We certainly need functions – they are essential for math (and computing, science and engineering), but the notation has certainly got to change!

You’re probably right about my square brackets suggestion. I was going for a “different, but not too different” solution, on the grounds if it is way off, no mathematician will ever want to consider it.

The real problem though is psychological regarding the person faced with a new idea. Learning something new is inherently intimidating for most people and therefore motivation is very important. Math teachers are challenged at every step of the way to make their subject interesting. Notation that is visually confusing to the beginner emotionally upsets him and makes him feel stupid and frustrated, even sleepy. I got my wife through her AA math requirements by teaching her to keep her scratch paper as neat and orderly and uncramped as her answer page. She was failing math for one reason only, she was visually overwhelmed and that made her too tense to think clearly.

When a student sees a lot of unfamiliar gibberish it is just plain scary and that makes it easy to quit. On my part, I have yet to find anything difficult about calculus except the way it is being taught, and that includes notation.

As for new notation that can be typed, many brilliant people must have worked on this in order to create computer programs that do math. Most of the comments above are as scary looking to me, a beginner, as f(x) which gives me a rush of adrenalin that puts me into flight or fight syndrome. Maybe it would be better to invent completely new symbols and draw them up in a font creation program and distribute them for free. I disagree with square brackets, for example, because they’re already used for something. You have to use something that’s not already got dibs on it or it will be automatically rejected by minds already taught to use it for something else. That means current math symbols should not be changed into something else.

No one has ever explained to me why there should ever be any such thing as function notation, and I haven’t been able to guess. Maybe it helps in higher math, but “Calculus Made Easy” by Silvanus Thompson never uses it.

Anyway, this is a fascinating topic and I hope to see something come of it.

(define (g x) (+ (^ x 1/3) (log 2 x) (sin (cos x))))

(define (h x) (? (i 1 (length x)) (+ 1 (* 2 (element i x)))))

(define z (list (g 7) (h 13)))

This kind of notation is called symbolic expression or S-expression. It is quite convenient to work with in an editor that understands S-expressions and provides navigation, manipulation, auto-parenthesis balancing, and auto-indentation facilities. My example is in prefix notation, which makes the rules for interpretation simpler, but S-expressions don’t have to be prefix-style.

Some people complain about all the parenthesis, but I’ve always liked them. There are alternative syntaxes which retain the same logical properties but reduce the usage of parenthesis, but they’re not as clean and simple IMO. I’m not claiming S-expressions are the end-all-be-all, and there are other notations I like.

When I heard that there are old math formulas that were discovered much later to be incorrect (via interpreting with a symbolic calculation computer program), I had to laugh, because even the expert inventors of the formulas had problems with the notation.

I’m looking forward to a renaissance of innovation and improvement of math notation, which I think is especially necessary as the world’s use of math becomes more complicated, more depended on, and more people need to understand it.

I’m wondering about such liberal use of passive verbs, though – this is generally a no-no when trying to improve readability. But I can see that once you read past each verb, you are then talking about the next algebraic term.

The human brain can always find ambiguity.

“(x) 2 power 2 multiply” could be

(x^2)*2

or

(x^2*2) = x^4

Couldn’t it? The first describes a single algebraic entity, the second indicates a requirement to do a numerical process, resulting in a single entity.

(x) variable
[1] subscript
{x,y,z} array or set elements

converting infix functions:
f(x) = 2x^2 + abs(16x) + 1
g(x) = thirdroot(x) + log2(x) + sin(cos(x))
h(x) = sigma(i=0,infinity,2i+1)

to postfix functions:
f: (x) 2 power 2 multiply 16 (x) multiply abs add 1 add
g: (x) 1 3 divide root (x) 2 log add (x) cos sin add
h: (i) 2 multiply 1 add 0 infinity sigma

This would be harder to read at first but it would be unambiguous and one could evaluate it (aka a computer could) just looking at it going left to right.