An Arithmos is a fundamental yet flexible concept. Probably the Greek or Pythagorean concept was not so flexible. An Arithmos derives from a Mosaic, which is a fixed embedded array of various forms , a manifold by another name, a coordinate reference frame, a vector space et al.
We tend to embedd the Cartesian space as or reality, but the Greeks embedded the Arithmos into each form, so they had a Formenlehre. It is not clear whether they had a universal frame as we culturally do,mbut the Socratic philosophy of Form would be the best candidate if they did. Parmenides and Zeno, in chasing the Monas, confused and obscured the introductory notion of the Arithmos or Arithmoi, but by Analogy it underpins all their philosophy.
To embed one reference frame as reality may be culturally acceptable, but it denies nd belies the flexibility of the Arithmos, and the Arithmoi. The Grassmanns struggled to derive a Formenlehre, which gave them or Herrmann at least the most general flexibility of mind and reference and analogy. He expresses this in saying that we no longer are restrained to think in only 3 dimensions!
Well to those who bond themselves in 3 dimensions, this makes no sense. Yet if they bound themselves in n dimensions this would make perfect sense. Because the ontological mature of dimensional thinking is cultural, it is perhaps hard to realise the anti ontology.
Suppose there were no dimensions what would that be like? Now from that point view decide what would be 1 dimension. Immediately you intuitively have 2 degrees of freedom, but you forget or perhaps did not realise that you had infinite choice as a point! You had an infinite choice of how you defined 1 dimension. Thus you exist in an n dimensional space. It is your culture that restricts you to 3!
The Arithmoi as forms or rather embedded forms are the forms we perceive everyday. The more curvaceous ones I have named Shunyasutras! They are all Arithmoi, and they are all dynamic.
We have a technology based on this deep understanding of the Arithmoi. It is called CGI. Computer generated images, typically have what is called a wireframe form, or an underlying net. These are the dynamic Arithmoi. If the Net incorporates Bezier curves as some vector graphics do, then they are what I have called Shunyasutras.
Arithmoi and Shunyasutras are clearly of great sophistication in the CGI domain.
Now I have consistently said the Modern Mathematics is no longer a viable subject boundary. I have subsumed it into computer science, and specifically into CGI.
In this domain we can actually make sense of sequences and series as iterative processes of construction. The things they construct are both spatially and video frame significant. The temporal concept consistently derives from record keeping, so I do not refer to it as a fundamental. The spatial record or frame is fundamental.
I have explained how Hamilton conceived of this in his "science of PureTtme", which is a fundamental reconstruction of the Algebra of mathematics of his day, embedding Arithmetic into these lineal or vector forms, and creating the first fully functioning vector algebra before Grassmann published.
Hermann on the other hand published the most complete lineal Algebra with a method of how to go to any number of " basis vectors.", thus in 1844, the only 2 people who could have understood each others work were Hermann Grassmann and Sir William Rowan Hamilton.
Things have slightly improved since then, but the combined labour on their work is done by the Clifford Algebraists and they still number but a few. One of the reasons is that they have had to spend a great deal of time on Taxonomy! The field is so rich that it is easy to get lost in it. Taxonomy is a way of keeping in contact with one another.
Hermann advised careful consideration of Begriffe. This is a wonderful German word for keeping a handle on things! The translation " terminology" does nt do it justice. Labelling might be a better translation.
Now if you give deep thought to how you label some sequence or series you can in fact make analogies in any " dimension". Thus deep and thoughtful labelling can mean that certain results drop out naturally and easily, and have general applicability.
It is the identification of sequences, and series which is crucial to this property of the Grassmann method of Analysis and synthesis. Thus a proper grasp of sequences and series is fundamental to progress. Series, as it turns out rely on a fundamental grasp of sequences, because series are patterns of sequenced actions of aggregation or combination, but distinguished by a fundamental algebraic rule: simplify by combining like terms! The general rule is simplify by combining like labels, and proportion dissimilar labels.
We collect together like labels: we simplify. We collect together all types of labels into a sequenced proportion. These fundamental actions determine our algebra and our geometry and our Astronomy. They are the construction of our Manifolds, that is the construction of our Arithmoi, and our fundamental sequencing of proportions within the Arithmoi..
The sequencing guides whether I can model a scatter of recorded elements of the Atithmoi as a record, or an iterative sequence based on internal or external structures among the Arithmoi..
For example I can proportion the squares in a plane Arithmos as an iterative sequence based on the count of the sides. The Fibonacci sequence requires me to follow a more complex proration ign action: some of these actions are mono tonic increasing or decreasing, some are modulo increasing and decreasing, some have to fulfill proportion constraints, some count constraints . All these different constraints have to be tested for, and it soon becomes a complex nightmare. Thus a processing system that can carry out a single defined function ( a notion derived from a functionary in the Prussian Holy Roman Empire) can be labelled, and those functions called in sequence to complete a modelling task..
Alan Turing with his early experience of such Machines developed the theoretical or coding context for these types of functions, and the engineers struggled to build autonomous systems that would carry them out.
These functions can be done mechanically, but electro mechanics became the preferred engineering frame. Eventually this became renamed as quantum Mechanics, for which Dirac devised a whole new st of labels to work on the Arithmoi. You have to understand, the Arithmoi have not changed! The labels we use have developed and changed. We now look at specific proportions in the Arithmoi, or in an Arithmod, to model some aspect of the Arithmos or the Arithmoi.
For example. The proportions for a straight line of elements in the Arithmos can be labelled by 2 sub Arithmoi to this particular Arithmos. Sounds crazy until yo take out a piece of graph paper and realise that the Fractal pattern in front of you is an ancient Mosaic called an Arithmos, and it is divisible into fractal Arithmoi!
When Mandelbrot said we did not have a geometry for clouds he was only 99% true. The 1% where he was false was where the fractal patterns he could see recorded were recorded on an ancient embedded form called an Arithmos! We had a "geometry" basis all the time, it is just that the geometers ignored or shied away from it!
We start at the Arithmos. We record the data, we analyse the data to find sequences and series. We proportion those sequences and series to see if we can iterate them or make them recursive. If we can't, we habpve a machine that can store the data as records and display it on a screen in somany ways including as a mosaic photograph of the original ,or a time lapse movie.
The Arithmos has come a long way baby!