In the first chapter of _The Language of Mathematics_, Keith Devlin writes
about the emergence of mathematics as a field of study and how that depended
on the development of more and more abstract concepts.
Here I quote from the section entitled "Symbolic progress":
As an illustration of the distinction between the *use* of a mathematical
device and the explicit recognition of the entities involved in that device,
take the familiar observation that order is not important when a pair of
counting numbers are added or multiplied. Using modern algebraic
terminology, this principle can be compressed in a simple, readable fashion
by the two commutative laws:
m + n = n + m, m * n = n * m.
In each of these two identities, the symbols m and n are intended to denote
*any* two natural numbers. Using these symbols is quite different from
writing down a particular instance of these laws, for example:
3 + 8 = 8 + 3, 3 * 8 = 8 * 3.
The second case is an observation about the addition and multiplication of
two particular numbers. It requires our having the ability to handle
individual abstract numbers, at the very least the abstract numbers 3 and 8,
and is typical of the kind of observation that was made by the early
Egyptians and Babylonians. But it does not require a well-developed
*concept* of abstract numbers, as do the commutative laws.
(end quotation)
When I read this what immediately popped into my mind was the following J
statement:
v~ -: v
Here we find a natural extension to the trend of abstraction Devlin was
emphasizing. At the point where one can read this simple bit of J, the
prevailing ways of stating commutativity (m + n = n + m) seem about as
clumsy as the use of examples with particular numbers. After all, the
particular functions are no more relevant to the point of commutativity than
are the values that fall within their domains. What makes a dyadic function
commutative is an insensitivity of argument order, which is precisely what
the additional abstraction of the J phrasing brings to the fore.
Tracy Harms
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