The notation nobody teaches you, and the thirty marks it is made of
Most people who stall on a formula have not failed at mathematics. They were never given the vocabulary the page assumes: what a letter is standing in for, which letters are local, and where the declaration hid.
A reader stops at a formula and concludes the subject is beyond them. Almost always the problem is smaller and more fixable than that: the page is written in a notation of about thirty marks and a dozen silent conventions, and nobody ever sat them down and read it aloud.
What follows is four of those conventions, in the order they cause trouble. They are the opening of a course in the library, and you can read that lesson in full without an account.
A letter has one of three jobs, and the symbol does not say which
Every letter in a formula is doing one of three things.
- A given quantity, which you are expected to know the value of even though it is not written: the number of rows in your data, the learning rate you picked.
- An unknown, which is what the formula exists in order to determine.
- The name of a whole collection, which is the one that trips people up, because it looks exactly like the other two.
The letter x might be a single number or a list of nine hundred of them, and nothing in the mark tells you. The job matters because it decides what you are allowed to do: you can divide by a given number once you know it is not zero, and you cannot divide by a collection at all.
There is a convention about which letter means what, and it is the only help on offer
Nobody states it, and any author may break it, but it holds often enough to be worth learning.
| Where in the alphabet | What it usually is |
Near the start, a b c | fixed quantities |
The middle, i to n | counting and indexing |
The end, x y z | the data and the unknowns |
| Capitals | collections, matrices, sets |
| Greek letters | quantities somebody chose rather than measured |
A reader who expects this will be right about four times in five, and being right four times in five is the difference between reading a line and decoding it.
Some letters exist only inside one expression
The i underneath a sum sign is not a quantity at all. It is a placeholder that takes each value in turn while the sum is carried out, and when the sum is finished it does not exist.
That is why the same i can appear in three unrelated sums on one page with nothing colliding, and why an i appearing in the answer of a sum is a mistake rather than a result. The same goes for the variable under an integral and the one in a for-all claim. Everything else in a formula is the opposite: it means the same thing everywhere on the page, and renaming it means renaming it everywhere.
The letter was declared somewhere, and finding it is faster than guessing
Mathematical writing declares its letters, usually in a sentence beginning "let", or a clause beginning "where", or a phrase like "for a dataset of n examples". The declaration carries more than the name: it says what kind of thing the letter is, which is exactly what tells you whether it is a number, a list, a function or a set.
A great deal of apparent difficulty in a paper is a declaration three paragraphs above the formula. A great deal of genuine badness in a paper is a declaration that is nowhere at all. Looking for it first is quicker than guessing, and finding that it is absent is a result rather than a failure.
What to do about it
The honest answer is that this is a vocabulary problem, and vocabulary responds to being taught in order and then used. Four marks a day for a week, each one read out loud in words, and then a real formula taken apart to the last symbol.
From the library
Reading the Notation
Eight lessons: what a letter stands for, something in and something out, the sum sign as a loop, numbering things, products and powers and roots, the four different equals signs, the alphabet everyone assumes, and one real formula read all the way through.
It is the first course on the track that ends with being able to open any course here without the mathematics being the thing in your way. The first lesson is free to read, and if you ask for a course of your own, the first one is free from beginning to end.