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interesting point  

One important corollary of Axiom1 is that Everything i construct is necessarily relative to me in "my" model. Hence:
Axiom 3:- i can construct any given number of other reference points within "my" model by iterative processes.

This among other things allows me to rotate {in fact all affine transforms} my model relative to myself so i can gain insights from different vantage points within my model/ experiential continuum. The number of reference points is unbounded. The iterative processes are the basis of "trial and error" within my experiential continuum as i construct the re configured model from my initial assumption of cultural paradigms.

Axiom 4: The "context" in Axiom 1 is not constructed or definable within my model but is perceivable by "me" by a iterative process of negation of all elements within my model.
Basically i can't say what the context is but i can say what it is not by recursive means.

Axiom 5: I stabilise "my" model by an iterative process of "acceptance".

"My" experiential continuum changes with what i "accept" as a basis for the iterative processes of perception and recognition. the cultural forms which i accept from conception are numerous and pervasive and as i alter these my perceptions change as does my experiential continuum. "not altering" then is a nascent notion of acceptance.

Axiom 6: The Set FS is the universal set within which my model/ experiential continuum is defined and has a rule:- all processes on its elements are iterative /recursive and all its elements are determined by iterarive processes. notFS will be the recursive definition of the context in axiom 1. However there is a mapping from notFS onto FS such that FS is a model of notFS.

Axiom 7: Iterative/recursive processes operating on notFS are perceivable.

These processes will be compared with enegetic transfomations within FS.

I just wanted to briefly discuss the important iterative process of metrication, to highlight how we have always been using fractal patterns practically. In establishing a metric we are doing something very simple and repetetive and recursive in fact. By taking a bounded object as a standard for length we quantified our metric for lenghth. By an iterative process of disection we were able to establish a pattern of marks from this standard length which form the patterns found on all measuring rules or tapes, and this we can continue until the thickness of the marks used become significant. By using finer and finer marks we can produce infinitesimal regions on our standard length. but we can also extend the pattern in an infinite collection of standard lengths which are self similar to every other division within the collection and which have the same structure in orders of 10 throughout. So the metric we use every day is a region within an infinite fractal pattern. It is worth noting that the problem of the thickness of the marks on a boundarised object limiting accuracy has led to a laser beam of a certain wavelength being used as the standard but again the boundary conditions for the bouncing beam are iteratively determined.From this the standard has become the space time constant itself, but again the accuracy of this defintion is subject to an iterative process because of the uncertainty principle. So as i indicated we can see that our standard scientific metric is based upon a fractal pattern which is infinite as well as infinitesimal.

There are five fundamatal processes in mathematics: boundarisation, enumeration, operation, mapping, iteration. Of these iteration is fundamental to all the others.
Boundarisation, is a perception led process in which the region of attention is differentiated by a perceived boundary from the entire perception field, such that i am able to perceive one side of the boundary as distinct from the other side.The boundary itself is arrived at by iterative processes within my sensory perception faculty and truncated ones at that. Boundaries are limit surfaces in general and are abstracted from from my experiential continuum by a cultural process of naming and or definition. The iterative process that is involved in boundary formation is highlighted by the simple game of determining if a region is within or on one side of a defined or perceived region called a boundary. As the region gets smaller and closer to the boundary region, the perception of the boundary becomes more and more precise. The boundary may or may not be the limit of this process. If it is not the limit of this process we often define it as such,which tidies things up but hides the iterative foundation of boundaries in general.

The region we use as a boundary may be a symbol of an abstract notion of a boundary; for example a pencil line is often used to represent a boundary called or defined as a line. The pencil line is in fact a region whose boundaries are no more definite than any other region because they to are perceived through an iterative process.

Our perception of boundary is a complex mix of mass, colour, tone, shade and region.

Certain boundaries are named culturally such as line, circle, ellipse etc. others are exemplified such as leaf or petal etc. The abstract concepts of boundary often used in mathematics of a certain type have tended to queer the pitch in favour of non "real" forms which of course have obscured the iterative nature of boundaries and boundarisation.

Enumeration of bounded regions is a cultural process that we are all exposed to. We call it counting. This along with boundarisation forms the basis of our nascent notions of quantity. Quantification is one notion that follows from enumeration,but also order and rank are closely  involved with it. Finally quality arises from the process of enumeration and quantification and rank.

The cultural interation "+1" called counting leads not only to notions of quantity but also of quality. The iteration process is not finishable, therefore not finite. It is not only not finite it is not bounded above. So i am given an unboundable increasing process which contains my notion of infinite. The enumeration of quantities in the form of a metric is the basis of our measurement schemes, and any metric is the result of an iterative process.

Quantification is one of the cultural things that we have done to enable enumeration of things we perceive but which are not bounded. We take a bounded object that is affected by this perceived event. This is used to then quantify these unbounded events. These quantities again are the result of recursive processes. For example the notion of length is quantified by a standard which was at first a proportion of a supposedly fixed distance, but then a standard fixed length of a given material at a given temperature. The uncertainty of this standard led to the use of a certain frequency of electromagnetic radiation to define the length of a meter. Finally the universal constant the speed of light in a vacuum is used to determine the standard length of a meter using a very accurate clock cycle of a given element of caesium. This iterative process will no doubt continue as we find out more refined measures of the quantities we hold as basic.









 

Mathematics has always been a collection of useful calculating tools , methods/procedures and language or notation for representing these aspects of cultural interest and enquiry. So has philosophy. The obfuscation came when the two were combined by western philosophers/ students of the arts. We now have a semi mystical approach to one of the most practical aspects of cultural enquiry namely iterative investigation of boundarised regions in space. The quest for precision and proof for example are philosophical concerns disguised as mathematical. The true nature of mathematics is fun and games! PLAYING WITH REGIONS OF SPACE. Discovering their relationships and working out methods to record the different aspects of those relationships. Geometries despite their seeming abstractness have their root in boundarised regions. The internal properties of boundaries within these regions have occupied diferent cultures and schools of students for those cultures own specific drivers. But the operations on and within boundarised regions have tended to be pushed outside the general conception of mathematics only slowly being identified and included over time in many guises. So addition /subtraction is not questioned, but at one time not now, translation rotation reflection were. Certain properties like symmetry and self similarity cannot be adequately defined without these operations. we have no signs for these operations yet we have marks for stages within an iterative process (addition, counting) which have been given a status not connected to the process itself, a status called number. The numerals are given philosophical status by tradition for example the pythagorean school gave them a whole other mystical aspect, but the iterative operation is overlooked as are/ were many other operations on boundarised regions. When Descartes in particular mixed philosophy with the methods and tools of algebra he created the basis of modern mathematical requirement for rigour.x² +y²=1 the equation for a circle is only that on the region [-1,1] on the x axis, outside of that we have not defined the operations to be performed on the mapping. Whatever those operations are they result in locatons off the plane x,y. The operator i such that i² = -1 is not a number except in a philosophical sense. In this same sense the operators and are numbers, However they are not numerals as they do not mark a stage in the iterative process of counting.

What ontological attributes does this "I" have?

FWIW,

db

Certain things are undefinable in axiom one. So axiom one is merely a convenient starting point for any ontology. Ontological is to me simply explained by " on to logical". We move from an undefined experiential conglomeration on to a logical sytematic arrangement of experiences" This is in part the unavoidable constructionism that is referred to in axiom 1. More linkable to mapping directly are the inate patterns that we recursively perceive. So for example "straight", "curved" , "crossing" , "angle" , "shape", "boundary" etc are all experiences which to describe we have to exemplify,because we inately perceive these instinctively, but not instantly. Recursion or iteration is necessary to make these perceptions distinct.
Notions like orientation cannot be defined without these inate forms. In this regard "fixed" is an inate iterative experience vital for the concept.
Mapping is a relationship notion with many exemplars, not the least of which is symbolic notation or language itself. For example a word can be related to an object by pointing and uttering the sound or pointing to a symbolic representation of a sound or pointing to an object and the symbolic representation. This by the way is magic in the oldest sense and writing is and always has been the casting of spells as has the spoken narrative or injunctive.
A little thought reveals that our whole experiential continuum is a mapping of sensory data to culturally valued sensory data in culturally valued arrangements or forms for the most part, although we exclude many experiences in this pocess or rather ignore. Those who do not ignore every experience they are culturally encouraged to  may be hailed as a genius or a mad person depending on the utility of what they attempt to map into their society's collective unconscious ie their language. Mathematics borrows from this cultural mix such mapping as mathematicians find useful to illuminate or elucidate the relationships they are exploring whether they be quantities, numerals, permutations,series and sequences etc. Descartes ( following al hourisin) for example perceived a mapping between geometry and algebra for example. This is a descriptive mapping which facilitates certain methods of solution for geometric problems and vice versa algebraic ones. What is often laid to one side is the recursive nature of mapping. This is not to say that this is not pragmatically realised, but rather to say that it is not recognised for what it is: Iteration.