And this inherent feature can in principle be extended to all numbers.
So far from representing abstract independent entities with an absolute interpretation in quantitative terms, the very nature of number is inherently dynamic and relative, representing the two-way interaction of both its quantitative and qualitative aspects.
And the fundamental nature of addition and multiplication is revealed through this new understanding, where these two operations likewise operate in a dynamic complementary fashion enabling understanding to ceaselessly switch as between the quantitative and qualitative type aspects of number.
In a way I find it utterly surprising that this fact is not more widely appreciated for I have been keenly aware from childhood of an obvious problem (that is steadfastly avoided in conventional mathematical interpretation).
Thus in the most basic sense, there is a valid sense in which we can quantitatively distinguish (as independent members) the individual units of number.
However equally there is a valid sense in which we must likewise qualitatively relate (as shared members) these same units.
And to put it bluntly in conventional mathematical terms this latter qualitative aspect of number is inevitably reduced in a merely quantitative manner.
In other words from this limited perspective, no clear distinction can be made as between the twin notions of number independence (where units have a separate identity) and number interdependence (where units have a shared identity) respectively.
And it is this fundamental problem that lies at the root of the problem of properly relating the operations of addition and multiplication.
In an attempt to clarify this point further let us look at the simple example where we multiply the first two primes i.e. 2 and 3.
So in conventional terms 2 * 3 = 6. Now expressed more fully we could represent this
as 21 * 31 = 61.with the three numbers represented as points on the same real line.
However, as well as a quantitative transformation through multiplication (i.e. to 6) a qualitative (dimensional) change in the nature of the units also takes place.
So if we imagine a rectangular table of width 2 and length 3 metres respectively, then the area of the table will be expressed in square metres.
So we move from 1-dimensional to 2-dimensional units. However in the standard treatment of multiplication, this qualitative transformation is then reduced in a quantitative manner, with the result i.e. 6, represented in 1-dimensional terms (as a point on the real number line).
Though in dynamic relative terms, base and dimensional numbers are quantitative and qualitative with respect to each other, both are understood however through this geometrical representation in a somewhat linear manner.
So we switch in other words from the analytic use of number representing (actual) finite quantities to the corresponding analytic use of number representing (actual) finite dimensions.
However in dynamic interactive terms, the switch is always of a truly complementary nature (i.e. from analytic to holistic and holistic to analytic aspect, respectively).
Thus there is a hugely important holistic aspect to the interpretation of multiplication (without which it has no strict meaning).
To more easily illustrate this point, imagine that we have 3 objects (say coins) placed in two rectangular rows!
Using addition in standard analytic fashion, we could add up the 3 independent coins in row 1, = 3 and then proceed to add up the 3 independent coins in row 2, = 3.
And as both rows would likewise be considered as independent, the total number of coins represents the sum of the 2 rows i.e. 3 + 3 = 6.
From the conventional mathematical perspective, multiplication can then be used to short-circuit the process of laboriously adding separate rows, through the recognition that a common similarity exists as between each row. So now, the realisation that we have two similar rows leads to the use of the operator 2, which is then multiplied by the number of objects in each row.
So from a multiplicative perspective, the total number of objects is 2 * 3 = 6.
Thus in conventional mathematical terms, multiplication is represented as a form of short-hand addition.
So, 2 * 3 = 3 + 3
Of course, with just 2 similar rows, multiplication does not offer any real benefit over addition. However, say with 100 similar rows, multiplication would then offer a much simpler way of expressing the total no. of objects (than the successive addition of many rows).
However there is a hugely important - though largely unrecognised - flaw to this interpretation of multiplication.
Again, the conventional mathematical approach to cardinal number is based on the assumption that unit members are independent of each other in an absolute fashion.
However, the very process of multiplication entails the recognition - as in the example above - of the common similarity as between the various rows (or alternatively various columns) and likewise the common similarity as between members of each row (and column).
Thus in our example, the use of 2 as an operator depends on recognition that the two rows are similar in number terms, i.e. share a mutual interdependence.
And from an ordinal perspective, it does not matter which row is identified as 1st or 2nd, for by definition they are mutually interchangeable.
Therefore, in this example, the recognition of the similarity of the two rows implies the holistic - rather than the analytic - interpretation of 2, where the mutual interdependence of both unit rows is recognised.
Expressed in an equivalent manner the recognition of the similarity of the two rows, whereby they are recognised as mutually interdependent, entails the qualitative aspect of 2 (as “twoness”). By contrast the recognition of the independent nature of each row entails the quantitative aspect of 2 (as two).
This recognition of similarity as between the two rows equally implies recognition of the similarity i.e. shared interdependence of the 3 items within each row (where 1st, 2nd and 3rd units are potentially interchangeable with each other). Equally it implies recognition of the similarity of the 2 items within each column (where 1st and 2nd units are potentially interchangeable with each other).
So all in all, the operation of multiplication implies recognition of both the analytic independence (where units are viewed as separate) and the holistic interdependence of number (where units are viewed as similar).
And this operation can only be properly understood in a dynamic interactive manner, where both the analytic (quantitative) aspects of number as independent and the holistic (qualitative) aspects of number as interdependent, are viewed as complementary.
This likewise applies to addition.
It is all very well treating numbers as independent in an analytic manner, but the consequent result from addition requires a transformation to a new holistic identity (not apparent in the independent units).
So, for example, we can start by viewing 1 + 1 as independent units in individual quantitative terms. However the very act of recognition, enabling the new whole collective identity of 2, requires corresponding realisation of the interdependence of unique units (as interchangeable). So addition and multiplication properly entail both analytic and holistic aspects.
Thus the crucial point here is that addition and multiplication are analytic (quantitative) and holistic (qualitative) with respect to each other with both operations complementary in dynamic interactive terms.
This implies that when - as is customarily the case - addition is identified directly with the quantitative transformation of number (in default 1-dimensional terms), multiplication is then - relatively - identified with its qualitative transformation (where the dimensional aspect changes) and vice versa.
Thus, in the simplest case, from the former perspective of addition, 11 + 11 = 21. So here we have a quantitative change in number in linear (1-dimensional) terms. Thus the base aspect of number changes while the dimensional aspect (implicitly) remains fixed.
However, from the latter perspective of multiplication, we have 11 * 11 = 12. Here in inverse relative fashion, a qualitative change in number occurs. So in a complementary manner, the dimensional aspect of number now changes while the base aspect remains (implicitly) fixed.
In conventional mathematical terms, it is customary to represent the natural numbers (without reference to base or dimensional characteristics), once again reflecting the reduction of qualitative to quantitative characteristics.
Therefore the natural numbers are listed, representing solely number quantities, as 1, 2, 3, 4, …
However from the enhanced interactive perspective, number possesses two distinct aspects (base and dimension respectively) both of which alternate as between analytic (quantitative) and holistic (qualitative) meanings in the dynamics of understanding.
Alternatively, we could say, using a very close physical analogy, that number keeps switching as between its particle (independent) and wave (interdependent) aspects, with respect to both base and dimensional characteristics.
So again rather than the natural numbers being given in absolute fashion as 1, 2, 3, 4, …, we now have two twin interacting aspects,
11, 21, 31, 41, … and 11, 12, 13, 14, …, which I customarily refer to - in relative terms - as the Type 1 and Type 2 aspects of the number system respectively.
And when understood in the appropriate dynamic manner, experience of these two aspects keep dynamically switching in complementary fashion, as between analytic (quantitative) and holistic (qualitative) aspects with respect to both base and dimensional characteristics, depending on which reference frame is employed.
So, if the base aspect of number is interpreted in a quantitative manner, then the dimensional aspect, relatively, is qualitative; however if the base aspect is interpreted in a qualitative manner, then in this context, relatively, the dimensional number is now quantitative.
Thus all numbers can be given - depending on relative context - analytic and holistic interpretations with respect to both base and dimensional characteristics.
Both Type 1 and Type 2 aspects, in this interactive appreciation, are intimately related to the key distinction as between addition and multiplication as complementary opposites.
When, for example, we multiply 11 * 11 to obtain 12, a qualitative change in the nature of number as dimension takes place. So 2 in this context, represents the holistic notion of dimension, relating to the interdependence of both unit members. Though in direct terms the transformation applies to dimensional units, implicitly this likewise entails the unit base numbers.
Then, when we add 11 + 11 to obtain 21, a quantitative change, relatively, in the nature of the base units takes place. So 2, by contrast, now represents the analytic notion of number (as base), relating to the independence of both unit members. However, though in direct terms the transformation now applies to the base units, implicitly this likewise entails the dimensional notion of 2.
Therefore the remarkable fact remains that whenever we use any number, with respect to base or dimensional aspect, in the standard analytic quantitative manner, this likewise implicitly requires the corresponding holistic notion of that same number, which has now become implicitly embodied, as it were, in the number.
So from a psychological perspective, the conscious rational interpretation of number always carries the distant echo of its unconscious holistic counterpart.
However, in the main, we remain completely deaf to this unconscious echo.
So we cannot explicitly define addition (without implicitly requiring multiplication); and we cannot explicitly define multiplication (without implicitly requiring addition).