This bears remarkable comparison to the wave/particle complementarity of quantum mechanics.
So for example if we define a number in Type 1 terms e.g. 5 (51) as representing the particle aspect, then in Type 2 terms 5 ( 15) represents the corresponding wave aspect. So here 5 switches as between both its particle and wave aspects in Type 1 and Type 2 terms.
Remember again that when the Type 1 aspect is associated with the cardinal aspect, then - relatively the Type 2 aspect is thereby associated with the ordinal aspect!
Because in the dynamics of experience, we continually switch in two-way fashion as between cardinal and ordinal notions with respect to natural numbers, this likewise implies therefore that we keep switching likewise in two-way fashion as between particle and wave aspects.
And these aspects themselves can switch depending on the point of reference. So the Type 1 equally can be associated with wave and the Type 2 with particle aspects respectively.
In fact, what is not all realised - and which will cause utter consternation when eventually grasped - is that the quantum mechanical behaviour that is apparent at the sub-atomic regions of matter, is an inherent aspect of the true dynamic nature of the number system.
We will now illustrate this wave/particle like behaviour of number with respect to the recognition of 5 objects - say 5 chairs. Now in quantitative terms, the recognition of 5 implies the recognition of 4 which implies the recognition of 3 which in turn implies the recognition of 2 which finally implies the recognition of 1. So 5 ultimately represents 1 + 1 + 1 + 1 + 1.
However such recognition in fact is very subtle, in that through experience we constantly switch as between whole and part notions (and vice versa).
So implicitly in recognising the 5 chairs as independent part items, we must also recognise the overall collection of these chairs as a whole group (= 1).
Therefore the switch from the 5 individual chairs to the collective recognition of the 1 set of chairs (as the whole group) entails the corresponding switch from part to whole aspects respectively. And then in like manner, to switch back from the recognition of the 1 set of chairs to the 5 individual chairs, requires the corresponding reverse switch from whole to part aspects.
Now, this switch from part to whole (and whole to part) recognition of number, strictly entails the dynamic interaction of both the quantitative and qualitative (and qualitative and quantitative), in the two-way interaction of the Type 1 (particle) and Type 2 (wave) aspects of number respectively.
However in conventional mathematical explanation a reduced interpretation is given solely in terms of the Type 1 quantitative aspect.
So in effect the notion of the unitary whole (with respect to the group of five chairs) is reduced to the part notion of the 5 chairs in a merely quantitative manner.
Now if we look at a group of five chairs, what we see in quantitative terms are the 5 individual chairs.
However the very ability to see this group of 5, constituting in this context a unique whole (as a set) directly entails qualitative - rather than quantitative recognition. Thus in the dynamics of understanding, an intuitive recognition of the interdependence of whole and part is required.
This then enables the switch from the part recognition (of the 5 individual chairs) to the subtler whole notion of these chairs representing a group.
Now without this implicit recognition of whole/part interdependence, which is directly intuitive in nature, there would no way of making this important switch in recognition, with the important connection as between the part individual chairs (as 5) and the whole collective group (as 1) impossible to make
Now we can equally see this in reverse. We could start with the five chairs as 1 collective unit through imagining them perhaps wrapped up together in a transparent bag. So this bag (of chairs) now represents 1 in a quantitative whole manner. Now to recognise an individual chair as a part unit (as 1/5) of the whole, we have to make the opposite transition from whole to parts notions which implicitly involves the switch from quantitative to qualitative. This again is provided through the intuitive recognition of the interdependence of part and whole, enabling the decisive switch in recognition to be made.
However in explicit terms we now recognise the individual 5 chairs again in a merely quantitative manner.
Thus the frames of reference with respect to whole and part (and part and whole) recognition keep changing in the dynamics of experience. This implicitly requires the intuitive recognition of the interdependence of whole and part (and part and whole) for these switches to be made in a two-way fashion. However explicitly this is quickly reduced with both whole and part notions interpreted merely in a rational quantitative manner (as independent).
So in effect the whole is reduced to its parts in a quantitative manner.
And this gross reductionism is the most fundamental problem imaginable which pervades the entire field of Mathematics and all its related sciences!
Dealing with this problem will entail the most radical intellectual revolution in thought yet in our history.
So rather than number - all all its related mathematical notions - being understood in a merely reduced absolute manner amenable to the conscious use of (linear) reason, we will have to move to a new approach, inherently dynamic in nature. This will entail the balanced use of both (conscious) reason and (unconscious) intuition in a manner where both the quantitative and qualitative aspects of all mathematical notions can be explicitly recognised.
And then the knock-on effects of this new appreciation of number (and extended mathematical relationships) for all the sciences will be truly enormous!
Thus to follow on from our example, though we can refer to 1/5, 2/5, 3/5 4.5 and 5/5 in an absolute type quantitative manner i.e.as (1/5)1, (2/5)1, (3/5)1, (4/5)1 and (5/5)1 , as we have seen this completely ignores whole/part interaction entailing the qualitative aspect.
Thus strictly the interpretation of all fractions - as all numbers - is of a dynamic relative nature.
Thus the absolute nature of these fractions should be viewed as a limit to which the truly relative interpretation approximates. Put another way a necessary uncertainty principle applies to all numbers.
Thus an irrational number such as √2 has a relative value that cannot be represented in an absolute discrete manner.
However, all absolute numbers (of a discrete nature) are in turn representations of number interactions of a strictly relative nature!