RE: RS - Mathematical representations

From: Chris Lofting (
Date: Fri 19 May 2006 - 01:26:04 GMT

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    In the context of recursion and meaning creation the following post of mine to another list may be of interest - the original focus was on mathematics in biology/physics etc. Note here the use of +/- dichotomy introduces a
    'boundary' even if we consider the number line as 'infinite':

    ------------------- Lets review this from an engineering perspective in the form of our use of Mathematics to represent sensory perspectives.

    A major emphasis is that of 'orthogonality', expressed in the use of Cartesian coordinates. The most common format is the establishment of the X, Y, and Z dimensions. These dimensions, being determined in Cartesian format, are in fact representations of dichotomies of +/-. The establishment of the number 'line' is the setting-down of a basic context and so we set the BASE context as the X axis and so a +/- dichotomy. WITHIN this base context we introduce the Y axis and WITHIN that XY context we introduce the Z axis. This focus manifests self-referencing where it is the LABELS that differentiate +/- dichotomies. (through use of those labels I don't need to
    'see' anything)

    This derivation of XYZ gives us the standard logic relationships of Z <= Y
    <= X. In logic operators this represents the IMP (implies) operator where we work backwards - given Z it IMPLIES the presence of XY. Given Y it implies the presence of X. Given X we have X. There is hierarchy here reflected in the improvement in PRECISION as we move from X to XY to XYZ and onwards.

    The three dichotomies will give us eight categories (2^3) where the establishment of a 'clear' perspective means we distance the categories from each other but retain the 'whole' that they all represent. This is done VISUALLY through the formation of a meaning space that is a cube/octahedron etc. Keep adding dimensions (aka +/- dichotomies) and our precision goes up as a power of 2 - thus with 12 dimensions, when maximised in distance from each other, form a dodecahedron (2^12 = 4096 possible categories)

    The addition of SCALES to each dichotomy is equivalent to RECURSING the dimension such that the '+/`' sense is quantified where the categories derived are in the form of 'numbers' WITHIN the particular dimension category (2Y is not the same as 2Z etc but we can also 'cut' the dimension to give us finer and finer levels of precision within the dimension)

    The DEGREE of scaling shows us recursion at work - this is the issue in number line representation where at the level of WHOLE numbers we have
    -4,-3,-2,-1 0 1,2,3,4... Add in PART numbers (and so rationals) and we have a new line with finer distinctions. ADD in RELATIONAL numbers, irrationals, and we have a new line with even finer distinctions. This is all examples of self-referencing.

    For the representation of cyclic/morphic dynamics we have to add a PLANE perspective through the use of imaginary numbers that allow us to
    'transform' numeric representations over the plane as whatever we are representing is changing form/spatial-orientation etc.

    Thus the XYZ nature of dimensions as lines shifts levels to that of
    'dimensions as planes' etc (thus we go from real to complex to quaternions to octonions - the REPRESENTATION of these perspectives are in REAL number formats that develop along powers of 2 - and so that recursive focus:

    REAL - (a,0) COMPLEX - (a,b) QUTERNION - ((a,b),(a,b)) OCTONION - (((a,b),(a,b)),((a,b),(a,b)))

    EACH level repeats the previous but with finer differentiations and that allows for 3D+ representations where the maximising of distance between all categories gives us a lot of room to derive meaning from!

    I emphasise all of this so one can try and get the 'picture', although that can lack precision ;-) (the discretisation of Mathematics is in the conversion from geometric perspectives to algebraic perspectives - and so a shift from the dimensional to the dimensionless. At the level of the POINT so we have high precision where, for example, a tangent is the sharing of a point by line/circle. This is all representative of information encoding by PHASE (and so constructive/destructive interferences and so holograms) as compared to AMPLITUDE (photographs).

    To get a 'picture' one will need to go one level past the picture to make your picture really 'clear'. And that means clearly understanding the dynamics of meaning derivation through the self-referencing of dichotomies. The 'truth' as such is not 'visual', it spans all senses and so can be represented in each but each is then expressed all at once and so in parallel. Its like Astronomy - telescopes cover many perspectives of the visual range but also move into radio etc and THAT information is needed to get a 'full spectrum' understanding of things.

    In IDM we identify the core units of meaning and that also covers the expression of these units through number types in Mathematics:

    Wholes - sense of blending Parts - sense of bounding Static relatedness - sense of bonding (share space) Dynamic relatedness - sense of binding (share time)

    These QUALITATIVE mappings allow one to FEEL meaning in the quantitative and so build-up an intuitive experience. Since the same qualities apply to specialist perspectives within Mathematics - e.g. algebra vs geometry - so develops an intuitive sense of which specialist perspective is the 'best suited' to describe some situation.


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