Mathematical Notations are Hell

The problem with mathematics is simple: notations. It make everything hard. Specially for the student that freshly joins the university and has to deal with a lot of similar yet incompatible notations and corresponding way of thinking. There are a lot of loosely defined conventions (when not an article's author ones) that make understanding mathematical text .

For instance:
f: is it a function, is it a scalar?
a: is it a function, is it a scalar?

What a strange notation system where letter of item of the same set have implictly different type, unless explicitly stated otherwise.

For instance:
f: should be a function
a: should be a scalar, or a vector
m : shoud be an integer index

if the value of the character is so important to its type then why changing case, changes types, but in unrelated and incompatible ways?

For instance:

F: integral of f
F: set named F
F: function named F
A : matrix
A : vector
A : set
M : matrix (and only a matrix)

What a strange notation system, where a space and lack of it have a signification?

For instance:
aa = a*a
a a = a*a

What a strange notation system where symbols have very different meanings:

For instance, what do you read here?

f(a) : apply the function f with the parameter a
f(a) : multiply the scalar f with scalar a
f(a) : multiply scalar f with vector a

f^a : scalar f to the power a
f^a : function f in a set of family of functions indexed by a?
f^a : scalar noted f^a

What a strange language where the semantics of operation can change:

For instance:

f(a) : apply the function f with argument a
f(a) : declare a predicate f for argument a
f(a) = : define the value of f for a variable called a (unbound variable, ie generic)
f(a) = : define the value of f for a value called a (bound variable, ie given)

Maths are hard to understand because mathematical notation are notation, they are not a language. Which means that with every field, subfield and individual research paper tou have to learn from scratch what the scrible you read mean.

In a sens they are a spreceise a writing "todo: buy stuff" on piece of paper: when you'll read it you'll have to struggle a lot to understand what the author meant.

That's too bad to crush the opportunity of the masses to understand an use mathematics on the whim of using notations. Like the dreadful conventional direction of current in electrical engineering... which is like calling a fish a pig an then trying to understand how do pig swim.


A great source of confusion come from the fact that sometimes letter denote vectors rather than scalar, and operations apply differently to them. Because summation and combination operations are implicit, it makes it more difficult to understand. For instance when considering matrices, it is important to know it it is a product or a tensorial product that is applied.

When multiplying two letters, possibly:
- we multiply a vector by a scalar which yields a vector,
- we multiply a line vector by a column vector, which yields a scalar
- we multiply a column vector by a line vector, which is an error
- we multiply a matrix and a vectors,  or a matrix and a scalar, etc.