To become or not to become. LO23954

From: AM de Lange (
Date: 02/11/00

Replying to LO23921 --

Dear Organlearners,

Greetings to you all.

I prefer to think of the mind of a person as a "university" with many
"faculties" rather than one. Many fellow learners may prefer to specialise
in one faculty. But even more of them had such bad experiences in other
faculties of their minds that they rather prefer to avoid these faculties.

Perhaps the mathematical faculty is the one which most fellow learners
avoid -- let it be. However, I want to help those who wish to stop
avoiding this faculty so as to develop it once again -- let it become.

In LO23887 I connected the "let it be" to the mathematical relation "is
equal to" with symbol "=". Hopefully it is the symbol which you have used
most and perhaps fear least. I also connected the "let it become" to the
mathematical relation "is smaller than " with symbol "<". Hopefully it is
the symbol which you have used least and thus, perhaps, will fear most.

Fear is a good thing because it makes us very cautious when we have to
overcome that fear so as to succeed in our goals. So what is common to all
our goals -- to be or to become -- or by some strange irony/harmony, both?

In LO23921 I took the mathematics one step further by looking at the
mysterious X of mathematics, the symbol feared so much because it can also
symbolise what is still unknown. Most people think of X as X = ..... where
the empty .... has to be filled in. Such thinking is the result of
eliminating X from its context. Later it requires adding X back into its
context. Context? Which one? This is where most people experience a
serious problem.

However, it is also possible to think of X as something of value in a
context. This value then has to be substituted by other values so that the
ramifications have to be observed and compared. I did it in the earlier
contribution LO23887. Hopefully it showed that we can indeed also follow
the "one step substitution" process rather than only the "two step
ellimination and addition" process.

I have ended that last contribution (LO23921) with
>Finding harmony between becoming and being is an art.

Let us see what it means.

The grand problem with the relationship for being, namely the "="
(equivalence relation) is that on its own it has very little use for any
kind of evolution.

Try to equate a primate with a human and see what you get!

Learning, for example, is a kind of evolution which happens in the mind.
Think of your knowledge and let it be symbolised by K. Think of two
successive states (values, instances) in your knowledge and symbolise them
by K(1) at an earlier time and K(2) at a later time. Please study LO23921
once again should this K, K(1) and K(2) confuse you. There I used X, X(1),
X(2) and X(3) to explain the ideas behind them. It is very likely that
this explanation will be useless so that you will have to ask me to
explain it again. But try to tell me what facet needs explanation.

Let the being become.

Which of the following two relationships apply?
        K(1) = K(2) --- is equal to
        K(1) < K(2) --- is smaller than
The first relationship tells that your knowledge is static.
The second relationship tells that your knowledge is
growing positively. Your intuition would immediately pick
the second one as the "most becoming".

There is nothing wrong with this immediate response. But remeber the art
-- to let this becoming evolve into a harmony with the being. How? By
finding also the mediate (distant) response. What? When will the first
relationship make sense? The first (equivalence "=") relationship applies
only when you are not a learner!


Perhaps some of you will argue that since all humans are learners for
every second of their lives, the first relationship does not apply so that
there is actually no harmony. I will agree to that just to keep on
arguing. I will now argue on a deeper level for the harmony. I really do
not intend any tricks, but sincerely try to tell you about something which
is very important. Let me first do it and then tell about it.

The second (order "<") relationship applies when you are a
learner, slow or fast. When you are a
    slow learner, K(1) "is SLIGHTLY smaller than" K(2),
but when you are a
    fast learner, K(1) "is MUCH smaller than" K(2).

Compare these two indented sentences. Remember that we previously used the
words "is .... smaller than" to indicate an order relationship "<". But
now they are exactly the same in the two sentences so that they form part
of an equivalence relationship "=". What have we done to them -- destroyed
their sense? No. They used to distinguish between form, but now we have
made them into content. This is a creative collapse. By doing so, we made
it possible for a new order relationship to emerge, indicated by
        SMALLER < MUCH
Thus we again have harmony between "=" and "<".

This emergence of form from form again and again is a key feature of
mathematics. Please note that it is something which has to happen in your
mind. It can never happen in the mathematical articulations on paper or
screen. If Fred Nichols want an example of tacit knowledge which can't be
articulated, this is perhaps the best one ever. I can type a zillion of
words to try and tell it to you, but actually I cannot tell it. You have
to seek its becoming in your own mind. Keep on seeking until, with a
"bright flash", you become aware that it is actually happening.

Let us go into an even deeper level of form -- or should
I say higher level of form? Because what I want you to look
at is not
        K(1) = K(2) --- is equal to
        K(1) < K(2) --- is smaller than
        K(1) > K(2) --- is larger than
Since the mark (1) refers to an earlier time than mark (2),
the knowledge value K(1) decreases into the knowledge
value K(2). "Impossible" you might claim.

The older we become, the more we become strange to the youth.

There is something which troubled me since my first year as a teacher in
1972. Assume a learner writes tests on successive sections (1), (2), (3)
and (4) of the course. Assume further that these sections involve equal
amounts of work to be mastered. Assume that there is no stochastic
(random) behaviour in the student's responses. Assume that the learner
writes standardised tests, one on each section, with NON-ZERO outcomes
T(1), T(2), T(3), T(4). Which one of the two relationships
        T(1) = T(2) = T(3) = T(4)
        T(1) < T(2) < T(3) < T(4)
ought to apply?

The first one has such a nice scientific ring to it -- "reproduceable
results", even on different sections of the work! Should we drop the
assumption that there is no stochastic behaviour so as to allow variations
among an average, the ramification will be to compute the average as
        T(av) = [T(1) + T(2) + T(3) + T(4)]/4
(Get the total and divide it by four.) Dropping another
assumption, for example that the sections have equal
amounts, we can even introduce so-called weights in
the averaging process.

But what about "double learning" -- learning from both learning successes
and learning failures. Is this not yet another way in which form can
emerge from content which previously was form itself? Is this "double
learning" not "evolution in learning"? If it is indeed the case, which of
the two expressions above apply? Not the first, but the second
        T(1) < T(2) < T(3) < T(4)

So, what troubled me as a teacher? I wanted learning to follow the
        T(1) < T(2) < T(3) < T(4)
but the formal educational institution forced me to conform
to its tradition with the equation
        T(1) = T(2) = T(3) = T(4)
As a result, neither of the two expressions applied, but rather
the distressing case
        T(1) > T(2) > T(3) > T(4)
What does the form of this symbolic expression tells us?
A persistent impairing prevails in the learning of most pupils!
In a few of them it was so vicious that within one year they
became failures. In some of them it was very slight, barely
noticeable over five years. But here at university many of
these fortunate successes soon become unfortunate failures
themselves too. The system has few favourates, if any.

I tried to save where I can and there are many pupils and students who can
witness to this. But eventually I became convinced that when the system is
wrong while the majority believes the system cannot be wrong, it is like
whisteling in the wind. Far more get lost as a result of the system than
the few who can be saved. So what remains? Go for the very heart of system
and substitute it with the heart which it should have had. So what is
wrong with the heart of the present system?

I think that by now you can intuitively guess it. Too much of the
equivalence "=" relationship and too little of the order "<" relationship.
Too much being and too little becoming. What education needs, is the
emergent transformation

Since education is concerned with learning in all subjects, it will be
presumptious of education to go for this very emergent transformation
while not one of the subjects has yet manifested it. However, in the
previous two contributions I have indicated that in at least mathematics
and thermodynamics this transformation is happening. It is beginning to
happen in many other subjects too. Thus education will stay behind with
horrific consequences should it transform too late.

How much intellectual inertia will there be which has to be overcome with
this transformation? How much vested interests are there in the old
paradigm of being? How much is science (and its offspring technology) a
victim of excessive "=" rather than highlighting the harmony between "="
and "<"? How much plain fear for the unknown will we have to reckon with?

How much is a couple on their honeymoon in the desert the victims of it?

Let us go back to even before 1687 when Newton became famous though the
publishing of his Principia. Today Newton is becoming infamous because
some people reckon that he led humankind into linear thinking. But is it
the case? Was he himself not the victim of excessive "to be" thinking?

In 1660 Robert Boyle (27 years before Newton) discovered the law
        P x V = constant, or
        P(1) x V(1) = P(2) x V(2)
for gases kept at a constant temperature and amount of
substance. I have explained the mathematical meaning
of these symbols carefully in LO2392. The physical meaning
of P is that it symbolise the quantity pressure and V the

What does the mathematics symbolise in this law? Try to
learn yourself to explicate it some day, for example, as
        "when the pressure on a gas is decreased, its
         volume has to increase and vice versa".
I am not now at all interested in this explication. If you want
to learn how to do it, buy an outdated edition of a textbook in
chemistry -- it will cost only an apple and an onion. I much
rather want you to focus your attention on the form of Boyle's
law. What is the thing which relates the left side to the right

Boyle's law make use of the "=" (equivalence relation). Does it strike
you? Not yet? Let me then tell a story with seemingly no mathematics to

When I drive in the desert, the surface of the dirt roads often becomes
very rough with humps, potholes and corrugations. I am not only a
scientist, but also have a vivid imagination. I picture in my mind how
the shape and the volume of the tyres become contorted by this rough
surface. So does the volume of the air in the tyre change too, bigger and
smaller, bigger and smaller, every second. Every second Boyle's law is put
to the test -- and it magnificently succeeds. Does it strike you too?

Then suddenly, I see a sharp piece of rock sticking out of the road's
surface like a nail. No time left to swerve out of its way. Stop. Damage?
Puncture. Air is blowing out of the tyre. Technology fails. Suddenly
Boyle's law does not apply anymore. So is there anything which still

Yes, indeed. Let us carefully symbolise the form of what is becoming after
the moment of puncture. Should I just do it, it will be a "two step
elimination and addition" process. Why? Because I will eliminate much of
my experiences and then add to what remains so as to impress you. But in a
"one step substitution" process I must tell you why I am able to do it.
In 1968 I got stuck with my research in soil science. I had hundreds of
equations using the "=" which I could apply. None worked because soils
are not merely beings, they also become. I searched for a solution for
months until the winter of 1969 when I finally stumbled on a book
(published in 1962) with many order relationships "<". In that book the
author mentions the year 1947 as his own decisive turning point.

Now you will have to paint your own picture, trying to recall where you
got stuck with the excessive use of the equivalence relationship "=".

Let us think about pressure P:-
The air in the tyre has a pressure which we can measure at
an auto service station. Let the value of this pressure be
symbolised by P(2). The atmosphere also has a pressure
which we can measure with a barometer in a laboratory. Let
its value be P(1). These two values are not equal. But since
they are both pressures, they are ordered (if not "=", then "<").
Since the pressure in the tyre is greater than atmospheric
pressure, the order relation is symbolised by
        P(2) > P(1) --- "is larger than"
This order relation means the same as that the difference
P(2) - P(1) is positive. This positive difference itself can be
symbolised by
        P(2) - P(1) > 0
Here is a numerical example. When 5 > 2, then 5 - 2 > 0
since 5 - 2 = 3.

Let us think about volume V:-
The layer of atmosphere covering earth has a volume, just as
the peel of an orange has a volume. The volume of the
atmosphere is very large. The air inside the tyre has also a
volume. It is small in comparison to the atmosphere's volume.
Air which blow from the tyre into the atmosphere increases
by a tiny amount the atmosphere's volume. Since that tiny
amount is BLOWING into the atmosphere and thus
CHANGING its volume, let us symbolise "change in volume"
of the atmosphere by /_\V. We do it because our symbol for
change will always be "/_\". The value of /_\V is positive
since the volume of the atmosphere increases. This may be
symbolised by
        /_\V > 0

Let us finally think of the pair:-
Let us multiply the "difference in pressure" with the
"change in volume". This mental task
        "difference in pressure" x "change in volume"
will be symbolised by
        [P(2) - P(1)] x /_\V
where [P(2) - P(1)] is the "difference in pressure" and /_\V is
the "change in volume". The product of two positive quantities
is itself also positive. It means that the product of two positive
quantities is never equal to zero, but always greater than zero.
Should we symbolise this last sentence, the following symbolic
expression will do:
        [P(2) - P(1)] x /_\V > 0

This symbolic expression is an order relation and not an equivalence
relation. Does it strike you? Not yet?

Let us go back to my story. I am in the desert in some valley of
desolation. Imagine that I have no spare tyre left and that my air pump
got broken by my kids. I can lose my mental control and run wildly away in
any direction. I can stay and listen dumbfounded how the tyre is blowing
off. But what about trying to uncover the form of what is now becoming of
the tyre? How can I become with form myself according to the becoming in
form of the tyre? Am I not allowing by this incident for mathematics to
emerge within my mind?

Nice way to go on a psycho trip and forget about all my troubles!

Now, is the following not very strange indeed? Since Boyle
in 1660 up to 1947 (almost three centuries later) no scientist
ever gave attention to the
        "order relation of becoming" in gasses
        [P(2) - P(1)] x /_\V > 0
In the mean time, hunderds of millions of learners had to
cram into their heads one of the
        "equivalence relation of being" in gasses, namely
        P x V = constant --- Boyle's law
or go branded as learning failures. Some caring teachers
tried to ease the hurt by saying that those who did their best
and still failed have no talent for science and mathematics.
It is nothing to be ashamed of.

Let us forget about shame and blame because they cannot ever solve
problems. My problem is to show you in an "recogniseable" way how my
mathematical faculty works so that you can begin working on your own
mathematical faculty, not in my way, but in a way which is fitting to you.
Your problem will be to decide whether you will begin working on your own
mathematical faculty or not. One thing neither you nor me can avoid -- we
have to live by our decisions.

Let us go back to me there in the valley of desolation. Do I get
frightened because air is blowing out of the tyre? No. If there was as
much air in the tyre as in the atmosphere, there is nothing to worry
about. I can drive to Cairo and back without worrying. This is the
inventive idea on which the hovercraft works -- the rest is innovation.
However the tyre has only a limited amount of air -- essentiality
"quantity-limit" (spareness). Is this the shocking part of the
experience? Perhaps. But I knew formally about "difference"x"flow" > 0
long before I knew anything formally about the seven essentialities.

What is frightening to me is that while the air is blowing out, the tyre
becomes visibly FLATTER and the sound of the blowing audibly SOFTER. These
qualties are changes in form. A new form emerge. This calls for my
mathematical faculty. How will I express this new emerging form which
concerns "flatter" and "softer"?

Let us think again of the order relation
        [P(2) - P(1)] x /_\V > 0
Since we want to go to a deeper/higher level of form, there is
a lot of notation in it which becomes unnecessary baggage.
I will have to get rid of it. How? By a creative collapse rather
than a brute application of Occam's razor. This is how I will do
it. When I created the expression [P(2) - P(1)] x /_\V, I worked
on two parts of it, namely P(2) - P(1)] (difference in pressure)
and /_\V (change in volume). They are the two parts which form
together a pair. So let me simplify [P(2) - P(1)] x /_\V by giving
it the symbol PAIR. I now use a name for a symbol. Yes, names
are symbols. Thus I can rewrite the order relation as
        PAIR > 0
Except for the creative collapse ( and thus minus the baggage),
it is still one and the same thing as
        [P(2) - P(1)] x /_\V > 0

So what frightens me there in the valley of desolation?
When I first observed the air blowing out, it is event (1). The
expression [P(2) - P(1)] x /_\V had a certain value at event (1).
If I want to symbolise this value in the expression, it will have
to look like {[P(2) - P(1)] x /_\V}(1). Horrible! Is this what
frightens me? No. I know that when I want to indicate the first
value with the symbolic expression PAIR, it becomes PAIR(1)
by way of substition. A minute or so later -- called event (2) --
my observation as to the value of [P(2) - P(1)] x /_\V will lead
to the value PAIR (2). At event (3) its value will be PAIR(3), at
event (4) it will be PAIR(4), etc.

What frightens me there in the valley of desolation is
the ordering
        PAIR(1) > PAIR(2) > PAIR(3) > PAIR(4)
Let us seek harmony between the "<" and "=".
Do you still remember the distressing case of learner
performances above, namely
        T(1) > T(2) > T(3) > T(4)
It is in mathematical form exactly the same thing! Is there
more to it than merely the mathematics? We will tackle this
question in my next contribution by thinking of the car's

Meanwhile, how can I simplify what is so frightening? Shall
we use Occam's razor? No. We will do something which
we already have done a couple of times. We will collapse
creatively. We will have to find the form in
        PAIR(1) > PAIR(2) > PAIR(3) > PAIR(4)
How? We have done it several times already.

        PAIR(1) > PAIR(2)
        PAIR(2) < PAIR(1)
the difference between a value and an earlier value like
        PAIR(2) - PAIR(1)
will be negative. For example, if 5 > 2, then 2 - 5 = -3
so that 2 - 5 < 0 because -3 < 0. Thus the "change in PAIR"
which is symbolised by /_\PAIR will be negative. The
symbolic expression of this is:
        /_\PAIR < 0

Any equivalence relation like
        P x V = constant
is called a "zero order" relation. The "zero" tells that it
is actually not an "order relation", but an "equivalence
relation. Any order relation like
        [P(2) - P(1)] x /_\V > 0
        PAIR > 0
emerging from it, is called a "first order" relation. It is the
first proper order relation. Any order relation which emerge
from the latter like
        /_\PAIR < 0
is called a "second order" relation. This second order relation
is called a "minimisation ordination" because it involves a
"< 0", i.e. /_\PAIR decreases. However, the first order relation
        PAIR > 0
is called a "maximisation ordination" because it involves a
"> 0", i.e PAIR increases.

The frightening thing to me is that both
        PAIR > 0
        /_\PAIR < 0
apply to the air blowing out of the tyre -- maximisation for
PAIR > 0 and minimisation for /_\PAIR < 0. They tell me
that my future is boxed in between them so that "something"
definite in the future is attracting the present. What is this
"something"? A completely flat tyre with the pressure P(2)
in it equal to the atmospheric pressure P(1). In other words,
the becomings
        PAIR > 0
        /_\PAIR < 0
tells me that the being
        P(2) = P(1)
will result, sooner or later. Any further journey with this tyre
will become impossible SHOULD I WANT TO USE THE

I once encountered a young married couple stranded in the desert in
exactly the same way as I have describe above. They planned on a
"way-out" honeymoon. They were so much in love that they forgot their
common sense at home. They were in a state of great anxious because they
did not have any of their little food or water left over. When they set
out for the desert, they imagined that love would see them through and
money will buy the rest. When I reached them, they had already been
walking for an hour or so in an attempt to save themselves. But they were
heading towards the Skeleton Coast!

I took them back to their car. When I got there, I asked the young man why
he did not drive the car back into the direction from where they came.
"What, and ruin the tyre and the rim?" Perhaps I should have left them
there because that one question ought to have done more than enough to get
them thinking.

In the cities we can buy repairs or replacements. In the desert we cannot
buy them. So we have to plan and buy in advance. We have much time to do
that. In the valley of desolation we see a natural "nail" in the road's
surface. We have no time to avoid it. Hence we are forced by nature itself
to contemplate what "flatter" and "softer" means. The honeymoon couple
learnt that it can happen to a tyre too.

Mathematics IN CONTEXT helps us to focus on important
issues so that we can make better choices. What does
       PAIR > 0 & /_\PAIR < 0 ==> P(2) = P(1)
means without the story in which it emerged?

Perhaps we have to think of the period 1660 to 1947 as the
honeymoon of humankind with science. Most couples have
experience their honeymoon as a case of
-- an equivalence relationship "=". The stark reality of marriage
is that it involves order relationships "<" too. Married couples
who cannot deal with the "<" end up in divorce. I become very
uneasy when I read that people recommend such a divorce
between humankind and science. Perhaps, upon such a divorce,
humankind will like that honeymoon couple head towards the
Skeleton Coast once they are confronted in the Valley of
Desolation with the true order of things when technology fails.

We need science to tell us where technology will fail as sure as the sun
will rise tomorrow. Stop asking scientists only the good part of the story
so as to make money from it. Ask them about the bad part too after having
promised not to judge them for being responsible for the bad part. Shame
and blame will not do us any good.

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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