Linear Thinking LO22881

AM de Lange (
Thu, 14 Oct 1999 11:46:04 +0200

Replying to LO22861 --

Dear Organlearners,

Leo Minnigh <> writes:

>Particularly in the discussion between At and Winfried, linearity
>has been linked to the mathematical formula of a line: y = ax + b.
>Important to realise is the meaning of all the elements in this
>formula: both axes y and x; the ORIENTATION of the line by 'a',
>and the POSITION of the line by 'b'.
>By defining lines in this way, there must already exist a framework
>of linear axes. If the x-, and/or y-axis is non-linear, the line will
>non-linear too. In a Cartesian framework both axes are rectangular
>(they are perpendicular to eachother), but that is not a prerequisite
>for the linearity of the line.

Greetings Leo,

Thanks for you valuable comments on the GRAPHICAL REPRESENTATION of a
straight line.

Fellow learners can experiment themselves with the consequences of your
comments. Let us do it.

Fellow learners need to prepare a "straight line in straight context"
so that they can conduct their experiments. In other words, they
have to prepare a completely (consistenly and coherently) linear
picture. First they have to draw two axes x and y as straight lines
perpendicular to each other. Then they have to represent at EQUAL
intervals the numbers .......-4, -3, -2, -1, 0, +1, +2, +3, +4,
on each axis. Put the zeros where the two axes cross each other.
Finally they have to draw any straight line in the plane defined by
these two strainght axis.

Now take a new piece of paper, draw the same two STRAIGHT
axis x and y, but DO NOT represent the numbers at equal intervals
for BOTH axis. Since an interval can be interpreted as a change,
equal intervals are changes without changes. In other words, by
making the intervals not equal, we bring changes in changes into
the picture's context, i.e. we dance changes in the background.
Use ANY NON-LINEAR PATTERN to make these changes in the
intervals of ONLY ONE AXIS, NOT BOTH. There are a thousand-
and-one patterns which you can use. For example,
* make each interval half as large as its preceding one
* make each interval one unit larger as its preceding one
Again plot the numbers at these unequal intervals. Now, finally
determine the x and y values of any number of points on your
original straight line. Plot these (x, y) values, point for point, on
the new set of axes (one with unequal intervals) to get a number
of points. Connects these points to form a line. This line will not
be straight, but have some curvature to it. Its non-linear form
will depend on the non-linear pattern which you have used to
make the nonequal intervals on the one axis.

Now repeat the experiment, but use DIFFERENT non-linear
patterns to produce unequal changes in the intervals of BOTH
axes. Use one pattern for the x axis and another pattern for
the y axis. Plot points from the original straight line on the new set
of axes (with unequal and dissimilar intervals). Connect these
points to form a line. This line will be non-linear as well.

Now repeat the experiment, but use the SAME non-linear
pattern to produce unequal changes in the intervals of BOTH
axes. Plot points from the original straight line on the new set
of axes (with unequal, but similar, intervals). Connect these
points to form a line. This line will be straight once again like
the original line!!!!!

In the experiments above we have changed the magnitude of the
intervals, but kept the axes as perpendicular straight lines. We
can also experiment by keeping the intervals equal, but changing
the axes from straight lines to non-linear curves. These
experiments are slightly more difficult to perform than the previous
ones. However, you should try them to grow in your experiences
and hence insights.

Also try to work with straight lines in other kinds of representations
than only graphical ones. For example, the following 2x6 matrix
represent a straight line
03 06 09 18 27 36
04 08 12 24 36 48

Think about a monotonous obsession. As soon as we make
persistently the same changes right through the entire background,
a straight line will remain a straight line. In other words,
SYMMETRICAL changes in our Systems Thinking will not help
us to escape from linear thinking. But when we restrict changes
to some specific facet or dimension of the background, a straight
line will transform into a non-linear curve. In other words, only
ASYMMETRICAL changes (allowing even more than one of a
kind) in our Systems Thinking will help us to escape from linear
thinking. Consequently it will be beneficial to focus on
asymmetrical changes.

Leo, you write:
>We should write a book with the titel:
>"On form and content".

Any change from content to form, or from form to content, is always
non-linear. In other words, dancing on content-form is non-linear.

The topic of "birth" is central to the change from content to form and
the topic of "death" is central to the change from form to content.
"birth" and "death" are non-linear phenomena.

There is but one physical birth and one physical death. How linear
is our thinking when we remember little, if anything, of our physical
birth and contemplate little, if anything, on our physical death? How
linear is our thinking when we have not yet questioned how many
spiritual births and spiritual deaths we may have -- none, one or

Since "birth" and "death" apply to all LIs (learning individuals),
do "birth" and "death" apply to all LOs?

How much role do "birthing" and "dying" actually play in our thinking?
Is the act of "dying" a necessary consequence upon the act of

What happened to the biblical persons Enoch and Elia?

How much do "creative ascent" and "creative collapse" figure in our
thinking? Is a "creative collapse" prerequiste to a "creative ascent"?

What happened to Abraham on Moria and to Jesus on Golgota?

>The word 'linear' is a form-description. But the moment we
>introduce the question 'WHY' , the content comes in play.

Leo, what a brilliant insight you have given us here! I want to
put a question to that insight.

Which content?

To help you with your lateral thinking, I have prepared the
following questions. The content from which the form has
emerged? The form which has collapsed into content? The
content+form as the system SY? The system SY as form
and the surroundings SU as content? The universe UN as
content with the systems SY and SU as its form?

Leo, you refer to your contribution LO20686 in which you wrote:

>But thoughts move through a multi-dimensional space ......
>..... the 7-D creativity space of the seven essentialities.....
>..... And to make things even more complicated, the seven
>.... essentialities all together influence each other in a dynamic

Why do the seven essentialities influence one another? Because
they concern the seven different dimensions of the form of one and
the same content.

Here is another experiment which fellow learners can make. Draw
any non-linear curve on background defined by straight line axes
with equal intervals on them. The task now is to make that curve
a straight line -- to perform a reductionistic excerxise.

The long path is to find the formula of which the graphical
representation is the non-linear curve. We can, for example
use non-linear regression analysis to arrive at a formula. Say
the result is y = 3x^3 - 2x^2 + x - 6. We then use this
formula to transform the linear x axis with equal intervals into
a linear x axis with unequal intervals. We then plot on the axis
1 at 3.1^3 - 2.1^2 + 1 - 6 = 3 - 2 + 1 - 6 = -4
2 at 3.2^3 - 2.2^2 + 2 - 6 = 24 - 8 + 2 - 6 = 10
3 at 3.3^3 - 2.3^2 + 3 - 6 = 81 - 18 + 3 - 6 = 60
etc. Finally we represent the original graph on the new system
of axes to get a straight line.

The short path is to realise that we view at the graph from a 3-D
background. The graph itself is 2D, i.e containing only x and y.
So what we now do, is to reduce the third dimension z in our
viewing of the grpah. Thus, rather than looking perpedicular along
the z-axis at the graph, we begin to lower our eyes away from the
z axis towards the x-axis so that the angle of sight reduces from
90 degrees to finally 0 degrees. At precisely 0 degrees we are not
looking AT the paper anymore, but ALONG its flat surface in the
direction of the x-axis. The non-linear curve will now be a straight

In the long path we took the "becoming" of liveness out of the
curve (system) by letting changes of changes in it collapse into
the x-axis (surroundings). In the sort path we took the "variety"
of otherness out of the curve (system) by letting the z-axis
collapse into the x-axis (surroundings). In other words, in both
cases we took an essentiality (liveness of otherness) out of the
system SY to let it disappear into the surroundings SU. In both
cases the reduction caused the transformation of the system
into a linear behaviour.

This experiment tells us how to think about all seven
essentialities (the 7D space of creativity) and not only one or
two of them. As soon as we take one of the seven essentialities
out of the system, it will cause the system to behave more
towards a linear fashion. The more the essentialities we reduce
from the system, the more the system behaves linearly.

Leo, you end with:
>I like to conclude:
>A line is a human invention.

I would like to add to it with:
A line is a human invention -- from which we can emerge
by the intervention of all seven essentialities.

Best wishes


At de Lange <> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa

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