I remember in primary school - at about the age of 10 - dealing with the areas of rectangular fields in a class on arithmetic.
The term "acre" is widely used in this regard and we were informed in class that 1 acre = 4840 sq. yards.
I remember thinking to myself then that a field of length 80 yards and width 60 yards, would thereby have an area very close to 1 acre (i.e. 4800 square yards).
But then I realised something much more significant. The area of this field (of 80 * 60 yards) relates to square (i.e. 2-dimensional) units.
However, when we express the product of 80 * 60, the answer is conventionally given in linear (i.e. 1-dimensional) units. So I could already see that there was something seriously lacking with such conventional practice.
And this realisation proved far from a passing concern at the time as I tried to come to terms - literally - with the missing dimension of multiplication.
In particular, I wondered for a long time regarding the simplest case of the multiplication of 1 by 1.
Here, no change takes place in quantitative terms. But a qualitative change takes place in the nature of the units (which are now 2-dimensional).
And I was greatly puzzled at what happens when one then obtains the square root, for now two answers are seemingly valid i.e. + 1 and – 1.
It was only later when seriously engaged with philosophy at University, that I could provide the deeper qualitative explanation for such strange mathematical behaviour.
From a qualitative perspective - and I am now using mathematical symbols in a holistic rather than analytic manner - Conventional Mathematics is characterised by a strictly linear (i.e. 1-dimensional) approach in rational terms.
What this implies is that the dynamic interactivity of complementary opposite poles - that necessarily characterise all mathematical relationships - is ignored with relationships, in every context, reduced in an absolute quantitative manner.
This is very true with respect to the multiplication of numbers e.g. 2 * 3.
We can easily see - especially where 2 and 3 relate to concrete type measurements - that a switch takes place - through the operation of multiplication - from 1-dimensional inputs (with respect to the individual two numbers) to a 2-dimensional output (with respect to the collective result).
So both a quantitative change with respect to the starting inputs and a qualitative change with respect to the nature of the dimensons is involved (through this operation of multiplication).
However in conventional mathematical terms, a merely reduced quantitative interpretation of the result is given. Through the operation of multiplication the result of 2 * 3, represents 2-dimensional (rather than 1-dimensional units). However it is conventionally given in a merely reduced 1-dimensional manner. So the resulting number 6 is treated as another point on the number line!
Now what complicates the matter somewhat here is that we likewise use a number to refer to the dimensional (as well as the base) units!
So in base terms 2 * 3 involves a transformation in quantitative terms;
And the change from 1 to 2 (with respect to dimensions) relatively involves a transformation in qualitative terms.
But 1 and 2 - as dimensions have a quantitative meaning in analytic terms. So mathematicians for example are quite happy to work abstractly in n dimensions (from a quantitative perspective).
However the true holistic meaning of dimension indirectly arises when we attempt to express its meaning in the standard 1-dimensional manner (associated with conventional linear logic).
So in the case of 1 * 1 = 1, each side can be expressed as x2 = 1, with x = + 1 and – 1 respectively.
Once again, we illustrated all this in relation the crossroads example. So with a single independent frame of reference when approaching a crossroads (from either a N or S direction), we can identify left and right turns in an unambiguous manner.
So if for example (heading N) a left turn at the crossroads is designated as + 1, then this can be unambiguously separated from the designation of – 1 (as a right turn). Likewise if (heading S) a right turn is designated as + 1, this can be unambiguously separated from the designation of – 1 (as a left turn).
So within each isolated pole of reference we have unambiguous (1-dimensional) interpretation in an analytic manner.
However, when we simultaneously view the approach to the crossroads, from both N and S directions, we have complete paradox in terms of former designations. For what was + 1 (heading N) is – 1 heading S; and also what is – 1 (heading N) is + 1 (heading S).
Equally what was + 1 (heading S) is – 1 (heading N); and what was – 1 (heading S) is + 1 (heading N).
So this latter manner of "seeing" properly represents 2-dimensional appreciation in a holistic manner (which then is rendered paradoxical when indirectly interpreted in 1-dimensional terms).
And what is not all appreciated in conventional mathematical terms, is that the various roots of 1 (with an inherently circular relationship of interdependence with respect to each other), indirectly represent the true holistic significance of mathematical dimensions!
However whereas at the 1-dimensional level (of analytic reason), these roots appear as separate (in an either/or manner), at the direct 2-dimensional level (of holistic intuition), they now appear as fully united (in a both/and fashion).
Therefore with respect to the meaning of 2-dimensions, these appear as independent (either + 1 or – 1) at a reduced (1-dimensional) analytic level.
However, they then appear as fully interdependent (both + 1 and – 1) at the enhanced (2-dimensional) holistic level.
And the reason for this contrast is that we thereby move from a true holistic perspective (entailing the interdependence of opposite polarities) through direct intuitive appreciation to a reduced analytic perspective (entailing the independence of each pole), when subsequently interpreted in rational terms.
So mathematical dimensions can be equally given both analytic and holistic meanings. Thus the analytic meaning of 2-dimensional equates with the notion of a rectangular body with both length and width (as separate dimensions).
However the corresponding holistic meaning of 2-dimensional relates to the direct appreciation of interdependence with respect to complementary opposite polar pairings. So the ability to - literally - "see" ( in a directly intuitive manner) left and right turns at a crossroads as purely relative, represents the holistic meaning of 2-dimensional.
And again the key importance of all this is that our very experience of all mathematical relationships is necessarily conditioned by twin opposite polarities that continually interact in a dynamic relative manner, i.e. external (objective) and internal (subjective) and part (quantitative) and whole (qualitative). So both analytic and holistic interpretations always arise in ths dynamic context.
And just as dimensional numbers can be given both analytic and holistic meanings, this likewise applies to base numbers.
So for example in the standard Type 1 representation of 2, i.e.21, 2 here represents the base number (with the default number of 1, the default dimension).
Once again in standard mathematical terms, the analytic meaning of 2 is solely recognised (in formal terms), where it is interpreted in a merely quantitative manner (as comprising independent homogeneous units).
However 2 equally has an important holistic meaning in this regard, which in fact has a vital bearing on true interpretation of the multiplication process.
Now imagine that we have two rows of some item - say peas - with 3 in each row.
Now if we were to approach the total number of peas involved (using addition) we would add up the 3 peas ( understood as independent items) in the 1st row, and then continue to add up the 3 items in the the 2nd row (again as independent items) and then combine the two rows (with both sub-totals likewise considered as independent).
So we would thereby obtain 3 + 3 = 6.
Now the key to moving to multiplication with this process is the recognition that we have repetition with respect to the sub-total in each row. So therefore by counting the number of rows and using it as an operator to be multiplied by the number of rows involved we obtain the answer.
So 2 * 3 = 6.
Though multiplication might not seem of much advantage here, imagine if we had 100 rows (with 3 peas in each row) it would be very tedious adding up separately the 100 rows (of 3 items).
So using 100 as an operator, 100 * 3 offers a convenient shorthand for this process.
However what is missing entirely from conventional appreciation of multiplication is that when we - as in this example - use 2 as an operator it carries a distinctive holistic meaning (that cannot be identified with the standard analytic meaning)!
So the key to recognition that the 2 rows of peas are in fact identical with each other is a common shared similarity. So even though we count (as in addition) through recognising the individual independent identity of each item, multiplication requires the complementary recognition of the common shared interdependence of all items (whereby we are able to "see" the various rows as "copies" of each other).
So the use of 2 here - in relative terms - in the multiplication operation 2 * 3, has a holistic rather than analytic meaning. However because reference frames continually switch in dynamic terms, 2 equally can be given an analytic meaning, with 3 - in relative terms - a holistic meaning.
However this is my very point!
When we multiply two numbers such as 2 * 3, both quantitative (analytic) and holistic (qualitative) aspects are inevitably involved.
Therefore when rightly understood in a dynamic interactive manner, we must conceive of these numbers as possessing aspects that are relatively independent and relatively interdependent with respect to each other.
And of course this intimately applies to the primes and the unique factor compositions that generate the natural number system.