
Ask HN: How can I learn to read mathematical notation? - cursorial
There are a lot of fields I&#x27;m interested in, such as machine learning, but I struggle to understand how they work as most resources I come across are full of complex mathematical notation that I never learned how to read in school or University.<p>How do you learn to read this stuff? I&#x27;m frequently stumped by an academic paper or book that I just can&#x27;t understand due to mathematical notation that I simply cannot read.
======
QuadrupleA
Sort of copy-pasting one of my earlier comments, apologies, but I highly
recommend the book "Mathematical Notation: A Guide for Engineers and
Scientists" by Edward Scheinerman (Amazon link:
[http://a.co/c1jcvLH](http://a.co/c1jcvLH) ). It's short, and perfect for
looking up symbols, brackets, subscript/superscript conventions etc. for
science and engineering math (for me, the stuff I encounter in my programming
life, when I read papers & books that deal with graphics programming, DSP,
cryptography, etc.).

It's difficult to google mathematical stuff, especially since each symbol has
many different meanings depending on which branch of mathematics you're
dealing with - this book solves that problem nicely by letting you look up by
the roman letter a symbol is similar to, by mathematical discipline, etc.

~~~
sabertoothed
You put a referral code in the URL? I find that somehow objectionable.

~~~
adaszko
Even if so, this is an acceptable incentive for people to comment with Amazon
links. If find this a problem for services like
[http://hackernewsbooks.com/](http://hackernewsbooks.com/) (there are few
others). If someone mentions just a title of a book, it doesn't get indexed.

------
woodson
John von Neumann once said, "Young man, in mathematics you don't understand
things. You just get used to them." While probably not the originally intended
meaning, you can take the quote as saying that, after reading a lot of papers
in a given field, you will “get used to it” and recognize recurring patterns
and what they mean. This is particularly important as more often than not,
part of the math is left out (presumably because it’s too trivial for the
authors or the want to fit the page limit). So, part of being able to “read
the math” is to infer what’s going on in between.

Many papers, especially im engineering, use a lot of mathematical notation
that doesn’t benefit the reader, it’s just there to show off. Often, there are
mistakes in there too, because no reviewer typically goes to the trouble of
checking every single equation. When reading a paper, don’t get bogged down by
all the equations. Read it once or a few times before getting down to that
level. Often it’s helpful to read other descriptions of a particular
algorithm, for example in a student’s thesis, which contain more detail and
contextualize some of the math.

While you may not find this comment particularly helpful, as I’m not pointing
to a guide or something, you could take away from it that it takes practice
and that one shouldn’t be discouraged when you don’t understand the math in a
paper, as I guarantee you there are maths professors that couldn’t make sense
of it either.

~~~
Koshkin
> _it’s just there to show off_

That is true, unfortunately, in much of the mathematical literature. Way too
often authors use the formalized language where the plain language would
suffice. This kind of abuse is extremely widespread. It is one thing when a
formula is used to represent, say, a complicated integral; it is another when
a formula is used to express something just as easily said in a couple of
words.

~~~
SomewhatLikely
When I first began writing papers I tried to describe using prose. I found
that readers were confused and there was much more ambuigity than I
anticipated. By using mathematical notation, it becomes a lot less likely that
you are misinterpetted and often takes less space. It's kind of like the
difference between providing code versus a written description of the
algorithm.

~~~
kazinator
The problem is assuming that the reader is already familiar with the notation,
and consequently not explaining it.

~~~
ssijak
If it is a mathematical paper it is targeted at other mathematicans eho should
be able to understqnd the notation.

~~~
MRD85
My math professors would teach maths under the presumption we knew
mathematical notation. I felt I got thrown in the deep in during first-year
calculus units but I also now feel strangely adept at reading maths so it had
a positive effect.

------
tomrod
I've been reading mathematical notation for decades. You face three things
I've noticed:

1\. People are typically _really_ bad at writing mathematics -- notational
brevity is not a virtue when attempting to communicate ideas. You may find
value reading through notes on Knuth[0] and what he taught regarding writing
mathematics.

2\. There is a certain level of field/conceptual awareness needed when
translating the coded concepts in an academic paper/textbook. However,
consistent with (1), people are bad at encoding. Using several topical texts
at your disposal can help. For example, in econometrics your standard masters
year 1 texts are Greene, and Wooldridge. Wooldridge is expansive and simpler
to read, Greene is more fundamental and uses horrendous notation. I found
reading the same topics in Wooldridge helped me decode Greene, from which I
was able to deepen my understanding of Wooldridge.

3\. Very few people can read a paper or text once and know it immediately. The
most studied professors I know will take months to fully digest a seminal
article that incorporates new ideas -- if you're new to a field, _every_
article is seminal to you.

Don't give up. It takes practice to decode, to put the concepts into a
mentally straightforward order, and so on. The fact you're asking about it
shows that you care, and if you let that grow you'll get to the point you want
to be.

[0] [http://jmlr.csail.mit.edu/reviewing-
papers/knuth_mathematica...](http://jmlr.csail.mit.edu/reviewing-
papers/knuth_mathematical_writing.pdf)

------
cowboysauce
I think you've got this backwards. You don't need to understand the notation,
you need to understand the math. If you don't understand the notation then you
probably don't understand the math in the first place so learning notation
won't help. Learning how a matrix is written is useless without understanding
what a matrix is.

You need to find out exactly what type of math is used in the paper and learn
that. Learning notation will come along with that. If you're interested in
machine learning then you should probably start with:

* Proofs and logic (a general requirement to understanding math) * Linear algebra * Statistics * Some multi-variable calculus

~~~
lucb1e
Partly, perhaps. For me, a large part is the notation. I know conditionals, I
know loops (as in for loops), I know SQL... but if you show me logic, set
theory, a sigma, or, uh, what's this sigma but for multiplication, I wouldn't
have known what you meant a few years ago. It's not only understanding the
math, it's also the notation.

I still don't know how to read things like curly braces or matrices or a dozen
other things that I don't even know the name for, and figuring it out is
usually more work than it's worth. As a result, I don't understand Wikipedia
articles that are supposed to be introductory and understandable to the
masses, but because the article's writers are being fancy, it's only
understandable for a privileged few. Often the concepts, if I do bother to
understand them or look them up elsewhere, are rather simple.

It also doesn't help that people use things incorrectly or ambiguously.
Sometimes a line over some math will mean the average (or "mean", to use the
ambiguous lingo) and sometimes it's an infinite repetition.

Or the lingo: a friend of my girlfriend was baffled I didn't have X in school
and didn't know what it was (I think it's derivations). She kind of explained
it to me and I kind of understood, not recognising it. Later that evening I
used it, unknowingly, in programming. When showing my girlfriend what I made
she pointed it out to me.

Heck, I recently used logarithms fairly intuitively where they were never
properly explained to me. I just knew the formula for entropy calculation
(log(possibilities)/log(2)) by heart and managed to make something that
converts a number from any base into any base (e.g. base 13 to base 19). I can
do math, just nobody bothered to tell me about the fancy symbols. To me, the
symbols mainly seek to obfuscate, look smart, and ensure job security.

~~~
sn9
Pretty much every math textbook I've read has defined the notational
conventions early on.

(I'm not saying this is the case with every math text, just the ones I've read
that are intended to teach the reader the subject.)

I see all these comments about reading papers with confusion and I get the
impression that they didn't read the prerequisite information from textbooks
for whom they're the right audience as that's generally how you pick up
notation.

------
ashelmire
I've taken reasonably advanced math courses (I was, for a time, a math major),
and I have a great deal of difficulty with machine learning papers. They often
reference domain-specific formulas that are prerequisites, sometimes without
naming them, and rarely does a paper actually define every symbol used in a
formula (in fact, I don't think I've ever seen a paper do this - it's usually
none, and they only define a few elements of the notation). The worst part is
that they'll use a known formula, but change the notation slightly without
explaining why.

These are bad academic habits. If you're using a formula with a dozen
variables - at least state what they are, as well as any more unusual or
overloaded notation, subscripts, etc. If you're using a formula from
elsewhere, it's better to restate what these symbols mean in your paper,
rather than send the reader off on a hunt. Be clear, be explicit, leave no
doubt. You don't have to explain all of probability theory, but it takes only
a line or two to vastly improve the readability of your paper.

~~~
iguy
Machine learning papers are a particularly toxic brew of people in a rush, who
like to show off, and who spend all day fiddling with code & little time
reading or writing mathematics. Lots of formulas for quite simple things are
at best ambiguous and at worst simply incorrect.

------
dragon96
My recommendation is to focus on these two items and ignore everything else
until then:

1) Logical quantifiers: ∃ ("there exists") and ∀ ("for all"). Quantifiers can
get confusing when they get strung together. I like to think of them as
challenge-response games. For example, your real analysis textbook asserts
that a function f is continuous at x if ∀ e>0, ∃ d>0 such that if |x0-x|<d,
then |f(x0)-f(x)|<e. I think of two players, Ella and Daniel, with the ∀
player (Ella) trying to _disprove_ and the ∃ player (Daniel) trying to _prove_
the statement. Since the definition asserts "∀ e>0", all Ella needs is a
single counterexample e' such that no matter what d'>0 Daniel chooses, the
condition is false. Or mathematically written, Ella's objective is to prove "∃
e'>0 such that ∀ d'>0, the condition doesn't hold." Notice how taking the
converse flips the quantifiers.

2) Set notation: S = { x∈R | 0<x<1 } reads as "S is the set of real numbers
that are between 0 and 1". You can also think of this as {x ∈ domain |
filter_condition(x,y,z,...)}. You'll see many mathematical objects represented
as sets, so it's pretty important to know how sets are defined.

~~~
kraitis
>challenge-response games

This is standardly called "game-theoretic semantics" in the literature.
Enthusiasts can find more info here (or just by Googling around):
[https://plato.stanford.edu/entries/logic-
games/#SemGam](https://plato.stanford.edu/entries/logic-games/#SemGam)

------
jjgreen
From outside, one might think that the notation is universal and unchanging,
nothing could be further from the truth: each field has its own take, its own
conventions. One even finds groups within a field who "go their own way", and
there are national differences too. Sorry.

Realistically, find a paper that interests you, hopefully they will list some
expository text in the references (or a paper that they cite does). Get it and
read it. There's not really any alternative if you want to understand the
notation, it represents complex ideas compactly, so you need at least a basic
grounding in those ideas.

------
JesseAldridge
[https://en.wikipedia.org/wiki/List_of_logic_symbols](https://en.wikipedia.org/wiki/List_of_logic_symbols)

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

These might help a bit.

But as someone with similar problems, I'm beginning to think there's no real
solution other than thousands of hours of studying.

~~~
williamstein
"There is no Royal Road to Geometry." \-- Euclid

I've spent time studying number theory (Ph.D. Berkeley, wrote 30+ papers and 3
books), and it really is very deep. If understanding some notation or
mathematics doesn't come easily to you, that's normal. It often takes Ph.D.
students years of fulltime study just to understand a single research paper.
This is because mathematics is a very deep subject, certainly much deeper than
everything else I've encountered in academia. The good part is that pretty
much all mathematics does make sense, and can be truly 100% understood if
you're willing to invest enough time, unlike the case with many other things
in life! An added bonus is that much of mathematics is also incredibly
beautiful, when you understand it.

Listening to lectures by excellent speakers (many are on youtube now) helps a
lot.

~~~
rocqua
> pretty much all mathematics does make sense, and can be truly 100%
> understood if you're willing to invest enough time

Inter-universal Teichmüller theory seems to be a counter-example.

~~~
williamstein
To me it is an example, not a counterexample, because the IUT papers have been
completely understood and judged to be incorrect by recent fields medalist
Peter Scholze (and several other top mathematicians).

------
elf_m_sternberg
I recently wrote an article called "So You Want To Get Into Theoretical
Computer Science," which is mostly about how to read a bunch of papers to
figure out some obscure, heretofore unimplemented (outside of academia)
algorithm.

The biggest challenge was reading the mathematical notation, to whit: the four
biggest papers in the field I was studying used _four different_ notations,
and that you'll end up writing your own translation guide:

[https://elfsternberg.com/2018/10/27/so-you-want-to-get-
into-...](https://elfsternberg.com/2018/10/27/so-you-want-to-get-into-
theoretical-computer-science/)

------
ColinWright
I don't know of any easy way to learn to do this from the outside without
coming up through the ranks (as it were). The only thing I can suggest is to
get a few examples, and ask some friends, colleagues, or on-line forums what
they mean. Pick one or two, and ask in different places. The level of
assistance you can get from a well asked question in the right place can be
extraordinary.

But be aware that mathematical notation is not, without context, unambiguous.
So you might need to provide some context for what you're doing, or where you
get the examples from. Some people think this is a weakness of mathematical
notation to be solved, but I've found it tremendously powerful. There is no
doubt, however, that it is an accidental barrier to entry.

So post a picture of a piece of notation, and maybe someone will either
explain it, or point you at a book/website that you can read.

~~~
cursorial
Ah, I see, so it's just a matter of reading something, not understanding it,
figuring out what it means and learning from that rather than there being a
place where I can specifically read up on it.

I guess I just need more practice.

~~~
ColinWright
> _... so it 's just a matter of reading something, not understanding it,
> figuring out what it means and learning from that rather than there being a
> place where I can specifically read up on it._

Not entirely ... but partially. That's one way you can start to learn. Usually
formulas are presented with a broad gloss in English (or other) to let you
know roughly what it is saying, with the formula just being a precise way of
saying it that can subsequently be used in algebraic manipulations.

Example:

The force due to the Earth's gravity is represented by _g_ at the surface, and
falls off as an inverse square. Thus:

F_d = g(R/d)^2

where R is the radius of the Earth.

------
lmilcin
Well... perseverance is the name of the game.

The secret to reading papers in unfamiliar field is to not get discouraged
just because you don't get it immediately. Instead, I will research each
concept I encounter that I don't understand well.

Let's assume I want to understand something in a field I am not familiar with.
It is important to understand that this typically means I have to get at least
somewhat familiar with the field. I don't expect to read a paper and get it
just from that single read!

I will open the paper and first just read what I can from start to end without
stopping to research anything. This gives me some idea what the paper is about
and what are some of the concepts that are important and how they are used
later. I call this scanning.

I will go back to the start and slowly go researching EVERYTHING, ABSOLUTELY
EVERYTHING that I don't understand. This is most of the work. I will keep the
paper open and have something to mark where I currently am. I will research
meaning of every symbol and every piece of notation, every concept being
introduced, until I am confident I understand it.

This may take multiple days to advance a single line as I will frequently
spend time on tagents or just getting up to speed with the basics.

Once I go through entire paper I will get back to start and now read it again.
Now that I know the language of the paper I can focus on the matter at hand. I
may do this few times until I am satisfied I got everything I could.

Again, I want to stress it, don't expect to understand a paper in unfamiliar
field on the first reading. It is completely unrealistic. Use materials, ask
people to explain, research on the Internet, rinse and repeat.

------
rwilson4
I think “mathematical maturity” is an important aspect of this discussion.
Mathematical maturity is basically the ability to read a paper or textbook
without assistance. It takes years to develop, and you still have to be
familiar with the topic! For example, I feel comfortable reading statistics
papers, but I would be totally lost reading number theory. As others have
commented, familiarity with the foundational ideas is just as important as the
notation.

So how do you get mathematical maturity? A bachelor’s in math, or equivalent.
Meaning, approximately 4 years of focused, ideally guided, practice. I don’t
think there are any faster ways.

Edit: Also important to know: most papers are poorly written, so that
definitely doesn’t help. It’s especially important early on to identify who
the leaders are in a field, and focus on their writings. In statistical
Machine Learning, I recommend anything by Trevor Hastie, Rob Tibshirani,
Bradley Efron, and Martin Wainwright.

------
gmiller123456
Understanding the notation is only the first little step in understanding the
underlying principles. There are a lot of machine learning books/classes that
will cover the underlying principles you need to understand what's covered in
the book. Exactly which one to recommend depends on your background, but
Andrew Ng's Coursera Course [1] has been a staple for people without a lot of
math background.

[1] [https://www.coursera.org/learn/machine-
learning?authMode=log...](https://www.coursera.org/learn/machine-
learning?authMode=login)

~~~
sqrt17
Ng's Coursera course will teach you what happens in Deep Learning and Machine
Learning, but at least the Deep Learning course is very very light on the math
side and avoids scary mathematics rather than making it accessible.

Without having read it, I'd recommend the "preliminary maths" part of Bengio
et al's Deep Learning book - it teaches both the letters and the language, so
to speak, and if the language isn't for you, you'd better throw away the
papers and completely concentrate on reading and understanding the
implementations that are our there, using the implementation first and
foremost and the paper only as a backup to provide explanations when the
implementation does mysterious or unexplained things.

You can do deep learning productively without having a PhD, but you won't be
able (nor obliged) to read and understand PhD-level academic papers unless you
have a solid (i.e., maths or physics or math-rich CS BSc) maths background.

------
jg-prog
[https://github.com/Jam3/math-as-code](https://github.com/Jam3/math-as-code)

this might be a start for you

~~~
symplee
Nice, thanks. Do you know if any math textbooks have been translated into code
notation?

------
daniel-levin
I think the lazy and somewhat cynical answer is also the most accurate: learn
the mathematics.

~~~
nf05papsjfVbc
Indeed. I was thinking that the question is in some ways like asking "How can
I understand the chemical symbols used in these equations". One could perhaps
memorise that 'Na' is Sodium and H is Hydrogen but without knowing what they
are and how they behave etc. an equation in isolation is meaningless and won't
help to understand the reaction.

------
y7
While there is usually some notation that is implicit, and where the meaning
depends on the particular field, most papers should explicitly define their
notation.

Are you sure it's a problem of notation, and not just that you're not used to
reading slowly? Reading academic papers is very different from reading prose,
and I find that even though you might struggle understanding it at first read,
going through it line by line very slowly does help a lot.

Do you have an example paper you're struggling with?

~~~
vanderZwan
Not OP, but in my experience experts everywhere have a blind eye to the
conventions they are so used to that they end up skipping giving a definition
for them them.

------
pps
I'm also interested in this topic, here you can read some interesting stuff:

"Links to resources talking about how to understand mathematics, mathematical
language and mathematical notation."

[https://github.com/nbro/understanding-
math](https://github.com/nbro/understanding-math)

------
yhoiseth
Have you tried using a tool like Anki? Previous discussion:
[https://news.ycombinator.com/item?id=17846356](https://news.ycombinator.com/item?id=17846356)

~~~
ColinWright
While I agree that the "Spaced Repetition" technique is phenomenally powerful
- I've used it to memorise Ozymandias, the entire periodic table, English
Regnal dates, and more - there is very little point in memorising what the
symbol "is" (that's a signum, that's a small delta) because what a symbol
"means" changes from discipline to disciple, paper to paper, and author to
author.

------
AndrewKemendo
Frankly, I don't think you can without it being specific to the field you're
learning. That's why having a symbolic reference is always amazingly helpful
in textbooks.

For example, the lowercase sigma symbol is used for a half-dozen different
things depending on the field you're working in.

In Machine learning, if you use a logistic function as your activation
function you'll use the lowercase sigma symbol to represent the sigmoid
variable. However if you're working in Physics, the lowercase sigma symbol is
the Stefan-Boltzmann Constant. So any good text or otherwise will explicate
what the function symbols represent.

Some algebraic/calculus functions such as sum over set, divisor, sets
etc...should be the same, and I haven't been able to learn how to crack that
specific question as to generalizing all of those operations that isn't "go
back to get a mathematics degree."

[1]
[https://en.wikipedia.org/wiki/Activation_function](https://en.wikipedia.org/wiki/Activation_function)

[2]
[https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_const...](https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_constant)

------
kabdib
Many papers are just badly written. I've lost count of the number of times
I've been reading formula when the author suddenly starts using terms that are
not defined in the paper.

Okay, if you're reading a physics paper and you encounter 'c', that's probably
the speed of light, 'm' is likely mass, and so on. There's context and culture
involved, and that's usually fine.

But if you're wading through something less mainstream, say some denotational
semantics or queuing theory, and the author starts dragging in undefined
alphabet soup, what's the reader supposed to do? Sometimes the answer is to
get more culture in the subject area, or read some of the same author's
previous papers, or just forge ahead in the paper and hope that understanding
will come anyway.

Don't get me started on formula that use single character variables nearly
everywhere, but have zingers like 'hz' \-- which is not 'h' times 'z', but
Hertz. If math is supposed to be so rigorous, how come there's no agreed upon
grammar, with reasonable ways to extend it to specialized domains?

------
inverse_pi
There's a long version and a short version. The long version is you have to
learn to write mathematics by yourself. Start with an intro course and start
deriving theorems by yourself. Do not look at the proofs. At this stage,
details are very important and can't be overlooked. You need to be your own
critic and keep asking why and how to every single detail and step until you
can convince yourself that you would be able to naturally come up with the
theorem and proof. Continue doing this to higher level courses. This is how I
learned Math since middle school all the way throughout graduate school.

The short version is you have to ask the right questions. Naturally for every
theorem or equation, there are 3 big questions:

1) What does the theorem/equation say? What's the intuition behind it?

2) Why is it true?

3) How does one come up with it?

One must ask these questions in the exact order. To understand what the
equation really means, you should break it down further to smaller components.
What is this variable? What does it represent? What is the intuition behind
what it represents? What's the implication when the variable increases,
decreases, etc? Do that for every single component in the equation/theorem.
One should fully understand the intuition and clearly describe all quantities
before trying to look at the equation/theorem as a whole.

To understand why an equation/theorem is true you need to build up a
repertoire of theorems related to the quantities of interest. The bigger your
repertoire, the easier you can prove or disprove something. The more advanced
way is to build up intuition around the quantities of interest then come up
with intuitive hypotheses. The hypotheses are often easier to prove/disprove.
The process repeats.

~~~
earlgray
This answer matches my experiences very closely. When you feel you brain
begging you to gloss over the equation and move on - that's a red flag that
you need to slow down, exercise discipline over your concentration, and figure
out what's going on. It might take a few days for the intuition to settle.
Personally I've known it to take a couple of years.

edit: patience, self-forgiveness, and a willingness to accept frustration are
important traits. You might spend a whole week banging your head against the
wall, feeling like you're making no progress, and then one day everything
falls beautifully into place. That doesn't mean you did something correctly on
the final day - it means you did everything correctly for the whole week
before. Don't view a difficult and unrewarding day as wasted time. You're
building something very difficult and that takes a bewildering amount of time.

------
mindcrash
Since you say you know how to code:

[https://github.com/Jam3/math-as-code](https://github.com/Jam3/math-as-code)

You are welcome.

~~~
pier25
Woah, this is good.

Math is super expressive compared to actual code. It's like SQL compared to an
ORM like notation.

------
starbeast
I run into the same problem, am much better at understanding maths stated as
code than in mathematical notation.

3blue1brown's animated maths channel is pretty good. -
[https://www.3blue1brown.com/](https://www.3blue1brown.com/)

Khan Academy's maths section covers a lot of stuff -
[https://www.khanacademy.org/math](https://www.khanacademy.org/math)

math.stackexchange is excellent for specific questions -
[https://math.stackexchange.com/](https://math.stackexchange.com/)

One thing that works for me is to try and find something stated both in code
and in maths notation, then you can work out one from the other.

~~~
imjustsaying
I always wondered why math notation hasn't been replaced by code.

While math notations can be interpreted in different ways, code is going to
give an unambiguous result.

Why don't mathematicians, especially those writing in machine learning or
computer science domains, do this? Is it just a problem of agreeing on a
common language?

~~~
yesenadam
You mean, "Why don't they use a language you understand, instead of one you
don't?"... Why expect they should?

I don't think _math notations can be interpreted in different ways_ usually is
right. That would defeat the purpose of using them, if the meaning wasn't
exactly clear. And maths notation is more compact, a lot more. Even assuming a
translation between the 2 always is possible or makes sense.

~~~
fooker
If math notation was unambiguous, automatic theorem provers would take that as
input instead of inventing their own languages.

------
lucideer
I don't think it's possible to learn "mathematical notation", as a thing. This
is like asking "how can I learn human language".

There's different notation per field and subfield but I would argue that the
process of figuring out notation per example or per paper is generalisable and
not too difficult:

\- identify the field of the paper; keywords, general categorisation, etc.

\- take an example and break the notation into parts

\- Google each of the parts independently alongside field keywords (results of
this search should also give you some contextual info alongside each component
which should help your knowledge)

\- compile the aggregate of your research

\- reread the paper and see if your understanding has improved

\- repeat

\- try another paper in the same field

------
13415
1\. Try to get a mentor.

During my time as a PhD student in philosophy, I had the luck of having a
mathematician and physicist turned philosopher as my supervisor. He helped me
a lot in understanding notation and improving my own mathematical writing
style (~ getting silly notation ideas playing around with LaTeX out of my
head).

Notation heavily relied on conventions and these differ from field to field.
Mathematicians are usually very clear and define everything. They have to be
easy to understand, because what they are talking about gets very complicated
very fast. Unfortunately, in other areas like logic-oriented theoretical CS
there is an unhealthy focus on 'notational precision' (plus shortcuts, making
things worse) rather than being understood that can make papers rather
incomprehensible at first.

2\. Look for textbooks with good mathematical prose.

Get some recommendations. Some good introductory books are relatively short,
self-contained, define everything, and avoid symbols in favor of English in
many places. Try to get one of those in the area you're interested in.

3\. Check your reading style. (kind of obvious advice that is often given, but
maybe still helpful)

You may be reading the texts in the wrong way. I did that very often and
occasionally still do. Reading a mathematical text takes way more time than
any other text and requires active participation. Do the exercises and stop
every so often to check your understanding. Constantly use pen and paper and
check everything. Draw diagrams.

------
RossBencina
I'm still learning mathematics (I don't think the learning phase ends). I used
to worry about "reading" notation. I think what's changed is that I have
gotten used to not understanding unfamiliar notation. And I realize that if
the notation appears unfamiliar, the mathematics may also be unfamiliar. Being
able to "grok" notated mathematics is much more about understanding the
mathematics than about being able to read notation.

Therefore I would be cautious about jumping to the conclusion that your
problem is as simple as "simply cannot read" (unfamiliar notation). Maybe
that's true in some cases, but it's also likely that the notation stands in as
a kind of shorthand for elaborate, maybe even arcane, concepts that you don't
understand very well (unfamiliar mathematics). Working out which
concepts/theories you need to study may be a better place to start than
worrying about the notation per-se. Eventually you'll drill down to some level
where you do understand the notation, or it is explained in a way that you can
understand. Also consider that learning to read and write go hand in hand --
reading a notation gets much easier once you start writing your own sentences
in the same notation (e.g. do exercises, learn to write proofs.)

With few exceptions, don't expect the notation in one area to be useful, or to
mean the same thing, in another area. However there are a couple of generally
useful things to know: (1) Greek letter names (so you can recognize, write and
pronounce them without being confused) -- just learn the ones that show up in
your reading, and (2) set theory notation plus some basic set theory (read
Halmos' Naive Set Theory book up to the point that it becomes confusing).

Others have mentioned it, but be aware that it is common for one field to have
multiple notation conventions. Often the notations originate from seminal
papers or widely used books. Different authors frequently use slightly
different notation, even within the same field, and sometimes a single paper
may contain multiple contradictory notations. I've even had lecturers who
switch notation half way through a lecture. If you're studying multiple texts
you may want to translate all of the key theory into one consistent notation
-- but at a minimum you need to be able to keep track of the correspondence
between the different notations as you're reading.

------
nprescott
I was similarly stumped, specifically regarding discrete mathematics not too
long ago. I was interested in reading Lamport's Specifying Systems[0] but felt
I lacked some requisite background knowledge[1]. I ended up picking up
Stavely's Programming and Mathematical Thinking[2] which I enjoyed.

While I would recommend either as a primer into discrete math - do you have a
specific subset that you're interested in? That might make suggestions more
pertinent to your interests.

[0]:
[https://lamport.azurewebsites.net/tla/book.html](https://lamport.azurewebsites.net/tla/book.html)

[1]: I think, in retrospect, that Specifying Systems actually stands on its
own pretty well and doesn't require supplementary material when paired with
the video course.

[2]: It is subtitled "A Gentle Introduction to Discrete Math Featuring Python"
and it is _very_ gentle, but I'm also glad to have read it, it did wonders for
giving me a kind of reference into math terms from programming (which I
already knew).

------
jeremymcanally
I've been eyeing this book to help bridge some gaps in the same area for me:
[https://amzn.to/2OZUIlE](https://amzn.to/2OZUIlE) It talks more fully about
mathematical proofs in general, but it gives a really good introduction to the
notation and such in a much more compact space than most other ones I've seen.

------
diffeomorphism
Mathematical notation is not anything magical and mostly there to allow for
easy discussion. Also equations are only a small part of notation and words
and definitions are much more important, e.g. "isomorphism".

Since mathematics is abstract anything used necessarily has be precisely
defined. Thus, the best way to learn mathematical notation is to get some
beginner level rigorous mathematics book for mathematicians, e.g. Rudin's real
analysis or Lang's Linear Algebra, and read through chapter 0 and 1. Books
like these tend to start entirely from scratch: What is a number? What does
'+' mean? What is a sum symbol?

While each field tends to have some further notation, that is often explained
within the paper or in standard references and you should be able to read
mostly everything after reading up on basic notation sections in analysis,
linear algebra and maybe category theory.

------
hashr8064
I was just like you. Here's what I did.

1\. Picked up a High School Algebra Book. Read from beginning to end and did
all exercises.

2\. Repeat #1 for Algebra 2, Statistics, Geometry and Calculus. (Really
helpful for learning those topics fast was Khan Academy).

3\. Did MIT Opencourseware's Calculus and Linear Algebra Courses w/the books
and exercises.

Now, this took me about 2 years maybe you can get it done quicker, you're at a
level where you can pretty much pick up any book, I think I picked up Elements
of Statistical Learning, and actually start parsing and understanding what the
formulas mean.

One thing I always do is tear apart formulas and equations and play with
little parts of them to see how the parts interact with one another and behave
under specific conditions, this has really helped my understanding of all
kinds of concepts from Pearson's R to Softmax.

------
partycoder
Or you can have syntax highlighting in your brain, like Feynman:
[https://en.wikipedia.org/wiki/List_of_people_with_synesthesi...](https://en.wikipedia.org/wiki/List_of_people_with_synesthesia#Richard_Feynman)

------
gomangogo
Read this book : [https://www.amazon.com/Mathematical-Methods-Linguistics-
Stud...](https://www.amazon.com/Mathematical-Methods-Linguistics-Studies-
Philosophy/dp/9027722455)

------
newprint
I'm going through the books - 1.How to Prove It: A Structured Approach;
2.Mathematical Proofs: A Transition to Advanced Mathematics The are very
different books on how to read/write proofs, but complement each other nicely.
On top of that, I'm doing though few elementary logic and set theory books,
they also have what I would call "barebone proofs", since logic/set theory
don't even deal with numbers. My goal a very different from yours, in my case
I want to learn real and complex analysis and move to quantum physics, but
nevertheless, two books mentioned above are helping me a lot.

------
peakai
This used to stump me, but I went through the Cal1/2/3 series at a local
university and it made reading ML papers much easier, you could try attending
a local school if self-learning is proving too time consuming for you.

------
jlg23
Some things that help me a lot, in order of helpfulness (for me):

* Read formulae aloud, using natural languages for symbol names (it is not "sigma of k from i to j", it is a "summation of terms with k from i to j").

* Look at the proofs/construction of these formula and understand them.

* Look at translations of the formulae (applied physics textbooks, more often program code); sometimes it helps to work your way back to the formula from a starting point you are more comfortable with.

* Understand that making up symbol meanings ad hoc is one of the perks of mathematicians - you can find fights within a single math department on how notation should be.

------
Zolomon
[https://www.khanacademy.org](https://www.khanacademy.org) is a good place to
start. Otherwise, you can take a look here
[https://en.wikipedia.org/wiki/List_of_mathematical_symbols](https://en.wikipedia.org/wiki/List_of_mathematical_symbols).
There are free calculus courses available on
[https://coursera.org](https://coursera.org) that you can start on, otherwise
a local introductory math program might be available to you?

------
deytempo
The real problem is people who understand machine learning don’t know how to
describe it in terms of anything except math. You don’t really need math to do
it just like you don’t need math to understand neurons or how they function.
Every concept in machine learning can be demonstrated in simple programming
without any math. Why it is not is because most people who are doing it at the
experimental level are at a university and they don’t start learning it until
they already have a tremendous math background but to say it requires math is
ridiculous

------
bllguo
You have to spend time to build your mathematical foundations. You have to get
familiar with it through problems and exercises. I don't think there are
shortcuts.

If you don't have the foundations it's wasteful to spend time trying to work
out academic papers and such. Bite the bullet and start from the beginning,
it'll pay off over time. It will also allow you to go much farther. Even if
you manage to 'understand' what a paper is saying, without the background you
won't be able to do much other than replicate it.

------
erwan
Hard to answer without knowledge of prior background or exposure that you
might have.

As a rule of thumb, I would advise to start with elementary linear algebra,
statistics, and probability. Edit: As pointed out by a reply comment, these
require a good grasp of calculus.

Notation is only syntactic, most importantly you want to understand the
_semantics_ (i.e the significance of the construction your are studying). To
achieve this, you need to do the proper background work and the rest will
follow naturally.

~~~
arsalanb
This. I mean you can google stuff like "what does the curly L mean" and it
would tell you that it's a Lie derivative, but that wouldn't mean much to you
unless you understood that it did.

So you need to understand the work and the math behind it and you will get an
idea about the notations used. Not the other way round.

------
terrycody
I recommend you read this article about how to learn CS and become a computer
scientist or machine learning expert, from 0 ground.

In the article, author recommend u have to have a very strong knowledge base
of math, statistics, probability, and linear algebra in order to read those
things.

[https://www.afternerd.com/blog/learn-computer-
science/](https://www.afternerd.com/blog/learn-computer-science/)

------
idclip
my advice is learn haskell of youre a programmer.

haskell helped me think of math as a game of pacman, i understood arity, the
concept of "well defined" as ADTs and FOLDs, tail recursion helps understand
series.

learn you a haskell is a better math book than it is a haskell book.

i had extreme sifficulty with math, mainly because my temporal lobe is more or
less mush due to MS.

then all you have to do is substitute symbols.

oh and the #haskell channel on freenode is huge and VERY helpful.

Sent from my iPhone

------
bogomipz
The following is a decent and reasonably priced reference:

"Mathematical Notation: A Guide for Engineers and Scientists"

[https://www.amazon.com/Mathematical-Notation-Guide-
Engineers...](https://www.amazon.com/Mathematical-Notation-Guide-Engineers-
Scientists/dp/1466230525)

------
max_
I recommend you to have a look at "Hand Book for spoken Mathematics"

[http://web.efzg.hr/dok/MAT/vkojic/Larrys_speakeasy.pdf](http://web.efzg.hr/dok/MAT/vkojic/Larrys_speakeasy.pdf)

------
anxtyinmgmt
The first few chapters of
[https://courses.csail.mit.edu/6.042/spring17/mcs.pdf](https://courses.csail.mit.edu/6.042/spring17/mcs.pdf)
provides some good pointers on notation.

------
Koshkin
> _How do you learn to read this stuff?_

You learn while studying math. There is no way around it.

------
eximius
Other people have listed great books and resources.

What I don't see, however, is to ask a friend. Find someone who is familiar
with the notation, if not the material, and ask questions. You'll likely get a
more clear answer.

~~~
ColinWright
You say:

> _What I don 't see, however, is to ask a friend._

From
[https://news.ycombinator.com/item?id=18510564](https://news.ycombinator.com/item?id=18510564)
:

> _... get a few examples, and ask some friends, colleagues, or on-line forums
> ..._

~~~
eximius
My bad. Guess I didn't see it.

------
cpach
Perhaps Khan Academy could be useful?
[https://www.khanacademy.org/](https://www.khanacademy.org/)

------
empath75
To me it’s exactly the same as learning to work with an unfamiliar code base.

Just pick some small part of it and start tracing back until you find
definitions.

~~~
deytempo
Yea except you are supposed to compile it and run each line and iteration in
your head

------
GChevalier
For maths and machine learning, Khan Academy and Coursera can get you
somewhere.

------
earlgray
Could you give some examples of notation that you struggled to understand, and
explain what about it you find confusing?

One difficulty of notation is that the hierarchy of abstraction builds
dizzyingly quickly, and soon you're manipulating symbols that generalise a
whole classes of structures that were themselves originally defined in terms
of lower-level abstractions. When this becomes overwhelming, it usually means
that I didn't give my understanding of the lower levels long enough to settle
and mature.

Concise notation and terminology is only useful if the underlying ideas are
organised neatly in your mind, and the best way I've found of achieving this
is to study a subject obsessively for some time, then put it away for a few
weeks, and then go back and try to see the big picture and find out where it
doesn't fit together by trying to derive the main results from scratch. Then I
start again and fill in the blanks. After a few years things begin to make
sense, but this process takes time and it's difficult and tiring (or at least
that has been my experience of it).

In order to read research papers fruitfully it's crucial that you understand
the basics well, and the best way to do that is to work through books aimed at
undergraduates or young graduates. People don't read foreign literature by
jumping straight in and looking up every word and every grammatical
construction as they go. They become familiar enough with the language by
reading easier texts until the language is no longer an obstruction - then
they're free to appreciate what's happening at a higher level. The same for
driving: you wait until you're comfortable operating the car mechanically
before you drive on busy roads. The same, also, for mathematics.

It is not at all unusual to to find notation and technical terminology tiring.
Everyone does to some extent. I hate it. But it's necessary.

Some resources I found useful:

Naive Set Theory by Paul Halmos. One of the great mathematical expositors,
Paul Halmos here describes the fundamental language of mathematics: set
theory. This is a book for people who want to understand enough set theory to
do other parts of mathematics without obstacle.

How to Prove It by Daniel Vellemen. A nice introduction to logical notation
and common proof structures, aimed at helping incoming maths students to
become comfortable with the basics of formal language and notation.

Anything by John Stillwell. Stillwell is an inspiring teacher who insists on
including the practical and historical motivations for the abstractions we use
(this is, sadly, rare for modern teachers of mathematics). If you find
yourself wondering why people cared about a problem enough to solve it,
Stillwell might be able to help.

I suspect you'll also need resources on linear algebra (Halmos has 'Finite
Dimensional Vector Spaces') and analysis but I'm not sure as I don't work in
machine learning. I just sort of learnt linear algebra as I went and I avoid
analysis as much as a supposed mathematician can. (context: my undergraduate
degree was in economics and didn't carry much mathematical content other than
some basic linear algebra - now I'm a graduate student in mathematics and
computer science who uses a lot of category theory and abstract algebra. The
transition was painful. Really painful.)

------
eldavido
I love how there are these two parallel threads going in HNville: (1) COLLEGE
SUCKS BURN IT DOWN and (2) "I'd like to learn complex things like machine
learning and math that take more than reading 5 blog posts to master, how do I
do it"?

Sometimes this place just cracks me up.

I honestly think the answer is pretty simple: go to college. It doesn't have
to be expensive. Take a community college course in calculus or undergraduate-
level probability. Skip the gen eds and don't worry about the degree if you
want to learn something narrow like this.

In any case, just find a mentor. On-the-job if you can, otherwise pay for a
class.

What you shouldn't do is try to self-study by reading a book. You can perhaps
do this but only if you're smarter than average and more motivated than most.
Since you probably aren't, just take a class. Night school, maybe a MOOC.
Preferably something heavy on analysis or proofs.

But you should do it with others. Math is a very social discipline, it's good
to be able to discuss and have partners to work through things when you get
stuck. And if you're like me, you WILL get stuck on things. This stuff is
_hard_.

Another thought: this whole "college is great"/"college is terrible" dichotomy
seems to occur people people don't think enough about quality. I think bad
colleges are terrible and great ones are fantastic. I don't know any way I'd
have learned all the complex topics in math, stats, probability, etc., I did
without attending a big 10 engineering school (UIUC in my case)

~~~
elhudy
>What you shouldn't do is try to self-study by reading a book.

My experience has been that this is exactly how college math works; you pay
for self-study. The professor reads directly out of a book (in poor English),
or off of pre-made slides provided with the book, for 2-4 hours per week, and
then you are left to do the problems from the book on your own time.

Class populations are so large that if everyone had asked clarifying
questions, we wouldn't have completed the readings.

If your college experience was different I'm envious of that.

Edit: as a fun side-note, our calc professor was well known for taking up the
entire class to draw out a single proof on the chalkboard. As the chalkboard
got full, he would incidentally erase his previous writing with his giant
belly as he putzed across the room.

~~~
admax88q
There are other benefits beyond being able to ask questions.

Tests and deadlines provide motivation to do the actual work.

Having a curriculum means that the content is laid out in a logical order that
the professor believes should be achievable.

There is a stupid amount of information out there. Breaking it down into a
progression that students can follow in order to learn and understand it is
incredibly important.

If you're motivated enough then sure maybe you can just buy and read the
textbook, although sometimes professors deviate from that when it's wrong.

> Class populations are so large that if everyone had asked clarifying
> questions, we wouldn't have completed the readings.

Good thing not everybody asks, and those that do ask are generally asking
questions shared by a good chunk of the class.

~~~
ChuckMcM
> _Tests and deadlines provide motivation to do the actual work._

This is so true in my experience. As a young man I could listen to the lecture
and say "Yeah, yeah, I got this." and then try to work the homework problems
and realize that no, I didn't really understand it. It is hard to force
yourself to do the work if you don't have a negative externality for not doing
it.

------
m1cl
irc/freenode

/j #math

~~~
idclip
hood advice if youre masochistic. expect to be called names and made to feel
stupid.

but in their defense, i was a difficult student.

------
mlevental
here's a dirty seceret: we (people that have mathematical training) don't read
the notation very closely either. it's very much like reading code: i don't
look very closely unless there's a pattern i don't recognize or one that's
broken. so ironically reading notation is about ignoring notation :) the
problem for you is that you simply haven't had enough exposure to know what
the "patterns" are and hence don't have them chunked/mapped efficiently.
unfortunately there's no "royal road" here and you simply have learn to enough
math to know what these things are but i encourage you to skip notation that
isn't intelligible and continue reading. having done that enough times you
will eventually actually understand the notation.

~~~
analog31
I was a college math major, and I can't read math the way I can read text or
even sheet music. I familiarized myself sufficiently by writing out my own
derivations and treating the textbook as a reference while working the
problems. The act of writing the stuff out and manipulating it with my hands
is how I learned it.

~~~
Someone
I think that’s how everybody does it with stuff that’s new to them.

After enough slow progress (“one page per day” can easily be speed reading),
parts of what you are reading become what mathematicians call ‘trivial’, and
your reading speed of similar texts increases.

I think there’s an analogy with ‘reading’ a chess position. If you watch the
ongoing Carlson-Caruana match on
[https://youtube.com/watch?v=DgvqBjrusIA](https://youtube.com/watch?v=DgvqBjrusIA),
you’ll notice that the commenters can easily go through three or four variants
in a minute, and call one position an obvious draw, another clearly winning,
etc. The reason they can do that is that they have looked at thousands of
similar positions, and remember the essential parts of them.

------
BucketSort
This is the beguiling thing about mathematics: it is right there on the page
in front of you, yet often so far from your grasp. How could a statement said
so simply as "a^n + b^n = c^n has no solutions for n > 2" be so difficult to
reason about? This duality of having something and not having it is something
I've only experienced in mathematics and bad relationships.

It's not about understanding the notation, as others have said, it's about
understanding the principles expressed by the notation. You may learn the
grammar of a language, but that is a far journey from understanding its
poetry, which is full of norms and views beyond what is captured by its
grammar.

