Information Technology Reference
In-Depth Information
FIGURE 8.
on a function, let's say
y
=
x
2
, and produces a function:
Di[x
2
]
=
2x
, or in
Menger's elegant notation:
Di[( )
2
]
=
2
. Does
Di
have Eigen functions? Yes
indeed: the exponential function
y
=
e
x
, in Menger's notation:
Di[exp]
=
exp
,
and on account of the extraordinary relationship of the exponential func-
tion to the trigonometric functions
sin
and
cosin
:
Di
4
[sin]
=
sin
,
Di
4
[cos]
=
cos
, i.e.,
sin
and
cosin
are the Eigen functions of the differential operator
iterated fourfold.
One doesn't have to restrict oneself to mathematical expressions. Menger
developed these ideas for logical functions (1962), a generalization that is
significant here. For example, the algebraic expression:
S=S
()
=S D
(( )
D=D
()
=D S
()
(
)
of the composition of the two systems
D
and
S
in Figure 8 makes the recur-
sion of the two functors
D
,
S
clearly visible.
06. Compositions (the properties of the composition are not the properties of the
components)
Viewed historically, attention to qualitative changes that arise in the tran-
sition from aggregate to system was guided by an unfortunate formulation
of this transition that was given by “generalists,” “holists,” “environmental-
Search WWH ::
Custom Search