The average mathematician believes in numbers. They are real objects to them. Most of the rest of us follow suit. However the concept of a number is a notion fraught with tautology and not a little leger de main!
In the west the concept takes form over many centuries of intellectual turmoil, reflecting the business of Moines! War, overthrow, pillage and resettlement each contributed to the eventual meaning of the notion. The biggest and most obvious factor is translation between languages and mistranslation.
The western philosophers who are most influential in examining the notion were the Pythagorean school in southern Italy . They represented even at that time a cosmopolitan apprehension of the subject, drawing together influences from all the major cultures and civilisations around the world. And yet no other civilisations quite put it together as did the Pythagoreans!
The essential idea is that humans respond to the environment with a logos response. This is a complex of all types of signal responses in the human frame associated specifically to one type of channel: the vocalisation of sound.
Already this is an inadequate concept, because humans ” think” sub vocally, without making a sound also! The muscles involved in the speech act still fire but no major muscle group responds with the required intensity to make the air vibrate audibly.
The reflexive nature of the early languages captures this dichotomy, the action can be external and interactive or internal and reflexive.
This distinction is really rather fundamental. It means that tautology has a function. It is able to support many distinctions. In this case we can use the Logos response to represent the external speech act and an analogos response to represent the reflexive internal speech act.
Both of these are experiences, but one I might distinguish as objective, the other as subjective.
In short when I respond to an object and name it ” one” (1) I alo name an associated internal experience ” one”! It is this subjective experience that I can associate to every similar experience regardless of the actual form of the object responded to, even if it is an entirely subjective form!
Similarly I have an experience of multiple form which allows me o associate multiple form to any single or complex object.
The Greek concepts I will outline next.