Name of the numbers
Once learned the distinction between cardinal and ordinal aspects, after mastered the tokens, another great step forward was the creation of names for the numbers.
This allowed for much greater precision in speech and opened the path towards familiarity with abstract numbers.
Prior to the emergence of number-names, numbers were referred to by intuitive terms; often directly appealing to the natural environment. For instance, 1 might have been "sun", 2 might have been "eyes", and so on. Subsequently some kind of structure emerged from body counting, a sort of anatomical mapping: "little finger" for 1, "ring finger" for 2, "middle finger" for 3, "index finger" for 4, "thumb" for 5, and so on.
Oral symbols
However, the need to distinguish between the number-symbol and the name of the object being used to symbolise the number led people to make an ever greater distinction between the two names, so that eventually the connection between them was entirely lost.
As people progressively learned to rely more and more on language, the sounds superseded the images for which they stood, and the originally concrete models took on the abstract form of number-words.
The idea of a natural sequence of numbers thus became ever clearer; and the very varied set of initial counting maps or model collections turned into a real system of number-names.
Written symbols
Concrete symbols and spoken expressions were not the only devices that humanity possessed for mastering numbers. There was also writing, even if that did arise much later on. Writing involves figures, that is to say, graphic signs, of whatever kind (carved, drawn, painted, or scored on clay or stone; iconic signs, letters of the alphabet, conventional signs, and so on). We should note that figures are not numbers. "Unit", "pair of", "triad" are "numbers", whilst 1, 2, 3 are "figures", that is to say, conventional graphic signs that represent number-concepts.
A figure is just one of the "dresses" that a number can have: you can change the way a number is written without changing the number-concept at al.
These were very important developments, for they allowed " operations" on things to be replaced by the corresponding operation on numbersymbols.
Writing meant three extraordinary developments in abstraction that occurred in close succession, probably within the century between 3100-3000 BC. These abstractions concerned the creation of 1) two-dimensional signs, 2) abstract numerals and 3) phonetic signs.
The magnitude of these strides in the mastery of abstraction can be realized by comparing and contrasting the degree of abstraction between tokens and writing.
- The tokens were tangible but the signs of writing were intangible. They abstracted the tokens that abstracted goods. The awkward piles of three dimensional tokens could disappear.
- The tokens were used in one-to-one correspondence but writing abstracted numbers. For the first time, signs expressing numerals abstracted the concept of number from that of the item counted. For example, a sign for “one” was placed next to the sign for “jar of oil.” The invention of abstract numerals made obsolete the use of different counters and numerations to count different products. With the abstraction of numbers, counting had no limit.
- The tokens were strictly limited to representing concrete units of real goods, whereas writing abstracted immaterial sounds of speech. As a result, writing was no longer confined to recording goods but could strive to communicate the most abstract ideas.
