# Re: Fractal Foundations of mathematics: Axioms notions and the set FS as a model

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## Pollaplasios and Plethos: Fractal Formations

The notion of form is dealt with in a fundamental way in Euclid

Firstly form is plastic, that is the notion of form is dynamic and transformable. Thus the notion of formless is that of not being workable with . Typical plastic materials are clay, putty, wax and plasticine. Working with these materials and others leads to the development of distinctions. Firstly the material forms or is formed . The forms that arise are objects, shapes, images, shadows or moulds:all terms to describe distinctions of form, that is the resultant of the action of forming.

Thus a form comes about through a dynamic change of a magnitude or quantity of material, and plethos is the greek for a magnitude or quantity of any material or object. Thus plethos has an inherent meaning of Form: the result of a quantity or magnitude transforming.

Pollaplasios is related to plethos by the root notion pleissos, and i have referred to it as multiplefotm. Thus we have a fractal like relationship between a form that is a multipleform of lesser forms. Pollaplasios is made up of plethoi(a plethora of individual plethos!). This fractal formation is at the heart of the notions in the use of the tools of the arithmoi, because it is how the arithmoi, or an individual arithmos is formed using the common measure method. Thus this relationship is misleadingly denoted as multiplication when, as i say it is a process of formation, that is factorisation into constituent parts and structures.

This is even more evident when the method of common measure is applied as in Book 7 of Euclid. Factorisation is the goal of book 7, that is so a common measure may be found and an artios arithmoi constructed as in the process of protometreesi

Some other distinctions we develop for forms are how extensive and thin they become. These sharpen into length, breadth and depth.

The method of protometreesi is central to book 7, and it is an algebraic , formal method of analysing magnitude, especially as quantity, and specifically as fractal nets of monads(arithmoi). At this general level the arithmos and the arithmoi become tools for analysing relationships between magnitudes and forms.

The use of arithmos is symbolic, and the use of arithmoi is algebraic. The symbol arithmos is made up of unit symbols called monads. Thus the structure is not only algebraic it is also vector-like. Rather than defining arithmos in terms of vectors let us define the vectors in terms of the dynamic monads.

The monads can take on the name integer, the arithmos then becomes a net of of integers. We can count the integers and give the arithmos a name determined by the count. Each arithmos may have a different count name or the same count name, but we cannot impose "order" on the arithmoi by count names. The count names of the integera bear a sequence and we can arrange a sequence by matching the count names of the arithmoi to the count sequence. However, only by comparing the arithmoi in the common measure method can one derive a magnitude arrangement or sequence.

Comparing the arithmoi is recursing the protometreesi method at the level of arithmos rather than monas. At this level we derive distinctions between the arithmoi that enable a common integer monad to be chosen, the integer count name to become ordinal, and a count sequence to denote order and magnitude and quantity.

What was i doing when i counted? i was simply singing or matching rhythm to the monads in the plethos, arithmos net.

From the arithmoi we can construct a dynamic vector/ magnitude algebra, or an integer arithmetic, and that is just for starters!

All of these arithmoi derive from this common measure method. Their individual distinctions sit within the outputs of this process of comparison. A development of this process is the sieve of Eratosthenes, the factorisation of arithmos into multipleforms, the relationship of a common measure of a monad, the sum arithmos from several smaller arithmoi, the synthesised arithmos and the arrangement of formations of arithmoi into pollaplasia

Now an arithmos regarded as a multipleformed one is when whatever they are in it are monads, the multipleform is synthesised by the things themselves , and generated thus.

I have to go back to the definition of synthesis to understand how a multipleform is generated from the things(monads?) within itself. This particular definition is tautologically intense, introducing the term "tosautakis" which is a tautological adverb! However, bearing in mind the fractal nature of these structures aids understanding.

We do not have a definition of multiplication, but rather a definition of synthesis and thus factorisation.

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