
How to Learn Advanced Mathematics Without Heading to University - shogunmike
https://www.quantstart.com/articles/How-to-Learn-Advanced-Mathematics-Without-Heading-to-University-Part-1
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ska
The main thing to remember is that nobody learns mathematics to any
significant depth by reading. The only way to learn it is by doing it. Doing
it carefully, and in full detail, not falling into the trap of "and I
understand it from there".

So the biggest barriers to doing it on your own aren't source material (there
is lots of that) it's a good source of correction. There is also the usual
problem of self study, in that you don't have a roadmap and can waste time
easily.

That said though, one advantage if you are diligent is that you probably by
necessity learn techniques of checking your work (formally and informally)
earlier than typical students, which is a good thing.

~~~
chatmasta
This is probably the reason why there are so many more autodidact programmers
than mathematicians. When programming, you know if you made an error as soon
as you try to run your code. The program, by definition, must be "correct" in
order to execute. The "source of correction" is the error detection built into
any programming language runtime or compiler.

No such "source of correction" exists for mathematics, and that makes it an
inherently more difficult subject to teach yourself, because any errors you
make will "fail silently" unless you are capable of detecting them yourself,
which by definition you cannot do without experience. This is why a
mentor/professor makes learning mathematics so much easier; he/she plays the
role of mathematical compiler.

~~~
thegenius2000
Good point and great analogy, though I think EWD might be turning over in his
grave. While the error detection built into compilers and the testing process
are, in general, indispensable tools for developing stable software, a program
that executes without any evidence of error is a great distance from a program
that has been shown to be correct.

~~~
Retra
It's still easier to build up an understanding of computers under that
environment. You may not be proving your programs correct, but you are
probably proving them useful.

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noir_lord
I've been getting into math recently and found
[https://www.youtube.com/user/professorleonard57](https://www.youtube.com/user/professorleonard57)
Professor Leonard's videos are excellent, clear and he injects just enough
humour to keep it from been too dry.

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danharaj
Learn it in a group. Mathematics is a social activity. Meeting even once a
week to discuss the material you're trying to learn will vastly improve the
learning rate for everyone involved.

~~~
Patient0
Brilliant suggestion!

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lumberjack
There is no reason to shun university.

If you want to learn mathematics but do not have the time for a full time
degree do it part time.

You are most definitely not actually going to reach the same level of mastery
of mathematics as even the most mediocre graduate by studying alone. It
requires a huge amount of commitment and staying focused and discipled enough
is very hard when you don't have a set goal and good continuous guidance and
motivation.

It is also very boring and much less productive to study alone. I'd estimate
that 70% of my learning comes from the tutorial sessions. Reading the
literature is just the starting point. It only counts for 10% if that.
Homeworks and exams are also very useful because they make sure you have the
prerequisites to move on and start learning something new.

~~~
tanker
I disagree, at least at the undergraduate level.

There is no shortage of problems and solutions to check your knowledge for
much of undergraduate mathematics.

As to the value of tutorial sessions for you, it is important to remember that
people learn differently. Speaking with an expert is valuable, but not
required in my opinion.

To really learn a subject, you have to go beyond the minimum required to get a
good grade on homework and tests. You can absolutely do this in university,
but I found that I was rarely willing to do so. Studying on my own, I find it
easier to reach a deeper level of understanding.

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graycat
An early place where the OP dropped the ball:

> and some good experience manipulating continuous functions and their
> derivatives.

Nope. If a function is differentiable, then it is continuous, but continuity
is not sufficient for differentiability. So, we can't talk in general about
the derivatives of continuous functions.

E.g., each sample path of Brownian motion is almost surely differentiable
nowhere.

Just f(x) = |x| is continuous but not differentiable at x = 0.

Maybe the OP advice is okay in England, but here in the US I would advise
people wanting to learn to f'get about the OP and get better advice.

For job opportunities in _quantitative trading_ , I tried that on Wall Street
here in NY, and got nowhere. I came with a Ph.D. from a world-class US
research university with my dissertation research on stochastic optimal
control, which should have put my resume near the top of any stack. My
favorite prof was a star student of E. Cinlar at Princeton and, thus, about
the best there is for mathematical finance. I came with a long, solid
background in software, peer-reviewed publications in mathematical statistics,
optimization, and artificial intelligence.

I had a good background in second order stationary stochastic processes, power
spectral estimation, and the fast Fourier transform -- no interest.

Got nowhere. That was before I heard about James Simons.

One interview was by a guy who recruited for Goldman Sachs, and he didn't have
a clue about my background.

Another interview was at Morgan Stanley: The interview was in their computer
group, but I indicated that I'd like to get into quantitative trading -- they
acted like they had never heard of any such thing.

I got the impression that only a very tiny fraction of the people on Wall
Street had good backgrounds in measure theory and stochastic processes based
on measure theory, that my resume never got in front of any such person, and
that the other people didn't know measure theory, anything about Brownian
motion, power spectra, time series analysis, etc. and were looking to hire
people like themselves.

Candidate Lesson: Study all the math you want, but don't expect Wall Street to
be interested.

~~~
nmrm2
>> and some good experience manipulating continuous functions and their
derivatives.

> Nope

1\. "good experience manipulating differentiable functions and their
derivatives" sounds weird in prose.

2\. Some continuous functions are differentiable. Those ones have derivatives
you can manipulate. In fact knowing when a function is not differentiable is a
pretty useful skill.

 _> The interview was in their computer group, but I indicated that I'd like
to get into quantitative trading -- they acted like they had never heard of
any such thing._

Sounds like you interviewed for position X and talked about wanting position
Y, and were rightly rejected as "not a good fit; likely to leave at first
opportunity".

 _> only a very tiny fraction of the people on Wall Street had good
backgrounds in measure theory and stochastic processes based on measure
theory_

Probably true. Think of this as "calculus for engineers vs. analysis", and
imagine how well a civil engineering interview would go if you talked about
different types of integrals instead of talking about how to use the basic
stuff to build good bridges. Fact is most people in industry are looking for
"calculus for engineers" levels of formal understanding. Enough to be useful
and make money while avoiding expensive mistakes.

 _> and were looking to hire people like themselves._

Also probably true. This is a good assumption not just on Wall St but
everywhere.

~~~
graycat
> Think of this as "calculus for engineers vs. analysis", and imagine how well
> a civil engineering interview would go if you talked about different types
> of integrals instead of talking about how to use the basic stuff to build
> good bridges.

That analogy shouldn't apply to quantitative trading on Wall Street: That
challenge needs more than just _engineering_ math approaches if only to read
the literature.

E.g., apparently broadly the first cut way to evaluate _exotic options_ is to
use the Brownian motion solution to the Dirichlet problem, that is, the
subject of Markov processes and potential theory. The subject is awash in
measure theory, e.g., stopping times, the strong Markov property, regular
conditional probabilities, of course conditioning and the Radon-Nikodym
theorem. This isn't advanced calculus for engineers. E.g., the work of Marco
Avellaneda at NYU Courant, Steve Shreve at CMU, no doubt the work of E. Cinlar
at Princeton.

------
amai
A single book is enough to learn mathematics:

Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A
Comprehensive Guide

It has a whopping 1300 pages, but it has everything you need. And if that is
not enough for you get

Cahill: Physical Mathematics

This will give you advanced topics like differential forms, path integrals,
renormalization group, chaos and string theory.

~~~
diymaker
Thanks for the suggestion. Like you mentioned though getting to the end of
1300 pages will take ages. Hope I can find that sort of time.

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steQ
I know that many learn programming themselves but I'm curious if anyone have
learn advanced mathematics this way.

~~~
namelezz
What are the best ways for a non-genius, normal person to learn advanced
mathematics?

~~~
bmer
It is very easy to get most mathematics textbooks online, through slightly
unsavoury means. It is far more difficult to figure out which textbooks are
worth reading---this is a process that requires trial and error, and browsing
through recommendations (math.stackexchange and mathoverflow.net have many
good textbook recommendation questions, with many excellent answers).

Also, it is very easy to audit courses at universities! Get out there, ask the
professor if you can audit the courses (make friends with them too!), and
enjoy yourself a stress-free, and money-free, quality education.

~~~
kafkaesq
"Unsavoury", as in?

~~~
forgetsusername
Piracy.

Though, between price fixing and booksellers going under and not being able to
guarantee that you will retain access to the books you bought, I'd say that
this form of piracy is morally ambiguous.

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mathattack
Amazing how thorough the Math curriculum is in the UK! I got a minor at my
undergrad just by doing stats, linear algebra and calc through multivariate. I
think I was just 2-3 courses shy of a BA in the subject.

I'm curious how many people have met self-trained mathematicians in real life.
I know there are storied examples, but is it feasible? (Compared to say -
self-trained programmers or writers)

~~~
duaneb
> I got a minor at my undergrad just by doing stats, linear algebra and calc
> through multivariate.

I wouldn't call this a good thing; this just seems like a sane basis for any
maths work (including Comp Sci).

~~~
mathattack
That's my point. It was pretty much what was required for the CS degree. The
UK standard is much higher.

~~~
mng2
I don't think it's just the UK. At Berkeley, for example, the Math minor
requires 4 lower division courses and 5 upper div.

[https://math.berkeley.edu/programs/undergraduate/minoring-
ma...](https://math.berkeley.edu/programs/undergraduate/minoring-mathematics)

~~~
mathattack
Yes - 9 versus 5. Is Berkeley quarterly or semester? Either way, it looks like
a much more rigorous program. (I've generally been very impressed with
Berkeley Math and CS grads)

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goc
Any one know any books or resources that concentrate on invariant programming?
Translating recursive code into properly tail recursive or iterative code can
be pretty difficult.

~~~
danharaj
`Pearls of Algorithmic Design` by Bird - Beautiful little book. It's a series
of problems that are solved by first writing the naive program and then
transforming it rigorously to make it more efficient.

`Algebra of Programming` by Bird and De Moor - This a treatment of the theory
that is implicit in the methods of the previous book.

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analognoise
It's cute that quants think what they do is advanced, isn't it?

~~~
orbitingpluto
It's cute that the article is littered with Amazon referral links to
textbooks.

~~~
davidivadavid
I'm curious why people care about articles with referral links. It's not like
you're paying more if you end up buying something, is it?

~~~
fabulist
It is always good to know when someone is profiting financially from a
recommendation they're giving you.

~~~
davidivadavid
Sure, I'll buy that in principle. But wouldn't you agree that it mostly
matters if someone is recommending some shady financial product or otherwise
high-ticket item? What's the worse that could happen when someone makes
(text)book recommendations?

Affiliate links are also a good way to compensate people for curating valuable
information. So the sort of "it goes without saying affiliate links are bad"
attitude is kind of a mystery to me.

