The most famous sutra (teaching) in Mahayana Buddhism is known as the Heart sutra which contains - according to one commonly used translation - the following lines
"Form is not other than Void, Void is not other than Form".
I will here briefly attempt to explain the deep meaning of these lines and their potential great significance in mathematical terms.
Form relates to material phenomena, which in dualistic terms are largely understood as possessing a distinct independent existence.
Indeed this is what gives phenomena a customary rigid identity in analytic terms.
However the corresponding holistic perspective is to view phenomena not in terms of their distinct quantifiable identity, but rather with respect to the qualitative interdependence which ultimately connect all phenomena.
And such holistic appreciation relates directly to intuitive realisation that is the hallmark of advanced contemplative type awareness.
The culmination of such holistic awareness then leads to the realisation of the unity of all form (with respect to a common underlying spiritual identity).
However this ultimate appreciation of the qualitative interdependence of all reality requires corresponding detachment from the recognition of phenomena of form with respect to their separate phenomenal identity.
Thus the unity of all form coincides therefore with the emptiness (i.e. nothingness) of such form (in a separate phenomenal manner).
Of course in experiential terms this can only be approximated in a dynamic relative fashion.
Thus as the underlying spiritual unity of all creation becomes more evident, (distinct) phenomena of form become ever more transient as they arise and pass away from attention with increasing alacrity.
Eventually, the temporary dynamic nature of distinct phenomena will not even appear to be present in consciousness (though indirectly they must still be generated).
So at this stage, the unity of all form (as the actual realisation of the underlying spiritual nature of all phenomena) will approximate ever more closely to the (empty) void, as the pure potential basis for the subsequent emergence (in actual terms) of all such phenomena.
Now this has a direct relevance for mathematical appreciation.
The two most fundamental numbers are 0 and 1
From the customary analytic perspective, these two digits are given an absolutely separate independent identity.
Their great significance is demonstrated by the binary digital system on which the present IT revolution is based. So all information can be potentially encoded through the analytic use of the two digits 1 and 0!
However, what is not all clearly recognised is that all mathematical symbols and relationships, with a customary analytic interpretation (in quantitative terms), can equally be given a holistic interpretation (with immense potential implications from a qualitative perspective).
Therefore 1 and 0 have an important holistic meaning, which complements their accepted analytic interpretation.
And just as 1 and 0 are considered to be absolutely separate in analytic terms. they are considered as fully relative - and ultimately identical - with each other from the corresponding holistic perspective.
So, in holistic terms, 1 and 0 are seen - as it were - two sides of the same coin, which mutually imply each other.
Thus 1 (as the qualitative unity of all relatively distinct phenomena) implies 0 (as the corresponding nothingness with respect to a separate identity in quantitative terms) and vice versa.
Therefore in holistic mathematical terms, the lines quoted above from the Buddhist heart sutra, could be simply represented as,
1 is (ultimately) indistinguishable from 0, and 0 is (ultimately) indistinguishable from 1.
However the clear implications of such understanding is that we have to let go of the absolute identity of mathematical symbols in both analytic and holistic terms.
Thus both the analytic (Type 1) and holistic (Type 2) aspects of the number system can only be rightfully understood in a dynamic interactive manner. Thus notions of both quantitative independence and qualitative interdependence respectively, are now understood as complementary notions with a relative - rather than absolute - meaning.
Now we will briefly see how these analytic and holistic interpretations directly apply to the simplest of numbers.
For example when we understand "3" in the customary analytic manner, it can be defined as
3 = 1 + 1 + 1.
So the individual units are understood here in an independent homogeneous quantitative manner (that - literally - lack qualitative distinction). So we have no way of distinguishing the separate units from each other (which would require some unique qualitative feature).
In more complete terms, we can express this quantitative notion of "3" in Type 1 terms as 31.
Alternatively this can be expressed - using units only - as (1 + 1 + 1)1.
However, when we understand "3" in the unrecognised holistic manner, interpretation is subtly inverted.
So here "3" represents - not individual separate units of quantity - but rather the interdependence of all units in a collective manner.
Then in direct terms, just as analytic appreciation occurs in a rational, holistic appreciation occurs in a complementary intuitive manner!
This latter aspect of number is more fully expressed in Type 2 terms as 13.
Alternatively this can be expressed as 1(1 + 1+ 1).
So now both The Type 1 and Type 2 aspects have been expressed with reference to the number "1".
However when we appreciate these two aspects appropriately in a dynamic interactive manner (i.e. in Type 3 terms) it becomes apparent, like the turns at a crossroads, that the use here of 1 is inherently paradoxical.
So once again using our crossroads example in heading up a road (in a N direction) that a left turn at the crossroads can be unambiguously identified.
Likewise in a reverse manner, in heading down the road (in the opposite S direction) that the left turn at the crossroads can again be unambiguously identified.
However when, in a dynamic interactive manner, we attempt to embrace the approach to the crossroads simultaneously "seeing" from both N and S directions, then the identification of a left turn is rendered paradoxical. For what is left from one direction (say heading N) is right from the opposite direction (heading S) and vice versa.
It is quite similar in number terms. What is identified as 1 (from the Type 1 perspective) is in fact 0 (from the corresponding Type 2 perspective). Likewise what is identified as 0 (from the Type 1 perspective) is 1 (from the complementary Type 2 perspective).
Let us look at our example more closely to identify why this in fact is so.
Now again with respect to 31, I refer to 3 as the base and 1 as the corresponding dimensional number respectively.
Then, when we identify the base number 3 = ( 1 + 1 + 1) in quantitative terms, the corresponding dimensional number 1 (in Type 3 terms) should correctly be interpreted in a complementary qualitative manner.
In other words, whereas the base number 3 (= 1 + 1 + 1) refers to an actual number (in quantitative terms), the corresponding dimensional number 1 refers - in this relative context - to the potential for number existence (in a qualitative manner).
So whereas the actual number is finite (in quantitative terms), the number dimension is strictly speaking infinite in nature (potentially applying to any number).
Therefore the number dimension - having a qualitative meaning that provides the basis for subsequent relationships between numbers - is nothing (i.e. 0) in an actual quantitative manner.
So 1 as used in a qualitative context is strictly 0 (in corresponding quantitative terms).
Likewise with respect to 13, the meaning of 1 is subtly inverted, as now implying the base unit for all subsequent qualitative relationships (where interdependence is achieved).
Then in relative terms 3 (= 1 + 1 + 1) now carries a numerical significance in dimensional terms (as 3 related dimensions).
However what has a finite numerical meaning in a qualitative manner, strictly has no meaning in quantitative terms.
So again 1 (where 1 now numerically refers to qualitative identity) is strictly 0 in a corresponding quantitative dimensional manner.
In fact, we can indirectly show how this is so!
With respect to 13, we can indirectly express in a quantitative manner the circular nature of interdependence that attaches here to the 3 dimensional units by obtaining the 3 roots of 1.
So the 3 roots of 1 are 1, – .5 + .866i and – .5 – .866i respective, which geometrically can be expressed as 3 equidistant points on the unit circle in the complex plane.
And the collective sum of these roots = 0. Holistically this can be explained by the fact that these express (in an indirect quantitative manner) the qualitative notions of 1st, 2nd and 3rd (in the context of a group of 3 members).
Thus wheres cardinal identity relates to the quantitative nature of number (made up of independent individual units), in this context, ordinal identity relates by contrast to the corresponding qualitative nature of number (whereby the collective interdependent identity of unique units is expressed).
So if we are to properly understand the nature of number in dynamic interactive terms, we must recognise the complementary nature of both analytic and holistic aspects (where number is defined - relatively - in both linear and circular terms).
The implication of this is that the very nature of 1 and 0 now must likewise seamlessly switch as between each other. So what is 1 in a quantitative context is 0 in a corresponding qualitative manner; likewise what is 1 in a qualitative context is likewise 0 in a corresponding quantitative manner.
Therefore the ultimate nature of number approaches a state of pure ineffable mystery, where both linear and circular frames of reference are united in the pure marriage of both the quantitative and qualitative interpretation of mathematical symbols.
And here we have the holistic identity of 1 and 0 that ceaselessly change between each other.