Indeed if we were to use a close analogy with quantum physics, number keeps switching as between particle and wave aspects.

So in this sense the wave-particle duality that applies to matter (especially at the sub-atomic scale) equally applies to number. Indeed I would maintain that this observed physical phenomenon itself ultimately reflects the whole-part duality of number!

We have also seen that conventional mathematical interpretation is inherently unsuited to dealing with this issue. Because of its unambiguous (1-dimensional) nature, it inevitably reduces qualitative meaning in absolute quantitative terms (whereby in effect the whole is reduced in terms of its constituent parts).

Thus again, if we were to use a close physical analogy, the present position in Mathematics is akin to the attempt to understand quantum mechanical behaviour in terms of standard Newtonian concepts.

Indeed in truth the problem is even more fundamental than this!

Nothing less therefore than a radical reformulation of the nature of the number system - and indeed by extension all mathematical notions - is now required.

For the simple fact exists that at present one cannot give a properly coherent interpretation of the simplest example of multiplication - indeed the simplest example of addition - in terms of the accepted mathematical paradigm.

Thus with respect to the number system, the present static absolute approach urgently needs to be replaced with a new inherently dynamic interactive interpretation, whereby the distinctive nature of the part and whole aspects of number can be properly preserved.

Therefore, for many decades now, I have proposed that rather than one natural number system - interpreted in an absolute rigid manner - that we need to recognise that there are two complementary aspects to this system, which interact with each other in dynamic fashion.

I refer to these aspects as Type 1 and Type 2 respectively.

Initially the Type 1 aspect would appear to bear the closest resemblance to to conventional interpretation.

So, again in conventional interpretation the natural numbers are listed as:

1, 2, 3, 4, ..........

Now in Type 1 terms, these are listed in more refined manner as:

1

^{1}, 2^{1}, 3^{1}, 4^{1}, ……..
Again to simply illustrate in conventional terms,

1 + 1 = 2.

Therefore both units are treated in an independent fashion as quantitative parts, with the resultant total representing the sum of these quantitative parts.

Thus "2" - though referred to as a "whole" number - in this context, is given a merely reduced quantitative meaning (i.e. as the sum of constituent unit parts).

However in Type 1 terms,

1

^{1 }+ 1

^{1 }= 2

^{1}.

The dimensional number (i.e. power or exponent) here refers implicitly to the corresponding whole status of the number.

Therefore to explicitly recognise that 1 + 1 = 2 (in a quantitative manner), one is implicitly recognising that 2 is equally associated with a new unitary whole status.

In yesterday's blog entry, I illustrated this with respect to the two slices (of the cake).

So the ability to recognise that that the combination (through addition) of each (individual) part slice resulted (collectively) in two part slices, implicitly requires recognition of the total cake as a whole unit. Thus the very ability to recognise the two individual slices as being related to the overall cake would be impossible in the absence of this implicit recognition of the cake possessing both a part and whole status. So again its part status is represented by its 2 individual slices. However its whole status is then represented by its distinctive status as 1 cake.

And by including the dimensional number of 1, we are here recognising the corresponding whole identity of the number "2".

The deeper implication of this is that this whole identity (in 1-dimensional terms) implicitly enables an interdependent relationship as between the two individual units (of 2) to be maintained.

So in ordinal terms, we would look on the two slices of our cake as the 1st and 2nd slices respectively. However we have now moved from the notion of independence (with respect to the two individual slices in cardinal terms) to the complementary notion of interdependence (with respect to the "same" two slices in an ordinal manner).

And this equally applies to number. Thus in cardinal terms we can refer to 2 as 1 + 1 in quantitative part terms (where both units are independent in a homogeneous fashion).

However in corresponding ordinal terms, we can refer to 2 as 1st + 2nd in a qualitative whole manner (where both units are interdependent in a uniquely distinctive fashion).

And this whole nature of 2 comes from switching from its part status (as comprised of 2 independent units) to the new identity (as a unique whole in its own right).

Therefore, it is impossible to properly recognise the distinctive nature of the cardinal and ordinal interpretations of number, without also properly recognising the dual nature of number in terms of its part (analytic) and whole (holistic) aspects.

Thus there is an underlying paradox here:

In explicit quantitative terms, we attempt to define each number as the part combination of individual units.

So again for example, 2 = 1 + 1.

However this part total of 2 itself represent a single unit (in qualitative whole terms).

Therefore though we are indeed entitled to explicitly make clear quantitative distinctions with respect to the part nature of number (i.e. in analytic manner), implicitly we need to bear in mind the holistic qualitative nature of number, which makes these distinctions possible.

However, we equally have a Type 2 aspect to the number system.

Now in the Type 1 aspect we have separate number (quantitative) objects (defined within a 1-dimensional framework).

However with the Type 2 aspect we have the same quantitative object (defined within multiple dimensional frameworks).

The easiest way to appreciate this is in terms of a unit line (in 1-dimensional terms) which is now used to define a unit square (in 2-dimensional terms).

Therefore through the quantitative nature remains unchanged as 1, clearly the dimensional nature of the number object has changed (from 1 to 2).

Now if one reflects for a moment on the 2 dimensions of a square object, clearly they cannot be independent of each other but must be related in a very ordered manner.

Thus the crucial point about the Type 2 approach is that we now are adding related (i.e. interdependent) units. Thus these units now represent wholes rather than parts (as was the case with the Type 1 aspect).

Thus when we add for example 1 + 1 (now representing wholes) the whole status i.e. the dimensional nature of the object is directly changed.

Now the startling fact is that what represents addition with respect to this Type 2 aspect, represents multiplication from the Type 1 perspective.

So 1

^{1 }* 1^{1 }= 1^{2}.
And 1

^{2}^{ }= 1^{1 + 1}.
However, when adding numbers as wholes (representing dimensions) the new qualitative change (i.e. the dimensional status of the object) can only explicitly be understood, through implicitly recognising the quantitative nature of the base unit (as measured in 1-dimensional terms).

Thus we can only combine numbers (as parts) through implicit recognition of their corresponding whole status. Likewise we can only combine numbers as wholes (representing dimensions) through implicit recognition of the quantitative nature of each dimension (in isolation).

Thus the Type 2 aspect of the number system is listed as:

1

^{1}, 1^{2}, 1^{3}, 1^{4}, ……..^{}

Note how it is the inverse of the Type 1 aspect, reflecting the change is explicit focus from the part (quantitative) to the whole (qualitative) nature of number.

However just as explicit part recognition implicitly requires corresponding whole, equally explicit whole recognition implicitly requires corresponding part recognition respectively.

In fact both forms of recognition are dynamically complementary with each other in a two-way manner.

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