by Milton Dawes
Here is an attempt at an operational definiton of the natural numbers. Pick it to pieces. Please.
In Science and Sanity and elsewhere, Korzybski stressed the importance
of regarding mathematics as "a form of human behavior", and "the only
language, at present (1933), which in structure, is similar to the
structure of the world, and the nervous system." (pp.249 and 250)
And on page 258 he wrote, "Thus, a number in any form, 'pure' or
'applied', can always be represented as a relation, unique and specific in a given case; and this is the foundation of the exactness of dealing with numbers."
Definition of "1" - "1" is the mathematical symbol, representing a
collection or set of whatever, such that an operation performed on any
member of the set is identical with (indistinguishable from) that
performed on all members of the collection or set.
From this definition, one can generate all other natural numbers. For
example, "2" is the mathematical symbol representing a collection or
set of whatever, such that an operation, performed on one member of the
set, if similarly performed on one other member of the set, is
performed on all members of the collection or set.
A definition of the number "3" is demonstrably consistent with the
definition of "1" and "2". So "3" can be defined as "The mathematical
symbol representing a collection or set of whatever, such that an
operation performed on any 2 members of the collection or set, if
performed on one other member of the set, is performed on all members of
the collection or set."
We could also define "3" and other natural numbers from a higher order of
abstraction, in terms of collections, or sets and sub-sets. So on these
terms, "3" could be defined as the mathematical symbol, representing a
collection or set of whatever, which is exactly and entirely comprised of
a collection of "1" and a collection of "2". (The "and" here can be taken as an
operational "description" of the arithmetic operation called "addition.")
And we could also define "1" as "The mathematical symbol, representing
a collection or set, with no sub-set - or a set which is identical with
(indistinguishable from) its sub-set."
If you are wondering about "0", here are some operational definitions
to consider. "0" is the mathematical symbol representing a collection or
set, on which no operation except a mathematical one (multiplying,
adding, etc.) can be performed. "0" could also be defined in terms of
non-existence as "a symbol representing non-existence of a collection or
set of whatever, in a specified situation. Or, "0" is a symbol
representing what remains of a collection or set, after all members of
the set or collection have been operated on.
It should be noted that the operations mentioned in the definitions are
not mathematical operations, but ordinary everyday human operations such
as eating, selecting, touching, etc. As such, I am expecting refutations, criticisms, attempts at ridicules, etc., to be based on suggestions of operations, that would demonstrate the invalidity of the
definitions.
It might be interesting to note, that we can assign "multiordinal" and
"multi-meaning" characteristics to the natural numbers. A collection
symbolized by "1" could constitute "1" of "1"; or "1" of "100, or "1"
of 1,000,000 and so on.
