2.6.3 
WHERE NEUTRALITY DETERMINES THE MEAN 
To take the mean value m of
a special or universal collection of
catenals as the neutral value is tantamount
to assuming that k(m) = 0, that
______ ___
k(v) = 0 and that v^ = 0
[v with a caret over it is the symbol for the catena value].
It cannot be proved that the mean value is always the neutral value,
and vice versa. And to be able to disprove it, one would first
have to make clear whether such a claim is a linguistic,
psychological, physical, normative or other kind of claim (or
all at once). Moreover, one would have to agree on the type of
catenization function and the exact form of
catenality to be chosen in each case.
Here we shall treat the equality
of neutral and average or mean value merely as a postulate, albeit a
challenging one indeed. The simplest mathematical form of
this hypothesis of meanneutrality is:
___
v^ = 0
(the mean
catena value is 0).
It applies to every independent form of catenality, and both to
universal and to certain special catenizations. For
special catenizations the distinction between catenals taken
into consideration (and included in the mean) and all other
catenals (not included in the mean) must be relevant. As already
suggested, one may suppose that such a distinction is relevant
if the catenals belong to a closed system.
The hypothesis of meanneutrality can be read in two directions:
(1) given the neutral value (that is, its empirical
equivalent), the mean is equal to it in a closed system or in
the world at large; and (2), given the mean, the neutral value
is equal to it. We also assume that the derivations are
nonfactitious, if possible;
this in accordance with
the rule of nonfactitious priority. It
should be interesting to first have a look at one or two cases in which
the neutral value is given, before turning our attention to cases in
which the mean is given or calculable.
With regard to the energy increase catena increase of energy
(or gaining energy) is positive, decrease of energy (or losing
energy) negative, and neither gaining nor losing energy, but
having a certain amount of energy nevertheless, neutral. According
to the hypothesis of meanneutrality no energy will 'melt' into
nothingness in a closed system and no energy will 'spring'
from nothingness, since the mean energy increase in such
a system has to be 0. The independent form of catenality
concerned is, then, not energy as distinct from mass but energy
inclusive of mass. If there is energy (including mass) which
does seem to disappear into or to arise from nothingness, the
physical system in question is simply not closed. This impossibility
of a closed system's
increasecatenary mean not being 0,
and thus of energy melting into or springing from nothingness,
is precisely what the physical definition of closed system
rests upon. The hypothesis of meanneutrality (or a more
specific derivative postulate) thus underlies the very principle
of the conservation of mass and energy, and similar,
physical principles of conservation.
It might be objected that altho the hypothesis of meanneutrality
may hold in certain cases it does not hold in other
cases. Yet, if someone believes to know a counterexample, it is
imperative that all the basic assumptions are checked again.
Take, for example, the suggestion that it would be false or
absurd to assume that in a closed system all the objects are on
the average (if the catenization is linear) at rest. Why could
there not be more movement in one coordinate direction than in
the opposite coordinate direction (granted that a closed system
need not be spatially closed in all directions)? The flaw in
this reasoning is that the movement of objects in the closed
system is related to an external system, that is, a system in
which we live ourselves, or a system of which the first, closed
system would form part. But motion catenization with respect to
that different or larger system (even if 'universal') is not
relevant if the first system is a closed system at all. If it
is, then rest in this system is the mean displacement per
second; and the mean displacement per second in the two directions
of a coordinate is then 0 in this particular system with
respect to the coordinates of this system. That the average
change of place may not be 0 with respect to a system in which
we ourselves call things "at rest" or "in motion" is just not
relevant if we are really talking about a closed system.
