Yes, conventions have emerged, people tend to use the same sort of notation in a given context, but in the main, the notation should be regarded as an aide memoire, something to guide you.
You say that you're struggling because of "the math notations and zero explanation of it in the context." Can you give us some examples? Maybe getting a start on it with a careful discussion of a few examples will unblock the difficulty you're having.
One main cause for this belief is that in a programming there is one true noation (or rather, a separate one for each language) that is unambiguous and clearly defined.
I dislike maths notation as I find it lacks rigour.
I was filled with crushing disappointment when I learned mathematical notation is "shorthand" and there isn't a formal grammar. Same goes for learning writers take "shortcuts" with the expectation the reader will "fill in the gaps". Ostensibly this is so the writer can do "less writing" and the reader can do "less reading".
There's so much "pure" and "universal" about math, but the humans who write about it are too lazy to write about it in a rigorous manner.
I can't write software w/ the expectation the computer "just knows" or that it will "fill in the gaps". Sure-- I can call libraries, write in a higher-level language to let the compiler make machine language for me, etc. I can inspect and understand the underlying implementations if I want to, though. Nothing relies on the machine "just knowing".
It's feels like the same goddamn laziness that plagues every other human endeavor outside of programming. People can't be bothered to be exact about things because being exact is hard and people avoid hard work.
"We'll have a face-to-face to discuss this there's too much here to put in an email."
I see this a lot from programmers, but in essence, you seem to be complaining that maths notation isn't what you want it to be, but is instead something else that mathematicians (and physicists and engineers) find useful.
This is such a disingenuous take. How many of the source code files you write are 100% self contained and well defined? I'd bet not a single one of them are. You reference libraries, you depend on specific compiler/runtime/OS versions, you reference other files etc. If you take a look at any of these scientific papers you call "badly defined", did you really go through all of the referenced papers and look if they defined the things you didn't get? If not then you can't be sure that the paper uses undefined notation. If you argue that it is too much work to go through that many references, well that is what you would have to do to understand one of your program files.
If programmers would write code like that (even fortran programmers use 3 characters), noone would be able to understand the code...
I recall my disappointment when as an artist I started studying different maths for use in animation. I would open a book from a university library and expected to find a page with summary of notation used in the book. Maps have this, I would grumble, why not math books?
Yes this is why we all use Hungarian notation and GNU indentation.
And then there is SQL.
[1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
Just to back up this point: In probably every university-level math book I've read, they introduce and explain all the notation used. In the preface and/or as concepts are introduced.
There are lists at wikipedia [1] and other places, but I'm not sure how valuable it is out of context.
[1] https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo...
I wrote it many times already and am bit tired of it, so just a quick summary:
- programmers[1] also use cryptic notation and tend to think in concepts rather than syntax
- nevertheless, programmers spend a lot of time commenting the code, documenting it, specifying it, and so on.
- why can't mathematicians emulate it? What is so wrong about attaching additional few pages to every paper that nobody wants to do it? Pages with explanation of the syntax used, even the common bits. And you know what they could also do? Link to external resources with explanations! But no. This is not happening. Do their PDFs have a size limit or something? Is inserting a link into a paper considered some kind of blasphemy?
I don't know the reason, but in all the discussions on this topic mathematicians almost always underestimate the importance of knowing the syntax. It's much more important for comprehension than they tend to admit. And in the end they do exactly nothing to make the syntax more approachable for newcomers. And then newcomers are out-goers in a heartbeat. It's so obvious that I can't help thinking it's premeditated...
EDIT: [1] Among many others, of course.
Textbooks routinely have a list of symbols and their definitions.
But, from my experience, notation is rarely the problem. I’d bet that the root cause of OP’s frustration is lack of understanding of concepts, not notation. (But, of course, it’s hard to say more without specific examples).
There is a formula with a triangle and I don't get what's that about, for example.
The triangle, for example, is the upper-case greek letter delta, which in calculus represents 'change of'. You might have heard of 'delta-T' with respect to 'change of time'.
In calculus, upper-case delta means 'change over a finite time' vs lower-case delta meaning 'instantaneous change'. The practical upshot, for example, is that the lower-case is the instantaneous rate-of-change at an instant in time, whereas the upper-case is the change over a whole time (e.g. the average rate of change per second for time = 0 seconds to time = 3 seconds).
If you are trying to grok this, I would suggest an introductory calculus or pre-calculus resource. It doesn't have to be a uni textbook - higher-level high school maths usually teaches this. In this particular case, the Khan Academy would be my recommendation because it is about the right level (we're not talking esoteric higher-level university knowledge here) and it is eminently accessable. For example, this link may be a good starter in this instance:
However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle":
t_c = 5 \middot 10^{-5} \sqrt( V / Dt )
... where the "D" I've used is where the triangle appears in the formula.
But that can't be it, because just two lines above it we have:
"For a pulse of width Dt, the critical time ..."
So that's stating that "Dt" is the width of the pulse, and should be thought of as a single term.
So maybe that's the wrong formula, or maybe it was just a bad example. So trying to be more helpful, the "triangle" is a Greek capital delta and means different things in different places. However, it is often used to mean "a small change in".
https://en.wikipedia.org/wiki/%CE%94T
FWIW ... at a glance I can't see where that result is derived, it appears simply to be stated without explanation. I might be wrong, I've not read the rest of the paper.
One place to start is https://en.wikipedia.org/wiki/Impulse_response
https://en.wikipedia.org/wiki/List_of_common_physics_notatio...
1. You're reading a journal article. They will assume you know the notation not just of the broader discipline (e.g. physics/electrical engineering), but of the subdiscipline and at times the subsubdiscipline. Journal papers are explicitly written not to be easy to comprehend by beginners.[1] Notation will be only one problem you'll face.
2. As has been pointed out, this is not a mathematics paper. Mathematicians have their own notation, as do physicists and engineers. As I mentioned in the above bullet, they can have their own notation even in subdisciplines (e.g. circuit folks use "j" for the imaginary number, and semiconductor folks use "i"). There is a lot of overlap in notation amongst these parties, but you should never assume because you know one notation that you'll easily understand the math written by other fields.
3. Most introductory textbooks will explain the basic notation. Unfortunately, I often do find gaps where you go to higher level textbooks and they use notation that they don't explain (i.e. they assume you've seen it before), but is not covered in the prior textbooks.
4. Finally, sorry to say this, but "delta" (the triangle) for representing change is used in almost all sciences and engineering. It was heavily used in my high school as well. If you're struggling with this you really need to read some introductory textbooks in, say, physics.
[1] I'm not kidding. I've spent time in academia and I've complained how obtuse some articles are, and almost universally the response is "We write for other experts, not for new graduate students". One professor took pride at the fact that in his field, one can comprehend only about one page of a paper per day - and this coming from someone who is an expert. These people have issues.
If you're trying to get into signal processing, it'll involve calculus in complex numbers, and knowledge of that is often gained through plodding through proofs and exercises over and over.
No, that belief isn't the problem; that actual status quo itself is obviously the problem. There are numerous notations and authors don't explain what they are using, assuming everyone has recursively read all of their references depth-first before reading their paper.
This isn't a criticism, it's just that notations vary wildly in those areas, and there's lots of cross-over of notations, not all of which agree with each other.
I'm not an expert, but I've had some exposure to the problem(s).
some people literally make the notation up as they go along
Math notation is not math, any more than music notation is music. Notably, the Beatles couldn't read sheet music, and it didn't hold them back.
The best comparison would be is reading someone else's computer code. At its best computer code is poetry, and the most gifted programmers learn quickly by reading code. Still, let's be honest: Reading other people's code is generally a wretched "Please! Just kill me now!" experience.
Once you realize math is the same, it's not about you, you can pick your way forward with realistic expectations.
Also I’ve found the converse true. There are people who can manipulate mathematical symbols very well but actually don’t understand the big picture or general direction. The analogy would be that there are people who can write and read music notes (even transpose to different keys) without hearing it in their head (I was one of them).
If I may humbly add, try making your own notation and playing around with it. Very rapidly one realizes just how hard a problem good notation is.
The other day an EE grad student came to my office to show me his lab's research. After changing notation a couple of times, I was able to recognize that their novel combinatorial problem was an instance of graph coloring.
(define ((Lagrange-equations Lagrangian) q)
(- (D (compose ((partial 2) Lagrangian) (Gamma q)))
(compose ((partial 1) Lagrangian) (Gamma q))))As all the others already told you. you don't learn by reading alone.
[0,ξ[={x|0<=x<ξ}
Which was fun trying to figure out when written in handwriting where ξ,{,} all look the same.
If you can't figure out what it's supposed to be, this equation starts with a half-open interval denoted: [ξ,0[. This notation has some advantages but can be make things hard to read.
Learning the Greek alphabet pays off.
But it isn't just about the notation. You also need to understand the concepts the notation represents, and there aren't really any shortcuts to that.
These days there are online courses (many freely available) in just about every area of mathematics from pre-high school to intro graduate level.
It's possible for a sufficiently motivated person to learn all of that mathematics on their own from online resources and books, but it isn't going to be an easy task or one that you can complete in a few weeks/months.
Your response is to scold him for having the problem he already said he had and instead of recommending resources you told him to go look on the internet.
And you implied he doesn't have motivation.
Now, of course, you have the internet and it can tell you what the square root of 217 is. Consequently, the value of these used CRC handbooks is low and many are available on eBay for a few dollars. Pick up a cheap one and in it you will find many useless pages of tables covering square roots and trigonometry, but you will also find pages of formulas and explanations of mathematical terms and symbols.
Don't pay too much for these books because the internet and handheld calculators have pretty much removed the need from them, but that is how I first learned the meanings of many mathematical symbols and formulas.
You might also look for books of "mathematical formulas" in you local bookstores. Math is an old field and the notations you are stumbling over have likely been used for 100 years, like the triangle you were wondering about. (Actually the triangle is the upper case greek letter delta. Delta T refers to an amount of time, usually called an interval of time.)
Unfortunately, because math is an old subject it is a big subject. So big that no one person is expert in every part of math. The math covered in high school is kind of the starting point. All branches of mathematics basically start from there and spread out. If you feel you are rusty on your high school math, start there and look for a review book or study guide in those subjects, usually called Algebra 1 and Algebra 2. If you recall your Algebra 1 and 2, take a look at the books on pre-calculus. The normal progression is one year for each of the following courses in order, Algebra 1, Geometry, Algebra 2, Pre-Calculus, and Calculus. This is just the beginning of math proficiency, but by the time you get through Calculus you will be able to read the paper you referenced.
Is it really a year for each of those subjects? It can be done faster but math proficiency is a lot of work. Like learning to be a good golfer, it would be unusual to become a 10 handicap in less than 5 years of doing hours of golf each and every week.
Calculus is kind of the dividing line between high-school math and college level math. Calculus is the prerequisite for almost all other higher level math. With an understanding of Calculus one can go on to look into a wide range of mathematical subjects.
Some math is focused on its use to solve problems in specific areas; this is called applied math. In applied math there are subjects like Differential Equations, Linear Algebra, Probability and Statistics, Theory of Computation, Information & Coding Theory, and Operations Research.
Alternatively, there are areas of math that are studied because they have wider implications but not because they are trying to solve a specific kind of problem; this is called pure math. In pure math there are subjects like Number Theory, Abstract Algebra, Analysis, Topology & Geometry, Logic, and Combinatorics.
All of these areas start off easy and keep getting harder and harder. So you can take a peek at any of them, once you are through Calculus, and decide what to study next.
Wikipedia has been helpful sometimes but otherwise I have found reading a lot of papers on the same topic has been useful. However, this is kind of an "organic" and slow way of learning notation common to a specific field.
Tough love, kids.
I can read and understand undocumented code with relative ease. Reading math notation without any documentation seems pretty much impossible, otoh.
If you do this enough, the process becomes easier and the original notation becomes easier to understand. But it takes a lot of time and patience (as I'm sure it took for you understand undocumented code did as well).
So many answers and no correct one yet. Read and solve "How to Prove It: A Structured Approach", Velleman. This is the best introduction I've seen so far. After finishing you'll have enough maturity to read pretty much any math book.
Use the 3 line version of approximately equal (looks like tilde above an equal sign, ≅).
if it's not, try intuition
if that fails, email your mathematician friend and ask
don't have a mathematician friend? there's your next goal, go make one.
If it’s not, the book is badly written. Most of the time, you can’t rely on a specific bit of notation to be consistent across books or articles. Smart arses who try to impress the readers with their fancy unique notations are the bane of scientists doing literature reviews.
90% of the time, there needs to be a keyword when a symbol is introduced, e.g. “where Λ is the time-dependent foo operator” so you can get a textbook to find what the fuck a “foo operator” is. Then, the first time you spend a day learning what it is, and the next million times you mumble “what a stupid notation for such a straightforward concept”.
I've been repeatedly called a gatekeeper for this stance here on HN, but really: notation is a red herring. To understand math written in "math notation", you first have to understand the math at hand. After that, notation is less of an issue (even though it may still be present). Of course the same applies to other fields, but I suspect that the question crops up more often regarding mathematics because it has a level of precision not seen in any other field. Therefore a lot more precision tends to hide behind each symbol than the casual observer may be aware of.
That covers most of the basics, but I think your real question is how to learn all those concepts, not just the notation for them, which will require learning/reviewing relevant math topics. If you're interested in post-high-school topics, I would highly recommend linear algebra, since it is a very versatile subject with lots of applications (more so than calculus).
As ColinWright pointed out, there is no one true notation and sometimes authors of textbooks will use slightly different notation for the same concepts, especially for more advanced topics. For basic stuff though, there is kind of a "most common" notation, that most books use and in fact there is a related ISO standard you can check out: https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf#pag...
Good luck on your math studies. There's a lot of stuff to pick up, but most of it has "nice APIs" and will be fun to learn.
- better see the structure of the problem; or
- reduce the amount of ink they need to write the problem
Very similar to how programmers use functions, in fact.
To this end, mathematicians in different fields have different notation, and often this notation overlaps with different meaning. Think how Chinese and Japanese have overlapping characters with different meanings.
As others have stated, there is no "one true notation" -- all notation is basically a DSL for that math field.
Instead, choose a topic you are interested in, find an introductory text, and start reading. They will almost certainly explain the notation. Unfortunately, even within a field, notation can vary, but once you have a grasp of one you will probably grasp the rest quick enough.
I will mention, though, that some notation is "mostly" universal. Integrals, partial derivatives, and more that I can't recall right now all use basically the same notation everywhere, since they underlie a lot of other math fields.
Learning everything about math is nearly impossible like knowing everything about all code that exists.
That course should teach some basics for proof strategies. Ex here on page 2, there are definitions with examples: https://cs.uwaterloo.ca/~cbruni/pdfs/Math135SeptDec2015/Lect...
Specialized math tends to have specialized notation. For ex Linear Algebra, Calculus, Combinatorics. Any decent textbook will have an appendix or table with what the notation means.
I suggest finding contexts first, and exploring math within those contexts. Different subfields have their own conventions and notation.
For example, you might be working in category theory, and see an arrow labeled “π”. When I see that, I think, “Ah, that’s probably a projection! That’s what π stands for!”
Or you might be in number theory, and see something like π(x). When I see that, I think, “Ah, that’s the prime number counting function! That’s what π stands for, ‘prime’!”
Or you might be in statistics, and see 1/2√π e^(-1/2 x^2). When I see that, I think, “Ah, that’s the number π! It’s about 3.14”
Or you might see a big ∏ which stands for “product”.
The fact that such a common symbol, π, stands for four different things in four different contexts can be a bit confusing. So if you want to learn mathematical notation, pick a context that you want to study (like linear algebra), and look for accessible books and videos in that subfield. The trick is finding stuff that is advanced enough that you’re getting challenged, but not so advanced that it’s incomprehensible. A bit of a razor’s edge sometimes, which is unfortunate.
https://dlmf.nist.gov/front/introduction
Of course, if the real problem is that you need to learn some mathematical constructs, that is a different problem. The good news is that there's a lot of material online, the bad news is that not all of it is good... I often like Khan Academy when it covers the topic.
I wish you luck!
I also used get hung up on “mathematical notation”. But it turns out the problem wasn’t the notation. I was just bad at math. Well, out-of-practice is more like it.
Once you have the fundamentals clearly explained and you’re doing some math on a regular basis the notation, even obscure non-standard notation becomes relatively intuitive.
1) Search youtube for multiple videos by different people on the topic you want to learn. Watch them without expecting to understand them at first. There is a delayed effect. Each content creator will explain it slightly differently and you will find that it will make sense once you've heard it explained several different times and ways.
I will read the chapter summary for a 1k page math book repeatedly until I understand the big picture. Then I will repeated skim the chapters I least understand until I understand its big picture. I need to know the terms and concepts before I try to understand the formulas. I will do this until I get too confused to read more then I will take a break for a few hours/days and start again.
2) You have to rewrite the formulas in your own language. At first you will use a lot of long descriptions but quickly you will get tired and you will start to abbreviate. Eventually, you get the point where you will prefer the terse math notation because it is just too tedious to write it out in longer words.
3) You might have to pause the current topic you are struggling with and learn the math that underlies it. This means a topic that should take 1 month to learn might actually take 1 year because you need to understand all that it is based on.
4) Try to find an applied implementation. For example photogrammetry applies a lot of linear algebra. It is easer to learn linear algebra if you find an implementation of photogrammetry and try to rewrite it. This forces you to completely understand how the math works. You should read the parts of the math books that you need.
I was a college math major, and I admit that I might have flunked out had I been told to learn my math subjects by reading them from the textbooks without the support of the classroom environment. It may be that the books are "easy to read if a teacher is teaching them to you."
Talking and writing math also helped me. Maybe it's easier to learn a "language" if it's a two way street and involves more of the senses.
Perhaps a substitute to reading the stuff straight from a book might be to find some good video lectures. Also, work the chapter problems, which will get your brain and hands involved in a more active way.
As others might have mentioned, there's no strict formal math notation. It's the opposite of a compiled programming language. In fact, math people who learn programming are first told: "The computer is stupid, it only understands exactly what you write." In math, you're expected to read past and gloss over the slight irregularities of the language and fill in gaps or react to sudden introduction of a new symbol or notational form by just rolling with it.
Also remember that math notations are meant for people. If you learn the sigma summation notation, and if you wonder "So I understand what is \Sigma_{i=0}^{10}, but what is \Sigma_{i=0}^{-1}?" then you're wondering irrelevant stuff. If a math notation is confusing to use, good mathematicians will simply not use it and devise an alternative way to express it (or re-define it more clearly for their purpose).
Also, don't skip exercises. Try to solve at least 1/3 of them after each chapter. Exercises are the "actually riding a bike" part of learning how to ride a bike.
Think about it this way. A scientist, wanting to communicate his ideas with fellow academics, is not going to spend more than half the paper on pedantics and explaining notations which everyone in their field would understand. Else what is the purpose of creating the notations? They might as well write their formulas and algorithms COBOL style!
Ultimately mathematics, like most human-invented languages, is highly tribal and has no fixed rules. And I believe we are much richer for it! Mathematicians constantly invent new syntax to express new ideas. If there was some formal reference they had to keep on hand every time they need to write an equation that would hamper their speed of thought and creativity. How would one even invent something new if you need to get the syntax approved first!
TL;DR: Treat math notation as any other human language. Find some introductory texts on the subject matter you are interested in to be "inducted" into the tribe
2] dive deep into the history of math.
3] youtube… 3 blue 1 brown, stand up maths, numberphile, kahn academy. These channels are your friends.
4] don’t give up and make it fun. Once you’re bit by the bug of curiosity and are rewarded with understanding you’ll most probably be unstoppable but still, its a long road. Better to focus on the journey.
Lastly, the notation is what it is because of the nature of math itself coupled with the history of who was doing the solving exacerbated by the cultural uptake. There have been and will continue to be new notation. Its unfortunate that often to learn a new concept the barrier is with parsing the syntax. Stick with it and stay curious and those squiggles will take on new magical and profound meanings.
https://www.amazon.com/Mathematical-Notation-Guide-Engineers...
Try to read it aloud.
"The Probability Lifesaver" has a lot of good mathematics tips (which are not even mathematics related) most of which are not probability-specific. It's a goldmine.
As a sidenote I have MSc in Physics with a good dollop of maths involved and I am quite clueless when looking at a new domain so it's not as if university degree in non-related subject would be of any help...
Math notation becomes very readable, as soon as the teacher writes a example out on the black board, and that is why i will never forgive wikipedia / wolfram / latex for not having a interactive "notation to example expansion". They had such a chance to reform the medium - to make it more accessible to beginners and basically forgot about them.
Let me explain a little bit. Just like a foreign language you stopped learning and using after high school, what prevents you from using it fluently is not just the vocabulary and grammar, but also the intuition and the understanding of the language as a whole. Luckily, math is a human designed language, with linear algebra and calculus being the fundamentals. And again, learning them is about building intuition on why and how they are used, so whenever you encounter transformation, you think in terms of vectors and matrices, and derivative for anything relevant to rate of change. By using carefully designed examples and visual representation, Grant Sanderson greatly smoothed the learning curve in the video courses. Try it out and you'll see.
Beyond that, different fields do have slightly different notation. When you first encounter them, just grab some introduction books or online courses and skim over the very first chapters.
A lot of it is convention, so you do need a social approach - ie asking others in your field. For me it was my peers, but these days there’s Math stack exchange, google, and math forums. Also, first few chapters of an intro Real Analysis text is usually a good primer to most common math notation.
When I started grad school I didn’t know many math social norms, like the unstated one that vectors (say x) were usually in column form by convention unless otherwise stated (in undergrad calc and physics, vectors we’re usually in row form). I spent a lot of time being stymied by why matrix and vector sizes were wrong and why x’ A x worked. Or that the dot product was x’x (in undergrad it was x.x). It sounds like I lacked preparation but the reality was no one told me these things in undergrad. (I should also note that I was not a math major; the engineering curriculum didn’t expose me much to advanced math notation. Math majors will probably have a different experience.)
Almost all hand "proofs" in math papers have minor bugs, even if they're mostly correct in the big picture sense.
Even math designed to support programming (e.g. in computer graphics) is almost always incomplete/outright wrong in some meaningful way.*
But with a struggle, it's still largely usable/useful.
I've used advanced mathematics most of my career to do work (i.e. read a paper, implement it), but the ability to actually use math to do new things in computer science that mattered only to me only happened after I learned TLA+, which took a few weeks of solid study to click. Since then, it's been a pleasure. My specs have never been this good!
Lamport's video course on TLA+ is pretty good, but honestly I've read everything I can find on the topic so it's difficult to know what helped me the most.
*I think this is because, short of doing formal mathematics, there's no way to "test" your math. It's the equivalent of expecting programmers to write correct code the first time with no tests, and without even running the code.
This sounds somewhat abstract, as the math field is vast. If you consider the next level from where you believe your present standing is, I would try to revisit the college-level math which you probaby experienced back in time.
Generally, the textbooks rely on previous knowledge and gradually feed the new concepts, including the math notation as needed in the new scope.
I find it easier to get the feel for the notation by actually writing it by hand. Indeed it's just an expression tool. Also, you may develop your own way of making notes, as you go on dealing with math-related problems.
But in the core of this you are learning the concepts and an approach to reasoning. Of course, for this path to have any practical effect, you would need to memorize quite a bit, some theorems, some methods, some formulas, some applications. Internalizing the notation will help you condense all of that new knowledge.
Picking a textbook for your level is all that is needed to continue the journey!
When that fails, math.stackexchange.com is a very active and helpful resource. You can ask what certain notation means, and upload a screenshot since it’s not always easy to describe math notation in words.
If you don’t want to wait for a human response, Detexify (https://detexify.kirelabs.org/classify.html) is an awesome site where you can hand draw math notation and it’ll tell you the LaTeX code for it. That often gives a better clue for what to search for.
For example you could draw an upside down triangle, and see that one of the ways to express this in LaTeX is \nabla. Then you can look up the Wikipedia article on the Nabla symbol. (Of course in this case you could easily have just searched “math upside down triangle symbol” and the first result is a Math Stackechange thread answering this).
It's kind of a cop-out, but to be fair it's basically what I would say for programming as well. Try to simultaneously write code that clear to yourself and clear to others. There's no perfect method. Just constantly self-critique and try to improve.
Or that the notation differs from books to books ?
(In my case, I learned the notation via French math textbooks, and in the first day of college/uni we litteraly went back to "There is a set of things called natural numbers, and we call this set N, and there is this one thing called 0, and there is a notion of successor, and if you keep taking the successor it's called '+', and..." etc..
But then, the French, Bourbaki-style of teaching math is veeeeeeeery strict on notations.
Pick a direction (maybe discrete math, if you're trying to do CS) and get a book (I like EPP, as it is super accessible) and go, in order, through each chapter. Read, do the example problems, and do EVERY SINGLE PROBLEM in the (sub)chapter section enders.
Its a time commitment, but if you really want to learn it, this is one way to do so. IMO finding the right textbook is key.
I’d highly recommend this book. It’s what I had for my intro to proofs class in college and it was the best book I found for understanding. I found many other books on this topic to be kinda garbage but this one was amazing.
But you'll have to be a bit realistic when going through the book, it's going to take a good while.
Then some textbooks with exercises (e.g. Axler on lin alg).
The notation is usually an expression of a mental model, so just approaching via notation may cause some degree of confusion.
E Psi = H Psi
and we all joked you could just cancel the Psi and so E=H.
several very kind people explained vector calculus to me ("bold means a matrix, and this dot means matrix multiplication") but to be honest, I still can't read math notation but if you show me anything in numpy I'll understand it immediately.
Smart people often don't know the difference between an elegant abstraction that conveys a concept and a black box shorthand for signalling pre-shared knowledge to others. It's the difference between compressing ideas into essential relationships, and using an exclusive code word.
This fellow does a brilliant job at explaining the origin of a constant by taking you along the path of discovery with him, whereas many "teachers" would start with a definition like "Feigenbaum means 4.669," which is the least meaningful aspect to someone who doesn't know why. https://www.veritasium.com/videos/2020/1/29/this-equation-wi...
It wasn't until decades after school that it clicked for me that a lot of concepts in math aren't numbers at all, but refer to relationships and relative proporitons and the interactions of different types of things, which are in effect just shapes, but ones we can't draw simply, and so we can only specify them using notations with numbers. I think most brains have some low level of natural synesthesia, and the way we approach math in high school has been by imposing a three legged race on anyone who tries it instead.
Pi is a great example, as it's a proportion in a relationship between a regular line you can imagine, and the circle made from it. There isn't much else important about it othat than it applies to everything, and it's the first irrational number we found. You can speculate that a line is just a stick some ancients found on the ground and so its unit is "1 stick" long, which makes it an integer, but when you rotate the stick around one end, the circular path it traces has a constant proportion to its length, because it's the stick and there is nothing else acting on it, but amazingly that proportion that describes that relationship pops out of the single integer dimension and yields a whole new type of unique number that is no longer an integer. The least interesting or meaningful thing about pi is that it is 3.141 etc. High school math teaching conflates computation and reasoning, and invents gumption traps by going depth first into ideas that make much more sense in their breadth-first contexts and relationships to other things, which also seems like a conspiracy to keep people ignorant.
Just yesterday I floated the idea of a book club salon idea for "Content, Methods, and Meaning," where starting from any level, each session 2-3 participants pick and learn the same chapter separately and do their best to give a 15 minute explanation of it to the rest of the group. It's on the first year syllabus of a few universities, and it's a breadth-first approach to a lot of the important foundational ideas.
The intent is I think we only know anything as well as we can teach it, so the challenge is to learn by teaching, and you have to teach it to someone smart but without the background. Long comment, but keep at it, dumber people than you have got further with mere persistance.
It takes you from the concept of number.
Honestly, most math formulas can be turned into something that looks like C/C++/C#/Java/JavaScript/TypeScript code and become infinitely more readable and understandable.
Sadly, TypeScript is one of the languages that is attempting to move back to idiocy by having generics named a single letter. Bastards.