When one considers carefully in conventional terms the simple mathematical relationship,
1 + 1 = 2,
a strange paradox can be shown to arise.
Once again because these two units are treated in a homogeneous cardinal manner as independent (in a merely quantitative manner) this thereby creates an unavoidable uncertainty with respect to their corresponding ordinal identity (as qualitative).
Therefore, each unit can be ranked as 1st or 2nd in relationship to each other!
Thus the very ability to rank the numbers in a fixed ordinal manner unambiguously as 1st and 2nd respectively, requires some arbitrary qualitative criterion, that is not inherent in the cardinal identity of the numbers.
For example if I look at two cars in a driveway, to unambiguously order them as 1st and 2nd, some qualitative criterion must implicitly be used.
If I base my decision - say - on the age of the cars (ranking the most recently registered as 1st) then - provided they do not have the same registration date - this problem of ranking can be solved.
However according to an alternative qualitative criterion - say the sizes of the respective cars - this ranking could be reversed with what was 1st (on the previous ranking) now 2nd and what was 2nd now 1st.
The implication of this is that in the absence of any qualitative criterion, each unit can potentially be be ordered (i.e. related in an interdependent manner) as both 1st and 2nd.
Then in the more general case of the number n, each unit can potentially be ordered in terms of n ordinal positions.
So for example in the case of 1 + 1 + 1 + 1 + 1 = 5, each of the units here can potentially assume 1st, 2nd, 3rd, 4th and 5th positions in relation to each other.
Thus though from a cardinal (quantitative) perspective, each unit is given one fixed identity in actual terms, from the corresponding ordinal (qualitative) perspective, each unit can freely range over all numbers in a potential manner.
And once again from a psychological perspective, in mathematical terms, rational understanding - of an analytic nature - is directly identified with the fixed unambiguous nature of actual relationships; intuitive appreciation by contrast is directly identified with the corresponding extended nature of potential relationships (which appears paradoxical from the rational perspective).
So clearly once again both the cardinal (quantitative) and ordinal (qualitative) nature of number relationships can only be properly understood in a dynamic interactive context that entails both analytic (rational) and holistic (intuitive) type appreciation.
Put another way we can only hope to coherently understand mathematical relationships when both the conscious and unconscious aspect of understanding are properly integrated with each other in a balanced manner.
Because, as we have seen quantitative and qualitative aspects of number relate to distinct domains of understanding, the fundamental question then arises as to how they can be consistently related to each other.
Let's be clear here! This is by far the most fundamental issue that can be raised with respect to mathematical activity, yet it cannot even be recognised in conventional mathematical terms!
Once again in conventional terms the qualitative aspect is simply reduced in quantitative terms, so that this key issue of mathematical consistency cannot be approached from this restricted perspective!
And though we are running slightly ahead of ourselves at this stage, when properly understand this is of direct relevance to appreciation of the key significance of the famed zeta zeros!
For it is the very nature of the zeta zeros to provide this key interface as between the quantitative and qualitative aspects of mathematical understanding.
So looked at in an equivalent manner, if mathematical relationships can indeed be assumed to be consistent, we are faced with the key issue of somehow converting the qualitative type nature of number (indirectly) in a quantitative perspective.
Equally from the alternative perspective, we have the complementary issue of converting the quantitative type nature of number (indirectly) in a qualitative manner!
For example ordinal notions of 1st, 2nd and 3rd etc. do not possess a direct quantitative meaning. Rather they strictly relate to the qualitative relationship as between an interdependent group of numbers!
However, remarkably, we can indeed give these ordinal notions an indirect quantitative meaning in a fascinating manner.
Now once again, this issue does not arise in conventional mathematical terms, where ordinal meaning is quickly reduced in a merely quantitative manner.
I will explain precisely how this happens in future blog entries.