For that matter, what about the notation f(x) itself and the like for
functions? And sometimes just f, i.e. single letters for functions, as
well as for variables and constants?
Which suggests to me another kind of question about notation. What
notations have led to considerable confusion, even though they also had
advantages? For example, there was and still is the confusion between a
function f(x), which can be considered to stand for ordered pairs of
entities from the domain and range of a function, and the value of a
function at an arbitrary element x of its domain, which can be considered
to be stand for entities from the range of the function, or even sometimes
a single entity from the range of a function. I think it was Karl Menger
who sometime in the 1950s suggested using f(x) for a function and f[x] for
the values or a value of a function. I did that for a while, but the
practice didn't catch on. Menger also recommended the identification and
use of identity functions in elementary calculus teaching, and wrote bold
face letters for an identity function denoted by a letter like x to
distinguish it from other uses of a letter like x.
Another example would be the dx and dy/dx notation for derivatives, and
explaining such a thing as dy/dx = dy/dx in the case where this is not an
identity, but an assertion that a ratio of differentials is equal to a
derivative. Or problems with notation for integrals in which the integrand
is f(x)dx or the like, the integral is an antiderivative rather than an
antidifferential. Maybe problems with integral signs when using e-mail
should be included. Too bad the makes of the ASCII list of characters
didn't include an integral sign, right? Of course, this problem was
already there with typewriters, unless they were of special construction or
used special attachments.
Who was that algebraist, I think in the 1920s, who wrote a book about
groups which had besides letters from 3 or 4 alphabets (if I remember
correctly), also used musical notations such as clef signs, signs for flats
and sharps, etc.? And maybe also other notations from fields outside
mathematics that nig printing establishments had on hand?
Robert Tragesser has asked about the use of the various notational systems
of formal logic by mathematicians. They do indeed seem not to be used
systematically by most mathematicians, although I recall passing through a
period when I used on blackboards a backward E and an upside down A as
abbreviations for "there exists" and "for all" (and synonymous phrases), as
in Whitehead and Russell's *Principia Mathematica*.
And once more I will close with an anecdote. In a class on real variable
theory, a fellow student and friend of mine who had had a class from a
logician on Russell-Whitehead type logic tried at the blackboard one day to
prove a theorem by putting it into Russell-Whitehead notation and
manipulating the formulas entirely by rules of formal logic, especially
those applying to distribution of quantitifiers. He got nowhere to speak
of for some time, and the teacher finally interrupted and said with a note
of exasperation, "You can't just manipulate the notation, you need an idea!"
Gordon Fisher gfisher@shentel.net