Date: Nov 5, 2012 2:59 PM
Author: Dave L. Renfro
Subject: Re: An Interesting Point

Peter Duveen wrote (in part):

> A while back, it occurred to me that there are operator numbers
> which operate on other numbers, and since these seemed to be
> different from other numbers, and it never occurred to me
> before that there might be different types of numbers, each
> with different properties. so 5 x 75 is an! operation performed
> on 75, and thus, the "5" is a different type of number from 75.
> It sort of stopped me dead in my tracks, since I had never
> explored such an idea before. Wonder if you have ever explored
> such an idea. I would also like to know a simple source whereby
> I could review what you seem to indicate is "Indian" mathematics
> where the expansion of the number system and the operations
> involved are more carefully elaborated.

I don't know if this (what I discuss below) is what you're talking
about, but there are certainly different notions in mathematics
that are represented by the numeral '5'.

For one thing, there is '5' as a natural number, '5' as an integer
(formally, an equivalence class of pairs of natural numbers),
'5' as a rational number (formally, an equivalence class of
pairs of integers), '5' as a real number (formally, an equivalence
class of Cauchy sequences of rational numbers), and '5' as a
complex number (formally, an equivalence class of pairs of
real numbers). And yes, I'm aware that there are several ways
of proceeding through these steps. I know of at least half a
dozen ways of getting the reals from the rationals, for example.

The above are all essentially flavors of '5' as a number. You
can also have '5' as a function, namely the constant function
f(x) = 5. In the well known (in mathematics) book "Rings of
Continuous Functions" by Gillman and Jerison, constant functions
are represented by bold face numerals. Thus, in that book,
a bold face '5' means the constant function f(x) = 5, whereas
a regular face '5' means the number. I believe Karl Menger
also used numerals to represent constant functions in his 1940s
and 1950s attempts to reform the teaching of elementary calculus.

Functions represented by numerals can get more exotic than
this. For instance, when solving differential equations using
algebraic operator methods (google "differential equations"
along with "D operator"), numerals now represent multiplication
by constant operators on sets of functions. For example, 'D + 5'
represents the operation "d/dx + 5", which when you input
the function f(x), outputs the function f'(x) + 5*f(x).
In this setting, '5' would then represent a function whose
domain is a certain set of real-valued functions of one real
variable (all functions differentiable on a specified interval,
for example) and whose range is a similar (but not necessarily
the same) set of functions, which is defined by "5 evaluated
at f(x)" is equal to 5*f(x).

Dave L. Renfro