The concept of logos Analogos is explained in Book 5 and 6 of the Stoikeia. This is not the source of modern mathematics, but it is a major contributor.
The Pythagorean school of thought is the source of much of what we call mathematics simply because Mathematikos was a qualification attained by disciples in that school. The school itselph was devoted to natural philosophy , and Astrology and all that may be studied about our experience in this aspect of life. It is unclear what is genuinely documented evidence of Pythagorean philosophy and what is redacted through Plato, but all believed in different spheres of influence and experience and the interactions between them.
The simplest notion that Pythagoreans had was the mosaic they decorated their temples to the Musai with. These mosaics represented a blatant demo station that form could be fractured into mosaic pieces, that is deconstructed and nalysed, and then manufactured into whole mosaics that encapsulated a scene.; that is synthesised or resynthesized into some meaningful or significant whole.
The construction of a mosaic is a de facto experience of constructing a form of reality. The mosaic once completed may develop more and varied significances, as does any great work of art, but it's primary purpose is to record what has been either in experience or in imagination and to give it concrete form and expression.
One of the fundamental aspects of a mosaic is the relationships between regions and parts of the mosaic relative to each part.or region.
In Greek thought they apparently invented the word Logos to represent this relationship between 2 parts or regions. In any mosaic there are many Logoi, but comparing the logoi leads to an understanding of the form that is greater than the logos itself.
Formally Eudoxus showed that comparing 2 logoi , logos and Analogos leads to some kind of descriptive categorisation and the development of language . Rhetoric becomes intimately entwined in the process of comparing logoi, and one form of rhetoric is distinguishable as reason. It is yhis reason borne of rhetorical omparisons of logoi which developed in both directions: into more and more regional comparisons and ultimately geometrical forms; and into more and more countable relationships and inter relationships and intra relationships. . These relationships enshrined insights and apprehensions that require ways of handling, that is helpful terminology , agreed nd recognised signs symbols and words for defined concepts.
From the dimplest names for counting to the mst complex interactive expressions linking different consequences of counting, the Logoi facilitated this abstraction from the bare experience, into an analogous representation that hld the bare experience in a communicable comparison.
The bare experience was defined as magnitude! The experience of consciousness is a succession and sequence and series of magnitudes. These grestnesses of experience form he primitive ground of our consciousness. They represent the bare signal from our sensors. We cannot go beyond our sensory input. And yet our faculty of memory allows us to develop and explore imagination. Without memory I have no consciousness, and there is no I.
The incredible psychological power of order or sequence underpins our remembered experience of things. This order is imposed by the memory function or faculty.. The faculty is not do much sequential as finite. Once full no more information can be stored. However, this structure is too simple. Once one memory is full another takes it's place. It is this sequence that sequences memory recording and recall. It is this sequence which in turn defines all our subjective processing and imposes sequence on otherwise scattered data.
The memory aspect of consciousness is capable of huge parallel processing, but even so it is overwhelmed by the constant reams and reams of data. The encoding of his data is chemically and biologically influence, but the fundamental process is due to Fractsl processes involving rotations and translations nd radial bastions in these. . Thus much data is encoded physically while other data is encoded in spontaneous chemical reactions that are periodic .
The periodic nature of many of these systems also enforces sequential recording of data.
The mosaic of the Pythagoreans holds data in fixed spatial relations. These they mimicked in the Logos Analogos format. Order is essential for this structure. Thus the ratio structure itself is capable of encoding sprial form.
Writing it down on 2 dimensional sheets of paper obscures this , and in fact video technology is the best way of recording this kind of data. However, the camera does not replace the subjective processing that occurs when this data is manipulated to explorer relationships..
This same ordered structure, as used in ratios or Logoi are used in reference frames, in vectors and in many other uses.
Norman uses this one idea in many contexts, and in each context he gives the same structure different names!
However his development of number to polynumbers to polynomial to bi polynomial clearly shows that the fundamental object are these mosaics of data..
In constructing these mosaics or Arithmoi Normnan keys into the essential nature of counting, and thatis that it is iterative and fractal. There is no oe count, rather their are multiple counts of multiple distinguished forms. And the counts are meaningless without the forms. Nevertheless the forms give factorisation and aggregation to all pairs of conjugates.
Conjugates are a matter of focus. Shunya being everything is conjugated by my focus. The conjugates are pairs they aggregate to form a whole.mometimesvthe conjugates are artios that is exact , so they fit together perfectly and they factorise in multiples of the lesser. Most other times they are perisos, that is approximate and they do not factorise in multiples of the lesser but instead in multiples of the lesser Into multiples of the greater. Occasionally that does not happen, and then such conjugates are "relatively prime" literally the first positions in the Logos Analogos ratios.from which other logoi can be formed by multiplying only.
Norman starts his system with the nturl numbers. But these he shows as tally marks. The German word Zahlen is derived from old Dutch Tailen mening to tally. Tallying is as old as civilisation and it means to form a 1 to 1 correspondence between one thing and another. Some used scratches, some used impressions some used knots in cords nd some used pebbles . Where these tallies were recorded was as important as the tallies themselves, and in fact more so. Pretty soon the thing object or container on which or in which the tallies were recorded becme objects of value!. They soon developed their own marks, and so symbols of bundles of tallies were recorded in or on these media.
The symbolic mark took on a greater importance than the physical counter or scratch. The item used to tally was valued less because any item or items could do the actual physical recording. However the media on which the record was impressed had a greater value because it held possibly hours of physical effort in attempting to record objects. Human memory was not relied upon in general, because all animates tend to camouflage the bare facts.
Thus the different media for recording tallies became small or large objects of desire. And out of these kinds of traditions abstract mosaics grew, where each mark represented some symbol of the tally.
These mosaics are essentially records of where tally marks were positioned, so position becomes significant. At the same time, artisans could elaborate on these bad mosaics. Different tally marks could have different shapes. They could represent what the tally was actually corresponding to. Very soon the patrons hole wealth and status could be depicted in these elaborate tally charts called mosaics .
The Pythagoreans indeed developed elaborate mosaics as well as abstract ones, but the secret was to strip the mosaic back to the meagre functional role of 1 to 1 correspondence marked by position. Each position could have a particular shape, but the simplest shoe was the triangle. From the triangle the square, the pentagon and so on was utilised and explored. Parts were analysed and named, relationships noted, mythology created.
The circle became the ultimate form, although natural pebbles were more spheroidal than circular. Spheres were also investigated. Before the Arithmos concept could be synthesised and defined in a standard way, these primitive positional tallies and their elements had to be deconstructed analysed and reconstructed.. Thus the Arithmos generated Gematria that is measuring out things on the ground in order to analyse their elements and relationships and to tally or count their parts.
From these drawn marks in the earth, and the counting of the lesser into the greater, the parts into the whole, artisans were able to construct not only mosaics, but buildings, statues ans mechanisms by designs oe schematics, by sketches or Skesis, by patterns of drawn lines. The Arithmos therefore not only described a region of space, but lot the properties of its bounding elements, and the mythology of its use in architectural and synthesis use.
The use of this philosophy was evident to many artisans and Tekne or technological types. The Pythagorean school set out a pragmatic and utilitarian philosophy that linked very thing including the wandering planets in the spheres to Kairos, tht is the opportune time, or the tuned and attuned behaviour for each status in the heavens and earth.
Astrology was not mumbo jumbo, but a highly developed skill set based on empirical data, which among other things advised the individual, kings or nations what the status of their cosmos was at each momnt and when it was best to act in certain manners which would be in agreement with that status.
It is a cheap shot to belittle the mythological kconstructs of the ancients, decrying their belief in mythical Gods in the heavens. The influence of every object around us is not in serious question by any one. Some atheists or Gnostics or even those of a different mythology just want o be allowed to IRS the comos in thir own opinion. That urely cannot be an unreasonable request? But history shows that groups seek to oerce others for the want of Mammon! Reason then has little to do with much of human brhaviour when it becomes dominant!
Positioning quantities relative to each other and in sequences is therefore a natural human proclivity, and it underpins all our conscious apprehension of our experiential continuum.