Linear thinking LO22772

AM de Lange (
Fri, 1 Oct 1999 12:22:22 +0200

Dear Organlearners,

Have you also encountered the term "linear thinking"? Do you know what it
is? Do you know how to deal with it? Here is some thoughts on this very
contencious issue.

Some people use linear thinking as a synonym for logical thinking so as to
grind an ax against logical thinking. Logical thinking and linear thinking
are not synonymous to each other. In fact, trying to compare them is like
trying to compare a chicken with a potato. Although a chicken and a potato
are living organisms and thus have many things in common (DNA, enzymes,
cells, reproduction, ...., even KFC ;-), it is foolish to think of them as
the same thing.

So what is logical thinking? There are many fine books on this topic. Find
one which fits to your own world of experiences. A book which I
recommended to many people from diverse backgrounds and which was
beneficial to the majority of them, is "Symbolic Logic" by I M Copi.

What is linear thinking? For every thousand pages of information on
logical thinking, we will find less than one page of info on linear
thinking. What is even worse, in most of this literature on linear
thinking, the term "linear thinking" seems to have been used as the
description for the deliberate attempt to "deconstruct" traditional
thinking or grind an ax against something else.

The terms " line", "linear" and "linearity" play a major role in
mathematical thinking. Note that I am not saying that mathematical
thinking is linear. What I am saying is that mathematicians have studied
the "line" and "linearity" to a degree which dazzles the mind. Their best
known descripition of a straight line is the formula
y = mx + c

The majority of you fellow learners have been clobbered in the head by
this formula in shool. Perhaps you still have some nightmares on it. My
advice to you is to go to the mathematics section of a library and borrow
all the books on Analytical (descriotive) Geometry which appeal to you
while paging through them, despite your disposition. Then begin reading
them at home with a glass and bottle of wine close by. If you do not
understand something, just go on reading until you get to something which
makes sense to what you already know. In other words, do not try to import
concepts from the book, but try to recognise your own thoughts in what the
book has to say. At some stage you will get intoxicated, if not by the
little understanding of one those books, then by the wine.

I often get the feeling that a person's vehemenence against
linear thinking is subconsciously directed against the
uncreative encounters which that person had in trying to
learn mathematics at school. Here is a very carefully
contemplated formulation:
Trying to make mathematics your own by Rote Learning
cannot result in anything else than linear thinking!
I will use it to show to you what is linear thinking.

In what now follows it may appear that I am also clobbering your mind with
mathematical formulae. Since this is not my intention, you must try to
avoid this perception. My intention is to use these formulae as colours to
paint a picture. Each formula is a different colour. Should you then worry
that you do not understand a specific formula, it will be like that you do
not understand, say, the colour red. If you begin worry why red is red,
you will keep on worrying till doomsday.

I will now change the formula
y = mx + c
z = mx + c
just to change the monotony of your past experiences. This
change may appear to be insignificant to you, but it is not.
If this change reamins insignificant to you after you have
read this entire contribution, then clobber me over the head
because I have failed to teach you.

As I have said, I have read the past few years people bringing judgments
out against "linear thinking" as if time has already emerged into

I have never read any person ever objecting against "planar
thinking". Yet, the same "linearity" typical of the straight line
z = mx + c
also operates in the flat surface
z = mx + ny + c
We simply add another dimension (the y) to the original
dimensions x and z. A flat surface is nothing else than an
infinite collection of straight lines. In other words, we can draw
infinite many straight lines on a flat plane such as a piece of
paper. So, why the holy war against "linear thinking", but not
"planar thinking"?

A straight line becomes a curved line when "powers of x"
are included in the formula y = mx + c, for example
y = ax^2 + bx + c
The latter is the formula of a parabola (parabolic curve). The
x^2 is the "second power of x". It means "x squared", i.e.
x multiplied by itself. Any change in x is far different from any
change in x^2. Thus adding ax^2 to bx + c changes the way
in how it changes.

A parabolic surface would result if we make same change
(adding power terms) in the pattern of the formula for the flat
plane. The only difference to that of the parabolic curve is the
bringing in of another dimension indicated by an y.
z = ax^2 + bx + cy^2 + dy + e
Should we drop the cy^2 with as result
z = ax^2 + bx + dy + e
we will get the "long dome" typical of some factories. Looking
at the "long dome" from in front, its silouet appears to be
parabolic, but looking at it form the side, its silouet is merely
a slanting straight line.

Think about the parabolic curve
z = ax^2 + bx + c
once again. (You may easily extend the next thoughts to
the parabolic surface z = ax^2 + bx + cy^2 + dy + e.) If
"linear thinking" is so bad for us, why is "parabolic thinking"
not bad for us? It is certainly a case for non-linear thinking!
Does "parabolic thinking" exclude "linear thinking"? Yes, but
only if you cannot draw infinite many lines tangeant to the
parabola. Difficult to visualise? Take a pen and glue it
perpendicular to the side of a ruler so that its tip touches the
paper when the ruler is placed on the paper. Now draw a
parabola by handling the ruler. Euraka.

Does "parabolic thinking" excludes "dialectical parabolic
thinking"? The latter thinking employs two kinds of parabolic
thinking in opposition to each other. Take a piece of paper
and plot the graph of the formula
y = ax^3 + bx^2 + cx + d
to see how this "dialectical parabolic thinking" looks like!
It is like liking two opposite parabolic cirves to form a S (seen
on its side). Population dynamics (like any other digestor
mechanism) leads to such S graphs.

I can go on and on. My sole purpose would be to create
chaos (confusion) in your mind as to what "linear thinking"
amounts to -- to let you realise that people apply the concept
"linear thinking" without ever giving a though what the inner
detail of this concept is.

I want you to realise that most people use the "linear thinking"
as the very mathematical
itself (the symbol for an unknown) rather than the mathematical
y = mx + c
Once you realise this, you are in a much better position to make
sense from what they are writing about. To make sense you will
then have to use their "linear thinking" as the x (not the pattern
y = mx + c) and find in the rest of their writings a pattern depending
on x ("linear thinking")

In most of the cases you will find that there is indeed only one
other topic y which they also do not define internally, but which
they describe in a monotonous fashion in terms of "linear thinking"
(the x.) This y is usually some non-traditional topic which they
are obsessed with! What is even worse, the monotony of their
arguments indicates that they are actually setting up the
obsession = slant times "linear thinking" plus offset
where y = obsession, slant = m, "linear thinking" = x and
offset = c. In other words, they use their obsession y and
"linear thinking" x to set up a HYPER case of linear thinking.

This HYPER LINEAR THINKING concerning linear thinking is begining to
frustrate me beyond reason. That is why I have decided to write on "linear
thinking" before I finally become crazy through frustration.

I hope I am closer to point out to you what "linear thinking" is. It is a
monotonous obsession with an idea as if that idea is very novel. In other
words, poeple can think linearly whenever they get obsessed with something
(obviously important) as a novelty. That novelty can be anything, even
things such as "entropy", "creativity" or "learning". Yes, even such a
thing as Learning Organisation, Systems Thinking, Mental Models or any one
of the remaining three disciplines. etc. My greatest personal struggle
against "linear thinking" is not to get obsessed with "deep creativity" as
something novel.

In my series on the seven essentialities of creativity, I have come as far
as the fifth one. See Essentiality - "quantity-limit" (spareness) LO20541
<> Often I think I must stop
here so that the five matches the five of Senge's Fifth discipline. It
will then entail that you will have to articulate the sixth and seventh
yourself. Thinking about the "measurement problem" (advance reduction of
the wave packet by explication) of Quantum Mechanics, and the fact that
You have to grow in free energy by your won authetic learning, I am almost
sure that I should not write on one of the remaining two, otherness and

Otherness involves, among others, doing things in not only one way. Up to
now I have followed one way to give you the taste of linear thinking.
Linear thinking is a monotonous obsession. Its greatest danger is
preventing 'creative collapses" such as paradigm shifts. Allow me to give
you a different taste of linear thinking.

In mathematics we a very powerful discipline (once) called "infinitesimal
calculus". It hinges on the symbolic expression indicated by d/dx. The
d/dx may be called the "operator of change". (It is a name which I give to
it, but you will not find that name in any book.) There are now three
major versions for the d/dx -- the classical version of Gauss, the limit
version of Cauchy and the non-standard version of Robinson. As is usual,
even mathematicians argue among themselves which version is best (mononous
obsession ;-). In other words, they see the three trees, but not the
forest. What is the forest of d/dx?

To see the forest, we must open our eyes to the bigger landscape in which
the forest is situated. For example, consider quantum mechanics as an
example in which "infinitesimal calculus" finds application. The wave
formulation of quantum mechanics (Schroedinger) cannot be done without the
d/dx, the "operator of change"! It means that the creative emergence from
Newtonian Mechanics to Quantum Mechanics will be very difficult without
the d/dx of infintesimal calculus. (But not impossible as Heisenberg
showed us -- otherness at work.) In other words, when we fail to see the
forest of d/dx, even Quantum Mechanics may fade away.

Can the d/dx tells us anything new about linear thinking? Yes.
Assume that any topic y is a function of (or related to) topic x.
Our thinking about the function between y and x is linear
i.e. has the form y = mx + c when
(d/dx)y = m
In ordinary words, our thinking is linear when "changes in the
topic y as a result of any change in the topic x" [the phrase in
the quotation marks is symbolised by (d/dx)y at the left side]
becomes a "constant slant" [the phrase in the quotation
marks is symbolised by m at the right side].

Perhaps this is too complex to conceptualise. It took me many
minutes to construct the last sentence. It will take you many
more minutes to analyse this sentence. Let us not get
obsessed. Let us become crazy and go even one step further.
Apply the "operator of change" d/dx once again to the application
(d/dx)y -- leat us make a "double loop". When our thinking is linear,
the result will be
(d/dx)(d/dx)y = 0
It is very easy to desymbolise this result. It is simply:
Our thinking is linear when there is no second change in
the first changes of our thinking. In other words, our thinking
is linear when we fix the changes in our thinking.
It is very much like saying "We are ignorant to changes upon

Let us go back to my carefully formulated thesis in the
Trying to make mathematics your own by Rote Learning
cannot result in anything else than linear thinking!
What is so peculiar about Rote Learning (RL)? It produce
changes in our thinking, but not changes on changes in
our thinking! Add to this in RL the "monotous obsession"
with that information "processed by only the expert" and
hopefully you will understand just how linear RL is.

I am pretty sure that all you fellow learners have had many experiences
how creative solutions to problems seemed to be actively discouraged. If
you were not involved yourself in creating the solution, you were at least
a spectator to such spectacles. I want you now to think with concentration
trying to break through this perception of discouragement, failure and
destruction. Think about each case as far as your memory allows you. Did
you get the intuitive feeling that nothing happened because some person
vital to the outcome had a "mononous obsession"? Was it because of "no
change in change of thinking"? If your answer is yes to at least one of
these questions, then the root problem was LINEAR THINKING.

Obviously, the next step is to solve this root problem of LINEAR THINKING.
The guiding idea is to get rid of the "monotonous obsession" and bring in
"changes in changes of thinking". The rest is up to you.

Is "deep creativity" just another kind of LINEAR THINKING?

In "deep creativity" one makes a distinction between the content
(dynamics, phsyiology, semantics) of creativity and the form (mechanics,
morphology, syntaxis) of creativity. There is no chance for a "monotonous
obsession" with either content or form. The content of "deep creativity"
concerns "entropy production", something which we also could gave called
"positive change of entropy". The "entropy production" in "deep
creativity" meander between low values close to equilbrium where growth
(feeding/consumption) happens and high values at the edge of chaos where
bifurcations (emergence/immergence) happens. In other words, in the
"content of deep creativity" there is change (low/high) in "changes of
entropy" (i.e entropy production). Hence this meandering leaves no chance
for fixed changes to persist.

Well, perhaps "deep creativity" can be subjected to LINEAR THINKING. In
that case, just remember that learning is a first order emergent of
creativity. If creativity+learning still becomes a monotonous obsession
with no changes in changes of thinking, throw in religion and see the
bloody reactions taking place. Why? Because religion is concerned with
believing -- the second emergent of creativity and hence the first
emergent of learning. When thinking still becomes linear, try to
experience the emergence to the highest level of them all, namely
inconditional love, the one-to-many-mapping crown of life. By that time
you will surely become healed from linear thinking.

Is the change of the formula y = mx + c to z = mx + c still
insignificant to you?

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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