Cognates include Old Irish rím, Old English rīm (English rhyme), and perhaps Latin rītus. Also compare νήριτος (nēritos, “countless”).http://en.wiktionary.org/wiki/ἀριθμός
The ancients also connected rhythmos to arithmos (Number), but modern linguists trace arithmos to a different Indo-European root, rê(i)- (to reason, count), from which we also get such words as reason, rational, ratio, rate, and rhyme. Nevertheless, the ancient connection informs us about how Pythagoreans understand Rhea's responsibility for Number. This is natural, for the Indefinite Dyad is the principle of Plurality itself, which separates one thing from another, but also of the Matter that allows one thing to be different from another, by substantiating multiple instances of a Form.
When i initially researched the roots of arithmos i was lead to the notion or "ar". This of course is a transliteration of the PIE root using a particular transliteration scheme. There are many transliteration schemes, but the one above in read seems to be fairly widely accepted.
The etymology that extends from this root is extensive, so my aim today is to highligt the veins of it thar run into the greek linear B form of the language and underly the notion of aire,oire, kairos, harmonia and arithmos, rhythmos.
Essentially the root has to do with metered sound or expression. Thus it links or refers to a style of expression, consciously adopted by the speaker or originator or perceived by the listener or recipient. Thus in this regard , oratory or rhetoric derive from this root.
IN THIS SAME VEIN, THE RHYTHMICAL ARTS OF DANCING AND SINGING are related to this root and the notion of counting derives from these arts, not the other way round. Thus the association of this root with articulated movement in its etymological development.
The notions of "grasping" and "raising" are related to rhythmical motion and speech patterns in so many ways, from comprejending and delivering the dance sequence to modulating and raising the voice pitch and timbre. Thus Kairos refers to this proportionate art of engaging with others in communication through space.
To speak as in the case of setting forth propositions is a distinctive style of speech , and this is marked by using this associated root in oire. The difference between a proposition and a statement then is merely in the style of presentation! (and also the style of perception). Thus the term "force" is a reference to the rhetorical impact of a presentation style rather than to any validity of what is presented. Great "truths" may thus be opined simply and unobtrusively, while "the lie", the propaganda is trumpeted and believed all over the place!
Thus i find Euclid's use of ire as a distinctive stylistic device adopted to present his teaching material in an effective and probably concise manner. This style has had great influence over the millennia, and Grassmann returns to it fully in his Ausdehnungslehre 1844, but it is enshrined in the notion of the Axiomatic System of presentation.
Harmonia is a particular style that relates to presenting as one , both in movement and voice, thus the root is combined with monias /monos to convey this symphony and synteny. The Pythagoreans tended to think of this as Phusis a harmony of opposing natures.
Arithmos, in Euclid is defined relative to monas, and indeed it relies upon a prior notion of metron/ katametrese. Thus the ritualistic movement of placing a metron or a collection of monads is a particular style of dance, and the chnt that goes with it has or ever been recognised as "counting" or "numbering". The metron used is some region of bounded space, from a simple twig or pebble to a measuring bowl or bucket, and the motion of placing it carefully down on the form to be measured is inherently linked with articulated motion. To my great joy, while watching a wolf spider spin it web, this distinctive articulated motion was clearly recogniseable as i could see the octonion creature measure the place her next anchoring point on the web radials.
Now i did not hear the creature counting, but the grasping and carefully placing of an object, in this case silk strands was obvious. Euclid defines arithmos as a metron perceived as a monad, and arithmoi as the carefully placed collection of these monads, forming over the desired form a multiple form(pollapleisios) that covers. The distinction into a class called arithmoi is to delineate the ritual value of these forms over any other form, but at the same time to establish their complete generality. Thus an arithmos is anything used for the purpose of this ritual act which involves deciding on a unit, placing carefully, and rhythmically and counting etc.. This is a successive process, a sequenced process and a cyclical process, and clearly more complex than it is made out to be, and subjectively determined and evaluated.
The abstraction of the notion of number from this process has lead to many difficulties, further compounded by the belief that a subject called mathematics could be established distinct from philosophy and Theurgy. Today it is my opinion that the correct home for the idealistic topic called pure mathematics is within the philosophical realm of Computer Science, and Computer Science is to be founded among the empirical Philosophies if it is to have pragmatic utility s its basis.