One might say, the inventions of the unknown, and of the (real) variable.
I suppose these can be combined by regarding an unknown as an identity
function (e.g. on the real numbers), as the variable can also be
interpreted, and then interpreting solving an equation such as ax + b = 0
for x as finding the zeros of the function ax + b. This example brings to
mind the old terminology of "variable constants" (for a and b), an, I
suppose, "constant variables" (or just "variable") for x.
In any case, I expect one of the greatest innovations for algebra, as
distinguished here from arithmetic, was the use of letters for, let us say,
indefinite numbers or to-be-determined numbers.
Dominique Flament has indicated the importance of introducing the letter
"i" for, let's say, the pair (0,1) of real numbers, presumably together
with the algebra that goes with the complex numbers. This leads me to
wonder if this can be considered as the same sort of thing as naming the
base of natural logarithms with "e", or naming the ratio of circumference
to diameter of a circle with the Greek letter corresponding to the Roman
letter "p" (also known as "pi" :-)). That is, can naming of those few
transcendental numbers which are important constants, be considered an
important contribution to notation?
Come to think of it, which came first, naming of variables with letters or
naming of constants with letters?
James Landau has indicated the importance of the use of exponents. How
about the now not so common use of a radical sign for roots? Of course,
since the advent of computers, there's a tendency to get away from
superscript exponents, and radical signs, without too much loss of brevity
and clarity, don't you-all think?
What about notations for differentiation and integration?
Finally, let me relay a remark a mathematician (Bill Duren) made to me many
years ago, to the effect that the older mathematicians get, the less
specialized mathematical notation they use, and the more they write
mathematics in ordinary (natural) languages, as far as they can. If you
are suitably old, do you agree with this? Note: one *talks* mathematics
without nearly as much notation as one *writes* it, right?
Gordon Fisher gfisher@shentel.net