Few realise that both Aristotle and Plato never qualified as Mathematikos. By this i mean they were not approved by the Pythagorean School as masters of the Pythagoreab system. It is doubtful if Euclid received that honour, but what he did do was more faithfully pass on their teachings as recorded in the Alexandrian libraries.
Of course Eudoxus it seems without much Question was a Mathematikos approved by the Pythagoreab school. Joining with Plato meant that the Athenian Academy was more faithful than the Aristotelian Lyceum. In fact Aristotle tefutes, fatuously many of the Pythagorean principkes on Arithmoi. We find then Aristotelian scholars at odds with Pythagorean ones on the nature of Arithmoi, and as a consequence on the interpretation of Gematria as Mechanical Geometry.
Both Newton and the Grassman's knew that geometry was derived from Mechanics , not Euclid's Stoikeia/Stoikeioon, which i might reemphasise was an introductory course to Plato's Theory of Form/Idea.
The remarkable fact about Greek pythagorean teachings is that they deal with the notion of an arbitrary magnitude. It is the patterning of this arbitrary magnitude which distinguishes the primitive concept into other primitives which in brief i will label point, drawn line, , light catching surface, and finally shadow casting form. These primitive magnitudes are developed into ratioed and proportioned forms, the logos and the Kairos. These forms are then utilised by the observer subjectively as Monads. The Subjective concept of Monas is then used to develop an extensive Algebra of mosaic forms called Arithmoi. The concept of Arithmos is simply that of a mosaic. To define it too rigorously removes its amazing power.
One other subjective thing we do is pronounce names ove the Arithmoi, distinguishing them by name, but also by other criteria. Thus the Logos which strictly are ratios comes to denote generally grammar and linguistics and eventuallu logic and study, while proportion and proportioning comes to generally describe reasoning and rational subjective thought or thinking.
The pronouncing of names is universally known as counting and or sequencing, and the names are known as counts or Numbers. Initially he counts were identical with the developing syllabries in the growing civilisations. as these developed and codified into Abjabs and alphabets so a distinction became possible between scrips for te important phonemes and scripts as graphemes in general.
In a parallel development simple marks became synonomous, in a one to one onto relationship with objects, and these relationships intertwined with the forms of the mosaics /Arithmoi. Thus pebbles for counting and pebbles for Mosaics are indelibly linked.
The modern concept of Number actually develops during the turmoil of the Renaissance period, Where the mixing of Wisdoms led to Algebraic and Rhetorical confusion. This confusion has persisted to this day, especially with the introduction of Rational numbers in place of Ratios and proportions.
Would we benefit from untangling this mess?
Firstly Euclid defines a Logos as a magnitude which is compared with another magnitude, both visibly greater or lesser respective to one another, or kinaesthetically greater or lesser respective to one another.
The lesser magnitude is then utilised to measure the greater. That is the lesser is the Metron for the pair!
Now there is one proviso, there must be an unlimited supply of both magnitudes! Both must be capable of exceeding each other.
Because if the lesser does not measure the greater exactly, more has to be added until it does, and also their has to be a lesser and greater, or an equality or duality. We need enough of both magnitudes to be able to come to a precise decision about multiples of the Metron, whichever is chosen as the lesser.
Thus we see a principle of exact multiples, that is exact division that is artios is the standard. This is the basis for rational numbers!
There is another proviso: that the things compared should be of the same kind. Those who use the SI units are familiar with this rule, but it also appears in Algebra as the rule that like things only can be aggregated. This leads to different collections of different kinds, and the practice of comparing different kinds by ratios. Thus we may discuss the ratio of cats to dogs, not realising that such a comparison has no common Metron. The use of numbers as such a Metron disguises this fallacy.
However, the development of determinants and matrices and basis vectors has been instructed on these types of ratios, so we have made use of this analogous use of ratio to good effect, in ways the Greeks may have found very confusing indeed!
The proviso of having sufficient of each magnitude so one can exceed the other is because the measuring process is bilateral thus the lesser magnitude may not be an exact divisor of the greater, requiring an adjustment to get a perfect result. This could mean having more of the grater magnitude so the lesser may fit exactly or more of the lesser magnitude so the greater magnitude in multiple form may fit the lesser magnitude in multiple form precisely.
Having established a Logos as a comparison of 2 magnitudes, it is as well to realise that this is an entirely subjective notion! Thus Logos refers to that mental cognition of 2 magnitudes being comparable and distinctively so. Logos as this cognition or precognition is the source of that inner judging and comparison associated with distinguishing magnitudes, and it is associated with calling out and naming distinction. Thus Logos, here called Ratio is much more than numerals written on paper or magnitudes conceived as being quantities in comparison and contrast. It is the very idea of magnitudes the very experience of them in distinction.
So now when two ratios share the same characteristic behaviour, corresponding parts compared in a similar and congruent manner, then that experience of duality, of similar behaviours of deja vu is called analogia. That is proportion is formally analogy.
When reduced to numerals it is admittedly not as powerful a concept in appearance as it truly is!
Analogous thinking underlies all model making, and derives from comparison and contrast.
The machinery of obtaining a precise measures by counting Metrons serves to define the process of rationalising, and the repeats of that in an analogy serves to define proportioning.
But what of Kairos!
Tis is ;iterally the first time i have read book 5 in the greek and my surmising turns out to be based on the latinised translation of analogos! Thus proportionate and timely judgement which underpins Kairos does not directly derive from proportion as i thought.
Here is the reference to the resource on the topic followed by an excerpt
Perhaps Isocrates' emphasis on kairos is best summarised n Panathenaicus - one of his most ambitious discourses, since undertaken and published when Isocrates was ninety-seven, In this treatise He sums up the the goals of his rhetorical paideia:
" Whom, then. do i call educated?,,,,,,,,,,First those who manage well the circumstances which they encounter day by day, and who possess a judgement which is accurate in meeting occasions as they arise and rarely misses the expedient courseof action; next, those who are decent annd honourable in their intercourse with all whom they assiociate, tolerating easily and good-naturedly what is unpleasant or offensive in others, and being themselves as agreeable and reasonable to their associates as it is possible to be; furthermore, those who hold their pleasure always under control and are not unduly overcome by their misfortunes,......fourthly, and most important of all, those who are not spoiled by success,,,,but hold their ground steadfastly as intelligent individuals(30-32)"
This pragmatic, personal, and socially conscious recapitulation of what it means to be "educated" encapsulates the principle of kairos in all its nuances: the importance of living by phronesis or "practical wisdom"...
Thus for me Book 5 has become a game changer when i read the Greek words directly. Analogos may well be a distinct root idea for Kairos
Kairos (καιρός) is an ancient Greek word meaning the right or opportune moment (the supreme moment). The ancient Greeks had two words for time, chronos and kairos. While the former refers to chronological or sequential time, the latter signifies a time between, a moment of indeterminate time in which something special happens. What the special something is depends on who is using the word. While chronos is quantitative, kairos has a qualitative nature. Kairos also means weather in both ancient and modern Greek. The plural, καιροί (kairoi (Ancient Gk.) or keri (Mod. Gk.)) means the times.
In the sense of proportioning time, allocating it for different purposes which are appropriate, measured and balanced, but it is clear that time and circumstance is the main magnitudes in comparison. Thus opportunity is compared with time and an appropriate match made.
[quote[ analogon, proportio, ratio ] see also analogies of experience , as – if , hypotyposis , presentation , reason , regulative idea , schematism A foundational term of philosophy, ‘analogy’ has a continuous if underestimated history since Pythagoras. A general theory of analogy was first developed by Eudoxus (?406-?355 bc) in response to the crisis of incommensurable ratios ( logoii ) encountered by the Pythagoreans. The overcoming of this early crisis of Greek reason ( logos ) is codified in the definitions of logos and analogos in book five of Euclid's Elements. These situate analogy in terms of the similarity between the ratios of different magnitudes, as in Definition 6: ‘Let magnitudes which have the same ratio be called analogical’ (Euclid, Vol. II, p. 114). Euclid makes a clear distinction between an analogy of terms and one of ratios, focusing his attention upon the latter. As a result, his account of analogy stresses the similarity in the relation between the antecedent and consequent terms of at least three ratios, and not any similarity between the terms themselves. The philosophical implications of analogical similarity were first realized by Aristotle, who shows in the Topics how it might be used to relate ‘things belonging to different genera, the formulae being “A:B = C:D” (e.g., as knowledge stands to the object of knowledge, so is sensation related ... log in or subscribe to read full text[/quote]