One does not start combination theory by combining. One starts by dividing a form or idea.
The normal name for this sort of division is Analysis, and therefore the appropriate name for combination is Synthesis.
When one begins to synthesise one naturally uses the dimensions of Analysis, that is one synthesises in the way one has divided the form or idea. Thus a dimension is in point of terminology a boundary division made in the process of analysis. We tend to forget this when we discuss the dimensions or parameters of dimensions in space.
However, dimensions are some of the simplest cuts we can make in space, and as such they tend to skew our notion of the structure of space. Not one of us woul conceive of dimensioning space fractally, especially a boundary ; and yet this turns out to be the most natural and general structural arrangement of space!
At present we tend to conceive of combination in terms of juxtaposition or overlapping, but it is in fact more natural for combinations to involve fractal entwining of some lesser or greater degree. To dimension space fractally , therefore requires a paradigm shift in the motion of bound, and separation. Such a shift is toward statistical and probabilistic descriptions of boundaries and dimensions in general, reflecting a fractal disposition.
The analysis arises particularly at a boundary. Such a thing, a boundary pragmatically refers to a density change. The density change is usually perceived as sudden, and the usual change is to a lower density. Dimensions are usually "cleaved", that is a second density is introduced suddenly into an assumed uniform density, separating the continuity, into which separation a third lower density may rush to infill, or failing that a lower density of the cleaved density might remain. It is this variation in density which poses the fractal boundary description.
Traditionally one is steered to think in terms of continuity or at least contiguity, beyond this the boundary is defined as disjoint. However, this dimensioning of spatial density is pragmatically false, for when we look or measure these boundary densities we do not get this disjoint configuration, and there is some dispute about when two boundaries become contiguous. In the ideal formulation we constantly discount the intervening spatial densities. Thus a fractal description ofba boundary, indeed a process description that defines a boundary is in fact called for.
But if we have not a cleaving but a process of determination for defining a boundary, then this impacts on the notion of a dimension. In fact, to be serviceable the notion of a dimension has to necessarily be subjective, that is part of my processing ideation, not part of the external experience being processed. In such an external experience there are no dimensions. These are internal constructs which form part of a larger processing structure called a subjective reference frame.
Within my experience reference frames distanc as a prior art, an unconscious organisational structure made up of many cooperating sensations that help me define and orient my world and myself in that world. Thus allowing myself a unique reference frame means that I will eventually allow others theirs. This might take some training for an individual to realise the relativity of their reference frame. The construction of a common, cultural refernce frame allows for social communication, but also social control, and that means that the fractal paradigm would have to be deeply embedded in everything to make a paradigm shift, otherwise it will just be pushed to one Side.
Now our reference frame for dimensions length, breadth and height determine our synthesis, and form a basis for synthesis in space. However Grassmann looked analytically at just these things: what are the dimensions of form, and how do we use them to synthesise form, and all the attributes of a form. This is where the complexity of synthesis has been inadequately accounted for in algebraic terminology, and this is what grassmann addresses in his 1844 work.
So, starting with a reference frame that is entirely subjective, I apply a process of analytical comparison. Essentially that means I identify the proportions in space, the proportions of magnitudes, and then I focus on the proportion by typically comparing any pair within the proportion in more detail.
Now my initial proportion may simply be a sequence of relational labels, that is an algebraic proportion, which I piece wise modify by establishing measures of magnitudinal quantity. Within a comparative pair. Having established a successful Merton for a pair wise comparison, I am now by inductive logic, locked into a dimensional reference frame as I extend my focus to other pair wise comparisons. Therefore , my initial pair wise comparison is always fundamental, and contains the "spanning" reference frame for the whole of my experience, as well as the pair wise combinatorial system of logic.
It is a clear experience in my life, that this initial choice of comparison is subjected to review, and the results of the review may be accepted or rejected. This means that I have 3 possible results: review is identical to previous status, review differs from previous status; and finally review is indeterminate, prompting further review. By accepting the result of the review I mean I take action in line with the review, any other response to the reviews conclusions, and there are many other responses, I lump together as rejecting the review process and result to one degree or another.
Thus, there is a slim chance that over time I will develop my reference frame and combinatorial system to better reflect the data , empirical data as well as subjective processing data viewed as a formal system, with the aim of minimising inconsistencies and anomalies, improving the functional utility of my fundamental processing paradigms and noticeably improve the power and flexibility of them in expositing or explaining the interactions with space I observe, and enabling inductive predictions of greater accuracy.
The apriori reference frame is thereby update able, the proportion landscape is continually and iteratively updateable by the types of matrons I defund and develop for pair wise comparisons, and the metros I do develop, as quantity measures, become the dimensions of my analysis. As a collection, these quantity measures become a manifold describing my experience in terms of quantities of magnitude.
Now the manifold traditionally an unquestionably is situated within the structural and procedural frameworks , the grammars of our cultural language. Thus each language provides its native speakers with a manifold of quantities of magnitudes with which to describe their interaction within and with space in a cultural medium. Of communication. The grammars of these manifolds are idiomatic, but present a range of possible models of functioning manifolds.
In addition, within each cultural setting, there are usually specialist uses of the cultural language stock, some of which are transferable to the wider culture, but most of which are not, and these require a specialist cultural training or education to apprehend, comprehend and master. These manifolds have had various names in the past ranging from alchemy to jargon magic to witch crafty, science and technology, engineering, mechanics and mathematics, just to name a few.
Manifolds therefore exist in this form in specialist Argots, or languages cultures, with the Register of the vocabulary defining the metrons of the quantities of magnitudes. Quite often, these registers of metrons are poorly defined, allowing for imprecise communication, but nevertheless flexible generalities that rely not on one, but all the sensory meshes within the body. This often means that the metrons often become obsolete , arcane and mysterious as the cultural enclaves that use it develop in maturity and experience.
Etymology attempts to track this social churning of metrons and manifolds as embedded within cultural language frameworks.
Clearly, developed large scale established systems have not followed this trending, volatile root in establishing cultural metrons and manifolds. Thus we have to consider organisational structures at the level of empire to begin to understand the nation state organisational impositions. Empires enforce a wide cultural norm on many different cultural backgrounds, using the power of empire, and economic powers to do so. This quorum power extends down to individual subjective processing experience, by an effective mix of incentive and coercion, wrapped up under the terminology of socialisation or social education.
The most effective mixture of this sort was Hellenization! We still feel its effects even today. The establishment of empire wide cultural, scientific, literary and philosophical norms and mores, rhetorical styles and standards is still an amazing event to study even today. Arabic cultures which superseded roman and Greek ones still hum with Hellenistic philosophies in Arabic cultural garb. The truely mysterious Japanese and Chinese cultures may seem to be immune to this empirical phenomenon, but thst is not the case. Much of Hellenistic aspirations and notions wete transmitted to these cultures by trade and commerce, and we're it not for the strong reassertion of authoritarian government by competing warlords, centralising the power of empire and culture, these countries would be openly rather than covertly Hellenistic.
Much Chinese and nip obese Bhuddust wisdom is imported from a strongly hellenized Indian culture. Again, Indian Brahmans reacted strongly to this influence to reassertion traditional Brahman cultural mores. However, the traces of the Hellenistic underpinnings still remain.
Does this make Hellenistic culture the fundamental human manifold? On the face of it no. Culturally the everyday experience is vastly different, but scientifically, this cannot be doubted. The fledgling attempts to establish one empire of different cultures inevitably lead to disintegration of the empire, but not of the cultural elites, who typically retained a classically diverse educational background of humanities and sciences, retained educational and commercial links; travelled the empires to broaden there apprehension of each others cultural settings, and maintained libraries of ancient "wisdom" collected by the former empires for the very purpose of better administration, application, and technological understanding and advancement. To say that this was mainly encoded in Greek is only the half of it. From these Greek original conceptions, many were recoded back, warts and all , into traditional language for different culture, thus introducing Greek concepts in foreign language form, appearing as ancient cultural knowledges!
Quite often it would be the case that a Greek original derived it's power from a subservient cultural background in the empire, but the often thorough reworking of it by Greek mindsets meant that copying back into its native cultural setting was in fact a revolution of thought in these settings!
So the collecting of these cultural metrons into various manifolds in a language setting was found to hamper scientific and geometric thinking. While algebra, as a style of rhetoric was found to liberate scientific and analytical thinking. Leibniz apparently expressed the wish for a similar kind of algebra for the description of space, as mathematicians and geometers were now beginning to conceive as possible through Descartes and De Fermat's expositions in particular. Leibniz dream is expressed several times over the course of his life, but in general it was met with stupefaction. Descartes Cartesian system, though current had not yet even reached its full capability, and many were unfamiliar with algebraic rhetoric anyway. It is an expression of Leibniz genius for insight, that he felt the Cartesian system was capable of delivering so much more, if only scientists and philosophers could get together and map it out!
I have explained how the manifold intuitively exists within a language setting, and how algebra is a certain language style. So Leibniz was merely asking for a structured recasting of language into this rhetorical style of algebra, so one could talk precisely about space and geometry without all the continual and repeated exposition of what one was talking about! This was therefore a request to develop a geometrical argot or jargon, that would save space on the page, weariness of the mind from constant repetition to ensure the reader was keeping up, and time in terms of coming to the conclusions or applying the solutions.
Every algebraist from Cardano, Bombelli, Vieta to Descartes had introduced their own terminology to describe what they were doing. The problem was, these tended to be idiosyncratic, and often mysterious. Often the same term would be used in an opposing way by two different authors. Leibniz wish therefore was to remove this obstacle to understanding by developing a consensus terminology. He felt it was worth giving up a little academic freedom wo create a lingua Franca for geometry and space. The fact was, that despite not really being understood, the task he set eas in fact more complex than the simple Cartesian dimensioning suggested.
The task is clearly a social one, but also it is a cultural and empirical one. Gauss recognised the problem of different manifolds in the sciences and worked to establish the system of dimensions we use today. He tasked Riemann with that job of furthering that goal of an international scientific system of units, a scientific manifold. But how we're these units to be utilised? This also had to be defined and determined and agreed. Such a consensus was not easy to achieve, and required notable success technologically to even advance. In addition to the consensus in terminology and utilisation, it was fundamentally important to extend the magnitudes to which such a system might apply. Obvious quantities many may naturally come eventually to agree on, but philosophical qualities, how could they or should they even be quantified?
As usual, the solution to these difficulties were near at hand. Newton had established a philosophy of quantity, which in general was very well received, and grassmann had solved the translation issue of moving the manifold from its language setting into an algebraic setting. In addition the important subgroup of rotational functions had been solved by Rodrigues, but more importantly by Hamilton, establishing a functional algebraic manifold. The manifold was clearly limited and unfamiliar, not as rich as its language imprimatur but nevertheless it provided two powerful advancements immediately! The Quaternions themselves and the Maxwell Equations!
In point of fact, Grassmann's lineal algebra was the most important advance, but it was not recognise able at the time, mainly due to manipulation by Guss an Riemann for their own ends, possibly what they believed to be the larger good. In reality, it was not that the Ausdehnungslehre was ignored, but rather it was used by Gauss an Riemann without credit to Grassmann. It is the fundamental notion of Manifold in modern conception. In it we find grassmann referring back to Euclid as a prior art!
Quaternions are equally important, because thet established the usefulness of such formulations immediately, as well as establishing algebra, group theory, complex algebra, vector algebras and the new German notions of functions and mappings.
Because it did all this, quaternions were hard to. Grasp, especially in its expanded notation. Hamilton clearly showed that the theory of quaternions and conjugate functions could be exposited in a clean terminology. But when it came to application, all but the simplest cases lead to voluminous pages of text! It was enough to unnerve even the most ardent of mathematical geniuses, but not so Clifford, who took both Hamilton and Grassmann in his immense stride and simplified quite a number of intense applications.
Today work on the dimensions of the manifold continues, work on the biproducts of hamilton and grassmann continue , but work on the combinatorics of these quantities is in its infancy, especially the fractal dimensioning of thier boundaries. We have a probabilistic description of some boundaries, but I think the work needs to focuss on the Lagrangian constraints. The more these Lagrangian constraints can be defined as iterative fractal constraints, the better our manifold applications will probably be.
Thus a Lagrangian constraint will turn out to be the most general notion of a dimension/ parameter in space.
In my next post I hope to concentrate on how the generation of sequences in orientations impacts on the combinatorial processes of metron boundaries, and manifold assemblies.