Dear Organlearners,
Greetings to you all.
I dedicate this contribution to Andrew Campona. He is an artist who once,
as a child, had to play the role of Hamlet on the stage of mathematics.
Fortunately, he picked up a reptile rather than a skull. I admire him
because he is not playing forever the role of Hamlet.
The noble mind of Shakespear's Hamlet was troubled by the immortal
question: "To be or not to be?" To use the modern terminology of Senge,
every organisation to which Hamlet belonged (familyship, friendship,
courtship, statesmenship) ceased to function as a Learning Organisation.
As life seeped out of his organisational environment so that nothing was
pleasing any more, death itself became the pleasing thought so that
tragedy was inevitable.
To become or not to become? It is not an immortal question because in the
long run it questions immortality itself.
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.
I think that the mathematics in the indented paragraph above has already
made the majority of you fellow learners uneasy. What made you uneasy? Is
it the mathematics which we have already covered with respect to the "="
sign -- some mathematics of being? Or is it not that you anticipate some
mathematics yet to come and, knowing me, that it will indeed come? Ha! Do
you feel how knowing only a part about the future let the unease explode
into fright? Which one of the following two questions now rushes through
your mind before mental death might take over? BEING -- "What am I?"
BECOMING -- "What will become of me?"
Let us see what will become of you. Read the following indented paragraph
through once. Feel like Hamlet how life seeps away!
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!
No, it is not life which seeps away. By reading once through the last
indented paragraph you have not done mathematics. If you want to do
mathematics, you need to create the "form of content". How? Do the
following carefully. Go back and relate the two indented paragraphs to
each other. Compare the contents with each other, word for word and symbol
for symbol. Find out exactly where the contents are the same and where
they differ. Do it now before you go on to the next paragraph.
Have you done it? You have created mathematics! You have gained in
mathematical experience. You have experienced creating form out of
content. Wherever you have discovered that the contents are the same, you
have made tacitly use of "=" -- the "equivalence relation". Wherever you
have discovered that the contents are different, you have made tacitly use
of "<" -- the "order relation". In other words, you DO HAVE the tacit
knowledge on "=" and "<"!! If you did not have this tacit knowledge, then
you would not have been able to discover in what the two paragraphs are
the same and in what they differ.
Perhaps you are now more curious about "tacit knowledge" than the
"mathematical relationships = and <". If you are, then you will make me
very, very happy. Why? Because mathematical life is flowing back into you!
When we manage to express our "tacit knowledge" in "formal knowledge", the
"tacit knowledge" is the CONTENT and the "formal knowledge" the FORM
which we have given to it. In other words, the emergence from "tacit
knowledge" to "formal knowledge" is "deep mathematics". How "deep"?
Early this century mathematicians believed that mathematics is a building
which they have created with axioms as its foundation and logical
inferences as its tools. Both the axioms and the inference rules were
explicitly formulated. Thus they were part of the "formal knowledge" of
mathematics. It seemed as if all mathematicians (A, B, C, D, ....)
believed the same thing. Is it not like the "equivalence relation"
A = B = C = D = ......
in operation? David Hilbert even claimed that mathematics is the best
formal structure which can be created in this manner. Who will dare to
differ with this great mathematician? A child struggling with mathematics?
A student scared out og her wits by mathematics?
One Lutzen Brouwer dared to step out of the "equivalence relation". Step
into what? An "order relation"! He insisted that mathematics is not merely
a formal enterprise as Hilbert claimed. Mathematics reaches much lower
into intuition. In symbolic notation we may express it as
Brouwer < Hilbert
Brouwer used "intuition" and not "tacit knowledge" because Michael Polanyi
formulated this latter concept only some fourty years later. Perhaps
Brouwer would have used it himself should it have been available to him.
Nevertheless, then came the most stressing period in his life, perhaps the
most stressing ever to be experienced by any mathematician. Brouwer had
to walk his talk. He had to articulate what he meant by intuition giving
rise to formal mathematics. He had to do what nobody before him ever did.
He had to change the "can't" into a "can".
Brouwer's output was slow and fragile compared to that of Hilbert. The
mathematical community jeered at Brouwer. For them he wanted to overturn
the building which tens of thousands of mathematicians so carefully have
built through the ages. It seemed as if he was succeeding in overturning
only his own future. Well, the rest is history. Topology and Heyting
algebra made their appearance. Yet few expected the revolution which would
come and shake the formal building in its very foundations. It took many
decades for the revolution to come in the seventies.
The revolution was nothing else than that intuition has to deal with both
"being" and "becoming" in perfect harmony. The "order relationship <"
became as important as the "equivalence relationship =". I use "being" and
"becoming". Fritoff Capra of The Web of Life fame would have used
"structure" and "process". Mathematicians often use "set" and "functor"
or otherwise "object" and "arrow". It does not really matter when we focus
on the "tacit" (implicate) rather than the "formal" (explicate).
Does it mean that there are now as many "Brouwer"s in mathematics as there
are "Hilbert"s? No. And this is why mathematics still scares the wits out
of you. To understand this, you have to experience the following. It is
one thing to recognise your own tacit knowledge in the formal knowledge
expressed by another person. It is a sort of theoretical thing. It is
completely another thing to express self your own tacit knowledge into
formal knowledge. It is an ART because theory (the BEING) and practice
(the BECOMING) now have to join in harmony.
How will you experience it?
The two indented paragraphs far above are my own articulations based on my
own tacit knowledge. It is not and will never be the articulations of your
tacit knowledge -- unless I have the telepathic ability to enter your
mind. Now read again in quick succession the two indented paragraphs. If
necessary, do it again and again. Try to focus on how you recognise your
own tacit knowledge in them. If you can honestly say "there is nothing at
all in it which I do understand", then there is no recognition at all of
your own tacit knowledge in them. If you have to admit "not all of it is
Greek to me", then you are recognising some of your own tacit knowledge in
it.
Have you ensured that you did experience recognising your own tacit
knowledge?
So what is the next step? To articulate self your own tacit knowledge on
"=" (being) and "<" (becoming).
To become or not to become?
It really does not matter IN THE LO-DIALOGUE to show some technical
expertise in mathematics, science, learning or management theory. Andrew
has set us some wonderful examples in which he plunged ahead with his
artistry. I must tell that I admitted to Andrew in private that I feared
for his safety. He could easily have been clobbered by words. Artists
easily step out of any equivalence relationship. It is the very nature of
art to do so.
What does matter IN THE LO-DIALOGUE is that each of us can recognise some
of our own tacit knowledge in the articulations of the rest of us. When we
reply to that dialogue, should we not endeavour to express our own tacit
kwowledge in such a manner that in some of it they may recognise their
own?
This is also a main reason why it is senseless to judge or criticise the
articulations of somebody else IN THE LO-DIALOGUE with a blunt "You are
wrong" or a clever "You have not ...". Firstly, the articulation of a
fellow learner does not completely articulate that person's tacit
knowledge on the topic. Secondly, what we recognise in that articulation
is our own tacit knowledge. This is by far not the same as making sure
what that person's own tacit knowledge is since two completely different
persons are involved with words in between to confuse everything. Thirdly,
we have to allow for that person's becoming and not merely act upon being.
Perhaps we will then learn how to allow for our own becoming. Fourthly,
allow for symmetry-breaking. What?
Please, let us take care of each other's mathematical faculty.
Let us quickly tie up some loose ends. Look at "<" and ">". We may think
of each as two lines meeting each other at an angle. Each line has two
ends. Where the two lines join, the distance between the two ends are
"small" (in fact zero) so that on the other side the distance between the
two ends are "large". For example, in the symbol "<" the "small" is to the
left and the "large" is to the right. Hence, should we compare the numbers
2 and 4 with "<", we will write the 2 at the "small" side and the 4 at the
"large" side, i.e 2 < 4. But should we compare 2 and 4 with ">" rather
than "<", we will again write the 2 at the "small" side and the 4 at the
"large" side which gives 4 > 2.
The expressions "2 < 4" and "4 > 2" tell about the same thing, but in a
different manner. What is the difference? Asymmetry !!!!. There is no
symmetry-breaking in X = Y. But in X < Y we have the very epitome of
symmetry-breaking. Compare again the two indented paragraphs with each
other.
Louis Pasteur made many brilliant observations which no-one before made.
(He used to say that "discoveries only come to the prepared mind".) Among
other things, he noticed with a magnifying glass that crystals of
artificially prepared tartaric acid had two kinds of crystals -- the one
kind being a mirror image of the other. But crystals of tartaric acid
produced in nature consisted of only the one kind. They had no mirror
images. So he painstakedly separated the small artifical crystals by
tweezers. He made solutions of each kind. They still behaved chemically
the same. He remained curious!
Surely, they must have some activity different. So he searched for an
activity in which they differed. Eventually he found it. The one solution
will polarise light into the one direction while the other solution would
polarise it into the other direction. Chemists will say that natural
tartaric acid is optically active while artificial tartaric acid is
optically racemic.
Fisher took these investigations further. He discovered for many natural
acids, sugars and proteins that they are optically active. Furthermore,
they all polarise light into the left (laevo - L) rather than the right
(dextero - D) direction. It was now clear that natural organisms were not
behaving symmetrically like artifical laboratories. Nature acts with a
stunning symmetry-breaking! Nature does not have a mirror image. Thus
nature needs not to argue with itself like humans do in discussions and
debates when they search for the symmetries of equivalence relationships.
It is as if they still cannot believe that they are all humans and thus
can allow for symmetry-breaking.
Andrew, this contribution is actually the first part of a "rich picture".
I will continue the "painting rich picture" in a following one. Perhaps
you can already intuitively see where I am steering to. Is it
/_\E = 0
/_\S > 0
Who cares to paint too?
With caring 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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