We have seen how dealing appropriately with the ordinal (relational) nature of number requires going beyond the 1-dimensional qualitative approach (within which Conventional Mathematics is defined).
Again this is is necessary as thc customary approach has no means - within its own terms of reference - of satisfactorily distinguishing qualitative from quantitative type meaning.
Alternatively this entails that Conventional Mathematics cannot properly deal with the ordinal (qualitative) nature of number. Worse still because - in dynamic experiential terms - both cardinal and ordinal meaning are interrelated, it implies that Conventional Mathematics ultimately cannot even properly deal with its own chosen area of the quantitative nature of number (and by extension all quantitative notions)!
I have defined on numerous occasions the nature of the 3 Types of Mathematics (which are necessary in an overall comprehensive framework).
Once again Type 1 refers to the quantitative aspect (which is the sole specialisation of Conventional Mathematics). However even here there are two distinct approaches.
Unfortunately - as I would see it - Conventional Mathematics is very much rooted in an absolute Type 1 approach that is - qualitatively - of a linear logical (i.e. 1-dimensional) nature. At present it shows no openness whatever to a counterbalancing qualitative approach (that ultimatekly must be incorporated for full mathematical comprehension).
The alternative Type 1 approach - which I strongly advocate - is defined in a strictly relative manner. Though specialisation of the quantitative aspect of mathematical symbols is certainly legitimate in this approach, implicitly it recognises that a complementary qualitative type treatment of the same symbols is equally possible!
So the quantitatave aspect of Mathematics is understood here in a - relatively - independent manner.
And if we are to proceed to a truly comprehensive mathematical understanding (involving all 3 types) then the Type 1 must be defined in a relative - rather than absolute - fashion.
Just as the 1st dimension of interpretation (in this relative context) initially provides the (Type 1) standard for quantitative type cardinal interpretation of number (based on independence), as I demonstrated in a recent blog the 2nd dimension (Type 2) likewise provides the standard basis for qualitative type ordinal interpretation of number (based on interdependence).
Then combining the 1st and 2nd dimensions of interpretation - corresponding to the linear and circular use of logic respectively - one for example can give a complete explanation of the relationship which two objects (i.e. numbers) have with each other, where both cardinal (quantitative) and ordinal (qualitative) distinctions are both preserved. I used once more the - apparently - simple illustration of the two turns at a crossroads to illustrate this point!
So putting it simply! The 1st dimension of interpretation can be clearly associated with cardinal (quantitative) meaning (in a relatively independent sense); the 2nd dimension can be clearly associated with ordinal (qualitative) type meaning (in a relatively interdependent sense).
However rather like the particle and wave nature of sub-atomic particles, once we bring both aspects together, the wave aspect becomes also particle like, and the particle aspect wave-like: likewise with respect to number: once we attempt to combine both the cardinal (quantitative) and ordinal (qualitative) nature of numbers the cardinal nature acquires ordinal like features, whereas the ordinal acquires cardinal like features.
And the fascinating clue to what all this means is given by the higher dimensional numbers with respect to 1 (> 2) with their corresponding roots of 1 (> 2).
In other words once we go higher than 2, all roots of 1 combine both real and imaginary parts.
So the fascinating and important question then arises as to what the imaginary aspect means in this context (of cardinal and ordinal interpretation).
To appreciate this we need to go back to the 2nd root of 1, which is the quantitative counterpart of the number 2 as dimension (in qualitative terms).
This was defined as - 1. Now the corresponding qualitative interpretation was as the negation of independent conscious type understanding of a rational nature. This equally represents the manner through which interdependent unconscious type holistic appreciation of an intuitive kind takes place.
Therefore in all relationships, whereas understanding of the independent aspect of such relationships is strictly provided through (conscious) reason (1st dimension), appreciation of interdependence by contrast is provided through (unconscious) intuition (2nd dimension). However indirectly this latter aspect can be interpreted in a circular rational fashion (that is paradoxical in terms of conventional reason).
So - 1 in this rational context is given a 2-dimensional (ordinal) interpretation as both positive and negative (+ and -) which can also be expressed as the complementarity of opposites (i.e. opposite poles). It must be remembered that negating in a dynamic interactive context already presupposes a positive element (like anti-matter fusing with matter particles).
Thus we have established in qualitative terms, how - 1 thereby represents the fundamental nature of the 2nd dimension (through which interdependence with respect to opposite poles takes place).
However as we have qualitatively defined it here (in Type 2 terms), this represents the 2nd - rather than the 1st - dimension.
So of we are to reduce this notion appropriately so that it can now be defined in Type 1 terms, we thereby obtain the square root.
Thus the imaginary no. i = the square root of - 1, serves as the (reduced) Type 1 way of incorporating the ordinal (qualitative) aspect of Type 2 Mathematics in an accepted Type 1 cardinal (quantitative) context.
Equally from the opposite perspective of Type 2 Mathematics, i serves as the (reduced) Type 2 manner of incorporating the cardinal (quantitative) aspect of Type 1 Mathematics in an accepted Type 2 ordinal (qualitative) context.
The deeper implications of all this is that complex numbers - when properly viewed from a Type 1 or Type 2 perspective - necessarily incorporate both quantitative and qualitative type aspects.
However in each case one of these aspects remains masked (with its true nature hidden).
Thus from the Type 1 perspective, though we attempt to view both the real and imaginary aspects of number (in a merely quantitative manner), the imaginary aspect in fact represents the alternative qualitative aspect of Mathematics (that remains hidden however through being veiled in a quantitative mask).
Likewise from the Type 2 perspective, though we again may attempt to now view both the real and imaginary aspects of number (in a merely qualitative manner), the imaginary aspect now in fact represents the corresponding quantitative aspect of Mathematics (that remains hidden however in a qualitative mask)!
And once one clearly realises this dilemma, from the relatively isolated stances of both Type 1 and Type 2 Mathematics respectively, then one necessarily must start moving to the Type 3 approach (where both quantitative and qualitative aspects can be properly integrated).
All of this of course is deeply relevant to proper understanding of the Riemann Zeta Function. As it is defined with respect to the complex plane, with both (matching) quantitative and qualitative type interpretations, a Type 3 mathematical approach is very much required for its proper comprehension.