A continuum of points i try to not think in when looking at my experiential continuum. Rather i perceive a continuum of locations, and to reference those locations i have to construct a coordinate system relative to me, the metric i use being culturally determined. The advantgage of this to me is i can explore what is in or at the location in my experiential continuum rather than some abstract points. These apparently are regions with interesting properties which some call spatial. What happens within a region involves some iterative process or processes at many scales of the metric. These processes are involved with boundaries or within boundaries which are perceived. It is when I try to give the coordinates of the boundary that i find that i am engaged in an infinitesimal process of iteration. This is not strange as the metric itself is defined by an infinitesimal iteration.
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.
The basic operation or operation pair is addition/subtraction. The basic background to this operation is inclusion or exclusion. From this background I derive also associativity identity and other subtle operative parameters. It is the boundarisation process that enables me to generalise to inclusiveness and which emphasises the iterative nature of the operation . When I include one more region into a collection of bounded regions I am exhibiting the basic operation of addition and underpinning the iterative process of counting. If I exclude a region from a collection i have to specify a number of things: Is the excluded region already part of the collection- in which case i am performing subtraction; or is it being excluded from counting-in which case i may be performing some set operation on elements within a set, or some algebraic operation on elements within a set and of a certain type.
Multiplication and division are processes wholly derived from addition and subtraction as are integration of a function and diferentiation. These operations then and those derived/uncovered on particular sets are all iterative in form and implementation.
The discussion of nullity I am sure will be of suprise to many ordinary people but of course those who have to deal with the impracticality of some of our rules may benefit from following what Anderson suggests. I too once laboured under the false assumption that mathematics was a natural element of my world instead of part of a constructed model of infinite possibility space as i refer to it. The constructed arithmetic model of the transreals for computational arithmetic has much to recommend it. However the concept of nothingness emptiness etc is not to be confused with the requirements of a pragmatic computational arithmetic for dealing with distinct valuations. The inclusion of nullity and infinity as fixed entities on or off, in or out of the set of reals is a welcome clarification of the constructed nature of these fractal metrics.
With reference to the imaginary valuation i : it was always an operator which we were taught was a valuation at a primary level. For anything to be a valuation it must be a reference to a perceived value or quantity. The reals are so far our culturally neutral quantifiers as are binary representations of the reals. What we do is to try to devise operators and so methods to map our value exeriences onto the reals. Thus millions of colours are now specifiable by a real quantifier which our machine methodology is able to map onto a screen as a colour value. Like i the nullity is an operator and because of machine methodology we can realise both of these to practical advantage.
How does it affect my notion of reality? It further identifies the constructed nature of that concept and the cultural forces that determine or maintain it.
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 @ nullity and ∞ infinity are numbers, However they are not numerals as they do not mark a stage in the iterative process of counting.
Originally posted by author:
By definition every possible fraction can be written in an infinite iterative table of this form. We can devise a sequence which enumerates every fraction in this infinite iterative process. This means that the fractions are co untable but of course we are in an infinite process. A little thought convinces that all the fractions which are powers of ten in the denominator are also within this table. Therefore they by syllogistic logic are countable. Since these fractions form the basis of our decimal system it follows that every number representable using these fractions will also be countable. It seems therefore that Cantors description of uncountable infinities may be one of form and not of substance. If we engage in this infinite process we will encounter what we might call infinite fractions. Pi for example or e. Their are infinite fractions which in a limit process equal a simple fraction 1/3 is an example. The infinite fraction is 333../1000... Instead of countability being the criteria we should look at self similarity, and in particular the ability to match or self reference. So for example every fraction in the table could be matched against a fraction with a power of 10 numerator thus illustrating self similarity, magnification, etc.The problem of assuming that we can list every number in some categorisation scheme and that assumption being proved false is in fact a non problem as indicated above. The problems lie in a lack of rigour around the notion number and the notion of proof, two notions we could possibly do without in favour of numerals and convincing evidence.
Axioms 1,3 through 5 are the fundamental ontological attributes for "i" for this system.
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.