Linear Thinking LO22861

Leo Minnigh (
Wed, 13 Oct 1999 12:04:24 +0200 (MET DST)

Replying to LO22849 --

Replying to the general discussion of Linear thinking


Dear LO'ers,

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

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 be
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.

But there is even another prerequisite which we should realise: the
regular spacing of the units along the axes. It is not necessary that
these units are equal in length on different axes, but they should at
least be regular at each individual axis! It sounds trivial, but we will
see that this is not. You may construct beautiful bended and curved lines
with linear and rectangular axes but with irregular unit spacings. I
recommend the book: "On growth and form" by d'Arcy Thompson. This very
creative author has played with all sorts of frame works to decipher and
compare the directions AND speed of growth of an enormous variety of
animal and plant species.

And please hold the grips of your armchair (or fasten your seat belts, as
At would say), these units along an axis should either add-up, or
subtract, but not both! So we can give them names like 1, 2, 3, 4,
5......., or -1, -2, -3, -4, -5,.....

We probably all know of those rulers with the zero symmetrical in the
middle, counting to both directions in positive numbers. That gives some
real trouble, since nearly all numbers/names (apart from the
zero-position) refer to TWO positions on the ruler.

Is it not strange that we define the linear line y = ax + b with the use
of TWO other linear lines, e.g. the axes? The dog biting in two tails.
And is it not strange that we tend to forget that we live in a time-space
world with four dimensions, with four axes. Thus the line we are looking
for in our real-life, should be defined as:
y = ax + bz + ct + d, where x, y, z and t are the axes ('t' as the axis
of time) and where b and c are zero.
In the case of b = 0 AND c = 0, we have defined our line in two
dimensions, a flat plane without time.

However, the moment we discuss 'linear thinking', a PROCESS is introduced.
And with this process (the thinking), we invited the time-axis in our
system. That means that there are at least 3 axes involved: x, y and t.
All these axes themselves linear with regular spacing of the units, either
increasing, or decreasing. And time is a strange thing; we have discussed
this in the past several times. I recon that this intriguing axis will
appear also in our future discussions. Is the axis of time linear, and is
the spacing of time regular???

Where am I going with all these explanations of the trivial? Let me write
the following sentence:


It is a reasonable linear sentence, product of linear thinking. But is
this so? You and I are thinking forward and backward to comprehend this
sentence; you and I are checking the content by counting the characters.
Maybe you are lazy, you believe immediately that the content of this
sentence is true. But even then your thoughts were moving backward for a
moment. Backward and forward along a linear line. Is this linear
thinking?? In my mind it is not. Linear thinking is unidirectional. It is
a vector, not a line; it is a line with an arrow. We should write a book
with the titel:
"On form and content".

So now I hope that the picture becomes clear: we may think along a line -
which is one-dimensional - but we could go forth and backward. And we
could go with different speeds forth and backward.
The word 'linear' is a form-description. But the moment we introduce the
question 'WHY' , the content comes in play. This whole reasoning might be
linear, however the content of this reasoning is far from linear. And as a
matter of fact, we could see this non-linearity by digging in the archives
of this list. After several meanders, our river of thoughts refound an
earlier track. Dear readers please consult the archive of February 1999
and look to the contributions on the thread 'non-linear thinking'. I
copied a part of LO20686, where I wrote:

"Thoughts follow a track. Maybe the future of this track is uncertain. It
could be linear, it could be zig-zagging, it could be curved, it could
form a loop. But thoughts move through a multi-dimensional space. We may
take a snap shot of the thought and even the direction of that very
thought on that very moment could be frozen in the snap shot. However the
dynamics behind this movement are not possible to fix.
Snap shotting directions is also possible in the 7-D creativity space of
the seven essentialities. All of them have a kind of direction included
(the most obvious is the first one: becoming-being/liveness). And to make
things even more complicated, the seven essentialities all together
influence each other in a dynamic way. But the space is not suitable to
decipher the dynamics. 'Why' is a question which lives in an other world."

I like to conclude:
A line is a human invention.

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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