Things are not always what they seem!
I mentioned this yesterday with reference to the standard analytic treatment of number 1 + 1 = 2.
Indeed based on our common-sense intuitions of reality this statement seems so obvious that it is frequently used as a metaphor for self-evident truth.
In fact however it reveals the reduced nature of accepted mathematical understanding, which through millennia now of unchallenged usage has become so deeply ingrained in our thinking that we have become completely blind as to its limitations.
The conventional understanding of this simple number relationship is based on mere analytic appreciation (where the “whole” number 2, represents the quantitative sum of its part unit members).
So the unit numbers here are viewed as absolutely independent of each other (in a homogeneous manner). In other words, as there is no way to distinguish as between the two units, they thereby lack qualitative distinction.
However if the individual units were indeed absolutely independent in such a manner, there would be no way of combining them to obtain the new collective identity of 2!
Therefore the very ability to relate the two unit numbers presupposes a distinct qualitative identity of relational interdependence.
In the conventional treatment of addition, this key fact is completely overlooked.
In other words the qualitative aspect of number (i.e. relational interdependence) is simply reduced in an absolute quantitative manner.
This then enables the convenient fiction to be maintained that numbers can be combined with each other (while preserving their absolute quantitative identity).
Thus from this perspective 1 + 1 = 2 (understood in absolute quantitative terms).
However when one addresses the gross inconsistency entailed here, then the very nature of number is thereby radically changed.
For when we accept that all numbers necessarily possess a qualitative aspect of relational interdependence, then the corresponding quantitative aspect (of distinct identity) can only be properly conceived in a relatively independent fashion.
Thus all numbers necessarily possess both an analytic aspect (of quantitative independence) and a holistic aspect (of qualitative interdependence) respectively, which are dynamically related to each other in complementary fashion.
So let us now return to our simple identity, 1 + 1 = 2.
Again, in terms of appropriate dynamic appreciation, each unit is relatively independent in a quantitative manner, while equally sharing a qualitative identity of shared relational interdependence.
Whereas the quantitative aspect is directly related to the cardinal, the qualitative aspect is directly related to the corresponding ordinal nature of number respectively.
So again from the quantitative perspective, the two units are given a separate individual identity (in actual terms); however from the qualitative perspective, they are given a shared collective identity (in a potential manner).
So with respect to number recognition, each individual unit must be quantitatively distinguished (in an actual manner).
However equally, the two units must be given a qualitative identity of shared relational interdependence (in potential terms). From this perspective ordinal notions of 1st and 2nd with respect to the two units are viewed as fully interchangeable with each other.
At a deeper psychological level, whereas the analytic (quantitative) aspect of number relates directly to rational (conscious) recognition, the holistic (qualitative) aspect relates by contrast directly to intuitive (unconscious) appreciation.
So in conventional mathematical practice, the reduction of the holistic (qualitative) aspect of number recognition, in analytic (quantitative) terms, parallels precisely the corresponding psychological reduction of intuitive (unconscious) appreciation in a rational (conscious) manner.
Properly therefore, a comprehensive understanding of number entails both conscious and unconscious aspects of appreciation that both need to be explicitly recognised.
However once more, in conventional mathematical terms, the holistic (unconscious) aspect is formally reduced in an analytic (conscious) manner.
Therefore the true nature of number is inherently dynamic with quantitative notions of relative independence continually interacting with qualitative notions of relative interdependence respectively.
And in this new dynamic context, the two relative notions of number independence and interdependence are in turn intimately linked with the mathematical operations of addition and multiplication respectively.
Thus when we start with the quantitative aspect using the operation of addition,
1 + 1 = 2.
More fully we can express this relationship as,
11 + 11 = 21.
And using the Peano postulates all the natural numbers can ultimately be derived from successively adding 1 to a previous natural number (starting with 1).
So to combine numbers in a quantitative manner (through the operation of addition) we must explicitly define them with respect to the default dimension of 1 (which in this context remains implicit in understanding). In this way all natural numbers are viewed as points on the same real number line.
Thus in this context, 2 (i.e. 21) represents the quantitative sum of the two base units.
However when we now, in relative terms, combine the two units through multiplication, a different qualitative notion of number emerges, which can be expressed as follows.
11 * 11 = 12.
So in dynamic complementary fashion the focus is explicitly on the dimensional units (with the corresponding base units now remaining implicit).
Thus in this dynamically related context 2 (i.e. 12) represents the qualitative product of the two dimensional units).
What this means in effect is that these two units are now understood in holistic ordinal terms (as relatively interdependent) whereby 1st and 2nd positions are freely interchangeable with each other.
So just as we recognise that what are designated right and left turns at a crossroads can switch with each other depending on the direction from which the crossroads is approached, we likewise recognise that the two dimensional units can likewise switch ordinal positions (depending on context).
However as always in dynamic interactive terms, reference frames can switch.
Thus there is a valid sense in which addition can likewise be identified in qualitative terms with the interdependence of its unit members.
So 11 + 11 = 21. So here we are explicitly recognising in a holistic manner the qualitative interdependence of the two base units (with quantitative recognition now implicit).
And finally there is equally a valid sense in complementary fashion, through which multiplication can be likewise identified with the quantitative independence of its dimensional units (i.e. where each dimensional unit is recognised as separate).
Thus 11 * 11 = 12.
Therefore what we have demonstrated here is the following:
1) The number 2 - and by extension all natural numbers - can be given both analytic (quantitative) and holistic (qualitative) interpretations with respect to both base and dimensional usage. Whereas the analytic aspect corresponds directly with rational, the holistic aspect by contrast corresponds directly with intuitive type recognition.
2) The relationship between both is of a dynamic interactive nature with quantitative and qualitative aspects understood in complementary fashion.
So when the analytic (quantitative) aspect of number is made explicit, the corresponding holistic (qualitative) aspects remains, in this context implicit.
However, when the holistic (qualitative) aspect of number is now in turn made explicit, the corresponding analytic (quantitative) aspect is now implicit.
3) What is extremely important to bear in mind is that in this new dynamic framework, addition and multiplication become the very means by which the switching from quantitative to qualitative (and in turn qualitative to quantitative) recognition takes place.
So addition and multiplication operate as dynamic complementary partners in the very recognition of number.
Thus if we return to the number 2 for a moment, there is of course a valid sense in which we can recognise its two units as quantitatively independent of each other; however there is an equally valid sense in which we can recognise that these two units are likewise interdependent (and thereby interchangeable with each other) in an ordinal manner. So in this context 2 necessarily contains unique 1st and 2nd units.
And both of these interpretations are necessarily related in two-way fashion with each other.
Thus we cannot explicitly recognise the quantitative independence of the two units (without implicit recognition of their corresponding qualitative interdependence).
And we cannot explicit recognise the qualitative interdependence of the two units as, interchangeably, 1st and 2nd respectively (without implicit recognition of their quantitative independence).
Thus properly understood, the number 2 - and by extension every other number - is necessarily of a dynamic interactive nature, comprising analytic (quantitative) and holistic (qualitative) aspects in a complementary manner.