; typing it up in LaTeX is even longer. Any tips or insights here?
faitswulff 8y ago
Aside, but is there an SRS that will send me flashcards as notifications on my phone? I'd rather not have to set aside large chunks of time to memorize things, I'd rather refresh my memory throughout the day.
qwerty456127 8y ago
Does anybody know a good math course covering all the high school and college subjects implemented in a collection of flash cards? I have no idea what do "normal matrix", "diagonalizable by" and "unitary matrix" mean (of course I can look these particular up) mean so far but I feel like I would like to try learning up to this level (including what I've learnt in the college and completely forgotten right after the exams) this way.
- playdohscave 8y ago
  pretty much any good linear algebra course should do the trick
- colechristensen 8y ago
  Those are all linear algebra concepts, I would suggest Gilbert Strang's series of lectures which can be found on YouTube. They should be fairly accessible without prerequisites.
- unixhero 8y ago
  KhanAcademy
- barger 8y ago
  While "all the high school math" is somewhat clear, the "all the college math" should be clarified a bit. Are we talking about math for social studies majors, engineers, physicists, economists or mathematicians? For example, there are books like [0] for engineers which serve as a boot camp of sorts. There are no theorems, proofs or deep math in it. There are many different kinds of engineers, so books like this don't include everything an engineer needs to know. For instance, there's no automata or group theory in [0]. It has a part at the beginning called Foundation Topics which could serve as "all the high school math one needs". In fact, this whole book could probably serve as "all the math an advanced high school student needs". There are books like [1] for physicists. They are meant to introduce physics majors to a wide array of math topics in a relatively pain-free way. This book is much more rigorous and contains a LOT more material than [0]. Basically, a theoretical minimum for a physics major. Economics majors also get somewhat rigorous math load where measure theory features a lot more prominently than it does in other majors (except mathematics proper). List of subjects for math majors vary from place to place, but you'll have much easier time down the road if you master the (rigorous) rudiments of linear algebra (vector spaces), group theory, number theory and real analysis. Of course, knowing more math (say topology, complex analysis, category theory, combinatorics etc) is always good. Before you get started with math for mathematicians, you'll want to learn their jive. A good intro is [2]. It's free and a really nice book. Another really nice book with non-existent pre-reqs is [3]. A pitfall that awaits a lot of people new to math is a concept of "multivariable calculus". This concept is a mess and means everything to everyone. Oftentimes it means surface level discussion of concepts in scalar fields (functions from R^n to R) and a little bit of talk about differential geometry of curves and surfaces (functions from reals and planes to R^n). The treatment is often not rigorous and n is limited to 2 and 3. After this laughable bullshit, one is thought ready to jump straight into the rigorous analysis of manifolds, granted they know a bit of real analysis. This is like jumping from 3rd grade straight to 9th grade. Along the way the most important thing missing is the rigorous treatment of vectors fields (say, at the level of Rudin). Some nice books here include [4]. Since diff geo (in particular, that of curves and surfaces) is its own thing entirely and there are a lot of really nice books for that like [5]. Also, note there's a nice new intro to "manifolds and stuff" [6] which is like what's calculus is to analysis. Before I forget, most intro to stats books are written for science majors and are entirely inadequate for math majors, but there are elementary intro to stats books for math majors like [7]. Originally, I wanted to write a more fleshed out and huge comment, but I am running out of time. Good Luck. [0] Engineering Mathematics by Stroud/Booth https://www.amazon.com/Engineering-Mathematics-K-Stroud/dp/0831134704/ref=sr_1_1?s=books&ie=UTF8&qid=1547410342&sr=1-1&keywords=stroud https://www.amazon.com/Engineering-Mathematics-K-Stroud/dp/0... https://www.amazon.com/Advanced-Engineering-Mathematics-Kenneth-Stroud/dp/0831134496/ref=pd_sim_14_1?_encoding=UTF8&pd_rd_i=0831134496&pd_rd_r=8ad90dba-176f-11e9-b24d-05ff6573df22&pd_rd_w=1UxKi&pd_rd_wg=qemWA&pf_rd_p=18bb0b78-4200-49b9-ac91-f141d61a1780&pf_rd_r=QPGTVRJQ7ESXHHQRB34Q&psc=1&refRID=QPGTVRJQ7ESXHHQRB34Q https://www.amazon.com/Advanced-Engineering-Mathematics-Kenn... [1] A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekers https://www.amazon.com/Course-Modern-Mathematical-Physics-Differential/dp/0521829607/ref=sr_1_3?s=books&ie=UTF8&qid=1547413650&sr=1-3&keywords=modern+mathematical+physics https://www.amazon.com/Course-Modern-Mathematical-Physics-Di... [2] Book of Proof by Richard Hammack https://www.people.vcu.edu/~rhammack/BookOfProof/ https://www.people.vcu.edu/~rhammack/BookOfProof/ [3] Linear Algebra: Step by Step by Kuldeep Singh https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/0199654441/ref=sr_1_7?s=books&ie=UTF8&qid=1547413772&sr=1-7&keywords=linear+algebra https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/... [4] An Introduction to Analysis by Wade https://www.amazon.com/Introduction-Analysis-4th-William-Wade/dp/0132296381/ref=sr_1_fkmr0_1?s=books&ie=UTF8&qid=1547413883&sr=1-1-fkmr0&keywords=multivariable+analysis+wade https://www.amazon.com/Introduction-Analysis-4th-William-Wad... [5] Differential Geometry of Curves and Surfaces by Tapp https://www.amazon.com/Differential-Geometry-Surfaces-Undergraduate-Mathematics/dp/3319397982/ref=sr_1_4?s=books&ie=UTF8&qid=1547414043&sr=1-4&keywords=differential+geometry+of+curves+and+surfaces https://www.amazon.com/Differential-Geometry-Surfaces-Underg... [6] A Visual Introduction to Differential Forms and Calculus on Manifolds by Fortney https://www.amazon.com/Visual-Introduction-Differential-Calculus-Manifolds/dp/3319969919/ref=sr_1_1?s=books&ie=UTF8&qid=1547414114&sr=1-1&keywords=fortney+manifolds https://www.amazon.com/Visual-Introduction-Differential-Calc... [7] Statistics for Mathematicians: A Rigorous First Course by Panaretos https://www.amazon.com/Statistics-Mathematicians-Rigorous-Textbooks-Mathematics/dp/3319283391/ref=sr_1_1?s=books&ie=UTF8&qid=1547414255&sr=1-1&keywords=statistics+for+mathematicians https://www.amazon.com/Statistics-Mathematicians-Rigorous-Te...
turadg 8y ago
Anyone wanting to innovate in tools to optimize learning, I’m building a team doing just that and I’d love to chat: turadg@quizlet.com. (Some of my earlier work: http://j.mp/aleahmad-thesis http://j.mp/aleahmad-thesis )
aoki 8y ago
in case anybody's curious, the Einstein letter mentioned in the article is Appendix II of Hadamard's "The Psychology of Invention in the Mathematical Field" (ISBN 0486201074).

256 points by uijl 8y ago

ohum 8y ago

There are defined ways of simplifying mathematics, computationally, such as via the CSP-NT computational laws

yshklarov 8y ago
I'm not sure what this has to do with the content of the article. Could you elaborate?

PacifyFish 8y ago

I love Michael Nielsen. He's so earnest and thoughtful. His recent forays into spaced repetition have been an interesting glimpse into his thought process. I am a bit of a spaced rep fanatic, but use it to internalize more general "Mental Models" (a la Farnam Street), useful tidbits e.g. how to horizontally align using flexbox, or surface-level concepts like product management frameworks. I love the effort Michael puts into grappling with ideas to strengthen his intuition. This is something I'd like to do more of, but feel that few of the ideas in my day-to-day are sufficiently complex (I don't have any use for linear algebra proofs, for example). Anyways, that's my word salad on Michael Nielsen and spaced repetition.

mxstbr 8y ago
How do you form flashcards out of mental models you encounter and want to remember and use? Could you post an example?
- PacifyFish 8y ago
  Sure thing. I’ll typically make a few cards at first trying to test different aspects of the MM (general concept, example applications, given definition identify the MM, quiz any internal associations I have with it like if I learned it in a specific context, etc.) then I can delete ones that aren’t useful and/or add others later. Let’s use the mental model of inversion as an example. I found this one in a Farnam Street blog post about Charlie Munger. Here are some cards I might create: Front - what is the principle of inversion? Back - when you want to try to maximize something, instead try minimizing its converse. Or vice-versa. Front - what mental model might help if you’ve unsuccessfully tried implementing programs to increase innovation in your company? Back - inversion. Rather than thinking of ways to increase innovation, can you instead think of things that are decreasing innovation and eliminate those? Front - what’s another way to think about reducing time spent on work tasks? Back - invert the problem. Try to increase time saved on nonessentials (e.g. laundry service, meal prep, outsourcing) Front - what’s it called when you work backwards through a problem you’ve already tried to work through forwards? Back - inversion Front - what would Charlie Munger ask you if you came to him with a tough optimization problem? Back - “have you tried inverting it?” This is a bit contrived, but I hope this gives you a sense of how I think about creating cards for mental models.

tmyklebu 8y ago

> The converse is also true (and is much easier to prove, so we won’t be concerned with it): a diagonalizable matrix is always normal. It is actually more difficult to prove. [2, 1; 0, 1] is diagonalisable (via [1, -sqrt(2)/2; 0, sqrt(2)/2]) but not normal.

michael_nielsen 8y ago
I mean diagonalizable by a unitary matrix. Fixed.
- Escapado 8y ago
  I never knew you were on HN but I'll use this opportunity hoping that you'll read this. Recently I finished my masters thesis in quantum machine learning (I'm a physicist) and I think without the amazing book you wrote with Isaac Chuang I would not have been able to do it. So this is meant as a huge thank you! I tried other QC books but I found yours the most insightful and approachable. On another note I also enjoy your essays on SRS and it encouraged me to start learning Japanese using it.
- fhars 8y ago
  You might also want to mention early that when you talk about the length of a row or column, you do not mean its (thanks, Apple, for autocorrecting that to it’s) length (i.e. number of elements), but its L2 norm. That just caused me to lose half an hour...

branweb 8y ago

As a long-time SRS user, I've really enjoyed Michael's recent essays on the subject though I haven't had much luck using SRS to learn structured, abstract things, despite several attempts. These days I use it mostly for things that are naturally atomic--vocabulary words, quotations, etc. It seems he and I approach SRS very differently. I see it as a place to drop bits of information I want to remember, and once the info is in there, I no longer have to think about it, outsourcing the effort of remembering to the algorithm. He seems to use it as a tool to mentally digest a subject, constantly revising his understanding and readjusting his cards. That process isn't foreign to me, but I prefer to use something besides SRS for it. If I had to constantly revise my cards, that would seem to defeat the purpose of using SRS anyway. Very interesting to get a different perspective on this though, so I'll have to think more about his approach. I'm also interesting in the creation details of his math-cards. Does he use LaTex? Maybe he goes into it in one of his earlier posts...

michael_nielsen 8y ago
Glad you enjoyed the essays. On LaTeX - I sometimes cut-and-paste screenshots, but usually just write raw LaTeX or pseudo-LaTeX (i.e., something LaTeX-like that wouldn't compile, but is easier to read). It'd be much, much better to use some kind of plugin that turned it into images, but I've been too lazy to try. This has, incidentally, made many of my matrix-based Anki cards rather ugly. I'll probably update a bunch of answers with cut-and-paste from this essay. Not to put too fine a point on it, but it's pretty bad I'm using raw LaTeX in this way - I'm sure it hurts my understanding! On the other hand, I've written so much LaTeX over the years that reading it seems really natural.
- yorwba 8y ago
  You wouldn't even need la plugin. Anki supports [latex][/latex] out of the box, provided you install LaTeX in a way that Anki can find it. https://apps.ankiweb.net/docs/manual.html#latex-support https://apps.ankiweb.net/docs/manual.html#latex-support
  - sbmassey 8y ago

deleted 8y ago

[deleted]

glaberficken 8y ago

>"You might suppose a great mathematician such as Kolmogorov would be writing about some very complicated piece of mathematics, but his subject was the humble equals sign: what made it a good piece of notation, and what its deficiencies were. Kolmogorov discussed this in loving detail, and made many beautiful points along the way, e.g., that the invention of the equals sign helped make possible notions such as equations (and algebraic manipulations of equations). (...) (I found Kolmogorov’s essay in my University library as a teenager. I’ve unsuccessfully tried to track it down several times in the intervening years. If anyone can identify the essay, I’d appreciate it. I’ve put enough effort into tracking it down that I must admit I’ve sometimes wondered if I imagined the essay. If so, I have no idea where the above story comes from.)" Can anyone find the essay mentioned above? Maybe a Russian speaker could track down the original (as presumably what Michael Nielsen is referencing here is an English translation).

e12e 8y ago
Judging from http://kolmogorov.com/ http://kolmogorov.com/ there seems to be a fair chance the original is in French. But I can't find anything that looks like the essay in question... Might be a bit easier if the name of the essay was known...
hopler 8y ago
A great article on a relatively similar topic, equality and isomorphism: http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
drsopp 8y ago
Perhaps it is the article On Sense and Reference by Gottlob Frege? http://www.scu.edu.tw/philos/98class/Peng/05.pdf http://www.scu.edu.tw/philos/98class/Peng/05.pdf
jacobolus 8y ago
There are a bunch of Kolmogorov essays that might be relevant, judging by their titles. https://www.turpion.org/php/reference.phtml?journal_id=rm&paper_id=1991&volume=43&issue=6&type=xrf https://www.turpion.org/php/reference.phtml?journal_id=rm&pa... Perhaps "On the school definition of identity", Matematika v shkole 1966 * * * Maybe a Russian speaker wants to go search? There doesn’t seem to be an English translation of most of Kolmogorov’s essays online. https://www.google.com/search?tbm=bks&q=Математика+в+школе+Колмого́ров+1966 https://www.google.com/search?tbm=bks&q=Математика+в+школе+К...

tankenmate 8y ago

This article seems the describe what it means to grok something; and that maybe one (easy?) way to reach that is via spaced repetition systems.

tomxor 8y ago

> You might suppose a great mathematician such as Kolmogorov would be writing about some very complicated piece of mathematics, but his subject was the humble equals sign: I've only just discovered Kolmogorov.. not being particularly mathematically talented myself I'm happy to find his work conceptually fascinating from an amateur's perspective, I hope the essay is found.

burtonator 8y ago

You guys might like this: https://getpolarized.io/ https://getpolarized.io/ Polar is an SRS based document management platform and personal knowledge repository I've been working on. It syncs with anki and allows you to create and read flashcards directly in the reader itself. Supports text annotations, highlights, comments, keeps track of where you're reading in documents, supports tags, rich formatting of notes, etc. It's also Open Source and supports cloud sync across your desktop machines.

verletx64 8y ago
Looks pretty good for building a first pass of more atomic Anki notes. I've found that over time, I need to make multiple editing passes on Anki Cards, to try to boil them down into something actually recallable. How has your experience been?
harperlee 8y ago
Wow, thanks a lot for this. I was considering building something similar but barebones, and now I can use this and hopefully contribute! Particularly happy about offline first!
- burtonator 8y ago
  Thanks.. I think one of the cool things is it supports HTML capture so you can take pages you see on Hacker News and store them for posterity!
  - harperlee 8y ago
    Sure! I was using evernote and later pocket for this but they ended up as a bin for past read content. With polar that can further go to anki and then to my memory :)
IanCal 8y ago
I've not really used this beyond just installing this now, but I'd encourage anyone that sounds a bit interested from this comment to go checkout the website. If the features it talks about are smooth, it looks absolutely wonderful.

Schiphol 8y ago

I wonder if that Kolmogorov essay on the equal sign might not rather be Frege's Begriffsschrift (the Wikipedia page links to a facsimile version: https://en.m.wikipedia.org/wiki/Begriffsschrift https://en.m.wikipedia.org/wiki/Begriffsschrift).

philip-b 8y ago

Hello Michael, I've read your 2 articles about anki. I am an avid user of anki myself and I use it for math as well. In your 2 articles you tell about using it to understand and retain memory of an Alphago paper and to do the same for the theorem about orthogonal diagonalizability of normal matrices. I wonder, how do you organize anki cards and in which order do you study them? Personally I put all cards in one deck and anki shows them in kinda random order, so I might get a question about convex optimization, then a question on numerical linear algebra, then a question on some dance moves. I take it you do it in a different way? Because you create many small cards for a topic I think you would spend too much time context switching if you did random order like me. For this reason I make larger cards, e.g. "Prove that for any linear operator on a finite dimensional complex vector space there is a basis such that the operator has upper triangular matrix in it"; and when preparing for an exam in institute I might even create a card "Definition, existence, uniqueness, and computational complexity of SVD". Also I think creating small cards might be bad for chunking, i.e. you won't get large chunks of all the related knowledge about a theorem and instead you will have small chunks - a chunk per card. Another question is how do you add all this information to anki? Anki obviously sucks as an exploratory medium. I often find that even clearing up my paper notes, taking photos of them, cutting them, and putting them into anki takes a lot of time; typing it up in LaTeX is even longer. Any tips or insights here?

faitswulff 8y ago

Aside, but is there an SRS that will send me flashcards as notifications on my phone? I'd rather not have to set aside large chunks of time to memorize things, I'd rather refresh my memory throughout the day.

qwerty456127 8y ago

Does anybody know a good math course covering all the high school and college subjects implemented in a collection of flash cards? I have no idea what do "normal matrix", "diagonalizable by" and "unitary matrix" mean (of course I can look these particular up) mean so far but I feel like I would like to try learning up to this level (including what I've learnt in the college and completely forgotten right after the exams) this way.

playdohscave 8y ago
pretty much any good linear algebra course should do the trick
colechristensen 8y ago
Those are all linear algebra concepts, I would suggest Gilbert Strang's series of lectures which can be found on YouTube. They should be fairly accessible without prerequisites.
unixhero 8y ago
KhanAcademy
barger 8y ago
While "all the high school math" is somewhat clear, the "all the college math" should be clarified a bit. Are we talking about math for social studies majors, engineers, physicists, economists or mathematicians? For example, there are books like [0] for engineers which serve as a boot camp of sorts. There are no theorems, proofs or deep math in it. There are many different kinds of engineers, so books like this don't include everything an engineer needs to know. For instance, there's no automata or group theory in [0]. It has a part at the beginning called Foundation Topics which could serve as "all the high school math one needs". In fact, this whole book could probably serve as "all the math an advanced high school student needs". There are books like [1] for physicists. They are meant to introduce physics majors to a wide array of math topics in a relatively pain-free way. This book is much more rigorous and contains a LOT more material than [0]. Basically, a theoretical minimum for a physics major. Economics majors also get somewhat rigorous math load where measure theory features a lot more prominently than it does in other majors (except mathematics proper). List of subjects for math majors vary from place to place, but you'll have much easier time down the road if you master the (rigorous) rudiments of linear algebra (vector spaces), group theory, number theory and real analysis. Of course, knowing more math (say topology, complex analysis, category theory, combinatorics etc) is always good. Before you get started with math for mathematicians, you'll want to learn their jive. A good intro is [2]. It's free and a really nice book. Another really nice book with non-existent pre-reqs is [3]. A pitfall that awaits a lot of people new to math is a concept of "multivariable calculus". This concept is a mess and means everything to everyone. Oftentimes it means surface level discussion of concepts in scalar fields (functions from R^n to R) and a little bit of talk about differential geometry of curves and surfaces (functions from reals and planes to R^n). The treatment is often not rigorous and n is limited to 2 and 3. After this laughable bullshit, one is thought ready to jump straight into the rigorous analysis of manifolds, granted they know a bit of real analysis. This is like jumping from 3rd grade straight to 9th grade. Along the way the most important thing missing is the rigorous treatment of vectors fields (say, at the level of Rudin). Some nice books here include [4]. Since diff geo (in particular, that of curves and surfaces) is its own thing entirely and there are a lot of really nice books for that like [5]. Also, note there's a nice new intro to "manifolds and stuff" [6] which is like what's calculus is to analysis. Before I forget, most intro to stats books are written for science majors and are entirely inadequate for math majors, but there are elementary intro to stats books for math majors like [7]. Originally, I wanted to write a more fleshed out and huge comment, but I am running out of time. Good Luck. [0] Engineering Mathematics by Stroud/Booth https://www.amazon.com/Engineering-Mathematics-K-Stroud/dp/0831134704/ref=sr_1_1?s=books&ie=UTF8&qid=1547410342&sr=1-1&keywords=stroud https://www.amazon.com/Engineering-Mathematics-K-Stroud/dp/0... https://www.amazon.com/Advanced-Engineering-Mathematics-Kenneth-Stroud/dp/0831134496/ref=pd_sim_14_1?_encoding=UTF8&pd_rd_i=0831134496&pd_rd_r=8ad90dba-176f-11e9-b24d-05ff6573df22&pd_rd_w=1UxKi&pd_rd_wg=qemWA&pf_rd_p=18bb0b78-4200-49b9-ac91-f141d61a1780&pf_rd_r=QPGTVRJQ7ESXHHQRB34Q&psc=1&refRID=QPGTVRJQ7ESXHHQRB34Q https://www.amazon.com/Advanced-Engineering-Mathematics-Kenn... [1] A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekers https://www.amazon.com/Course-Modern-Mathematical-Physics-Differential/dp/0521829607/ref=sr_1_3?s=books&ie=UTF8&qid=1547413650&sr=1-3&keywords=modern+mathematical+physics https://www.amazon.com/Course-Modern-Mathematical-Physics-Di... [2] Book of Proof by Richard Hammack https://www.people.vcu.edu/~rhammack/BookOfProof/ https://www.people.vcu.edu/~rhammack/BookOfProof/ [3] Linear Algebra: Step by Step by Kuldeep Singh https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/0199654441/ref=sr_1_7?s=books&ie=UTF8&qid=1547413772&sr=1-7&keywords=linear+algebra https://www.amazon.com/Linear-Algebra-Step-Kuldeep-Singh/dp/... [4] An Introduction to Analysis by Wade https://www.amazon.com/Introduction-Analysis-4th-William-Wade/dp/0132296381/ref=sr_1_fkmr0_1?s=books&ie=UTF8&qid=1547413883&sr=1-1-fkmr0&keywords=multivariable+analysis+wade https://www.amazon.com/Introduction-Analysis-4th-William-Wad... [5] Differential Geometry of Curves and Surfaces by Tapp https://www.amazon.com/Differential-Geometry-Surfaces-Undergraduate-Mathematics/dp/3319397982/ref=sr_1_4?s=books&ie=UTF8&qid=1547414043&sr=1-4&keywords=differential+geometry+of+curves+and+surfaces https://www.amazon.com/Differential-Geometry-Surfaces-Underg... [6] A Visual Introduction to Differential Forms and Calculus on Manifolds by Fortney https://www.amazon.com/Visual-Introduction-Differential-Calculus-Manifolds/dp/3319969919/ref=sr_1_1?s=books&ie=UTF8&qid=1547414114&sr=1-1&keywords=fortney+manifolds https://www.amazon.com/Visual-Introduction-Differential-Calc... [7] Statistics for Mathematicians: A Rigorous First Course by Panaretos https://www.amazon.com/Statistics-Mathematicians-Rigorous-Textbooks-Mathematics/dp/3319283391/ref=sr_1_1?s=books&ie=UTF8&qid=1547414255&sr=1-1&keywords=statistics+for+mathematicians https://www.amazon.com/Statistics-Mathematicians-Rigorous-Te...

turadg 8y ago

Anyone wanting to innovate in tools to optimize learning, I’m building a team doing just that and I’d love to chat: turadg@quizlet.com. (Some of my earlier work: http://j.mp/aleahmad-thesis http://j.mp/aleahmad-thesis )

aoki 8y ago

in case anybody's curious, the Einstein letter mentioned in the article is Appendix II of Hadamard's "The Psychology of Invention in the Mathematical Field" (ISBN 0486201074).