Emmanuel Attard Cassar wrote:
> Which were, in your opinion, the notations that have permitted the greatest
> advances in mathematics?
>
I think that this a great question, but a very difficult one, not only since
one has to decide which advance is bigger (in an abstract sense) but also
because one would have to insist in considering them (the advances) as isolated
data, not drawing one from another. Also, one would tend to consider very basic
and old stuff (like the wheel) because of its global importance, and then try
to contrast it against "modern" and more specialized advances (like
sophisticated technology).
So, I will not give an opinion but just an example: Numbers and their notations
are really important - no doubt about it. Then letters for constants, for
variables. What about functions and even categories?
Saunders Mc Lane, in "Categories for the working mathematician",
(Springer-Verlag, 1971, p. 29) says:
"The fundamental idea of representing a function by an arrow first appeared in
topology about 1940, probably in papers or lectures by W. Hurewicz on relative
homotopy groups. ((Hurewicz, W.: On duality theorems, Bull. Am. Math. Soc. 47,
562-563))
His initiative immediately attracted the attention of R. H. Fox and N. E.
Steenrod, whose ... paper used arrows and (implicitly) functors... The arrow f: :
X ((arrow)) Y rapidly displaced the occasional notation f(X) ((subset of )) Y
for a function. It expressed well a central interest of topology. Thus a
notation (the arrow) led to a concept (category)".
Arturo Mena