Linear Thinking LO23007

Leo Minnigh (
Tue, 26 Oct 1999 15:14:27 +0200 (MET DST)

Replying to LO22881 --

Dear LO'ers,

Now the discussion on linear thinking is meandering between form and
content it becomes complicated. I have also used all my brain power to
bring the discussion in other waters than the fairly clear water of
mathematics. I sense that a lot of members like to see a stronger
relationship with Learning Organisations in general. But again, this is a
very hard job.

What I have in mind is the following: 1) again some words on form and
content; 2) a tempt to react on the mail of At (LO22881); 3) linearity and
contents of (learning) organisations; 4) epilogue.

1. Form and content
As is mentioned earlier, linearity is a form. But as soon as we start to
describe the why of linearity, we need many words. We need so many words
to describe AND understand the contents, the character.
I like to give you a little piece of puzzling. It is a puzzle from the
famous Sam Lloyd.

One likes to have a chain of 30 links; the chain is circular like a
necklace (without a beginning or end, without a slot). One can buy such a
chain for $22. However you have six pieces of chain, each composed of five
links. A goldsmith will help you to make the desired chain of 30 links; he
asks $1 to open a link and $3 to make a closed connection. The question
is: what is cheaper, asking the help of the goldsmith who makes one chain
of the six pieces, or buying a new chain?

If you are linear thinking, you will buy a new one and spend $22. In that
case, you only think on the form. The content didn't matter so much.
However, if you were thinking non-linear, that is on the contents, you
will find that it is cheaper to compose a new chain from the old pieces
for only $20.
I hope you have found why and how.

I have used this problem several times when teaching creative thinking for
managers. I used the chain as a metaphor for organisations and
reorganisations. I will discuss this in paragraph 3. But first the
questions of AT (see LO22881):

2. Questions of At
>>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?

If we remember what At has written about the graphical representation of a
linear line in that very contribution, we may conclude safely that the
intervals on the axes are crucial for linearity. If the spacing of the
intervals on one axis is non-equal, the line (represented by the formula y
= ax + b) will NOT have a linear FORM. These intervals are the CONTENT of
the axis. I cannot find a better example to show the difference between
content and form. It is experiment (or excercise) 1 of At in his posting,
which gives the answer to "the content from which the form has emerged".
Also experiment 2 illustrates the influence of the content to the form.
In experiment 3 (both axes have symmetrical contents, e.g. intervals) the
FORM of the linear line has forced the CONTENT of the axes to be
symmetrical ("the form which has collapsed into content").

If you have done also the additional experiments (where the form of the
axes is non-linear), you have seen that the form of the axes has also its
influence on the form of the line.
And finally, if the intervals are non linear and/or non-symmetrical of the
non-linear axes, it obviously has its influence to the form of the line
too. It is in extreme circumstances possible to manipulate these intervals
in such a way that the non-linear line will be transformed into a linear

We may think also in another way. Did you noticed that for defining a
one-dimensional thing (the linear line) we needed TWO dimensions (the two
axes)? Did you noticed that we could call the line as a system SY, and the
framework of the axes as the surroundings SU? Did you noticed that this is
NOT enough? Because that the CONTENT of SU is crucial as well? If this
content of SU is equally spaced or symmetrical, the form of SY is linear
and that thus the content has a direct influence on the form.

Now, we must make another jump. If we try to imagine and comprehend this
two-dimensional world of SY and SU, we need at least another extra
dimension. Your eye which observes your piece of paper with the graphs of
the experiments, is ABOVE the paper, hence three dimensions. If the eye
was inside the paper, inside the two-dimensional world, we were not able
to see SY and SU together. So we were not able to sense the complex
interaction of the form of SY and form/content of SU. However, we were
able to sense some of the content and form of a part. Imagine an ant (as
for simplicity a two-dimensional animal) walking along the straight
x-axis. Since the ant lives in a two-dimensional world, it could leave
this path. So the ant is able to find out that the form of the axis is
linear. Suppose that the intervals on this axis are indicated by some
spots of honey. If the ant walks along the axis counting its footsteps
(difficult with six feet :-)), it is able to find out that the honey
spots are at regular intervals. The content of the axis is in this case
However if the honey spots are at irregular spacings, the ant will count
irregular numbers of steps. The ant is able to sense the content of the
The ant could leave the path of the axis and wanders in the
two-dimensional plane. With some luck, the ant will encounter another
line. If the line has no content, it is the constructed one; if the line
has honey spots, it will be the other axis. But the ant could hardly
decipher the relationships between axes and line, and form and content.

3. linearity and contents of (learning) organisations
Did you solved the chain problem?
Think of the six pieces of chain as departments of a company. If this
company will act as a total unity, the departments should be linked to
each other. Some companies, particularly those that behave as a production
line, may be seen as a linear organisation. Department 1 produces a
halfproduct for department 2 that produces some product for department 3,
etc. If the company is balanced, all the in- and outgoing products of all
successive departments are in balance. The products flow regularly through
all departments. As we know, it is a risky organisation, since each
department depends on the former department in the chain. The strength of
the total depends on the strength of the weakest. Changing the order of
depaprtments will create chaos in the company. It sometimes is good
thinking excercise to change the order during your daydreaming. Especially
to think of the consequences of such change. You may do this for every
proces in a chain. For instance the food chain in the animate world, the
workings of a combustion motor, etc. If there is something around you with
the adagium: first this, than that, you deal with linearity.
If we like to increase the production of the production line in our linear
company, what shall we do? If we like to double the production, should we
double the number of people of each department, double the incoming flow,
so that the outgoing flow is doubled too? If we think of these kinds of
questions, we are forced to think of the contents of the departments. You
should think of illness and holydays of employees, you should think of
environmental issues, you should think of capacity of machinery, etc. We
enter the world of Deming.

On the other hand, if we should reorganise and reduce departments, or
think of more efficiency of this linear company, what shall we do now?
Shall we use the 'kaasschaaf-methode' (the cheeseslicer-method) as we call
it in the Netherlands: take a tiny bit of every part. The linear method of
Back to the chain problem. We had six pieces of chain, each consisting of
five links. If we try to connect these independent departments (each
consisting of five persons), we could use one person of each department to
use as link to the next department; a demonstration of the
cheeseslicer-method. We think that each department could easily make one
person free for communication purposes with the next department. We use
this method even if the departments are unequal in size; you just needed
one person for the communication.
However in the special case of the chain problem, there is another
possibility: we could enforce five departments (each with one person) with
the people of the sixth department. We sacrifice one department and use
the five people to use as the links between the other five departments.

Ofcourse, the chain puzzle is just a metaphor. But in this discussion
about linearity and form and content, it shows very nicely that the
content played a crucial role. If the number of links of the pieces were
not five, the puzzle lost its character.

4. epilogue
I like to come back to something which I have mentioned in the earlier
paragraphs. It is my personal puzzle.
I have explaind that two dimensions (2D) are needed to define a
one-dimensional object (the line). I have explained that the ant, living
in a 2D-world, could hardly understand the relationships between form and
content of his world. No emergence will happen in his mind. His mind
should escape from the two dimensional world into the 3-D world to get the

My puzzle is: is an emergence a change from a nD-world towards a
(n+1)D-world? Is the emergence an ordering in the next dimension?
Maybe I am too misty. If I compare the emergences in the mind with
emergences of the physical world around us, I think of a laminar (2D) flow
that emerged into a vortex (3D); it will hopefully be clear for you what I
have in mind.
If we will fully understand the 7D-world of the seven essentialities,
should we create the clearing overview with our mind in the 8D-world?

I am afraid that my mind jumped too fast from the line (2D) into a
multidimensional space. I hope I have not lost it.

dr. Leo D. Minnigh
Library Technical University Delft
PO BOX 98, 2600 MG Delft, The Netherlands
Tel.: 31 15 2782226
Let your thoughts meander towards a sea of ideas.


Leo Minnigh <>

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