OTHER DERIVATIVE CATENAS
From a catena like the strongness or strength catena we
can directly derive the strongness-moreness or 'strongerness'
catena. But besides the derivations stronger or more
strong, weaker or less strong and equally strong
or equally weak we also know derivations like strengthening and
weakening. There thus exists a strengthening catena as well. In other
systems we will find an honoring catena, betterment catena, heating catena,
and so on. All these comparative catenas are increase catenas as
explicit triads or differentiation catenas as quasi-duads.
Differentiation stands to difference or otherness and different
or other as strengthening stands to strongness and
stronger, and as increase stands to moreness and more.
Also differentiation or increase catenas can be subdivided into
positivity differentiation and neutrality differentiation catenas
(or any other type dependent on the aspectual value taken).
And, analogously to the case of difference catenas, the increase
catena of an explicit triad is a positivity increase catena, and
the increase catena of a bipolarity catena a neutrality increase
catena. These similarities are obvious. They hold for every
increase or differentiation catena and the corresponding moreness
or difference catena.
Roughly speaking, the
difference-catenary approach is primarily
nontemporal, whereas the differentiation-catenary approach is primarily
temporal, at least on the assumption that every comparatively
catenal thing exists over a period of
Differentiation, increase or decrease can be active or
For example, active positivity differentiation means
making less or more positive(ly catenal) and passive
positivity differentiation becoming or growing less
or more positive(ly catenal).
Active neutrality increase is making-more-neutral(ly catenal), and passive
neutrality increase, becoming- or
(Literally speaking, one cannot make something positive or neutral,
or become positive or neutral. A primary predicate just is or is not
positive or neutral, while it is a nonpredicative primary thing which
can or cannot be made or become positively or neutrally catenal.)
Each differentiation catena is, as it were, the common denominator
of two catenas: an active (transitive) and a passive (intransitive)
one. Both catenas are each other's isorelative but --as already
explained in 2.3.3-- the passive variant
consists of pseudo-attributes only.
The fact that a catena is a difference or differentiation
catena, whether active or passive, says almost nothing about its
position in the total derivation system, except that it is not a
basic catena. Only if the original catena is basic is the
comparative catena a catena of the first level of derivation. In
general, it is of the (n+1)-st level, if the original catena is
of the n-th level. (Basic catenas are of the 0-level of
Catenas of the same basis are interchangeable. For example,
the longitude, latitude and altitude catenas are interchangeable
in that the spatial co-ordinate which is regarded as longitude
could also be latitude or altitude instead, and vice versa.
Catenas derived from two interchangeable catenas in the same way
are interchangeable themselves. Interchangeable catenas are
always equidimensional, but equidimensional catenas need not be
interchangeable. The difference and differentiation catenas of
the longitude catena, for instance, are equidimensional to it
but not interchangeable with it (with the exception of the
monovariant positivity difference catena for which c=0, which is
identical to it). Monocatenal and bicatenal difference catenas
are also equidimensional, but not interchangeable either: the
length of an object, for instance, or its shortness catenality,
is something else than the distance between this object and
another object, or its closeness catenality.
Unlike difference and differentiation catenas, which are
equidimensional to the original catena from which they are
derived, differential catenas have their own dimension.
Consider, for example, velocity.
If v represents an object's velocity, then v=∂s/∂t (with
∂s as the distance traveled and ∂t as the time spent); v is
therefore s-differentiated-to-time, and its dimension is m/sec.
On the basis of a terminological
analogy between value collections and catena extensionalities,
we may say that one can derive the motion catena by differentiating
the longitude catena, or a catena interchangeable with it,
Hence, we shall call the motion catena, the quasi-duad corresponding to
velocity, "a time-differential catena".
The term velocity is also used in the more general sense of
rate of occurrence or action.
In that case it corresponds to any time-differential catena, not just the
one in the derivation system of the longitude catena.
The differential catenas to time of the longitude, latitude
and altitude catenas are mutually interchangeable. It may be
said that each of the three time-differential catenas is a
motion catena; it may also be said that the motion catena is
the common denominator of these and similar catenas.
This is not a fundamental issue.
Starting from a differential catena which is derived from a
basic catena, we can in turn derive a difference or other
comparative catena from it. For example, in the physical system
of the longitude catena, the slowness catena is a modulus catena
of such a differential catena. The dimension of this modulus catena
is the same as the one of the differential catena, namely
m/sec. But also when we now comparatively derive a new catena
from the slowness catena, for example, the bivariant differentiation
catena with extreme aspectual value, the dimension remains
the same. Thus the dimension of this retardation catena is also
m/sec. On the other hand, deceleration (and also acceleration)
has the dimension m/sec2 in physics.
The catena to which this deceleration belongs, however, is not the
differentiation catena of the slowness or speed catena, but the
time-differential catena of the speed catena.
Just as a derivative and a first derivative are the same in
mathematics, so a differential catena is the same as a first
(In the theory of catenas a derivative need not be a differential catena
A second differential catena is, then, the differential catena of a
differential catena with respect to the same form of catenality or
The deceleration catena is no such second differential catena:
it is the first differential catena of the modulus catena of the
first differential catena of the longitude catena or a catena
with which it is interchangeable.
In traditional mathematics the differentials ∂v or
DELTAv and ∂t or DELTAt are evaluated
positive, if the new v or t value is closer to the positive extreme than
the old one. The aspectual value is on this account supX or
+INFIN. Now this view is typically that
from the perspective of positivity moreness, based on the
positivity increases of v and t. What underlies this conception
is positivity differentiation. The differential catena thus
derived is a positivity-differential catena.
It is not possible to make a mathematical differentiation
universally comparative, because the aspectual value will always
determine the value of the derivative function. On the other
hand, we are just as justified of looking at DELTAv and
DELTAt from the perspective of neutrality moreness, and to
evaluate DELTAv and DELTAt positive if the
new v or t is closer to zero than the old one, and negative if it is
farther away from it. The catena thus derived is a
If û1 is the catena value of the original
the value of the positivity differential catena, the value of the
neutrality-differential catena is
( | f(û1) | ×
( | û1| ×
for û1¹0 and
If û1 ¹ 0 and
f(û1) = 0, then
f0(û1) = 0
regardless of the value of
(This is because we choose a point on the curve which
is a point on the tangent midway between the lower and the
higher triangular points.) If û1 = 0 and
then f is indefinite; and if û1 = 0
and both f(û1)¹0
f0 is infinite (even if
f+ is finite).
If A is the difference catena of B, then B may be called "a
sum catena of A". Similarly, if C is the differential catena of
D, D may be called "an integral catena of C". For example, the
quasi-monad of energy can be conceived of as an integral catena
of the physical force catena with respect to the length catena.
Differential and integral catenas have to be distinguished from
the related quotient and product catenas. The product catena of
the physical force catena and the length catena would be the
physical moment catena, while the integral catena is an energy
catena. Nevertheless, differential and quotient catenas, and
also integral and product catenas, have the same dimension,
granted that the original catenas are the same.
As the kinds of
catenical derivations are closely related
to the kinds of mathematical operation, it is worth our while to briefly
consider the position of the simplest mathematical operations.
The first pair of operations is, then, that of
addition (a + b = c) and substraction (a - b = c). There is no repetition
involved in these operations. In the theory of catenas it
is the derivation of a comparative and equidimensional catena
which is the analog of the mathematical sort of operation on
this zero-level of reiteration.
(All mathematical operations or functions may also be employed, however,
to express logically contingent relationships between catena values,
something that is not our concern here.)
The prototypical mathematical operation on the first level
of reiteration is multiplication (a × b = c). This is in the
first instance nothing else than a form of reiterative adding-up:
b × a = a + a + ... + a , or
b × a = SUM(i=1, b) ai
(ai=a for every i).
Its correlative is division (a ÷ b=c). The derivations of
nonequidimensional differential and integral catenas, and of quotient
and product catenas, are the catenical analogs of mathematical
operations on this level.
The prototypical mathematical operation on the second level
of reiteration is involution or the raising of a quantity to
power (ab = c or a××b = c). This is in the first
instance nothing else than a form of reiterative multiplication:
a ×× b = a × a × ... × a , or
a × b = PRODUCT(i=1,b) ai
(ai=a for every i).
The correlative of this second-level operation is evolution, the
extraction of a mathematical root ( b ROOT a=c ). In their original shape
the numbers were what mathematicians traditionally call "natural".
It would be naive to take it for granted now that the number
of sorts of mathematical operation is thus exhausted. For
we can continue ad infinitum by repeating the operation on
the previous level of reiteration. On the third level this would
involve reiterative involution with its related forms of operation.
But on this and higher levels ordinary, and also mathematical,
language just lack the terminology to express ourselves,
even if we would like to. Nevertheless, it is possible to
develop a universal notational system for reiterative operations
of all levels by means of novel mathematical symbols. Suffice it
here to recognize that there are not only other catenas on the
first level of reiteration besides differential catenas but also
catenas on higher operational levels of reiteration.
They are catenas of
primary predicates for which there is
even no name in scientific or technical noncatenical discourse.