Nature of Addition and Multiplication

The two aspects of the number system that I have identified i.e. Type 1 and Type 2 respectively, are intimately related to the distinction as between addition and multiplication.

To see this clearly, I concentrate on - what I refer to as - pure addition and pure multiplication.

Pure addition simply relates to the successive adding of 1 to the previous unit (starting with 1).

Pure multiplication by contrast relates to the successive multiplication by 1 of the previous unit (again starting with 1).

In this way, we are enabled to generate both the Type 1 and Type 2 aspects of the number system.

Thus through addition of 1 we obtain,

1¹, 2¹, 3¹, 4¹, ......

So in this Type 1 aspect, the base are the natural numbers, which are all defined with respect to the (default) dimensional number of 1.

This defines the conventional mathematical approach to numbers where the natural numbers - and indeed all reals - are represented as lying on the (1-dimensional) number line.

However through multiplication by 1, we obtain,

1¹, 1², 1³, 1⁴, ......

Therefore in this Type 2 aspect, the dimensions are now the natural numbers, which are all defined with respect to the (default) base number of 1.

Properly understood, the number system represents a dynamic interaction as between the Type 1 and Type 2 aspects, which are - relatively - quantitative and qualitative with respect to each other.

Expressed in an equivalent manner, this entails both relative independence and interdependence with respect to all number relationships.

Returning to the Type 1, the basic assumption is that units are independent (in an absolute manner).

So when we say for example that 2 = 1 + 1, both of these units are considered absolutely independent of each other. And this independence constitutes the basis of the quantitative cardinal type appreciation of number! Looked at another way both units are homogeneous in an impersonal manner, which thereby lack any qualitative identity.  

However if units were indeed truly absolute, this would beg the fundamental question as to how they can be subsequently related to each other!

So conventional mathematical interpretation - which solely recognises the Type 1 aspect of the number system - is based on the enormous reduced assumption that all number relationships, necessarily entailing a quality of interdependence, can be interpreted in an absolute quantitative manner! 

Now moving on to the Type 2, the basic assumption is that units are interdependent in a relative manner with each other.

As this point is vital for the true appreciation of the distinction as between addition and multiplication, let me spell it out in some detail.

So for example the number "2" (as indeed all natural numbers) is equally defined in both the Type and Type 2 aspects.

So in Type 1 terms "2" is defined as 2¹ . However in Type 2 terms "2" is defined as 1².

Now - as we have seen - in the Type 1 definition, "2" is associated with the quantitative (analytic) notion of number (where units are treated as independent of each other).

However in the Type 2 definition, "2" is now associated by contrast with the qualitative (holistic) notion of number (where units are treated as interdependent with each other). Indeed we could conveniently refer to the qualitative notion of "2" as twoness.

In dynamic experiential terms, both the quantitative and qualitative notions - that are relatively Type 1 and Type 2 with respect to each other - are intimately related. The explicit analytic understanding of "2" in quantitative terms, requires the implicit holistic recognition of "2" (as twoness) in a qualitative manner. Likewise the explicit holistic recognition of "2" in qualitative terms, requires the implicit analytic recognition of "2" (as quantity).

And what is remarkable here is how addition and multiplication - when properly understood - are intimately connected with both the quantitative and qualitative recognition of number respectively.

However this connection is totally missed through the merely reduced (Type 1) interpretation of numbers in conventional mathematical terms.

Though I have here initially associated the Type 1 (analytic) aspect with the base and the Type 2 (holistic) aspect with the dimensional number respectively, these reference frames can be reversed so that the base can be given a holistic and the dimensional number an analytic meaning respectively.

So when dimensional notions are used in conventional mathematical terms, they are understood in merely analytic terms.

For example it is easy to imagine 1² in geometrical terms as representing a (2-dimensional) square of 1 unit.

However, even here, a huge unrecognised problem exists.

As we have seen the quantitative notion of 2 properly requires the independence of units.

So once again when we say 2 = 1 + 1 (the assumption is made that both units are independent).

However if we now say that 2 = 1 + 1 (with respect to the 2 dimensions of a square object) these units are no longer independent, but rather related to each other (as length and width) in a precise ordered manner.

Therefore in a very real sense an unrecognised fundamental confusion is at work in conventional mathematical interpretation, where addition (of dimensional numbers) now properly implies interdependent (rather than independent units).

So once again a gross reductionism is in evidence, so that addition (with respect to both base and dimensional numbers) is considered as identical. 

Therefore, we really have two forms of addition, with one entailing the addition of relatively independent and the other the addition of relatively interdependent units respectively.

Likewise we really have two forms of multiplication - again with respect to relatively independent and relatively interdependent units - respectively. 

So as we have seen, we can attempt to understand multiplication in a merely quantitative manner (related to the independence of units). And we have demonstrated how this operates with respect to the interpretation of square (2-dimensional) objects!

Also, by extension, conventional multiplication really represents just a shorthand form of addition.

Therefore in quantitative terms 2 * 3 = 3 + 3 (so the 2 in the multiplication operation simply signals that the second number i.e. 3 should be added twice!  

However the understanding of multiplication in a true qualitative context (entailing the proper interdependence of units) leads to a new holistic appreciation of number. So we thereby can move, for example, from the quantitative notion of "2" (entailing independent units) to the qualitative notion of "2" as twoness (entailing interdependent units).

So when one is holistically aware of "twoness" in any context, this entails an appreciation that both units share the same common quality of "2" (i.e. as twoness).

By contrast, when one is analytically aware of "2" in any context, this entails an appreciation that both units are relatively independent of each other.

And once again the experiential realisation of number requires the interaction of both quantitative and qualitative aspects.

So the true nature of number is thereby inherently dynamic, entailing the complementarity of both analytic and holistic aspects. 

And in psychological terms, this entails that both the conscious and unconscious aspects of understanding be explicitly interpreted.

It has been recognised by key researchers into the mystery of the Riemann Hypothesis (e.g. Alain Connes and Brian Conrey) that it entails some deep unresolved issue regarding the relationship of addition and multiplication. And I will gradually proceed to directly confront the Riemann Hypothesis.

I would certainly agree. However the issue is so fundamental that it cannot be properly approached through conventional mathematical interpretation.

In fact as I have attempted to demonstrate, a new dynamic appreciation of number is now urgently required entailing the balanced interaction of both quantitative (analytic) and qualitative (holistic) aspects. 

This has not only the capacity to revolutionise the understanding of number, but ultimately all mathematical and scientific relationships!