
Using spaced repetition systems to see through a piece of mathematics - uijl
http://cognitivemedium.com/srs-mathematics
======
glaberficken
_> "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).

~~~
jacobolus
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&pa...](https://www.turpion.org/php/reference.phtml?journal_id=rm&paper_id=1991&volume=43&issue=6&type=xrf)

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+К...](https://www.google.com/search?tbm=bks&q=Математика+в+школе+Колмого́ров+1966)

------
burtonator
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.

~~~
archiepeach
I’ve also been working on something similar for the past couple of years.

[https://vocabifyapp.com/](https://vocabifyapp.com/)

Vocabify is a tool to help you remember the words you come across. It uses a
basic implementation of spaced repetition to help you remember the words and
phrases you add.

Having Polar sync with Anki is an interesting idea, thinking about it now, I
might have been able to get further with my own tool by leveraging Anki as
well. Food for thought.

~~~
quakenul
> Vocabify is a tool to help you remember the words you come across. It uses a
> basic implementation of spaced repetition to help you remember the words and
> phrases you add.

Cool idea. I suggest improving the onboarding experience of your product
dramatically with minimal effort by adding this sentence to your site. Like
so: [https://snag.gy/iqOlFC.jpg](https://snag.gy/iqOlFC.jpg)

~~~
archiepeach
Good shout! Added. [https://snag.gy/JHfXKT.jpg](https://snag.gy/JHfXKT.jpg)

The Vocabify "definition" is added for any new users, so you'll need a fresh
browser instance to check it out. Thanks!

------
branweb
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
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
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
Anki also supports Mathjax natively, so you can put the latex-ish code between
a \\(\\) or \\[\\] and you don't need the image file step.

~~~
philip-b
I think it doesn't work in Android version

~~~
jw_mc
I have this at the bottom of my card templates. Works on desktop and Android:
[https://pastebin.com/uqUrJBdZ](https://pastebin.com/uqUrJBdZ)

------
PacifyFish
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
How do you form flashcards out of mental models you encounter and want to
remember and use? Could you post an example?

~~~
PacifyFish
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.

------
qwerty456127
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.

~~~
barger
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/0...](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/Advanced-Engineering-Mathematics-
Kenn...](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)

[1] A Course in Modern Mathematical Physics: Groups, Hilbert Space and
Differential Geometry by Peter Szekers

[https://www.amazon.com/Course-Modern-Mathematical-Physics-
Di...](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)

[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/...](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)

[4] An Introduction to Analysis by Wade

[https://www.amazon.com/Introduction-Analysis-4th-William-
Wad...](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)

[5] Differential Geometry of Curves and Surfaces by Tapp

[https://www.amazon.com/Differential-Geometry-Surfaces-
Underg...](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)

[6] A Visual Introduction to Differential Forms and Calculus on Manifolds by
Fortney

[https://www.amazon.com/Visual-Introduction-Differential-
Calc...](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)

[7] Statistics for Mathematicians: A Rigorous First Course by Panaretos

[https://www.amazon.com/Statistics-Mathematicians-Rigorous-
Te...](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)

~~~
qwerty456127
Thanks. So many great recommendations. Not a single link to a flash cards
collection that I could use with Anki however.

~~~
yesenadam
It's better making cards yourself - not just better, using someone else's pack
seems utterly pointless to me. Why would you want to miss this step?

It reminds me of the situation with books of transcribed jazz solos, which at
first seem like gold. But they're actually worse than useless, as you miss out
on everything you learn from transcribing solos yourself.

~~~
qwerty456127
I get the point, yet..

> Why would you want to miss this step?

Because I don't really have much time for every step. I don't need it to be
fun, I need it to be efficient (requiring as little time and effort as
possible). In fact I had always wanted an app of a sort that would feed me
math in concise pre-digested twitterish pieces.

Of course I can't expect somebody to do this for me but if somebody has done
and shared I'd like to make use of that. I am certainly going to share if I
find time to make the cards.

------
Schiphol
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)).

------
faitswulff
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.

------
philip-b
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?

------
tankenmate
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
> 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.

------
turadg
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) )

------
tmyklebu
> 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
I mean diagonalizable by a unitary matrix. Fixed.

~~~
Escapado
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.

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

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

------
aoki
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).

