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.