Parables and Parabolic Thinking. LO26070

From: AM de Lange (
Date: 02/07/01

Dear Organlearners,

Greetings to you all.

How is possible that some text can have two or even more different
meanings? How can we uncover more than one meaning in such text? Sometimes
it easy to spot such text as an allegory, fable, metaphor or parable. Some
persons have used them exquisitely like Esopus used fables or Jesus used

The word "parable" comes from the Greek preffix "para-" which means
besides or beyond and the word "ballo" which means to throw. It seems as
if parable meant in the Greek to throw two things beside each other or to
throw one thing beyond another thing. But the word "parabolle" itself
meant to compare two things. By lending this word "parabolle" the Romans
created their own Latin word "parabolus" to refer to a comparison.

Today the word parable in English means a story with one meaning which can
be used to clear up a comparable meaning in another telling. The word
parable was created by Wycliff in the 14th century in his famous
translation of the Bible. It is almost as if the parable has two meanings,
its surface meaning which first will be formed easily and its deeper
meaning which may be formed secondly by tenacious contemplation. A
parabola itself refers to a particular shape. When a string is pulled
tightly between the two hands, it forms a straight line. But when the two
hands are brought closer to each other, the string will begin to sag, more
in the middle than closer to the sides. The shape of this sagging string
is almost like the shape of a parabola. When throwing a rock through the
sky, its path traces an almost perfect parabola. The name for this shape
was first given by nobody else than the famous Archimedes 212BC. He
discovered this shape and two other possible shapes when cutting a cone
into two parts.

Sometimes it is difficult to spot more than one possible meaning in worded
information, text and speach . For example, parents may talk with their
teenagers on a quality like true, linking certain behaviours to false and
others to true. However, the teenagers interpret such telling as judgement
on their own behaviours. As Leo Minnigh would say, the content and form of
the message get confused and thus the message gets confused with the

Here is an even more difficult answer. I may present a logical reasoning
on some positive and constructive activity. The truth inverter "not" will
occur often in the arguments as I proceed from the negative to the
positive. The not makes a true statement false, but also makes a false
statement true. However, someone else reading that reasoning may interpret
that with almost every "not" I have been criticizing something.

Here is a most difficult example. I may write on the history or evolution
of the quality true. It may range from "false" through various fuzzy
values like "uncertain", "perhaps" and "almost" up to the the value
"true". I will mention each value and then tell of my observations on it.
However, in truth judgement one of these values is selected as the one
only posible outcome of a particular judgement. Thus another person
reading the contribution and observing that these values are used one by
one, may think I have been extremely judgemental using them whereas I was
actually avoiding judgements by covering them all one by one.

It is like the writing of the prophets in the Bible. Seen from the point
in time in which the prophet lived, the prophecy is an futuristic attempt
to trace the evolution of character of some society -- for example, from
bad to worse and from good to better. Can God-Love do any better than
describe the various pathways from which one has to choose? If a prophecy
came out false in his time, the prophet had to be removed permanently from
his own society because the prophet spoke without God's inspiration. But
seen from a point long after the time on which the prophecy had a bearing,
the prophecy seems to be no more than a harsh judgement, devoid of any
love for that society it had bearing on. How could God Love inspired such
a harsh judgement?

What strange things are going on in any writing (or speaking) when it
allows two different interpretations? I think it is the %Mental Model%
which we now have of natural languages. In this %Mental Model% we think of
a language as primarily a tight rope connecting the minds of any two
persons using that language. The signs %...% are used to identify this
particular Mental Model. The main notion in this %Mental Model% is that a
language had been created primarily to accomodate linear thinking. It is a
Mental Model because since time immemorial language had been used in a
great variety of expressions (curved meanings) to also overcome the
bondage and slavery stressed by blunt commands (linear meanings). Language
may be used by the divine in us to become free and not merely be used by
the demonial in us to enslave others.

Imagine two persons whose bodies are connected with a rope. Imagine their
movements under the condition that the rope has to be pulled tight and
thus straight at all times. Their movements will be severely restricted.
But imagine them moving around with a sagging rope, forming a parabola,
between them. They will have much more freedom of movement. In modern
times it is like comparing wired telephones (connected by stretches of
straight wires which use less copper) to cellular phones. Cellular phones
allow much more freedom of movement than wired telephones. Similarly the
%Mental Model% reduces our freedom for expressing our minds. Every day we
have to walk the tight rope of minding what we are saying rather than
saying what we are minding.

Why? I think that it has to with the $Mental Model$ which I have described
in the contribution "The endless becoming of good, right, true, lovely,
etc." These qualities have to do with the character of individual persons
and the character of organisations and societies. We may call them the
qualities of character. The $Mental Model$ says that as soon as even
merely one quality of character is uttered in speach or writing, judgement
is automatically implied by the mere mentioning of such a quality. The
more the shades of such a quality are mentioned, the clearer the
judgement. This $Mental Model$ has many offspring of which %Mental Model%
is one -- language must be used under tight rope conditions so as to allow
one only interpretation.

The destructive effect of the %Mental Model% is far worse in the so called
"exact" sciences like mathematics, chemistry and physics which seems to
rely more on symbolic expressions than in the humanities which rely merely
on natural langauges. I know that it seems to be an incredible claim, but
I have observed too many students suffering a lack of understanding
because of this %Mental Model%.

Let me give an example, remembering from past contributions that NaOH is
the formula for sodium hydroxide (caustic soda), HCl for hydrochloric acid
(spirits of salt), NaCl for sodium chloride (table salt) and H2O for
water. Consider the symbolic formula:
. NaOH + HCl => NaCl + H2O
The first, easily formed, surface meaning in words is
. sodium hydroxide added to hydrochlic acid will begin to react. After the
. reaction the products will be a mixture of sodium chloride and water.
In this first meaning the focus is on the beings of the chemical reaction.
The second, arduously formed , deeper meaning in words is
. one disappearing unit of sodium hydroxide together with one disappearing
. unit of hydrochlic acid become one emerging unit of sodium chloride and
. one emerging unit of water.
In this second meaning the focus is on the chemical reactions as the
becoming of beings.

Students who allow one meaning, according to the %Mental Model%, end up
with the first meaning which is ontological (being-like). The %Mental
Model% constrains them not to seek for the second meaning which is
ontogenic (becoming-like). They subsequently have to make many kinds of
calculations based on the chemical reaction. Most of these kinds of
calculations (like stoichiometric and themodynamic) have to with the
ontogenic nature of the reaction. The mistakes which these students
persistently make in these calculations drives any teacher into dispair,
should the teacher not self know from what and how these mistakes
originate. The %Mental Model% prevents the students and the teacher to
learn from these mistakes. Eventually they all begin to memorise what is
formulated in the books as the only "true knowledge". Since the books tell
little, if anything, on the two possible meanings and what each involves,
they remain oblivious to the %Mental Model% and its ramifications.

The point which I tried to make is that not only are natural languages
open to more than one interpretation, but also technical or symbolic
languages. In other words, once a person has created some information with
his/her formal level of knowledge, this information itself is devoid of
meaning and knowledge when it leaves that person. When a second person
encounters that information, this person has to create within him/herself
meaning and further knowledge from that information. It would benefit that
person immensely to bear in mind that all information allows two or more
possible interpretations rather than one unique interpretation.

Two possible interpretations can always be found by parabolic thinking
while linear thinking allow for only one interpretation. How? I will now
have to rely very much on your own imagination to explain it. See my
introductory contribution on "Learning and Imagination". Try first to
imagine the demonstrations which I will now describe. Once your are
satisfied that your imagination is sound, then do the demonstration
yourself. The other way of repeating my demonstration as a recipe to aid
your own imagination might also work, but will have far less value for
your future learning. You need not do all the imagination and only then do
all the demonstration because it requires immense concentration ove a long
period of time. Just finish one or two steps of imagination and then
follow it up with the corresponding steps of demonstration.

Please bear in mind that what now follows is imagination which ramifies
into speculations. I am not implying that your mind actually works in all
respects like what we are about to imagine. To find out how your mind
actually works, you will have to try and falsify these speculations one by
one. That speculations which remain will be a closer indication of how
your mind works. Since you are the only person which has access to your
own mind, I would not mind too much when other persons try to falsify what
goes on in your mind. The only way how they can know what goes on in your
mind, is by information which you gave to them. And this information
always have more than one possible interpretation.

Take exterme care that I do not cause some mind programming in you. My
suggestions to aid your imagination will be given as commands, making them
as linear (one meaning only) as possible. Should you not like what will be
happening on this imagination trip, abort it immediately. I will use "you"
and "your" rather than the impersonal "person" and "person's". I will not
also describe the demonstration itself. I think that your are intelligent
enough to create a physical demonstration from my suggestions to aid your

Think of (find) four pieces of electrical wire, two pieces with black
insulation and two pieces with red insulation. Any two colours will do,
but I think that the red and black colours will connect much better to
your tacit knowledge. Imagine all four pieces are straight (linear). Bind
mentally the two red wires into a rectangular cross. Call the joint the
origin O. Direct the one red wire in a horisontal direction. Call it the
X'OX axis. The other red wire will become directed vertically. Call it the
Y'OY axis.

Make mentally divisions on the X'OX axis, as many as needed. Think of a
property of character like "good". Now think of all its shades like
"badest of badest", "badder than badest", "badest", "badder", "bad",
"almost bad", "somewhat bad", "slightly bad", "between bad and good",
"slightly good", "somewhat good", "almost good", "good", "better", "best",
"better than best" and "best of best". I have described here seventeen
shades of good, but you can imagine even more shades. Attach them in this
order from left to right to consequtive divisions on the X'OX axis,
letting the "between bad and good" coincide with the origin O. Between
"between bad and good" and "slightly good" to its right you can imagine
many even more subtle shades like "somewhat slightly good" and "almost
slightly good". You can imagine this more subtle shades also between any
two other consequitive divisions.

Also make mentally divisions on the Y'OY axis, as many as needed. Think of
any property of knowledge like "hirarchy". Now think of all its
stratifications like sensation, experience, intuition, tacit, implicit,
explicit, formal, symbolic, etc. Perhaps you have never done this before,
but try to imagine an ordered hirarchy in the levels of knowledge. In some
cultures a five or six level hirarchy is not imagination, but common
knowledge. Again attached these levels of hirarchy to the divisions on the
Y'OY axis in an ordered fashion. Let the lowest level ("sensation" in the
example above) coincide with the origin O and the others follow in the
upwards (positive) direction. I will not suggest to you to imagine
negative levels of knowledge in the downwards direction from the origin O.
But it is easy to imagine them by invoking negative judgements on
knowledge. I would rather suggest to you to imagine virtual levels of
knowledge, i.e. knowledge which still has to be gained in future.

With these X'OX and Y'OY axes you may now imagine your reference grid. It
means that you will use this grid as the connection of a property of
knowledge like hirarchy (whatever its level) with a property of character
like good (whatever its shade). The divisions of the X'OX axis together
with the divisions of the Y'OY axis form together a gigantic matrix. In
each matric unit (element) one specific level of hirarchy (as a property
of knowledge) will be connected to one specific shade of good (as a
property of character). Thus no two matrix units can ever have the same
combination of level and shade. We will denote both axes together and the
grid which they form by XOY.

Now imagine (take) the two black wires. They will both represent the same
lump of information like a story, a report or even a mathematical theorem
rich with symbolic formulae. Whereas the colour red above reminds us to be
on the alert, the colour black here tells us that the lump of information
has no meaning or knowledge in itself. Keep the one black wire straight
(linear). Bend in your mind the other wire into the parabolic shape (like
a sagging rope or a the path traced by a rock thrown into the air). Call
the turning part of the parabola (beyond which the second leg of the
parabola also points into the same direction as the first leg) its
fulcrum. The straight black wire will represent the lump of information
when your mind is constrained by the %Mental Model%. The parabolic black
wire will represent the lump of information when your mind has been freed
from the %Mental Model%. Remember that the parabola is one of many, many
possible shapes.

Take the straight black wire and place it in your imagination on the red
XOY grid in a slanted manner. It will cut (cross) the X'OX axis ("good",
property of character) somewhere. It will also cut the Y'OY axis
(hirarchy, property of knowledge) somewhere. If the slope of the wire goes
upwards in the positive direction of "good" (e.i., good => better => best
....), we may say that you have a positive atitude to good. Should the
slope of the wire goes upwards in the negative direction of "good" (e.i.,
bad => badder => badest ....), we may say that you have a negative atitude
to good. Swivel the wire from the one slope to the other slope to make
sure that either a positive slope or a negative slope is possible.

Should you have placed the black wire parallel to the X'OX axis, we will
say that its slope is zero. The black wire will then cut only the Y'OY
axis. We may say that you have a zero attitude towards character. But
should you have placed the black wire parallel to the Y'OY axis, we will
say that its slope is infinite. The black wire will then cut only the X'OX
axis. We may say that you have an infinite attitude towards character.
However, let us rather now assume you have a finite, non-zero attitude
towards character. Now make once again very sure in your imagination that
in this case the straight black wire cut the X'OX at one and only one
point. Also make sure it cuts the Y'OY axis only once.

Now for the crucial phase of this imagination. Put the wire so that its
slope is positive, i.e., it points upwards in the positive direction of
"good". Imagine exactly where it cuts the X'OX axis as well as the Y'OY
axis. Make a mental mark of each of these two points. Displace the wire
slowly downwards for some distance so that its slope stays the same. In
other words, move it parallel to itself a little bit downwards. Again
observe where it cuts the X'OX axis and also cuts the Y'OY axis. The new
Y'OY cutoff will be lower than the older Y'OY cutoff. The new X'OX cutoff
will be more positive than the older X'OX cutoff. As we delve deeper down
in the hirarchy of knowledge, we move in the positive direction of shades
of good of character. That is why I reckon it is silly to judge any
sensation or even experiences as good or bad. But remember that a positive
attitude to character is required. Should the wire have been slanted into
a negative slope and thus a negative attitude to character, it begs for
judgements on sensations or even experiences

Perhaps you have been fortunate to place the black wire exactly so that
when it goes down along the Y'OY axis (and in the positive direction of
the X'OX axis), it reaches the origin O simultaneuously along both axis.
Then and only then both the Y'OY and X'OX cut-offs will coincide as one
cut-off. This unique case means that zero knowledge affords a zero shade
in a property of character like good.
When you study with the straight wire a lump of information, you first
have to observe how this wire is orientated on the XOY grid. Think of its
slope -- is it a positive attitude or negative attitude? Then move your
mental eye and not the wire to where the wire cuts the X'OX axis. At that
point the value on the Y'OY axis is exactly zero. This means that you do
not expect any knowledge to be created from the lump of information. Where
is the X'OX cut-off -- is it to the "bad" side or to the "good" side of
character? Now move your mental eye and not the wire to where the wire
cuts the Y'OY axis. At that point the value on the X'OX axis is exactly
zero. This means that you do not expect to make a judgement on the
character to be created from the lump information. Now if the Y'OY cutoff
is above the origin O, it means you have the knowledge to create the one
only interpretation from the lump of information. But should the Y'OY
cutoff be below the origin O, it means you still have to acquire the
knowledge needed to create the one only interpretation from the lump of

I think that you want to have both a positive attitide to character and
create a positive shade of character out of this lump of information. Now
make very sure with your imagination, moving the straight line if
necessary, that the it will cut the Y'OY axis below the origin O. This
means, should you have imagined the part of Y'OY below O to be knowledge
still to be gained, you will have to learn more so as to gain in
knowledge. Is it not fantastic what a positive attitude and positive
shades of character can do to learning. Alas, should you have imagined the
part of Y'OY below O to be negatively judged knowledge, will you want to
learn so that your knowledge can increase negatively?

In this last paragraph I have mentioned two possible interpretations for
the part below the origin of the Y'OY axis (a property of knowledge like
hirarchy). It is now time to go deeper into two possible interpretations
by removing the straight black wire from the XOY grid and putting the
black parabolic wire on the grid. Imagine that the parabola has its own
axis going perpendicular through the centre of its fulcrum. Thus its two
legs will spread symmetrical away from its axis. Make sure with your
imnagination that you have placed the parabola in a vertical position. The
the axis of the parabola will be parallel to the Y'OY axis.

We cannot imagine here any "attitude to character" because the slope of
the parabola changes continuously as we move from the one leg through the
fulcrum to the other leg. When the legs of the parabola points upwards,
the slop will change from negative to positive as you move into the
direction of positive shades of character. Thus we may rather say the the
parabola with upward legs represents a positive "attitude-becoming of
character". Flop in your mind the parabola over so that its legs points
downwards. Make sure with your imagination as before that we now can
sensibly speak of a negative "attitude-becoming of character".

We will now imagine the same displacement excercise as for the straight
line. Use the parabola with its legs pointing upwards. Begin by placing
the parabola high up in the grid so that its fulcrum is above the X'OX
axis. Observe mentally that it cuts only the Y'OY axis. Make a mark where
the cutoff is. Make mentally sure that with this position the parabola
cannot cut the X'OX axis. Now displace the parabola a little bit downwards
"parallel to itself", i.e keeping it in the upright position. Note the new
cutoff on the Y'OY axis. You are delving deeper down in the hirarchy of
knowledge. Also the centre of the fulcrum is now closer to the X'OX axis.
Use your imagination to displace carefully the parabola until the centre
of the fulcrum touches the X'OX. We may also think of it as one cutoff. It
is a unique case.

As soon as the parabola is lowered even further, it will cut the X'OX axis
at two points. This is very much different to the straight line which
never cuts the X'OX axis at more than one point. Make sure with your
imagination that to the left of the fulcrum (where the slope/attitude is
still negative) the cutoff has a less positive or even negative shade of
character than the cutoff to the right. At the cutoff to the right the
shade of character is less negative or even positive. It means that two
interpretations are possible with parabolic thinking. This is impossible
with linear thinking.

You can now do self all sorts of imaginary movements and flips to discover
many patterns of parabolic thinking. But I want you to use your
imagination one final time with me to make very sure of the following
observation. When the legs of the parabola point upwards (positive
attitude-becoming) and the parabola cuts the X'OX axis at two points,
shift your mental eye to where it cuts the Y'OY axis. Should the X'OX
cutoff to the left have a negative shade of character while the cutoff to
the right have a psositive shade of character, then the one only Y'OY
cutoff will be below the origin O. Again this points to a lack of
knowledge and thus learning still to come. Only when both the X'OX cutoffs
have a positive shade of character, will the Y'OY cut-off be above the
origin O. Only then is the knowledge sufficient to make two
interpretations which have both a positive shade of character.

OK, you can now stop imagining (if you can ;-) and wipe of the mental
sweat (if you can ;-). If you are an authentic learner, you will now be
creating many questions. Perhaps one of them will be "How can I transform
linear thinking into parabolic thinking?" It will be equivalent to "How
can I get rid of the %Mental Model% -- the one only interpretation on

To answer that one swiftly (since it is time to stop) I have to rely on
some of your high school algebra. I hope (perhaps in vain ;-) that you do
not mind. The formula for generating a straight line as its graph on an
XOY grid is
. y = b.x + c
The formula for generating a parabola as its graph on an XOY grid is
. y = a.x^2 + b.x + c.
The straight line clearly lacks the higher power term a.x^2 The term b.x
is called the lowest power term because it is actually b.x^1, but we
seldom write the exponent of power when it is 1. (Mathematicians are lazy
writers ;-)

With this high shool algebra you might suggest "add the higher power term
when it is absent in linear thinking". Yes, this one correct answer to the
question. But are there also other correct answers? Parabolic thinking
suggest that we ought to seek for at least one other correct answer. Here
is an exciting answer.

Think of two different straight lines y = b*.x + c* and y = b**.x + c**.
We use the * and ** to indicate that the b's and the two c's are
different. We can now join them disjunctively. In logic a disjunction
makes use of the connective OR. But we can also join them conjuctively. In
logic a conjunction makes use of the connective AND. But how will we do it
algebraic? Well in algebra the addition "+" is disjunctive while the
multiplication "." is conjunctive. So let us see what we get when we
either add or multiply these two formulas

For disjunctive addition we get
. y = (b*.x + c*) + (b**.x + c**)
. = (b* + b**).x + (c* + c**)
which has the form
. y = b.x + c
In other words, the result of the disjunction is again a straight line.
But for conjunctive multiplication we get
. y = (b*.x + c*).(b**.x + c**)
. = (b*.b**).x^2 + (b.*c** + b**.c*).x + c*.c**
which has the form
. y = a.x^2 + b.x + c.

The conjunctive joining means, in terms of the AND, that we have to get
rid of exclusive thinking in which we apply LEM (Law of Excluded Middle)
indiscriminately. This is how it can happen in an open, LO-dialogue. It
needs a team of three learners discussing a lump of information. The first
learner, still suffering some %Mental Model% creates the interpretation
. y = b*.x + c*
The second learner, also still suffering some %Mental Model% creates the
linear interpretation
. y = b**.x + c**
The third learner now tries to harmonise these two interpretations
conjunctively by creating the linear interpretation
. y = (b*.x + c*).(b**.x + c**)
showing that it has the parabolic form
. y = a.x^2 + b.x + c.
and pointing out its two different interpretations, perhaps with moving
the parabola in the context so that it indeed has two X'OX cutoffs.

The properties "good" and "right" (and their shades) of character had been
called morality since the time of the Greeks. With some two milenia of
linear thinking (and thus only one interpretation) things went simple. But
as more people began to think non-linearly, they began to create more than
one interpretation for each lump of information on morality. Thus thinkers
began to get the notion that morality is relative. I do not think that
morality is relative. I think morality is under evolution or "endless
becoming" just like anything else. We are now becoming aware of the fact
that more than one interpretation can be created on any lumps of
information on morality. This makes only the information on morality
relative, but not morality itself relative.

Morality as an "endless becoming" is not only making a choice between good
and bad. It is also finding out self the two shades of "good", making sure
of a positive attitude in being and becoming of character, shifting the
parabola so that both shades of character are positive so as to get an
idea of what is not yet known and thus needs to be learned. All this has
to happen inside the mind of a person into which no other person can
reach, but still can be fathomed by using information created by the first
person. Sadly, whereas linear fathoming with its one only value for each
measurement is vital to physical shipping, it is more than often deadly to
spiritual shipping with its one only interpretation for each creation of
Parabolic thinking is exciting. Think of the parable which Jesus told of
the lost son. The surface meaning is about a down to earth family. The
deeper meaning is about the Kingdom of God (Kindom of God as Richard Seele
recently noted!). The Father loves boths sons despite all the meanderings
of each son's character. But let us apply some parabolic thinking on this
parable once again. Most people interpret the lost son to be the one who
went to the far away land -- lost from family life. But what about the
son who stayed behind. He did not learn by mistakes. He also had the
Mental Model that love is conditional, requiring work as condition. Was he
not the son lost in learning?

I have written nothing on the seven essentialities of creativity in this
contribution. But I have composed this contribution especially so that it
has a lot of implicit information on the essentiality otherness
"quality-variety". I do hope that it will aid you to become more aware of
this essentiality.

A last note for systems thinkers keeping a watchful eye on the systematics
of thinking. I think that parabolic thinking is the transition (that
elusive "thing" which is sometimes called a "quantum jump") from
interdisciplinary thinking to transdisciplinary thinking.

With care and 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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