Sequences, Brackets and the combinatorial " laws" of synthesis.
Laws is quoted because they are informed suggestions with a particular synthesis goal. The suggestions themselves reflect the rules of rhetoric in order to make communication clear and congruent, while at the same time enhancing mnemonic ally the subjective objective interaction during discourse.
The discursive nature of philosophical computations hides the consistent interaction with space and spatial representations. Through rhetorical language forms and grammatical constructions, the conscious mind is taken and transported to different points of view, or perception in general, whereby the subjective process may result in that "Aha!" or "Eureka!" moment when ones experiences convict one of the congruent certainty of a particular resultant outcome.
That one may find such moments delightful seems to be the main motivational distinction for those who are of a philosophical bent as opposed to a pragmatic or concrete constructive bent, in which the satisfaction is obtained from a full panoply of sensory feedbacks arising from the realisation of a project through the physical, mental and emotional interplay of creating a concrete subject..
It would seem that intellectual delights are a meager response to the fuller pragmatic delights, but each individual reflects the neuronal structures within them. Some have greater sensitivity in the Amygdaloa and hypo campus areas and so are more easily satisfied than others who may require greater physical engagement to achieve a satisfactory stimulation. Whichever it may be, there is room in the world and a role for each individual to contribute and be valued
Katameetresei and conjugation are the same active sequence except in katameetresei one holds a concrete metron over a focus region, while in conjugation the focus region is subjectively and unconsciously used to scale and in some way measure the background to the focus region. The factor algorithm actually places this process of comparison with its rhythmical response of counting at the foundation of the process of determining commensurability. The fact that we count while doing other actions of comparison is indicative of the complexity of and flexibility of the response in making sense of spatial and sequential arrangement.
When Euclid chose the segmented line notation it is easy to miss a class distinction: the extended lire is extensive in its handling of magnitude, or intensive if a fixed line is used. However, the notation for parallellogram is intensive because it describes the internal multiple form structure of the parallelogram.
The intensive ness of a multiple form structure is like a density or intensity map. This intensity or intensive ness when applied at different levels generates fractal structure. This kind of fractal structure was what logos arithmos were designed to explore. Exponential ion or logarithms were and are the first notation of fractal recursive ness or iteration , but this has been hidden behind the notion of multiplication of factors . This exponential ion of factors is the notation of fractal conjugation or Katameetresei. Multiple forms as bundles is in fact fractal structure in its various forms. The Euclidean notation for prime factorisation s again fractal structuring of space in notational form.
Fractalforums and scales have always been recognised but called prime factorisation or logarithmic factorisation or even exponential factorisation. Bearing that in mind, the factorisation algorithm is also the fractalisayion algorithm in intensive magnitude situations, and especially trigonometric ones based on circular forms. Circularity promotes intensive magnitude descriptions and scales.
When wallis introduced exponentation it was in the context of rhetorical expediency. It is an act of total misdirection to call the notation arrived at in the course of discourse the subject itself! Thus mathematics is and always will be rhetorical in its heartbeat and soulical structure. Thus the rhythms of the rhetoric , the rhymes and flows, the nuances and tones convey what they may, but a graphical mark serves to accent what otherwise must be conveyed by inflexion. And these accentuations convey the relatedness but distinctiveness of aspects of real objects. But pretty soon, as we distinguish the differences in multiple faceted situations we turn to counting and rhythm. Wherever ther is dynamic change we respond by counting.
Thus exponentstion is the consequence of counting accentuations, and these accentuations distinguish what? Levels of conjugation, levels of intensive factoring, sequencing of recursive actions, or finality/finiteness of intensivity.
Supnation and subnation of accents and thus counting is a stylised version of these distinctions. What they mean besides the distinction has to be defined, but it always relates to a process sequence, and that process sequence involves any choice of action or actions. The resultant is always some form/structure, but often mythologised into some hyperspace rather than some extensive arrangent of extensive metron or intensive fractal arrangement of reciprocal intensive metrons.
Fractions as a conception of ratios of of corresponding objects extends the notion of reciprocal. And reciprocal is pure intensive fractalisation. As the denominator increases so does the count of smaller and smaller parts: so we get more and more of less and less.
Norman wants to base and construct his "mathematics" on the so called Natural numbers. In so far as they are rhetorical marks representing the counting response I think this is a securely founded place to start, but of course not the foundation. The foundation of all conscious interactions with space is dynamic conjugacy of Shunya, which involves apprehending and comparing and contrasting relationships of magnitudes. These actions found the consequent and subsequent distinction processes and sequences which are the fundaments of conscious language or logos, recursive and iterative kairos or proportioning, whicg is recursive and tautological precisely because it is conjugation in dynamic action. The Sunthemata that inheres with this processing is the acceptance of the way things are as by agreement with oneself or if wished some external other, real or mythological. It is this acceptance that gives orientation in a perfectly symmetrical spherical reality experience, the summetria. The use of Sumbols as adjuncts and reminders and tools of rhetoric to this Sunthemata is not only a natural flow from the process but also a fractal necessity of the process. It is only by holding on to symbols thst a dynamic reality experience can be encoded at all.
Nowadays we can use video and auditory recording media to encode complex symbolic patterns of relationships which we naturally perceive as dynamic film and sound representations of past reality.
In the light of this enrichment of our rhetoric we no longer need to be tied to the old symbols of the past, and thus to an old symbolism of a so called mathematics. Many relations can now be directly visioned making 2 dimensional approximations an anachronism. Newtons fluents for example can be directly envisioned in CGI graphics, making it a matter of choice regarding annotation.
Much of the rhetoric of the past now has to return to replace the inadequacy of old symbols. For a presenter can communicate now by rhetoric and suitable animation what required delicately chosen symbols to encode! So, is "mathematics" just all those symbolic sequences and patterns or is it a communication by one professor to another about the relationships experience able in interacting with space?
For me it has to be something like the latter, where a professor can explain his take on relationships of magnitudes in space, his encoding sequences and processes, and her valued insights into spatial conundrums of form.
So how do we talk to each other about the reality we apparrently share?
As Herakleitos points ou, it must be on the basis that all things flow, nothing stands still. Ourinteraction with this reality is to freeze frame it, to discretize what is ostensibly continuous. Knowing this we do what we can mentally. We conjuate Shunya into discrete regions in multiple freeze frames that form an existential sequence of freeze frames as a basis for distinctions of language and perxeption. Which fames we choose unconsciously determines our experience of reality, and from this we construct our cut down models of reality. Such models are numerous and complex, but always discrete and sometimes perfectly static. most however exhubit a kind of slow motion film loop. a partial looping sequence from a presumably continuous experience.
These jittering freeze frames allow us to develop the notion of space as a proportioned filmscape conjugated by the powerful process of conjugation. Each sequential looped freeze frame is sequenced or combined together quite freely. the only guide being an ongoing store of the sequential process that produces these memes or fundamental components of memory.
Comparing within the freeze frames provides intensive magnitudes and quatities while comparing across freeze frames develops the notion of extensive and dynamic quantiies and magnitudes. But these concepts are wholly driven by and derived from the model that the subjective processing has created.
The conceptions of Lagrange fundamentally encode this dynamic reality, but it is only the work of Hermann Grassmann that provides the key conceptualisation of how to put it all together to describe a dynamic reality. While Grassmann is virtually unsung except in the most abstract circles, our modern conception of space time and quantum Mechanics rely wholly on his conception, with Sir William Hamilton providing a concrete reference frame that exemplified Grassmann's general concept. Both arrived at their conceptions independently in the sense they never collaborated, but both drew on the Lagrangian conception of Mechanical description of reality.