This initial or intuitive definition is fine in the context of a referent , but if their is no referent it becomes mystifying, or better still applicable to anything. It is this second tendency that leads to private interpretations or translations of a form of words.mit is this tendency or freedom, that allows an iterative definition to be performed or made, or a generalisation or Analogy or metaphor to be drawn.
That to one side, the weary student in his quest to learn and understand often seeks to cling on to a secure notion, while a master may be trying to explore the applicability of a rule in many contexts, and enjoy the shifting sands on which the form of words place him.
In exasperation a defeated student may give up an interest in a subject while a master may struggle all his life to pin down what precisely his life's work is really about,,staring at hopeless denigration and refusing to allow it to negate the meaning of all his effort.
The psychological necessity for "truth" and "certainty" drives many to breaking point anxiety, while some perceive all things as in a game. The situation defines the validity of any behaviour, the rules of the game are accepted for the tpurpose at hand and not used for any other life situation.
Overtime the game player builds up a vast repertoire, so much that the player finds it hard to distinguish which sets of rules it identifies itself under. This may lead to a sense of alienation or psychopathy.
Finally their is the master who does not require identity. Every situation is responded to from a range of responses which constantly grow and modify..every purpose is selectively chosen for each goal or expectation, each minute is new and alive with potential and as variable as the sea. Crashing up against shores is part of the learning curve, but strategies are learned that bring the master safely home to any port..
Such a person may have gravitas, a set of core principles that it accepts das defining boundaries nd interactions, but few limiting constraints, except the pragmatism that it cannot do everything!
For such a person a logos may indeed be perceived as a magnitude. Such a logos is made conformble to a magnitude by always referring to a whole. Thus a fractional magnitude is really a logos of a part to a whole.
What this means is we can return to Eudoxus definition and place fractional magnitudes that is logoi in the place where he defines magnitudes d be, if and only if those logoi are of the same type.
Thus any logos of the form a pros b can replace the magnitudes in the logos, and also in the Analogos. However a pros b must be perceived as a magnitude, and historically a Persian mathatician argued for logoi to be considered as magnitudes also. To back up his postulations he posited a set of rules that led to the adoption of logoi as magnitudes called fractions.
The rules of fractions are actually to be found in the rules of rational "numbers". The use of the word number here introduces a false notion.mthe rules of logoi as fractions are defined in terms of magnitudes.
Nevertheless, private interpretations, as I say, have lead amany into a lifelong endeavour based on a shifting concept. It is ot just a change in words. The notions add constraints to the thought process, which limits the applicability by threat of confusion, enforced by ruthless pedagogy willing to shame and blame any deviation.
This thought arose in the context of the radian. The radian is a magnitude of rotation, but it is also a logos of lengths.. Without a protractor marked in radian the notion of a rotation is not at all clear. With a protractor the fundamental ratio of turn to whole turn becomes apparent.
Radians, gradients degrees are all measure in the ratio of a turn to a whole turn.
So when I say an object turns I need to specify which Ratio I am using. However, the tendency to treat a logos as a magnitude means this is obscured, even forgotten .
One way it is obscured is by changing the second magnitude into a kind or a type. Thus 6 eggs is considered a half dozen type or kind or half of a dozen kind.
The principles sketched out by Eudoxus in books 5 and 6 , that is a skesis or schematic of drawn lines are developed for real objects in book 7. However instead of beginning with a part magnitude of a magnitude, Euclid begins with a whole unit, that is a Monas.
A whole unit is an extensive object, so book 7 is an extensive algebra. In that sense books 5 and 6 deal with intensive as well as extensive magnitudes represented by sketches or skesis. Typically a drawn line is segmented, which indicates an intensive approach. If a line is extended by some fraction of another, it is still an intensive approach. However if a line is extended by some multiple of another, that is by some multiple form of another line, then it is an extensive approach.
It is strange tha Monas is defined first, and a Metron is not defined at all. However, katametresee is the process of using a Metron as a Monad to count with. So perhaps the unit has to be first defined in order to make a sensible definition of the process. However the use of the lesser to make a multiple form of the greater magnitude sis a clear enough process upon which to define a Monas and a Metron. Euclid perhaps took this process as requiring stricter definition, but more likely the religious significance or the philosophical significance of Monas demanded it be treated first as the beginning of all things and the synthetic goal of all things.
Given that Neoplatonic perspective, Euclid still uses a pragmatic general observation which is entirely grammatical. Of things, whatever they may be, the first one singled out is Monas, and is called 1. The singling out process has to be a laying down of this thing!
Now of course any bunch of things can fit this description. So if I had a set of 2 things a cup and saucer say, then the first one I choose in the sequence of placing on the table is Monas. However to think of this as a unit in any dimensional sense, such as the SI units is clearly not what is intended. Thus this unit is called one, it can be placed next to the cup, but it is clearly not a cup. Nevertheless I can place this unit over the pair and decide whether I have an exact count or an approximate count.
The notion of exact ( artios) and Approximate( perisos) is defined not in terms of Monas, but in terms of pollaplasios, that is a multiple form. In this example I clearly have a complex form but it is not a multiple of either of it's constituents, the cup and the saucer.
Thus a Monas becomes a Metron when it is applied to a multiple form made from repeated copies of itself. In that sense a Metron is a specialised Monas in the context of multiple forms.
Now an Arithmos can be defined as a multiple form made up from a standard unit in the way just described. Thus an Arithmos is a figure or form made up from a Metron.
While I have adopted a grid reference frame as n easy example of a general Arithmos, in fact the best notion to understand the Greek is that of a Mosac. Such a piece of floor covering is generally constructed of many different shapes, none of which are uniform or Metrons, but nevertheless they are monads of the Mosaic. Thus a monad as a unit means no more than a singled out piece of the whole, and this is the definition that Euclid gives for Monas.
The notion of a unit is therefore an imposed idea on the notion of Monas, which may still be one without being a unit in the strict SI sense, or even in the number sense.
Number, as I say is imposed upon the Greek at this point.
Returning to the cup and saucer we have 2 magnitudes in a skesis or pattern. The fact that they are not multiple forms of each other does not prevent us from taking them as a whole, that is perceiving them as a Monas. In this way we can make multiple copies of this Monas, and thus we extend this magnitude to a greater multiple form by synthesis.
When we look back at what we have done, we see we have taken a logos, perceived it as a magnitude and used it extensively in a multiple forming process, which now allows us to use Katmetresee with a defined Metron as a standard unit!
We in fact do this all the time with our weights and measures, which are all ratios of some standard.
Because we do not think of these things, we become hung up on Numbers, as if they define our real experience. Rather our real experience is always used to define alone, an Arithmos, or a logos analogos. Our main response after that is to sing, dance or utter poetry, from which rhythms we derive our notions of counting.