Consider a form a
Let it consist of multiple subforms as combined together in some proportion #s so that I may describe the process of combination as
a = $s #s*as where $ is the combination process and s is an index of the different forms.
Now each subform as itself can be analysed in the same way and can be written as some combination process of smaller forms…..
We may continue this fractal analysis as far as we like, but where I stop I must decide to make these subforms the basis of my constructed Synthesis. Not only do I decide a basis, I also decide a formalism.
Now on the bare bones of that you may imagine that I have analysed the form down to its constituent "particles" and that may be as you like it. However the form is more general than that, for the subforms are not specified as any quantity or magnitude until I stop and decide the formalism and the basis.
Suppose like Lagrange I wanted to have only velocity and distance as my magnitudes, then my basis would consist of proportions of these magnitudes, that is quantities of these magnitudes, synthesised under the formalism rules, and for each magnitude a set of constraints in terms of all other magnitudes, that is a set of constraints on the quantities in that magnitude within the form in terms of the relationship with the quantities in other magnitudes within that form, would need to be defined as part of the full description of the form.
Because this is a fractal, it is complex to initially consider. But when you understand that fractals are synthesised often from iteration of very simple rules you begin to see the trees in the forest, and which trees to focus on to get a simpler picture or mental apprehension of the dynamics.
Hang on a minute, I thought we were talking about Form?
Many of us do not know what form is. We in fact have to go back to Plato and Socrates to even grasp that Form and Idea are the exact same referent in different language bases: Latin and Greek. It may seem disingenuous to suggest that "form" has not been put to different use than" idea " over the millennia , and of course it has, but these uses have been individual, even cultural reinterpretations of the Socratic Platonic theory of forms/ ideas. Thus what must be emphasised is Grassmann' return to the purest sense of the Platonic theory.
Again, many do not realise that Euclid's Stoikeioon is an introductory text book to the Platonic theory of Ideas/Form, and not at all a Geometrical text. It is an academic introduction to philosophy through the philosophy or theory of form Plato exposited, with a view to preparing thinking administrators and the ruling "elite" for the Greek republic utopia Plato envisaged. This elite would be the best of the best, the Mathematikoi, the graduates of an intense Astrological training, who would be able to govern wisely, with kairos competent to both men and the "gods" . For they would not only know things of the meanest pragmatics, but also the illusory ways of the gods, and they could forecast the position of the god,s in the heavenly god Ouranos, and advise the republic accordingly on the wisest courses of action.
Thus for Plato philosophy had a utilitarian dimension, it could even bring about Utopia in the hands of those rightly trained in the Platonic philosophy. I do not go into it here, but Plato clearly used all his skill to establish a "magical" spell that would bring this about, both in literature,theatre, academia, and myth making, in which possibly Pythagoras is his greatest mythological augmented reality. Certainly, the later Academy sought to counter the myths of Christ with the myths of Pythagoras. Any way we have both today from which we may choose as we wish and construct our own multiple form!
Thus Euclid's Eudoxian Arithmoi form the basis of his analytical methods based on Idea/Form which both Grassmann, Hamilton and Bill Clifford in particular relied on heavily to exposit the algebras of space and time they could sense in the Phusis, the physical arrangement of opposites!
Thus, returning to the fractal form at the beginning of the blog, the basis I choose is really up to me, providing I can comply with the formalism.
Hang about again! How can you choode whatever basis you like? Because the forms/ideas are perceived by me, i can describe them as i prceive them, providing i identify the constaints. Thus Hamilton in his Theory of couples perceived everything as moments in time, and he identified the constraints and relations as he devloped the theory. Similarly Grassmann did this in a more general way, allowing one the freedom to use any magnitude of choice. Because one is using all "types " of magnitude, that is al "kinds" of magnitude, the german word Fach is used, and the notion Mannig fach is used to describe this description of a form. We call it a manifold in english.(die Mannigfaeltigkeit)
The manifold is the "fundamental basis " of the basis of any form. That is, once i have chosen a basis for a form by stopping the analysis, i must choose a manifold for that basis, that is i must inhere to that basis my basic set of attributes for space and objects in space.
So now we can describe the formal structure of a form in general, to be a fractal synthesis of subforms, with the basis being a chosen fine structure of subforms in synthesis. This fine structure of subforms is based on a ultimate or final basis called a manifold, which is a some combination of all the magnitudes we attribute to these basis subforms. Thus the manifold is a multipleform, not necessarily lineal, but the basis subform is defined in a lineal combination.
The fractal structures are thus defined in a lineal combination, but the manifold may be any arbitrary combination of attributes.
Bill Clifford saw this immediately and so attempted to combine what he understood of Grassmann's lineal algebra with Hamilton's Quaternion algebra.
However this is not a rich enough manifold to describe the attributes of space, and the lineal algebra requires a fractal synthesis concept to really make it work. Computers were going to be needed to go much further with this model of space, and Benoit Mandelbrot's fractal notion.
The fractal conception , clearly worked through by Grassmann as a " layered" structure, very crystallographic, was a model fit for his times, when it seemed that photography, photometry and X-ray crystallography was revealing the "deep structure" of space, that is a crystalline substrate into which the nnewly theorised Dalton atomic structure, and Mendeleyevan periodicity could be dramatically explained. This was in 1844! By 1860 everything seemed to be confirmed, but electrostatics was just about to change everything!
This immense repetitive regularity dominated the concept of the manifold, but Gauss and Riemann truely suspected this was not the true or complete nature. Grassmann admitted to defects in his analysis, especially with regard to rotation, but he had solution rosdmaps laid out before him. The intervention of Gauss and then Riemann steered him wrong, and away from his original inspiration, insight and plan. So much so that even he could not recognise the same prima facia in his own work! He wrote " it seems thst perhaps ausdehnungsgroesse take the following form…."
The boldness and the certainty were gone. He was no longer as sure he had got something as groundbreaking( grundliegenden) as he had assuredly thought. At least, not in the 1862 mathematical clothes he had to dress the ideas in.
It was not untill after Gauss died that he began to be " lionised" and the revival of interest restored his convictions. Unfortunately, he had too little time to work it all out, even though he had kept abreast of things. His health, his teaching load and other commitments ,and his diminished mental faculties meant it took time to get back into it. He did so by revising, annotating and updating his original works called Ausdehnungslehre, and then trying to analyse the role or place of the quaternions within it. Just before he died he published what has been taken as a critique of Hamilton, but which in fact was a sober and considered assessment. He found the quaternions within his system and wrote down the product required for them.
Let's be clear. Hamilton discovered a set of 4 magnitudes, 3 of which were "imaginary" or complex which behaved geometrically like a description of quantised rotating forms. Grassmann analysed them and constructed a model of them in his theoretical framework. Thus Hamilton identified the form, Grassmann modelled the Hamilton form. We do not have one appearing magically in the others framework, we have a translation from one framework to another. Why bother?
Exactly what Grassmann thought. Why bother to learn quaternions when my own framework describes them!. Then he died.
It was Clifford who was bothered. Riemann's Habilitation speech and lecture had inspired him to look for this mannigfach, and he found it in the lineal algebra he thought. But then he realised a linear combination of attributes was not going to explain Maxwells equations. The only lineal system that could was the quaternions. It seemed obvious that the manifold would need to be extended from Grassmann's lineal system to include the quaternions. He set about doing that, only to find later that Grassmann had already done it!
As powerful as his new manifold was it was still not rich enough, but now he knew from Grassmann's detailed explanation how to Extend any basis to include any magnitudes you want! Grassmann's analysis gave one the tools and the quality checks and the combinatorial cases that needed to be verified each in turn before you could say: this new extended manifold works this way for producing and combining. A lot more work has to be done the more elements in the basis before a basis can be recommended.
Now Riemann said that while anybody could suggest a manifold, it is properly the work of physics to establish the best manifold for our experience of reality, that means for example if a quantum manifold is chosen what is the relationship between the units of all the attributes. It turns out that the relationships can be written in the form of a wave amplitude function, in which every attribute has a wave description, and the description of all attributes becomes a linear combination of wave functions. Thus the concept of an Undulatory manifold that Bill Clifford had, and hoped to capture with his biquaternions, has become the standard descriptor of the manifold in quantum mechanics, but they use a complex probability function as the attribute basis, and the wave potential (amplitude) as the form basis.
Bill Clifford was able to advance Grassmann's original conception, and Clifford Algebras represent the continuation of Grassmann's ideals, but the manifold concept is truely Gauss and Riemann's. Yes Justus was working with others on a manifold, but it was by no means the generalised notion put forward by Gauss through Riemann. Hermann would perhaps have been the first to understand what Riemann was saying in 1853 , but this would have been no surprise since Riemann was quoting Grassmann's work when he spoke about his conception!