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Replying to LO23887 --

Dear Organlearners,

Greetings to you all.

In my contribution LO23887 dedicated to Andrew Campona of Oxford, I began

to paint a picture on the " =" (equivalence relationship) and the "<"

(order relationship) of mathematics.

I know that maths causes pain to many of you. I know that many of you

would prefer a life with no maths to cope with. I know that you love to

use metaphors in explaining things. So why do I swim upstream, getting

myself involved in all sorts of criticisms?

Let me answer with history rather than "entropy production". Roger Bacon

(13th century), one of the first professors of the ancient university of

Oxford, said that physics will only come to age when physicists have

learned how to incorporate mathematics into their descriptions. Perhaps

this was one of his claims for which he had to spend some years in jail.

Never the less, 4 centuries later, Newton proved him to be right.

Many of you will rightfully argue that Physics (P) and Learning

Organisations (LO) are not the same thing. To use maths, your

argument means that the equation

P = LO

does not hold. But let me tease your intuition a little bit more.

Does the ordination

P < LO

hold? No At, now you have gone way above our heads.

(For the next, please remember that

< means "is smaller than" and

* > means "is greater than".)
*

Let us look at language and not physics. We all know about

"degrees of comparison". Think of rich, richer and richest.

Now which of the following two expressions hold?

rich = richer = richest

rich < richer < richest

The "=" is most unfitting, but the "<" blends into the comparison.

Just to complete this ordering with a nice twinge, for poor (the

opposite of rich) we will have

poor > poorer > poorest

rather than

poor < poorer < poorest

Is there not mathematics in language?

In LO23887 I created two "pictures" which you had to compare:

The "equivalence picture":

*> Closely connected to the question "To be or not to be?"
*

*> is the mathematical expression
*

*> X = Y
*

*> It symbolises the wording "X is equal to Y". We call
*

*> it an "equivalence relation" since it relates X and Y
*

*> through the equality "=". We may also think of the
*

*> equality "=" as a "mapping in be" since the X has "to
*

*> be" so as to map onto the Y and vice versa. This can
*

*> be expressed in symbols as
*

*> X = Y corresponds Y = X
*

*> This makes the "mapping in be" symmetrical.
*

and the "order picture"

*> Closely connected to the question "To become or
*

*> not to become?" is the mathematical expression
*

*> X < Y
*

*> It symbolises the wording "X is smaller than Y". We
*

*> call it an "order relation" since it relates X and Y
*

*> through the ordering "<". We may also think of the
*

*> ordering "<" as a "mapping in become" since the X
*

*> has "to become" bigger so as to map onto the Y.
*

*> But the Y has "to become" smaller so as to map
*

*> onto the X. This can be expressed in symbols as
*

*> X < Y corresponds Y > X
*

*> This makes the "mapping in becoming" asymmetrical!
*

What I did in the "equivalence picture" was to create a mathematical

context (or field Smuts would say) for Hamlet's famous words. Hopefully

Shakespeare will not turn in his grave. What I then did, was to

SUBSTITUTE the "be" in "To be or not to be?" with "become" and

subsequently to MODIFY the context as consistently as possible. This

resulted in the "order picture".

I could also have done the following. I carefully wipe out all parts which

differs between the two pictures, replacing each such changing part with a

**** -- the "black hole" of mathematics. Here is the result.

Closely connected to the question "To **** or

not to ****?" is the mathematical expression

X **** Y

It symbolises the wording "X is **** Y". We

call it an "**** relation" since it relates X and Y

through the **** "*". We may also think of the

**** "*" as a "mapping in ****" since the X

has "to ****" **** so as to map onto the Y.

But the Y has "to ****" ***** so as to map

onto the X. This can be expressed in symbols as

X **** Y corresponds Y **** X

This makes the "mapping in **** " ****!!

Is there any escape from this "black hole". Are we dreaming the nightmare

again? Let us wake up.

Here each **** function as a "placeholder" or an "unknown" which has to

be given a certain value according to its context. In other words, it

functions as the "mysterious symbol" which mathematicians invoke left,

right and centre. They often use x or X for this "mysterious symbol", but

they may also use any other letter of the Latin or Greek Alphabeth when

they want to refer to different mysteries.

They make it sound so easy: "Let X be the ... of the ...". Put the

correct value into an empty X and everything is OK. But they have learnt

that some students do not find it quite so OK and sometimes even find it

horrible. So they now say "Let X be the variable which range over the set

{.....} of possible values". Nice democracy. Choose a value from the range

and give it to the empty X and see if it works properly. Sadly, now even

more rather than less students abhor the mystery of mathematics. Why? Are

you one of those students who becomes scared out of his/her wits when

hearing the (in)famous phrases "give X the value ...." or "Let X range

over ...."?

I have had many such students. The majority of them believed that to "give

X the value ...." should be as easy and simple as "a sponge soaking up

water" or a mind absorbing information by rote learning. Yet they

experienced that it was far more difficult and complex than what they

believed. Rather questioning this "belief", they "divorced" themselves

from mathematics with all the hurt which comes with this "divorce".

What the student actually has to do, is first to create a sensible field,

context or "picture" for the X which contains X and all its possible

ramifications. Next the student has to SUBSTITUTE one value for X with

another value for X, carefully modifying the context because of this

change in X, trying to stay as close as possible to his/her own tacit

knowledge. Finally the student has to complete the mathematical task

originally given with the appropiate values. I have shown with the the

"equivalence picture" and "order picture" how it is done with an X having

two (binary) values. Its a "doing" rather than a "theory".

There is far more to this SUBSTITUTION than that which

meets the eye. Let me try symbolise what happens.

ABCDE:X + Y ====> ABCDE:Y + X

With the ABCDE part I try to symbolise the context of X.

I use the first five letters of the alphabeth rather than one

to indicate that the context is rich. With the Y I try to

symbolise a new value for X. Thus with the complete

expression

ABCDE:X + Y ====> ABCDE:Y + X

I try to symbolise the substitution of X by Y.

Is this "one step substitution" not the same as two steps,

firstly

ABCDE:X ====> ABCDE: + X

and then secondly

ABCDE: + Y ====> ABCDE:Y

Look at the first step. Is it not a "reduction"? Is it not a

"fragmentation"? Should you have answered "yes" to any

of these two questions, I will certainly not differ from you.

But as a chemist I have to point out to you that chemists

have their own name for it -- "elimination". The second step

they will call an "addition". Perhaps you will more easily

agree to their name "addition" than to their name "elimination".

I have brought chemists into the "picture" for a very definite reason. (Is

the "picture" already becoming a "rich picture" for you?) They have

extensive and profound experience that a "one step substitution" is far

different from a "two step elemination and addition". The same is true of

mathematics.

Students who assume that the "give X the value ...." should be as easy and

simple as "a sponge soaking up water" have to make use of a "two step

elemination and addition". They first have to think reductionistic and

then as an after thought correct it by a gross generalisation. This "two

step" thinking soon gets them into turbulent waters until they finally

drown. Think of the **** example above on mathematical relations to see

how easily one can get drowned.

I want to encourage you to think in terms of a "one step substitution"

such as I have shown by my "equivalence" and "order" examples above. But

you will have to do two things in advance. First you will have to create a

context for one value of the topic. Then you will have to modify that

context as you substitute the topic with each of its different values.

It is these two doings which in my opinion make maths a worthy art.

(Remember that art-expressing is an elementary sustainer of creativity.)

It is also these two doings which in my opinion make LO-dialogue so

unique! (The same applies to the other three elementary sustainers of

creativity.) It sustains our becoming in creativity and its higher order

emergents like learning.

The classical command "Do to others as you want them to do to you" is a

perfect example of a "one step substitution" rather than a "two step

elemination and addition".

Please do not go over board. Both the path ways "one step substitution"

and "two step elemination and addition" have roles to play in life. My

intention above was to show how both and not merely "two step elemination

and addition" play a role in mathematics. Let me give a striking example

where both play a role, each in its own unique manner.

An essential feature of all living organisms is to make more cells. Again

we find two path ways in eukaryotic cells (cell with nucleus) -- mitosis

for asexual and meiosis for sexual reproduction. In both mitosis and

meiosis the first phase is to replicate the double helix DNA molecules.

The helix "zip" is gradually opened because simultaneously its now two

loose strands are replicated by RNA back into a double helix DNA. This is

nothing but a vey complex "one step substitution" process leading to a

"double load" in genetical content.

The next phase in mitosis finishes the job by relieving the genetical

double load. First the cell nucleus and then the rest of the cell split

into two daughter cells with DNA content identical to the mother cell. In

meiosis the same happens as in mitosis, but the job is not finished. It is

followed up (without DNA replication) by the two daughter cells (diploids)

splitting into four granddaughter cells with half the DNA content

(haploids). This is the "elimination" step. It almost seems as if there is

an over-compensation for the initial double load in genetical content. It

seems as if the next "addition" step corrects this overcompensation. The

two haploids cells (egg and sperm/pollen) fuses back into a diploid cell

with DNA content different to the parents cells.

To summarise, the DNA replication with RNA is, even though very complex,

still a "one step substitution" process. In mitosis (asexual reproduction)

the genetical double load is compensated by division into two daughter

cells with identical DNA content. The same happens in meiosis (sexual

reproduction), but then continues with superimposing a "two step

elemination and addition" process on it.

Just think of it. Perhaps mathematics had too much meiosis (sex) in it for

you to appreciate it as a child. Perhaps it is time to look at the mitosis

in it.

Allow me to tie up two loose strings which I will need in the next

contribution.

The first loose string:

~~~~~~~

Mathematicans have a very clever way to refer to the values which the

unknown variable X may have while not knowing even what these values are.

(Speaking of dealing with the unknown in terms of unknowns -- it is almost

like talking about tacit knowledge!) But actually it means they have a

powerful way to refer to the various values which have to be substituted

in the context of X with subsequent modifications. What they do is to

mark X with a subscript to indicate a value for it. Thus they create a

context (picture) for the values of X. Unfortunately these subscripts are

not possible with the ASCII format we have to follow in this list.

But it is possible to do mark X in a number of ways friendly

to the ASCII format. For example, do the following. If X has

the three possible values, say 4, 7, 8, then we indicate them

by X(1), X(2) and X(3). Here the (1), (2) and (3) act as

different marks on the X. Thus we have

X(1) = 4

X(2) = 7

X(3) = 8

So, when I write

P(1)xV(1) = constant

I actually mean that the expression

P x V = constant

contains two variables P and V for which the P(1) and V(1)

refer to a specific value for P and a specific value for V at

one and the same event marked by (1). Thus the

expression

P x V = constant

is called a general expression while

P(1)xV(1) = constant

is called an instantiation (exemplification) of the

general expression at event (1). The general expression

is formulated by means of variables and the instantiated

expression by means of values for the variables.

The second loose string:

~~~~~~

When a variable X has at least two posible values X(1)

and X(2) in that order (meaning X begins with X(1) and

ends with X(2) ), then it is possible to think of a

"change of X" given by

"change of X" = X(2) - X(1)

This "change" of X is symbolised by the capital Greek

letter Delta. (Think of this letter as the first letter of the

word "Difference".) We cannot use this letter in ASCII

format. Thus we will have to approximate it. The best

which we can do, is to use the signs /, _ and \ together

as one symbol /_\. Thus,

/_\X = X(2) - x(1)

To think of /_\X as the "difference in the two values

of X" given by

/_\X = X(2) - x(1)

ought to cause little, if any, conceptual difficulties.

Simply substract the beginning value from the final

value and the difference becomes known.

But to think of /_\X as the "change of X" which can be equated to the

difference X(2) - X(1) asks for some pretty deep thinking. To experience

what I mean, assume that X is a variable for length and that X(1) = 5 cm

and X(2) = 8 cm. Now take a ruler and ask somebody else (because you will

need three hands) to point her two index fingers, the one at X(1) = 5cm

and the other one at X(2) = 8cm. She is pointing at two BEINGS by

simulating them with FIXED fingers. We can even record her simulation by a

static snapshot.

Now use one of your own index fingers and TRY to point

to the "change of X" which is given by

/_\X = X(2) - X(1)

= 8 cm - 5 cm

= 3 cm.

In other words, try to point to /_\X = 3 rather than to

X(1) = 5 and X(2) = 8. While she can point with FIXED

index fingers to X(1) = 5 and X(2) = 8, you will have to

make a sweeping movement from 5 to 8 with your index

finger to simulate /_\X = 3. You will not be able to point with

a fixed finger at it. You will have to simulate the BECOMING

with, for example, a MOTION. You will not be able to

record this with a static snapshot, but you will be able to

record it by a dynamic movie.

Consequently, what we actually try to do with

/_\X = X(2) - X(1)

is to use the "equivalence relation" to connect the

BECOMING /_\X on the left side with the difference in

BEINGS on the right side. Can we really equate a becoming

with a being? This question troubled the mind of Jean Paul

Satre up to the very end. Perhaps you are one of the few

people with an imagination so vivid that this difference

actually functions as a BECOMING rather than a BEING.

Mathematicians who are perhaps not so imaginative,

have invented a new way which avoid the equivalence

relation to represent

/_\X = X(2) - X(1)

when thinking of /_\X as "change of X". They call it a

"diagram". They symbolise the diagram as

/_\X

X(1) ========> X(2)

where /_\X is the mark of the arrow =======>. This arrow

is needed to transform NATURALLY object X with mark X(1)

into the object with mark X(2). Thus they think in terms of

objects (beings) and arrows (becomings), each indentified

with a mark, all connected into one commuting diagram.

The commuting means that objects connect by an arrow like

in

/_\X

X(1) ========> X(2)

Here the whole diagram is a complex object explicated by

one arrow and two objects. But it also means that arrows

have to be connected by an object like in

/_\X(1) /_\X(2)

========> X(2) =======>

This whole latter diagram is a complex arrow, explicated by

two arrows and an object.

Perhaps you will feel that these diagrams might be a clever

invention, but that they are not very intuitive. Then you will

have to find an even more expressive way to walk your talk

-- provided you believe that intuition can be formalised to

some extent. The fact is that these diagrams above are the

outcome after many years of Brouwer's untiring efforts to

show that mathematics also involves trying to articulate

intuitive (tacit) knowledge. This is part of the mathematical

revolution in the early seventies which I wrote about in the

previous contribution LO23887. What revolution? The

tansformation in mathematics

FROM BEING TO BECOMING.

Compare this with the title of Ilya Prigogine's most

remarkable book

FROM BEING TO BECOMING.

in which he uses mathematics to show how dissipative (or

"entropy producing") systems self-organise through entire

nature. Unfortunately, the mathematics he uses makes it

a closed book for the many who would be interested in

reading it.

On the other hand, you may believe that intuition can't be

articulated. In that case the equivalence expression

"change in X" = X(2) - X(1)

ought not to cause conceptual difficulties at all since it is

a crystal clear relationship. There are some serious

mathematicians who believe this is the case so that you

will not be without company.

Does Prigogine's book need to be a closed book for you because of your

past mathematical experiences? Do you need to repeat Hamlet?

It all depends on

(1) you and whether you want even mathematically

To become rather than not to become.

(2) a Learning Organisation with caring teams and caring

leaders in each team guiding the team to reach a

definite goal.

Finding harmony between becoming and being is an art.

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

Learning-org -- Hosted by Rick Karash <rkarash@karash.com> Public Dialog on Learning Organizations -- <http://www.learning-org.com>

**Next message:**Yekoutiel (Couty) SABAH: "Organizational Learning & Knowledge Management LO23922"**Previous message:**Don Nardone: "Leadership Assessment Tool LO23920"**In reply to:**AM de Lange: "To become or not to become. LO23887"**Next in thread:**John Zavacki: "To become or not to become LO23932"**Reply:**John Zavacki: "To become or not to become LO23932"**Reply:**AM de Lange: "To become or not to become. LO23954"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

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