It is fair to ask Is there a process that products quantities? Do we interfere with quantities mentally and physically to get these multiple forms, or are there processes that naturally produce these forms and naturally factorise?
When we set up a notation that shows 2 factors represented by lines, in the context of a geometric form called a rectangle, are we justified in measuring , that is setting a metric to the lengths, and then multiplying these measurements and calling it area? And then to go on and treat the lines as dynamic and the area as dynamic as a consequence of the dynamic sides?
Actually no, because the underlying process is completely obscured. To justify this method or extension of use i need to see it develop from an underlying process. If such a process does not support an application the likelihood is of misapplications, misinterpretations and misunderstandings.
Euclid presents his material in this symbolic form, without number just a general notion of magnitude, made specific by Eudoxian proportion theory. This notion of magnitude is dynamic and suitable for all notions of quantity. But this notion of quantity took different clothes as time went on, dividing into number and form. Number being some abstract concoction and form being mysteriously generalised.
Newton was the last great synthetic analytical geometer who worked in a Euclidean format. His philosophy of Quantity is a deep analysis of Euclidean Philosophy and thus Pythagorean Philosophy as mediated by Plato. Thus dots and Quantity a Pythagorean assignment to form, precedes Geometry, informs gematria and denominates arithmoi. The style and poetry and rhetoric in Pythagorean philosophy about the arithmoi is lost in thee modern notions of number , geometry and form. Thus for many applications the process was abstruse and sometimes deliberately so. The power was not the public method, which may have some merit , but the private hidden method that may intimate deep mystical truths.
After Newton and particularly Kant, the fashion was to provide logical, scientific and Mechanical methods and processes to underpin applications, but mixed in with these were still many obscure conventions and practices hat wer taken as given, or for granted.
Yet still, Euclid based many of his notions on interacting with space and form, although he did not go into metrication in his Teaching material, probably because there were so many competing units or metrons(metra). The other processes he drew on include the arts of ropemakers, builders and architects.
In Book 2 Euclid draws upon his method exposited in Book1. The important concept that he used and highlights is the parallel Postulate. He begs ud to sccept it for this very reason. Without it we cannot establish a simple axiomatic system of combinatorics or comparison, especially of dynamic forms. In this system Euclid introduces the notion of the product of sides as dynamic variables, and as a notation for form. Thus xy stands for the form and also for the 2 lines that contain it between parallel lines. It is important to note that when we use graph paer, we actually apply the parallel line disposition in space that Euclid postulates. However, we tend to be too rigid in or application s Euclid allows the parallels to be omni directional, from which we selesct the appropriate setting or arrangement. Thus even in the same figure, such as a parallelogram we have 2 systems of parallel lines, and we may have as many as is necessary.
Thus in book 2 we find the system that Both Newton and Grassmann accepted and utilised especially in the dynamic situation, including the notational convention. Many recognise this today and in fact explain it as a geometrical algebra, with the sentiment that of course it is not "modern"!
Book 2 is rightly called "a classic" if it is not to be called "modern"!
What we see demonstrated in this book is the gnomonic method, analysis and notation of combinatorics and comparison, and the introduction to Eudoxian proportion which is elaborated in books 4 and 5. Books 1,2 and3 are intended to demonstrate the importance of the circle, its attributes and consequential properties and the reliance upon it for all consistent and reliable method. Although it may be said that the postulates and proposition and definitions are in a system, the system itself is not often reveled to be the system of spheres, and the method of spheres and the importance of the God Isis in making this known to mankind, "who should in fact show due reverence".
Although i am explaining to you the mindset of Euclid and Plato in the motivation for this work, it must also be pointed out the immense pragmatic and practical base of these methods and concepts. Thus anyone who builds temples or public buildings would recognise the soundness of these principles. It is the Theurgical content that moderns do not pick up on. No matter. I am not an apologist for the god Isis,but i am explaining why some notions may appear curious to our modern mindset. In fact, the Moral and religious nature of the Euclidean text did not escape the attention of our ancestors even in Newton's day, and a good way into the early 19th century. It was Lobachewski, Bolyai and Gauss who particularly made Euclid seem less relevant to modern concerns and scientific and Technological endeavours, rebranding his principle theorems as the subject Geometrie , they left the rest of his material to the classicists.
In fact after Establishing the circle, he hoped, Euclid goes on to Establish the Sphere, but for this he draws upon Eudoxian Proportional theory, without which we have no similar figures by which to make comparisons and thus estimates by logical deduction alone, but only after substantial induction demonstrates a law, only after millenia of experience measuring, supports a formal link between observation and formal proposition. Ptolemy for example used and corrected the observational records of the Babylonians to establish the Methods in his Almagest. Hipparchus used the information Thales brought back from Egypt to establish his methods of Trigonometry. These systems of inductive law from which principles may be stated and deduced are exemplified in the teaching material of Euclid. That we have this inductive material,is due to the empirical scientific method which religious and sagacious men inflicted upon their students and acolytes as a theurgical duty , to attain to whatever they expected of their god or gods
When we look clearly at the information before us we cannot avoid the relationship within us between Religious beliefs and the science an technology they inspire. That some religions may seem to be against Science and Technology, is in fact a mistaken view. The building of temples has in fact inspired the most sublime technical advances, and the very idea of building a temple on the moon will inspire yet more!
The symbolic use of the line, therefore has a long an illustrative history, but for further analysis of the foundation of "Mathematics" so called, but rather Pythagorean philosophical and Theurgical "science"/ knowledge i will focus on Euclid, and his main expositors Newton, Grassmann and Hamilton. To them the seemeioon and the gramme were potent sumbola of their sunthemata with their God, and with them they demonstrated their summetria, The summetria of their God.
In answer to my initial Questions i have turned to the biological understanding of the cell as one example. In this example we learn that not only must the product be replicated, but the producting process also must be replicated. When that is done we may then have a division and combination strategy that is based on binary mitosis or quaternary meiosis.
There also seem to be reactions in the chemical domain that due to their homeostatic nature, provide the cyclical basis for self organization into a combined whole, especially through catalytic chain reaction,
In the hydrodynamic domain fluidic self assembly using thin films is an industrialised process, and finally in the environment the cyclical weather patterns though random produce erosion patterns that are formative.
Thus i believe we have a good inductive base to expect to find some means of producting forms by dynamic factorisation, but that we must take care to address the contexts of the process and the style of the notation we use to distinguish the relevant parts. In particular, the use of the notions of "mathematics" must not be applied as if mathematics were a set of divine revelations rather than a collection of man made and man adjusted methods muddled and confused in places by too much reworking and too much self aggrandizement.
Although the word is relatively new, and thus did not describe what knowledge production went on before its invention in the Romantic Era it is still worth remembering
" Before mathematics was Science, and that science was driven by philosophy, the greater cousin to religion, and all technology was put to the service of the gods."