Deep Implications for Mathematics

The finding that every number - indeed every mathematical - symbol can be given a coherent holistic as well as analytic interpretation is of the very first magnitude.

In fact it reveals a fundamental unrecognised problem that lies at the very heart of Mathematics i.e. that its symbols cannot in fact be coherently understood in a static absolute fashion.

Rather the number system - and indeed all mathematical relationships - should be properly interpreted in an inherently dynamic manner reflecting the interaction of complementary aspects.

Now for some considerable time, I had been aware that we can define the natural number system in this complementary fashion (using Type 1 and Type 2 formulations that are the inverse of each other).

So again the (analytic) Type 1 aspect is defined as,

1¹, 2¹, 3¹, 4¹,………

Here the base ranges over the natural number sequence (with respect to the default dimensional value of 1).

What this means in fact is that number is viewed in a merely independent quantitative manner (as parts).  

Thus 3 for example = 1 + 1 + 1 (with each unit a quantitative part of the total number)

The (holistic) Type 2 aspect by contrast is defined as,

1¹, 1², 1³, 1⁴,………

Here, in inverse fashion, the dimension ranges over the natural numbers (with respect to a default base value of 1).

The essence of this interpretation is that number is now viewed in an interdependent qualitative manner with respect to its relationship with other numbers.

To illustrate let us start with the default base number of 1. Now, we can keep relating this number with itself (without changing the quantitative value) by moving into "higher" dimensions.

So we start with the 1-dimensional line (with a quantitative value 1) Then we move to a 2-dimensional square (again with a quantitative value of 1). And continuing we can next move to a 3-dimensional cube (once more with an unchanged quantitative value of 1).

So - quite literally - while the base unit remains unchanged as 1, we keep moving into higher dimensions with respect to this unit. These dimensions are possible precisely because we keep using 1 in a related ordered fashion, where it can represent the length, breadth and height of the resulting 3 dimensional cube.

Now in conventional physical terms, we cannot go beyond  3 dimensions (as our very notion of space dimensions is greatly limited by conventional quantitative notions of measurement).

However there is a deeper more fundamental meaning of dimension here which can indeed be extended indefinitely.

Imagine I make the statement. "There are 3 cars in the (same) car park".

This can be treated in the conventional Type 1 manner, where each unit is then defined in an independent manner.

So 3 = 1 + 1 + 1.

Imagine I now make the following statement "There is one car in three (different) car parks.

In conventional mathematical terms, this would be treated in exactly the same manner 

i.e. 3 = 1 + 1 + 1.

However, in truth a very subtle change has taken place involving the shifting with respect to complementary reference frames.

In other words to add the 3 items in this case (recognising each as a component of a total number of cars) we have to establish the commonality, through relationship, of each of the three cars that lie in different car parks.

Now in the first case, this commonality (or interdependence) is implicitly provided through the recognition that the 3 cars are in the "same" car park. In this way we are then able to view each car in a - relatively - independent manner.

However in the second case,  because the 3 independent units are in separate classes (i.e. car parks) we would not be able to recognise their common membership (of a group) without the holistic recognition of each unit as a whole.

So this is the crucial difference! In the 1st case we looked at each unit as a part. However in the 2nd case we must now look on each unit as an intrinsic whole (in its own right).

Now it is true that once we have established these units as wholes that we can again add each unit to obtain a total of 3.

However properly speaking ,we must now express this in terms of the Type 2 system.

So if we initially arrive at the total of 3 in Type 1 terms through the addition of parts (i.e. 3¹ = 1¹ + 1¹ + 1¹), then in relative terms, we must arrive at the total of 3 (with respect to the 2nd case) in Type 2 terms through the multiplication of  parts.

So 1³ = 1¹ * 1¹ * 1¹ .

However this latter expression can equally be shown as the addition of wholes (with respect to the dimensional number)


i.e. 1³ = 1^{1 + 1 + 1} .

Therefore what represents addition in Type 1 terms (with respect to parts) equally represents addition in Type 2 terms (with respect to wholes).

However what represents addition in Type 1 terms (with respect to parts) also, equally represents multiplication in Type 2 terms (with respect to parts).

So in truth number keeps switching - depending on context - as between a part and whole status. It is very much akin in this respect to the way in which sub-atomic matter can switch between particles and waves!

But in Conventional Mathematics, a merely quantitative interpretation of the nature of this relationship is possible (where wholes are reduced to parts).
It is not surprising that mathematicians realise that there is something fundamental missing from their appreciation of the relationship as between addition and multiplication.

However the implications of facing this are truly enormous, for as we have seen the basic paradigm on which Mathematics has been built for millennia clearly needs to be changed, whereby both analytic and holistic modes of interpretation (of equal importance) are fully recognised, that are clearly understood to dynamically interact in complementary fashion with each other.