**Next message:**Winfried Dressler: "Organizational Learning & Knowledge Management LO23955"**Previous message:**John Zavacki: "Organizational Learning & Knowledge Management LO23953"**In reply to:**AM de Lange: "To become or not to become LO23921"**Next in thread:**AM de Lange: "To become or not to become. LO23986"**Reply:**AM de Lange: "To become or not to become. LO23986"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

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!

Dead?

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

but

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

ordination

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

FROM BEING TO BECOMING

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

volume.

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

side?

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

it.

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

apply?

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

battery.

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.

Since

PAIR(1) > PAIR(2)

or

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

or

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

TYRE OR RIM AGAIN.

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

you-are-me-and-I-am-you

-- 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 <amdelange@gold.up.ac.za> 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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