We have already looked at the bi-directional relationship internally (within each prime) of both its quantitative and qualitative aspects. Thus the unit members of the prime from the cardinal (quantitative) perspective are balanced by a set of natural number members from the corresponding ordinal (qualitative) perspective.
Put another way the
cardinal notion of a prime (in quantitative terms) can have no strict meaning
in the absence of its natural number ordinal members (from a qualitative
Equally, in reverse
terms, the natural number ordinal members can have no strict meaning in
the absence of the cardinal notion of the prime.
So for example 3 as a
prime has no strict meaning in the absence of its 1st, 2nd
and 3rd members; likewise these 1st, 2nd and 3rd
members in qualitative terms, have no strict meaning in the absence of the
quantitative notion of 3.
understood, each prime must be understood in a dynamic bi-directional manner
entailing both cardinal and ordinal aspects (which are, relatively,
quantitative and qualitative with respect to each other).
We saw in yesterday’s
blog entry that if s is a prime, that the sth position (with respect to s)
always reduces to standard cardinal interpretation and that this represents the
default treatment of ordinal numbers in conventional mathematical terms. This
in turn indirectly equates in quantitative terms, with the fact that in every case, one of the s
roots of 1 = 1.
We then saw that there exists
a Zeta 2 function that complements the well known Zeta 1 (i.e. Riemann) zeta
function, where its zeros provide an indirect quantitative means of giving
unique expression to all the non-trivial ordinal positions associated with each
So the crucial function
of the zeros - when appreciated from this dynamic (Type 3) perspective - is
that they provide the ready means of indirectly converting from a qualitative
to quantitative type interpretation.
So for example in the
case of the prime number 3, the Zeta 2 function is given as,
1 + s + s2
the two solutions to this equation provide unique quantitative conversion of
the qualitative notion of 1st and 2nd respectively (in
the context of 3).
And by extension the
Zeta 2 function can thereby be used to provide unique quantitative conversions
for (non-trivial) ordinal positions associated with every prime!
provides an important basis for appreciating the corresponding external
bi-directional relationship as between the primes and the number system as a whole.
Now, when we
look externally at this relationship we find that all natural numbers in
quantitative terms are uniquely composed of a combination of one or more
So for the
primes themselves only one factor is involved (excluding the “trivial” factor
of 1 from consideration).
Then for a
composite number such as “6” two factors are involved.
So 6 = 2 *
3 (which is the unique prime factor combination for this number).
course prime factors can be repeated.
So 12 for
example has 3 prime factors where 2 is repeated twice (i.e. 12 = 2 * 2 * 3).
Now this is
all well and good insofar as it goes, but unfortunately as we shall see, completely one-sided.
standard mathematical interpretation, the primes represent the unique
independent “building blocks” of the natural numbers in a merely
what is crucially overlooked in such conventional mathematical interpretation
is that - quite literally - a corresponding qualitative dimension arises whenever
the multiplication of numbers takes place.
This fact -
which I have often recited - hit me forcibly at the age of 10 when studying
simple concrete problems involving the areas of fields.
example, if one imagines a large field with length 3 km and width 2 km, the
corresponding area will be given in square (i.e. 2-dimensional) units.
a quantitative perspective the answer is indeed 6, a qualitative transformation
in the nature of the units has thereby taken place through the very process of
another simple way also of coming to appreciate this qualitative connection.
Imagine there are 2 rows - say of cars - with 3 cars in each row.
the perspective of addition, one would treat all the items as independent.
Therefore one could count up the 3 items in one row (= 3) and then proceed to the second
row again counting up the 3 items (= 3) and then add the two rows.
So we are
here treating the items in each row as independent (and indeed the rows
themselves as independent).
In this way
the total no. of cars = 3 + 3 = 6.
what is vital to carry out multiplication, is the corresponding recognition of the
interdependence of each row (which thereby assumes a common similarity as
between the 3 items in each row).
So the very
reason one can now relate the operator 2 with 3, though multiplication, is
because of the recognition of the common similarity (with respect to the cars
in each row).
such similarity applies to the interdependence of the items with each other (which
is qualitative in nature).
properly understood, multiplication necessarily entails notions of both
quantitative independence and qualitative interdependence respectively, which
can then only be properly appreciated in a dynamic relative manner.
conventional interpretation of multiplication reduces its nature to that of
addition (where the quantitative independence of unit members is solely
in conventional terms, in our example, we start with the quantitative “building
blocks” i.e. 2 and 3 which are defined in linear (1-dimensional terms) i.e. on
the number line.
when we multiply these two numbers, even though their dimensional nature has changed,
the result 6 is still represented on the same number line!
crucial necessary qualitative nature of multiplication is thereby missed.
Thus if one
is to appreciate the true significance of the Zeta 1 (i.e. Riemann) zeros, it
is vital to recognise clearly the qualitative nature of multiplication.
from a quantitative perspective, the identity of the composite natural numbers
is indeed based on the primes, strictly speaking this all reversed in
qualitative terms, whereby the identity of the primes is now based on their
unique relationship with the natural numbers!
words with respect to the composites, the primes have a new interdependent
relationship as factors. Therefore though the primes are indeed independent in
a quantitative sense (as separate “building blocks”), through relationship with
each other as the factors of composite numbers, they likewise share a
qualitative interdependence with each other.
Now I have
made this simple point before though in truth it is extremely subtle to
understanding, that the very nature of number keeps switching, as it were,
between both a particle and wave identity (without ever being noticed in
again the standard quantitative definition of number is based on the
independence of each of its unit members.
So 3 = 1 +
1 + 1 (with each homogeneous unit independent in a quantitative sense).
we now say for example that a number - say 30 - has 3 factors the very nature
of 3 has now changed (from a dynamic interactive perspective).
point about each factor is that its identtity implies that it is related to a
composite number. So in this context, 2, 3 and 5 assume a new qualitative identity
as factors through their common relationship to the number 30!
likewise when we now say that 30 has 3 factors, 3 = 1 + 1 + 1 (but now -
relatively - in an interdependent rather than independent sense).
again this is akin to nature of left and right turns at a crossroads, where
what is deemed left and right is merely relative depending on a reference frame
based on the direction of approach (which can keep switching).
So just as
we saw how the Zeta 2 zeros can be used to deal with the very important
internal question of how ordinal qualitative notions (with respect to the
members of each prime) can be indirectly converted in a consistent quantitative
manner, we now have the parallel problem of how the Zeta 1 (Riemann) zeros can
equally be used externally to indirectly convert the qualitative nature of the
primes (as factors of natural numbers) in a quantitative manner.
And I have
shown before how the frequency of the Riemann zeros is intimately linked to
the factor frequency of the natural numbers.
once again if one accumulates the frequency of the proper factors of the
natural numbers (up to n), the total will bear a remarkably close relationship with
the corresponding frequency of the non-trivial Riemann zeros (up to t) where
n = t/2π.
the key importance of the Riemann zeros (from this perspective) is that they
provide an indirect means of converting the qualitative nature of the primes
(through their relationship with the natural numbers as factors) in a
the Zeta 1 and Zeta 2 functions are themselves complementary, relating to the
bi-directional relationship of the primes and natural numbers (externally and
internally) in both quantitative and qualitative terms.
So the Zeta
1 (Riemann) function can be expressed:
1– s + 2– s + 3–
s + 4– s + …… = 0.
corresponding Zeta 2 function can then be expressed as
1 + s1 + s2 +
s 3 + …… = 0 (initially where s is prime).
Notice the complementarity! Whereas the
natural numbers represent the base values with respect to the Zeta 1, they
represent the dimensional values with respect to the Zeta 2; and whereas the
unknown (s) represents the dimensional values with respect to the Zeta 1, they
represent the base values with respect to the Zeta 2.
Also whereas the dimensional values are
negative in Zeta 1, they are positive in Zeta 2; finally whereas Zeta 1
represents an infinite, Zeta 1 represents a finite series respectively.
Of course ultimately both internal and
external aspects of this bi-directional relationship between the primes and
natural numbers are themselves fully interdependent.
From the quantitative perspective, it does
indeed appear that the natural numbers are derived from the primes; however
equally from the qualitative perspective the primes appear to now obtain their
positions (expressing their relationship with each other) through their identity as unique factors of
the natural numbers!
And if one thinks about this for a moment, without knowledge of the gaps between the primes, it would not be possible to list the primes (as quantitative "building blocks" of the natural number system); likewise without knowledge of their quantitative value it would not be possible to establish the gaps between the primes (that express their qualitative relationship with each other).
So rather than an absolute quantitative relationship connecting the primes with the natural numbers in a one-way static manner, rather we have a dynamic two-way interactive relationship that operates relatively in both quantitative and qualitative terms.
This implies that an incredible
dynamic synchronicity characterises the relationship between the primes and
natural numbers , where they
ultimately approach total identity with each other in an ineffable and utterly
The very ability to - literally - "see" in a
pure intuitive manner this remarkable synchronicity, where number as form
becomes dynamically inseparable from number as energy, represents the
holistic extreme of mathematical understanding (where quantitative can no
longer be separated from qualitative appreciation).
At the other extreme we have the totally
abstract rational understanding of number representing absolutely fixed forms
where quantitative is totally separated (in formal terms) from qualitative
The fundamental requirement for
Mathematics is then the consistent integration of both types of understanding.
However this will require that equal emphasis is given to both analytic and
And without any hint of exaggeration,
Mathematics at present, despite its enormous advances, is hugely unbalanced
(through a complete neglect of its vitally important holistic aspect).
Nothing less than a total revolution in
perspective will rectify this problem!
Indeed this may be slowly initiated through
the inevitable continued failure to prove the Riemann Hypothesis, which not alone cannot be
proven (or disproven) but much more importantly cannot be properly appreciated in
conventional mathematical terms.