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
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