It is not, strictly, an improvement on the concept. It's the same concept expressed two ways.
But you cannot symbolically manipulate that prose text, where as you can stick that integral statement into an equation and manipulate it there along with the other expressions of the equation. The ability to perform those manipulations is an improvement.
But, nothing about writing math in text prevents manipulating the text, right? You can do it, it would just be laborious and exhausting for people who are familiar with the use of symbols.
Try a "solve for x" problem with the equation described in prose (still using short variable names and numbers though):
x added to 4 all divided by 3 is equal to 16.
one third of x plus four is the same as sixteen
the result of dividing four and x by three is sixteen.
There's a reason a good portion of elementary mathematics was spent on "word problems", where we reduced problems like this down to their essential equations (a process of abstraction, though not described to us as such). Usually you'd see the above as something like:
Andrew has some number of apples. Billy gives him 4
more. He splits the apples evenly between himself,
Billy, and Charlie. Everyone has 16 apples, how many
did Andrew have to start with?
Maybe I didn't make my point clearly. I'm not arguing against the use of notation or symbols. I'm asking whether Greek specifically is slowing learning down or intimidating people from starting. I'm asking whether learning the symbols with the concepts is the most efficient way to learn math, not whether it's the most efficient way to do math.
Maybe the laborious way is better if more people are comfortable starting that way, and easing more slowly into symbolic syntax.
Maybe there's a more compelling story about math that will get more people interested in it than teaching what symbols do.
If Greek is intimidating enough that many people never start, it doesn't matter how actually laborious the symbols are or not, the battle is lost.
The whole point of education is to approach and embrace unfamiliar concepts.
By the way, arabic numbers are... arabic.
It's a fact that some people are intimidated by Greek letters and highly symbolic math, I'm not sure why that's being pushed back on. It's an open question, not a statement on my part, whether using more familiar symbols would speed up learning. Are you absolutely certain it would not? Why? Have you tried?
Lots of people on HN believe music education would be improved by updating the notation to be more "modern", and complain that music is more difficult to learn than it could be.
What you seem to be saying is Japanese is too hard to learn, so let's just discard Kanji and use the english alphabet instead. You can do that, but at some point you are not teaching Japanese any more.
I was talking about conceptually expressive power. Symbols have no conceptually expressive power at all, we assign meaning to them by describing what they do using familiar language, and previously defined symbols. The transitive definition of any symbol is completely and only defined from first principles via familiar language.
∑ - sigma - s as in sum
∫ - long s - s as in sum
It's not totally arbitrary.
I could use sigma - s as in subtract.
I could use iota - I as in integrate.
Of course it's arbitrary.
That is correct, but at this point it is standard to see sigma as summation (or in some other contexts as set or space, but it's also clear in those contexts which meaning is appropriate, certainly an algebra student seeing sigma notation will not also be seeing sigma used for "set" or "space" in that same course).
Also don't confuse arbitrary with random. Having a reason doesn't mean a choice isn't arbitrary.
Bigger picture, maybe I'm misinterpreting you, but I feel like the logical conclusion of your argument is that nothing can change now, because canon. And yet basic math notations have changed throughout history.
It would take a lot of effort to change the summation symbol from sigma to something else. But if something else made learning math faster, the canon is exactly the part that needs to go, not an excuse to stop improving.
The symbols we use in math aren't to be confused with the ideas, they are shorthand. Ideas and symbols are independent, and we still have to explain what Greek letters do using English anyway. We use Greek letters for convenience, not for expressive power. Writing "∫ x ∂x" is only faster and more compact than writing "The integral of the equation x over x", it is not an improvement of the concept, right?
Use of symbols is the act of trying to compress everything into a smaller set, not the other way around.
For the modern audience, the use of Greek letters may hamper learning because the symbols are unfamiliar and foreign looking. It takes cognitive effort to learn the Greek letters in addition to the cognitive effort to learn the mathematical ideas. What if we could remove the cognitive effort on letters, and focus on the pure math ideas? What if the Greek letters are a speed bump to learning? That we use ∑ for sum and ∆ for change are arbitrary - canonical, but completely arbitrary. History could have decided that a circle was the integral symbol, or the capital letter I. Maybe math would be easier to learn if we switch to capital letter I.