
Mathematics for the Adventurous Self-Learner - nsainsbury
https://www.neilwithdata.com/mathematics-self-learner
======
gavinray
I know this is going to be the case for likely nobody, but I have browsed most
of the self-study math threads that pop up here as a forever-on-my-todo-list
thing and I have a remark to make:

 _I have yet to find a guide that does not start with the assumption that you
graduated highschool._

That is a very reasonable assumption to make. We are in a community of
technology and engineering, it would be a bit ridiculous to assume the people
you are surrounded by did not have a fundamental base of mathematics.

But the times I have tried to go through these _teach-yourself_ materials, it
went from zero to draw-the-rest-of-the-fucking-owl real quick. [0]

I have been programming for 14 years, but stopped doing schoolwork around age
12, and never did any math beyond pre-algebra.

Does anyone know of materials for adults that cover pre-algebra -> algebra ->
geometry -> trigonometry -> linear algebra -> statistics -> calculus? At a
reasonably quick pace that someone with a family + overtime startup hours
could still benefit from?

[0] [https://i.imgur.com/RadSf.jpg](https://i.imgur.com/RadSf.jpg)

(Also, curse the Greeks for not using more idiomatic variables. ∑ would never
pass code review, what an entirely unreadable identifier)

~~~
dhimes
I hope I'm not too late here, but if you are in the US I would _highly,
highly_ recommend signing up for developmental classes at your local community
college. _You are exactly whom those classes are for._ If you've tried on your
own before and struggled to stay motivated, doing it in a structured way, in
15 week "sprints", may be just the kickstart to your self-study program you
need.

Disclaimer: I was a full-time community college professor for a decade. I had
no idea what a resource they were. It's small money compared to either the
alternative of a university or not succeeding. If you use them you _will_
succeed. It's what they do, and they've been doing it for a very long time.

~~~
klyrs
Seconding this. After high school, I worked in industry until I got bored, and
went back to school starting with community college. I'd never thought I was
good at math, and I placed into pre-college algebra. Programming had taught me
to think methodically, so I crushed those early classes. Fast-forward a
decade; I've got a PhD in math (nothing like what I set out to do; I just
followed my passion).

~~~
hackernews7643
That is extremely inspiring. Can you speak more to how you went from industry
to a community college to getting into and completing a PhD program? Did you
quit your job to return back to school? What area did you end up specializing
in and what do you do now?

~~~
klyrs
Long and short, I got sick of the tedium in web development, quit my job and
went back to school. The dot-com bubble had just burst, and I had been taking
occasional classes including a very inspirational data structures course which
planted the seed with formal proofs.

After I went back to school, I tutored in the math study center to pay the
bills, which really helped cement not just the learning but also the notion
that I could survive academia. I'd gone in with a plan to study engineering,
but after I transferred to university, I kept dawdling on the math
prerequisites and not taking the engineering courses that needed them. So it
kinda gradually dawned on me that math was what I loved, and away I went.

I never strayed too far from computers. I'm a graph theorist, specializing in
computation; had I known better I'd have gone into computer science because
that's where I see the most progress being made.

~~~
landonxjames
> had I known better I'd have gone into computer science because that's where
> I see the most progress being made.

Interestingly this is similar to what my two advisors (one from the math
department and one from CS) suggested to me. It would be easier to do the math
I like in a CS department than it would be to do the CS I like in a math
department. Do you feel like math departments are more conservative when it
comes to working outside the discipline?

------
laichzeit0
I pretty much followed the same route as OP re-studying mathematics seriously
after 10 years in industry after initially doing a CS degree and doing mostly
software engineering but transitioning into Data Science the last 3 years.
When I saw Book of Proof then Spivak then Apostol on his list I chuckled
because that’s exactly the route I ended up following as well. Studying from
04:30 to 06:30 in the week and about 8 hours split up over the weekend, Spivak
took 8 months to complete (excluding some of the appendix chapters) but if you
can force yourself to truly master the exercises - and Spivak’s value is the
exercises - then you’re close to having that weird state called “mathematical
maturity” or at least an intuition as to what that means. You can forget about
doing the starred exercises, unless you’re gifted. Spend a lot of time on the
first few chapters (again, the exercises), it will pay off later in the book.
It was a very frustrating experience and I had so much self doubt working
through it, it’s an absolutely brutal book. Some exercises will take you
literally hours to try and figure out.

If you do Book of Proof first you will find Spivak much easier, since Spivak
is very light on using set theoretic definitions of things. Even the way he
defines a function pretty much avoids using set terminology. Book of Proof on
the other hand slowly builds up everything through set theory. It was like
learning assembly language, then going to a high level language (Spivak) and I
could reason about what’s going on “under the hood”. Book of Proof is such a
beautiful book, I wish I had something like it in high school, mathematics
would have just made sense if I had that one book.

I read a quote somewhere, think it was Von Neumann that said, you never really
understand mathematics, you just get used to it. Keep that in mind.

~~~
p1esk
So, you've made all that effort, how does it help you in your new role as a
data scientist? Is there anything you do now that requires "mathematical
maturity"? Or is it something that can be learned much quicker on as needed
basis?

~~~
laichzeit0
There are a lot of charlatans in the Data Science space who lack the necessary
mathematical background for their roles. For me it was necessary to get a
rigorous understanding of probability theory, and applied probability theory
is basically what mathematical statistics is about. My background was CS and
software so R, Python, data visualisation and ML operationalization is by and
large the easy part of Data Science to me. If you pick up any book like
Bishop's or ESL you will be extremely frustrated if your mathematical
background is not there. I didn't feel comfortable creating predictive models
for production use that I didn't completely understand what was going on
"under the hood", the assumptions being made and how they could fail. It's the
only ethical thing for any engineer to do.

~~~
p1esk
I do deep learning research for a living. I've taken graduate classes in
probability, stochastic processes, optimization algorithms, and signal
analysis (ECE PhD). I almost never completely understand what's going on under
the hood of my models as soon as they get larger than a single neuron XOR
mapper. That does not prevent me from finding ways to improve the performance
of very large models (millions of parameters and dozens of layers). I agree
that there are some papers (or the two books you mentioned) that can be quite
dense and heavy on math, but I can't say I've ever felt like I needed any math
other than basic calculus, linear algebra, and prob/stats 101 to understand
almost all ML methods that people actually use in real world. Obviously if you
want to make breakthroughs in theoretical ML, then sure, you do need the
mathematical maturity (mostly because you will need to be formally proving
things), but if you're a regular data scientist? Can you give some example
what kind of math is involved in your predictive models?

------
melling
A couple prior HN discussions with great comments related to learning
mathematics:

[https://news.ycombinator.com/item?id=19811715](https://news.ycombinator.com/item?id=19811715)

[https://news.ycombinator.com/item?id=20446796](https://news.ycombinator.com/item?id=20446796)

[https://news.ycombinator.com/item?id=11469494](https://news.ycombinator.com/item?id=11469494)

------
hackernews7643
One thing I don’t think is discussed enough is the process of how self-
learners in math get critical feedback. Most advanced level math textbooks do
not have solutions to check their work against nor do they have a way to get
feedback by an expert and this is essential for learning. Least with
programming, you can get immediate feedback and know whether what you did is
correct or not.

~~~
chobytes
Learn proofs well and you get pretty good and knowing when you’re right.
Enough for almost any problem you’ll be likely to encounter in a math textbook
anyway.

~~~
TheTrotters
I strongly disagree.

It's a little like saying "learn programming well enough and you'll know if
some piece of code works as expected without running it."

~~~
inimino
> "learn programming well enough and you'll know if some piece of code works
> as expected without running it."

That is 100% true.

Obviously the code needs to be self-contained (not calling into other unknown
code) but so do mathematical proofs.

------
deostroll
Initially I heard about Euler's famous Basel problem. Years later I got to
solving it for my self (for curiosity and fun). I guess what intrigued me was
to think of trigonometric sine as an infinite polynomial...After I worked it
out, I had indeed seen the fire in Euler's own eyes...I could see how excited
he was at having discovered something amazing...But this got me into hooked
into math history. What I really wanted was how people came about discovering
the Taylor's series...the intuition behind it. So that is how I came across
John Stillwell's book. I have to warn people it is rather academic. But if,
you, as a self-learner, is excited about mathematics, I would suggest Norman J
Wildberger's youtube lectures on mathematics history. I find the buildup to
calculus quite fascinating. J. Stillwell's book was the recommended reference
in those lectures...

------
daxfohl
This is so difficult. I've been doing it off and on for twenty years and not
made much of a dent in things.

The hardest part I think is understanding and measuring your progress. In
school you've got exams and classmates to compare against, profs to talk to.
Alone it's much harder. "Do I understand this well enough?" "Did I do the
problems right?" (Especially with proof problems, how do you _know_ you're
right?). "I can work through some problems one by one, but it feels like
something fundamental I'm missing. Am I, or is this chapter really just about
some tools?"

Then it's way too easy to say well I'm never actually going to use any of this
so why am I doing it ... and take a few months off and come back forgetting
what you'd learned.

~~~
l_t
I've tried a few things recently that help with that:

1\. _Don 't do exercises_ unless you want to. Completionism is a trap.

2\. _Take notes_. Rewrite things in your own words. Imagine you're writing a
guide for your past self.

3\. _Ask questions_. Anytime you write something down, pause and ask yourself.
Why is this true? How can we be sure? What does it imply? How could this idea
be useful?

4\. _Cross-reference_. Don't read linearly. Instead, have multiple textbooks,
and "dig deep" into concepts. If you learn about something new (say, linear
combinations) -- look them up in two textbooks. Watch a video about them. Read
the Wikipedia page. _Then_ write down in your notes what a linear combination
is.

Anyway, everyone's different of course, but these practices have been helping
me get re-invigorated with self-learning math. Hope they help someone else out
there. I welcome any feedback!

(edit: formatting)

~~~
throwlaplace
this is excellent execellt advice. seriously anyone interested in learning
math, chancing on this comment, should write it down. i wish i could upvote
many more times. i have a bachelors in pure math and am 10 years out. i have
time and again revisited things and didn't make good substantive progress
until i came to these same exact conclusions.

especially the part about skipping the exercises. if you're not trying to
write a dissertation or pass a qual (and you're just interested in learning
and being exposed) then you don't need to do them. a lot of exercises are a
hazing ritual or imagined by the author to be a dose of bitter medicine (i'm
looking at you electrodynamics by jd jackson) since they mistakenly believe
all readers are formal students.

the most important exercise is to mull over and consider what you're
reading/learning. naturally dovetails in to asking question: what happens if i
remove a hypothesis from a theorem, what happens if i add one, is there an
analogy to another object/group/measure/etc, etc.

also read multiple books ([http://libgen.is/](http://libgen.is/) is your very
very good friend and generous friend). a lot of math authors (no matter how
esteemed they are) are terrible writers or make mistakes (look up errata for
previous editions of your favorite book).

the only thing i'd add is to learn to use LaTeX to take notes - it is much
easier and faster and neater.

~~~
ColinWright
> _this is excellent execellt advice ... especially the part about skipping
> the exercises. if you 're not trying to write a dissertation or pass a qual
> (and you're just interested in learning and being exposed) then you don't
> need to do them_

I think this is deeply mistaken. In a well-chosen book, such as the ones in
the submitted article, doing the exercises is not to test your memorisation,
it's to develop your understanding.

Math is not a spectator sport. Reading _about_ math is fine, but it will not
take root and develop unless you engage with it, and the exercises are the way
to do that.

Ignore the exercises if you want, but you almost certainly will end up knowing
about the math, but not able to do it.

~~~
throwlaplace
>Ignore the exercises if you want, but you almost certainly will end up
knowing about the math, but not able to do it.

Isn't that literally exactly what I said?

> if you're not trying to write a dissertation or pass a qual (and you're just
> interested in learning and being exposed) then you don't need to do them

~~~
ColinWright
The submission and this entire thread is about learning math. That, to me,
implies learning to do, not learning about. Yes, you said:

> _if you 're not trying to write a dissertation or pass a qual (and you're
> just interested in learning and being exposed) then you don't need to do
> them_

There's ground in the middle, and this thread is about that. This thread is
not about learning for tests and qualifications, nor is it about "being
exposed", it's learning _how to do the math._

And for that you need to do the exercises. You don't need to do all of them,
you don't need to be completionist about it, but if you don't do the
exercises, if you don't actually _do the math_ then you won't actually be able
to do the math.

Specifically, you said (quoting again):

> _if you 're ... just interested in learning ..._

There's a difference between learning about and learning to do. If you meant
just "learning about" then you are at odds with the entire thread. True, in
that case you don't need to do the exercises, but I don't think that's what
people are talking about here. I think people are talking about being able to
do the math.

And if you meant "learning to do" then in my opinion you are wrong, and one
needs to do a large slab of the exercises.

Otherwise it's fairy floss, and not steak.

My apologies if all this seems overkill, but there's a real danger of talking
past each other and being in violent agreement, and I wanted to state
explicitly and clearly what I mean, and why I thought you said something
different.

~~~
throwlaplace
> you won't actually be able to do the math

but i'm not a mathematician. i don't need to be able to __do math __anymore
than i need to be able to __do history __(while reading serious history
books).

>And if you meant "learning to do" then in my opinion you are wrong, and one
needs to do a large slab of the exercises.

no i didn't. that's precisely why i used the word "exposed".

>violent agreement

we don't agree but i'm not being violent. but my responses are short and yours
are long.

i do not see the exercises as essential for anyone other than practicing
mathematicians. i have read a great many serious math books (i just recently
finished Tu's Manifolds book and am now reading Oksendal's SDEs). i read them
without doing absolutely any exercises but following the rest of the
guidelines in the post i responded to. the experience is gratifying because i
learn about new objects and new ways of thinking about objects i've already
learned about. that's absolutely the only thing that matters __to me __.

but let me ask you something

>That, to me, implies learning to do, not learning about.

here's a fantastic explanation of the topological proof of Abel-Ruffini

[https://www.youtube.com/watch?v=zeRXVL6qPk4](https://www.youtube.com/watch?v=zeRXVL6qPk4)

would you say that I don't understand that proof if i haven't done any
exercises related to it? and therefore would you say I didn't learn any math
by having watched that video?

~~~
ColinWright
We agree that if you want actually to be able to do the math then you need to
do the exercises.

Do we agree that if you don't do the exercises then you probably won't
actually be able to do the math?

You are discussing learning _about_ the math, and not eventually being able to
do it, because you say that you don't care about becoming a mathematician,
therefore you don't need to do the math. Fair enough.

But my reading is that that's not what this thread is about. This thread, and
the original submission, is about learning how to _do_ the math.

> _i do not see the exercises as essential for anyone other than practicing
> mathematicians._

I think you're wrong. Knowing how to actually do the math has proven useful to
many people for whom it is a tool in their craft/job/employment. Learning
Linear Algebra properly, being able to actually do it rather than just talk
about it, can be enormously useful in Machine Learning.

>> _That, to me, implies learning to do, not learning about._

> _here 's a fantastic explanation of the topological proof of Abel-Ruffini
> ... would you say that I don't understand that proof if i haven't done any
> exercises related to it? and therefore would you say I didn't learn any math
> by having watched that video?_

Understanding a single proof implies very little about one's ability to
actually do the math. I've met many people who are math enthusiasts and who
have watched hundreds of math videos. They say they understand all of what
they've seen, and yet they are unable to do the simplest proofs, or the most
elementary calculations.

My experience of people's abilities is that if they haven't done the
exercises, they usually can't actually do the math.

But you complain about the length of my replies, so I'll stop. I think I've
made my position clear, and I think I understand what you're saying, even if I
don't agree with it.

~~~
throwlaplace
>You are discussing learning about the math

You keep repeating this but you're evading the question about abel-ruffini and
the question about whether reading a history book is "learning about history"
as opposed to learning history.

You're making a weird distinction. People learn in different ways. Some by
doing exercises and some by just playing with the objects. I wonder how you
think actual research mathematicians learn new math from papers that don't
include exercises lol.

You edited your response.

>I've met many people who are math enthusiasts and who have watched hundreds
of math videos

There's a difference between watching numberphile or whatever and essentially
watching a lecture on a proof. Very few people are watching/consuming rigorous
expositions. I think that's the difference not the lack of exercise.

~~~
ColinWright
Learning about history is not the same as then being able to do research in
history, nor being able to apply the principles learned from it in context. So
no, reading a history book is learning about history, not necessarily being
able to "do history".

> _You 're making a weird distinction._

As someone who has done a PhD, done research in math, done research in
computing, worked in research and development in industry, taught math, and
headed a team doing research in technology, this is a distinction that I can
clearly see. My inability to explain it to you is regrettable.

> _People learn in different ways._

Yes they do.

> _Some by doing exercises and some by just playing with the objects._

Doing the exercises _is_ playing with the objects to try to answer specific
questions. Good exercises are carefully constructed to help the reader learn
how those objects work in an efficient manner.

> _I wonder how you think actual research mathematicians learn new math from
> papers that don 't include exercises lol._

In my experience research mathematicians learn now math from papers by, in
essence, constructing their own exercises based on what they're reading. In
general it takes significant experience and training to be able to do that.

Clearly you don't think one needs to do the exercises subsequently to be able
to do the math. Good for you.

I disagree.

~~~
throwlaplace
>As someone who has done a PhD, done research in math, done research in
computing, worked in research and development in industry, taught math

Me too so now what? I don't think your credentials give you any real authority
but just make you look like you're gatekeeping.

>Doing the exercises is playing with the objects to try to answer specific
questions.

Great so then we're in agreement: playing with the object _is_ doing the
exercise.

The funny thing is that at one time I actually did all of the exercises in
volume 1 of apóstol's calculus. You know what effect on me it had? I was so
bored I didn't read volume 2. And today I'd still need to look up the trig
substitutions to do a vexing integral.

~~~
ColinWright
> _I don 't think your credentials give you any real authority ..._

It wasn't intended to, it was to provide a context for my opinion.

So let me state my opinion as clearly as I can, and then I'll leave it.

* Math is a "contact sport" ... you have to engage with it;

* Reading books is not, of itself, engaging with the math;

* Watching math videos is not, of itself, engaging with the math;

* Well designed exercises are a valuable resource;

* If you can easily do an exercise, skip ahead;

* If you can't do an exercise, persist (for a time);

* Ignoring the exercises is ignoring a resource;

* For the vast majority of people, doing the exercises is an efficient way to engage with the material;

* To say "ignore the exercises" is, for the vast majority of people, an invitation to not bother engaging with the subject;

* Doing _all_ the exercises is probably a waste. Doing _none_ of them is an invitation to end up with a superficial overview of the subject, and no real understanding.

~~~
daxfohl
See? It's pretty hard. This is what I've been dealing with for the last 20
years of on and off trying to get through the bigger Rudin book and a couple
others.

 _Just_ reading doesn't get much at all. Not even a superficial overview. I
tried it. It's essentially a meaningless combination of words after a certain
point.

Reading extremely thoroughly is actually marginally useful. Stopping to think,
do all these assumptions matter, why, what if one of them changes, etc, pencil
in hand, making notes, testing things out. I've managed to "understand" the
topics when doing this, and so far it's been the highest ROI method. But it
does still leave one feeling like something is missing. Just because you can
sight read music doesn't mean you're an expert on the piano.

Doing exercises is a huge jump on investment, and the return on that
investment is a bit questionable from my experience. A couple reasons: first
you don't know if you did them right. If you did them wrong then that's
negative ROI. Second you don't know what a "reasonable" workload is. It varies
by author. Is it three problems per chapter, is it all of them, are some
orders of magnitude more difficult than others? Without some guidance it's
hard to know if your difficulties are due to not understanding basic material,
or due to that problem being a challenge geared toward Putnam medalists. So
they may cause you to question your understanding and thus mentally roadblock
you unnecessarily. And finally with proofs (and this may be a me thing), it's
pretty easy to say "I guess this is okay(?)" and move on, even if you're not
sure. Since nobody is ever going to review it, and it's _just_ a homework
problem, it's very very hard to will oneself to make sure every assumption is
correct and you're not missing anything, even if you feel like there's a good
chance you are. Or perhaps I just don't have the constitution to do so.

So while I think doing exercises is necessary for a deeper understanding, I
don't know whether the ROI is worth it outside of a classroom perspective. You
need feedback for exercises to be beneficial. At least, I feel like I do.

Finally, is even taking a class that useful if the end state is that two years
from then you'll have forgotten most of it and so what was the point. Can you
claim knowledge of a subject that you've never actually used beyond some
homework problems and exam questions, or is this _still_ a superficial
understanding? Having an ends where that knowledge gets used seems critical.

I feel like I have some knowledge but I don't feel like I'm _there_ yet. But I
don't know if I know where _there_ is. Maybe that's the biggest challenge.
Does completing a Ph.D. even get you to _there_? No idea. But, I guess it's up
to the individual to decide what they want out of it. Nobody can determine
that for you.

------
integerclub
For all the adventurous self-learners out here, we would like to invite you to
our self-study group named Integer Club.

IRC:
[https://webchat.freenode.net/#integerclub](https://webchat.freenode.net/#integerclub)

Slack:
[https://bit.ly/integerclubslackinvite](https://bit.ly/integerclubslackinvite)

Mailing list:
[https://groups.google.com/d/forum/integerclub](https://groups.google.com/d/forum/integerclub)

We pick up old concepts from popular textbooks and literature as well as new
stuff from new literature in both mathematics and computer science. We plan to
have online meetings periodically to share what we learn, work through popular
literature, and have a few talks on interesting topics.

It is a tiny community right now that hangs out at Freenode IRC but the Slack
channel is there too if you are more comfortable with that.

------
dorchadas
I think it's great that people are posting book links like this, however, what
I've found most helpful is actually having _someone_ to help guide you.

I realize how lucky I was that I found a Discord server ran by a math PhD
graduate who is willing to help us guide our learning. From this, I've started
learning Algebra and Analysis (just starting with the latter). It's nice to
have someone to discuss problems with when you get stuck and to guide you.
Likewise, he can suggest exactly which problems I should do for a give
chapter, that way I don't spend my time doing ones that just repeat the same
simple things over and over and can focus on nice, conceptual ones. So, if you
can, please try to find someone to help guide you, or be that guide for
someone else! Having it has made me seriously consider going back for a
mathematics masters (and maybe PhD), switching from my physics background.

~~~
nubb
Could you share the discord server? Thank you.

~~~
dorchadas
I'm not the owner, and it's a small server, so I don't feel comfortable.
Sorry. I think I found the owner through /r/math, so you might be able to find
his old post there though.

------
wyqydsyq
As someone who dropped out of highscool after 10th grade and never went to
university/college one great way I've found for learning mathematics without
any foundational basis is trying to learn CG/3D programming.

I always felt like maths was too abstract to keep me engaged, but when the
output of your work is immediately observable visually it becomes a lot more
engaging. There's just something so much more satisfying being able to "see"
the results.

Plus as a self-taught programmer, I find it much easier to learn front-to-back
by deciding on a desired outcome and working towards it, rather than
progressively building up abstract fundamental skills that can later be
combined to achieve a desired outcome (which is essentially the traditional
academia path for learning STEM fields)

~~~
ducaale
This is why I love to do game development without using a game engine. It
gives you a reason to learn math, optimize your code down to the metal, all
while having fun playing your game.

~~~
wyqydsyq
Yeah I've been self-learning 3D "the hard way" and have been really enjoying
it.

Keeping it as low-level as possible, I'm using CycleJS for dataflow management
and Regl.js for drawing via a CycleJS-Regl.js driver.

All state is explicitly managed observables/streams in CycleJS, which maps out
to Regl.js draw commands, which are basically raw frag/vert shaders with some
bindings mapping my state from CycleJS to appropriate uniforms/attributes.

I probably would be able to produce some usable output much faster if I used
an engine like Unity or a framework like Three.js, but I feel like I would
have missed out on gaining so much knowledge by only working with high-level
abstractions and never having to even touch GLSL code.

------
tildedave
I've been pursuing mathematics as a hobby for the last 2 years or so. I got a
mathematics major in undergrad so my motivating factor was mainly to explore
some areas that I hadn't done coursework on, primarily algebra and number
theory. (I focused more on logic in undergraduate/grad.)

I really enjoy how the subject is divorced from a lot of the modern attention
demands and encourages more of a 'zen' thinking style.

As others have highlighted, it can be difficult. I work full-time as a
software engineer and at the end of the day there's usually not much left in
the tank in terms of "creative work". The morning is usually more productive
for me - generally I'll spend 10-15 minutes on the commute in reading over the
proof of some lemma or working through some computational exercise.

Things that have helped me:

\- Focusing on a particular problem area rather than just "mathematics". The
classical problems of Gauss and Euler tend to be more my speed than the modern
mathematical problems of Hilbert or beyond. What started my journey was
looking into the insolubility of the general quintic polynomial equation,
something you learn in high school as a random factoid but has a lot of depth.

\- Studying from small textbooks that I can fit in a backpack, so I can "make
progress" during my commute. Dummit + Foote might be a great algebra reference
but it's just too bulky to transport.

\- Limiting the scope of how I think about the activity - my goal isn't to
master these concepts on the level of a mathematics graduate student, it's
more on the order of Sudoku. If I don't get something, that's okay. People
spend their whole lifetimes learning this material and I'm just trying to fit
this into whatever creative time I have left after the full-time job is done.

------
kevstev
Do any of you all have some tips for understanding mathematical notation? I
feel this is often poorly explained, and it feels like a language all its own
that just does not speak to me. I did pretty well in calculus, but I still
don't really understand what the dx was supposed to represent and in reality I
was just really good at pattern matching when it wasn't supposed to be there
anymore.

I try to read papers now and again with a math orientation, and I quickly get
lost when trying to translate the concepts into cryptic formulas, and often
when they make the "obvious" transition from step 3 to step 4 I just have no
idea how they got there.

I feel this is by far my biggest barrier to understanding most mathematics,
and I have thus far found no way to overcome it.

~~~
bo1024
I think usually the problem is "almost getting it" and trying to move forward,
which means small uncertainties add up and all the sudden one is totally lost
without being sure exactly why. So it's important to go back and make sure
each piece of notation is crystal clear before moving forward.

Any statement in math is meant to be directly translatable to human language.
You should be able to read it out loud in English and know exactly what you
mean when you say it.

Unfortunately, sometimes math uses awful notation. For example, df/dx. This is
a case where df doesn't mean anything (or at least it's not normally well-
defined), and dx doesn't mean anything either (same comment). But the notation
as a whole means something. If we write g = df/dx, then we can understand that
g is a function whose input is x and output is the slope of f at x.

------
zerubeus
I came to the IT industry after a bachelor degree in math, 5 years in and all
the math I know is gone I still remember some Fourier, signal processing and
probability statistics that I never used in my day job, or anywhere else.

Time is valuable, it's the most valuable thing a human being has, I understand
it's the hobby of OP to learn all this math, but unless you are going to use
it why wasting all the time?

------
angry_octet
The most key piece of advice is to take walks. Walking is essentials for
mathematics. Many times when walking with my father he would turn for home and
start walking faster, and by that sign I knew that he wanted to get home and
write down a lemma.

~~~
injb
I'm glad you posted this, because I use walks this way too. And because it
reminds me of William Rowan Hamilton and the quaterions!

------
jcurbo
This is a solid read, with good book recommendations. After several years of
tinkering with self-learning I bit the bullet and applied to a MSc in Applied
Math program. (via ep.jhu.edu) I've had to take some pre-reqs to get started
since it's been almost 20 years since I have been in a college math class, but
it's been an enlightening journey re-learning calculus and now dipping my toes
into differential equations. I don't think I could have gotten this far with
self-learning, but I realize YMMV.

I will say I don't feel like single-variable real number calculus tells the
whole story. I had taken that and linear algebra in undergrad but never any
further, and now that I've taken single and multiple variable calculus, with
real and complex numbers, plus integration of linear algebra ideas, the
mathematical model feels a lot more like a cohesive whole to me, highlighting
fundamental ideas that only barely peek through in a typical Calculus I class.
I would encourage anyone talking to calculus to at least do the typical Calc
II class, if not Calc III/multivariate. There is a beauty and structure to
building up from calc I through III that I was missing before.

------
peatfreak
I'm pretty skeptical about these "best of" lists of books for self-directed
mathematics education.

I have my own "best of" list that is very different to this list, although
there are a couple of crossovers.

If you are fortunate enough to have access to a university library (or
libraries) I would _highly_ recommend inquiring about access to their general
collection. I was also fortunate enough to study mathematics to a university-
level three-year degree at a research university. So I had an excellent head
start.

A HUGE part of my journey of collecting my "perfect library" of mathematics
self-tuition and reference books (and course books) was to do my own research
on collecting the perfect titles. I started when I was in the early days of my
mathematics degree and I used resources like Amazon, Usenet, libraries
(already mentioned), and ... that was about it.

Another important question to ask yourself is the following:

"Why am I doing this?"

Life is short and by the time you hit middle age, if you have a family or
bills to looks after, are you REALLY going to want to lock yourself away in
your study room to learn Lebesgue integration instead of focusing on the rest
of your life?

Consider that people fail to emphasise is that mathematics is a social
activity much more than many people realize.

Exercise: Find the topics of mathematics that are important to your goals and
are missing from the list and find your favorite books or two that cover/s
these topics.

Exercise: Consider whether your interest in (self-directed) mathematics is so
sincere such that you have a serious application in mind, that you might be
better off enroling in a course? Even if it's a night course that last a
couple of years, you will meet a LOT of people who can help in ways that are
immensely more productive than trying to do this all by yourself.

I recently purchased volume 1 of my favorite calculus and analysis book. It's
an incredible masterpiece. The coverage of topics is much broader and more
interesting than Aposotol or Spivak. The latter books are both very good but
they also have myopic, one-track pedagogical approaches and limited themes in
their coverage.

Exercise: Find your own favorite introductory calculus book that is suitable
for the motivated student.

~~~
laichzeit0
> I recently purchased volume 1 of my favorite calculus and analysis book.

Which book would that be if I might ask? I'm wagering... Courant? ;)

~~~
peatfreak
Indeed :-)

Courant & John, to be exact.

------
dwrodri
I recently had the experience of taking my first graduate-level probability
course. It assumed quite a strong familiarity with real/complex analysis, and
I suffered quite heavily. Something of note was that once I finally managed to
"peel back" the analysis, the underlying intuition made a lot of sense for the
simplest cases in probability (e.g. hypothesis testing between two normal
distributions is a matter of figuring out whose mean you are "closer" to).

I am of the opinion that notation is a very powerful tool for thought, but the
terseness of mathematical notation often hides the intuition which is more
effectively captured through good visualizations. I would really like to take
self-driven "swing" at signal processing, this time approaching it through the
lens of solving problem on time-series data, since as a programmer I believe
that would be quite useful and relevant.

~~~
watwatinthewat
In my opinion, the issue here is notation and a bit more. I did about eight
years of college in math, changed paths, changed careers, changed careers
again to ML/DL research, and now will finish a CS undergrad degree this month.

I put it in context because it's not quite a direct comparison since I have
been in greatly different situations and ages between studying math and CS,
but putting that aside, I have to say I have greatly enjoyed the computer
science means of teaching more than math, doubly when it comes to self-
learning. Concepts in math are generally taught entwined with the means of
proving those ideas. That's important if you're a grad student looking to be a
math researcher, but (IMO) it is not so great if you're a newer student or
learning on your own and trying to grasp the concept and big picture. A proof
of a theorem can be (and too often is) a lot of detail that really doesn't
help you grasp the concept the theorem provides or is used towards, often
because it involves other ideas and techniques from higher levels or just
different types of math, both of which are out of the scope of the student
learning the topic. Worse yet, it is standard for a proof to be written almost
backwards from how it would be thought out. Anyone from a math educational
background has the experience in homework of solving a problem, then rewriting
it almost in total reverse to be in the proper form to submit. This means not
only is the proof of the theorem not useful towards conceptual understanding,
reading the proof doesn't show you chronologically how you would discover it
yourself. That is a lot of overhead cost to break through to get to real
understanding, real learning. As you mention, notation as well is another
thing you need to break through.

I have found computer science and related classes to be taught more
constructively. Concept is given first, and then your job as student learner
is to construct it. Coming from the ML field, I love comparing math and CS
proofs of topics here. Explanations from CS people of back propagation, for
example, are always visual, and books/courses will have you construct a class
and methods to do the calculations. Someone with a bit of programming
knowledge can follow along in their language of choice. Math explanations get
into a ton of notation from Calc 3+, and it's going to take a lot of playing
around and frustration to get a working system out of the explanation. Even
the derivation section on Wikipedia is not something most people will
understand and be able to turn into useful output.

The more I see other ways concepts are taught, the more I wish math had been
taught a different way. There is a lot to break through in order to get to
real understanding, just by the way it's formed and taught.

------
thorn
I am always astonished to learn that there are such self-learners in the
world. I wonder how it is even possible to have a family and spend whole day
building a startup - I cannot imagine that startup work is less than 8 hours a
day - and then at evening they learn math or other complicated branch of
science. What time and more especially how much energy they have for the
family? Are these guys superhumans? I never was able to achieve such level of
daily energy spent without trapping in burn out. I am not critiquing or being
jealous here, just having genuine interest. How is it possible to be
sustainable across so many years?

~~~
FranzFerdiNaN
I have my doubts that the people who write these kinds of posts truly did
everything they say. As you say, it just does not seem possible to thoroughly
work through all those math books + raising multiple kids + maintain
friendships + working out daily (as he claimed he did) + work full time +
things like cleaning and shopping and other chores.

------
emmanueloga_
Wow that's a brutal list of books... I'm impressed the author could work
through all of that in just six years! I feel like math is a subject you need
to get back again and again to refresh in order to retain. I got some pretty
good grades in linear algebra back in the day... but I don't really remember
much about it right now, sigh.

My strategy to get back to study math these days is getting to learn Wolfram
Mathematica and Sage. Once I can move around those two, I feel like I will be
able to create a tighter feedback loop on whatever Math subject I'm happen to
be studying at the time.

------
leto_ii
Does anybody have any experience with _How to Prove It?_ by Velleman? Recently
I was thinking of starting on it, but I'm not sure about the level of
commitment necessary.

~~~
strls
I worked through this book to learn how to do proofs. It turned out way more
fun than I expected. The book really did demystify proofs for me. It took
several months of studying - there are many exercises. But completely worth
it. I'm glad I have read this book before studying Group theory and Real
analysis.

~~~
leto_ii
Thanks!

------
mikorym
I would recommend _Conceptual Mathematics: A First Introduction to Categories_
by Lawvere.

It is written by a true pioneer. And also, you will impress your friends by
your hipster foray into category theory.

However, this book is far from being hipster. Also, I would not be surprised
if a high school student would be able to follow this book over the course of
a year or two.

If you titled the book: _Sarcastic introduction to how simple set theory is_
then I would actually be fooled that it were the correct title.

------
sampo
There is 3 books listed that essentially cover the freshman (first year)
courses in Calculus. And 4 for Linear Algebra. If you work your way through
even 2 different books for one topic, you are going to have a broader
foundation in the topic than a normal math student in a normal university
after completing the corresponding course. And you will have spent much more
time, too. University courses don't usually cover everything that is in a
textbook. And students don't usually read books through. In fact, students
usually try to skim the course notes just enough so that they can solve the
weekly problem sets.

There is maybe nothing wrong with being thorough with the elementary topics if
you're studying for fun. But if you're studying for applications, I think you
should cover the basics only adequately, and then quickly move on to more
advanced topics. Basic Calculus is only the foundation, stuff that is actually
useful in applications comes later. Basic Linear Algebra can be useful in its
own right, but the advanced stuff is even more useful.

I suggest building an adequate foundation, not a comprehensively thorough
foundation, and then moving on to the more powerful stuff. Which varies
depending on what you actually want to use math for.

~~~
nsainsbury
To be clear, I don't at all advocate that people work through all the Calculus
books. Likewise for the Linear Algebra books. My aim was to provide
alternative options (which are easier, cheaper, etc.)

------
gshubert17
Another book similar to Morris Kline's _Mathematics for the Nonmathematician_,
which the OP mentioned, is Lancelot Hogben's _Mathematics for the Million_.
Originally published in 1937, it has been in print ever since, through several
revisions. This also takes a historical approach, beginning with numbers and
counting, measure and Greek geometry; and eventually covering calculus,
matrices, probability, and statistics.

You can take a look at it, at the Internet Archive,

[https://archive.org/details/HogbenMathematicsForTheMillion/m...](https://archive.org/details/HogbenMathematicsForTheMillion/mode/2up)

------
ww520
I found the Real Analysis course to the really really hard back in university.
I thought it was like the CS information analysis when looking at the course
title. It was nothing like that at all. It didn't help that the professor
teaching it was pretty bad. I remember he used to jog into the class carrying
a tennis racket and in tennis sporty dress with headband, seemingly just
coming back from a tennis practice and acting to be cool. People just rolled
their eyes. The teaching was just reading off the book. Darn, it was one of
the top schools. How did this clown get in?

------
ipnon
The perenial self-learning mathematics curriculum is hbpms.blogspot.com.

------
jdkee
This is fantastic. I recently been studying set theory and discrete
mathematics as a self-directed learner and it is incredibly helpful to see
others hewing the same path.

------
chrischattin
I love stuff like this. I graduated with an MS and never understood Calc II+.
It was always memorization and repeating various theorems etc on the test.
But, I didn't truly grok the fundamentals. I was just a good test taker and it
bothers me to this day. So, learning math has been a continuing project of
mine and things like this are beautiful.

So grateful. The world is wide open to the self learner in this day and age.

We are very lucky.

------
codeisawesome
Completely unrelated question but: how do you go about finding out that there
is an opportunity to build a small business selling Microsoft teams and slack
integration apps? I’m stuck in the mindset that software companies make
billions of dollars or no money at all. I’ve not seen the right kind of indie
hacker post that talks about how exactly people size their ideas and how much
money is possible to make.

------
tinyhouse
It was a great read. I bookmark this post hoping one day to buy a couple of
the recommended books. It's just too hard for me right now to find the time.
Life is too busy with a family, a full time job, and all the distractions
around. Self learning is one of the most enjoyable things.

------
Koshkin
Practice makes a master. More importantly, the _true_ understanding only comes
through practice. Also, more often than not to "understand" something comes
down to simply _getting used_ to it (which is how we learn things in the
elementary school). Practice is the key.

------
t_mann
Fyi, for those who like working with lecture notes, much of the material for
the math course at Oxford is online as well (no solutions, though):

[https://courses.maths.ox.ac.uk/](https://courses.maths.ox.ac.uk/)

------
tmpmov
Not sure if mentioned, another good book: Analysis by its History

------
mhh__
[http://www.goodtheorist.science/](http://www.goodtheorist.science/) this is a
step by step for theoretical physics.

------
dr_dshiv
Does anyone know a good historical approach to maths? Like, start with
Pythagoras?

Even something purely in the modern era, learning about Fourier and Weiner,
harmonic analysis, etc.

~~~
dr_dshiv
The best historical treatment I've yet found is the Time-Life book on
mathematics. [https://www.amazon.com/Mathematics-David-
Bergamini/dp/B0007G...](https://www.amazon.com/Mathematics-David-
Bergamini/dp/B0007G5WYG)

I know that Freeman Dyson attributes his proficiency in math to his love of
math, which he claims was kindled as a teenager by reading Bell's "Men of
Mathematics"

------
dustfinger
Buy a chalk board! There is no more enjoyable way to work through problems.
This is especially true as the problems become more complex.

------
kumarvvr
Is there a list that inclines towards abstract math that can be helpful to
solve programming problems?

------
piggybox
Thank you for such an inspiring piece

------
baby
Aren’t applications like Brilliant actually really good to do this?

------
wendyshu
Not quite sure why everyone needs to learn so much math.

------
polyphonicist
I am going to suggest something that might go against this idea of self-
studying math.

Do not do it alone. I mean, it is okay to self-learn mathematics as much as
possible but don't let that be the only way to learn. Find a self-study group
where you can discuss what you are learning with others.

I think the social-effect can be profound in learning. I realized this when I
used to learn calculus on my own. My progress was slow. But when I found a few
other people who were also studying calculus, my knowledge and retention grew
remarkably. I think the constant discussion and feedback-loop helps.

With round the clock internet connectivity, it is easier to find a self-study
group now than ever.

~~~
dentalperson
It's not super clear to me how this actually works in practice. I've seen
there is one public math meetup in SF, but the topic is usually different from
the one I want to study.

I'm glad to see there are online options for groups like Stack Exchange or
tighter group's like the one integerclub mentions, but I still seem to run
into the same problem. For example, I'm not sure how to get a group of people
that are interested in reading book X when I want to start it. If anyone has
advice on that, please share.

~~~
nsainsbury
Yep, this has been the story of my learning experience. I've studied
mathematics pretty much entirely on my own, but it's not because I wouldn't
love to have company!

Having said that, I think it probably would be sufficient to find _just one_
other person who is at the same level of mathematical maturity and has the
same degree of commitment to change the entire learning experience for the
better. You don't need a big group.

------
nilsocket
In mathematical history, there is a lot of discredit to Indian mathematics and
their contributions.

Number System, Algebra, Geometry, Trigonometry, Calculus, ...

[https://en.wikipedia.org/wiki/Indian_mathematics](https://en.wikipedia.org/wiki/Indian_mathematics)

