I wish to comment again on the unavoidable paradox inherent in the attempt to define number in an absolute analytic type manner.
Once again, one may attempt to define a number, such as 3, as composed of independent homogeneous sub-units (with no qualitative distinction).
However, by definition this means than any of these units can be defined as 1st, 2nd or 3rd in an ordinal context. In other words, without some qualitative distinction, we have no means of distinguishing 1st, 2nd and 3rd in analytic ordinal terms.
Thus the starting notion of absolute independence in a quantitative cardinal manner, implies the opposite extreme notion of pure relative interdependence in a qualitative ordinal manner.
And within a rigid static framework, there is no way of coherently reconciling quantitative (cardinal) and qualitative (ordinal) number notions. Expressed equivalently in a psychological manner, there is no means therefore for the coherent reconciliation of holistic (intuitive) with analytic (rational) notions!
Thus both aspects can be only be given given proper interpretation within a dynamic interactive framework, entailing both aspects of relative independence and relative interdependence respectively, through which we can provide a coherent approach for number.
This problem with number is closely related to the physical fact that we must always - again for coherent interpretation - view number as existing in a relative dimensional context of space and time.
In fact there is an extremely close connection - indeed they are ultimately indistinguishable - as between both the physical and psychological understanding of space and time and the holistic mathematical notion of number (in dimensional terms).
This indeed is another key reason why this dynamic approach to number, that I have been consistently proposing is so radical, in that it automatically leads to a completely new holistic manner of understanding space and time dimensions (in physical and psychological terms).
So the current rigid physical notions that we live in a world of three space and one time dimension automatically follows from the reduced analytic attempt to view all relationships with respect to their merely quantifiable aspects.
Then when modern approaches such as string theory propose a 11-dimensional world (10 of space and 1 of time) it carries no holistic relevance. In other words, we can make little or no no intuitive sense of such notions within the accepted scientific i.e. merely analytic, interpretation.
In fact, properly understood - from a holistic perspective - current mathematical and scientific interpretation is radically 1-dimensional and is strongly identified with the manner we understand time as flowing with respect to reality - identified thereby in "real" terms" - in a positive forward direction. The customary three space dimensions are then separated from time and rigidly identified with perceived features of objects (as having 3 dimensions).
And this view of the world is ultimately untenable, as the physical features of object phenomena have strictly no meaning independent of the psychological mental constructs we use to interpret such objects. However the accepted analytic approach continually attempts - even while admitting problems with this approach at the quantum level of investigation - to abstract in an absolute manner the (subjective) knower from what is (objectively) known about reality.
So I will briefly deal here with the simplest case of how the true holistic interpretation of "2" (as number dimension) leads to a new understanding of the nature of space and time.
The holistic meaning of 2 relates to the understanding of 2 as representing interdependent - rather than independent - units. Now, clearly where complete interdependence is involved, we can no longer distinguish units as separate. So this would concur with the purely intuitive appreciation of 2 (as an energy state).
However in practice, the holistic (interdependent) aspect of number must be constantly balanced by the corresponding analytic (independent) aspect for meaningful interpretation. Thus notions of number interdependence are always of a relative nature, that also necessarily imply corresponding notions of number independence.
In this context, I have detailed at length how the two notions come together in the understanding of turns at a crossroads. So within a 1-dimensional frame of reference, when one approaches the crossroads - say - heading N, then left and right turns have an unambiguous meaning. Thus, these represent two separate independent turns (in analytic fashion).
However within a 2-dimensional frame of reference, when one envisages approaching the crossroads from N and S directions simultaneously, left and right turns have a circular (paradoxical) meaning. So what is left from one direction (heading N) is right from the other (heading S) and what is right from one direction (heading N) is left from the other direction (heading S).
Equally we could say in this holistic context, that what is + 1 is also – 1; and what is – 1 is also + 1.
So this circular (paradoxical) language arises through attempting to express the 2-dimensional holistic dimesnional notion of "2" (representing the interdependence of the two units) indirectly in a 1-dimensional analytic fashion.
And as we have seen, this concurs with the mathematical task of obtaining the 2 roots of 1.
So in correct dynamic terms, the two roots of 1, i.e. + 1 and – 1 have an analytic meaning as relatively independent; however they equally have a holistic meaning (when combined) as relatively interdependent.
So the sum of the n roots of 1 (in this case 2) = 0, indirectly expresses the holistic interdependent i.e. qualitative nature of the number n as - literally - without quantitative meaning.
However to see clearly the implications here for the understanding of space and time, let us consider the two dimensions as relating to the external (objective) and internal (subjective) aspects of experience respectively.
Now when one becomes externally aware of a number object, it is thereby physically posited in space and time. One is thereby enabled to separate its individual identity in space and time.
Then, when one becomes internally aware of its corresponding mental perception, it is thereby psychologicially posited in space and time. So one is now likewise enabled to separate its individual identity, internally in space and time.
In conventional terms, a very reduced interpretation is given, whereby the number object is assumed to possess an absolute identity (in an external objective manner). This implies that its corresponding internal mental perception - if indeed this is even recognised - is assumed to correspond in an absolute manner with the number object, which in view of the experiential relationship involved is strictly speaking absurd!
It is because of this assumed absolute correspondence as between (physical) objects and (psychological) interpretation that we believe in the linear nature of time and space.
In other words, movement is viewed to take place strictly in a positive (forward) direction. And - as we have seen - this is based on the 1-dimensional mode of analytic interpretation that is the bedrock of all accepted mathematics!.
However, once we recognise the truly relative nature of the two-way polar interaction of external and internal aspects with respect to experience, then space and time likewise assume a merely relative identity.
So again, if we identify the forward movement of space (and time) with the external recognition of the number object, then the internal recognition of its corresponding mental perception must move - relatively - in a backward (negative) direction.
Then when reference frames switch to the internal mental interpretation of the number object, the forward movement of space and time is now identified with this psychological direction; this then implies that - relatively - the external number object is now associated with the backward (negative) direction.
Thus from a 2-dimensional holistic perspective, space and time are understood in relative fashion (in complementary physical and psychological terms) as both possessing 2 real dimensions (that are positive and negative with respect to each other).
And far from this representing some abstract notion, this inherently explains the true nature of space and time (where one's experience is now sufficiently refined to explicitly operate at the 2-dimensional level).
In the past such experience has been largely identified with the great spiritual contemplative traditions (East and West), without however any sustained attempt to relate the implications of such refined intuition to accepted scientific and mathematical understanding.
So the point I am making is that the holistic mathematical notion of dimension equates exactly with the corresponding nature of how space and time are experienced at these "higher" dimensions.
Thus the dimensions > 1 not alone express increasingly larger numbers in numerical terms. They equally represent increasingly "higher" dimensions of intuitive understanding (through which their holistic nature can be properly appreciated).
And though again in the past, the possibility of such understanding was confined to the small group of people traversing an advanced contemplative path (as defined by the various religious traditions) future generations will slowly awaken to the immensely important mathematical and scientific implications of such refined intuitive wisdom.
The deeper realisation here is that - far from being abstract - the notion of number is embedded as its most fundamental encoding in all created phenomena (both physical and psychological).
Indeed from this perspective, all the great wonders of nature represent the subsequent phenomenal decoding of number (in both quantitative and qualitative terms).
However it must be again remembered here that we have now greatly enlarged the notion of number to ultimately embrace - not only the quantitative realms - but also in a manner, that is scarcely yet imaginable, all of the qualitative realms.
So to return again to our starting point regarding the paradox of number.
As I have stated, if the sub-units of each cardinal number were truly independent (lacking qualitative distinction) no means would thereby exist for the recognition of their corresponding ordinal aspect.
One cut also express this therefore by saying that number has no meaning apart from its relationship with space and time!
In fact it is only the implicit assumption of the spatial (and temporal) connections as between the sub-units of a number that enables their ordinal nature to be recognised (from the customary analytic persepctive).
So when we represent - say the number 3 - on a number line as 3 units, these are extended in linear fashion in space.
With respect to time this likewise entails that recognition must take place in a clear unambiguous sequence In other words, the first unit is identified with the unit that spatially starts at 0; then the 2nd unit is identified with the unit beginning at 1; then finally the 3rd unit is identified with the unit beginning at 2. Thus for number to be ordered in this accepted linear fashion, it requires an ability to spatially distinguish the 3 units in a restricted manner.
Therefore, the units are not truly independent but intimately depend on a certain arbitary relationship with space and time dimensions.
And of course the very notion of such a relationship, implies - literally - a qualitative dimension to number!