Dear Organlearners
Winfried Dressler" <winfried.dressler@voith.de> writes:
>First week at university, first lectures on calculus, first lesson,
>the math professor came in and stated (I say: informed us
>about his motivation to spend time on ignorants like us - full
>expert mode): "About 50% of you will pass the exams. Only
>less than 10% will understand what I am going to teach you.
>But those surely not more that 2 of you have the potential to
>guide our society in future. And this potential should not be
>wasted."
Greetings Winfried,
How horrible the "tyranny of the experts", is it not?
>Since I met At de Lange here in the list, he is doing a good
>job in curing that wound slowly. But it takes time.
No Winfried, I am not doing the job of curing. You are doing it yourself.
I am merely pointing out that you (and everybody else) are capable of
doing the curing job by self-learning, emergently and digestively.
>Then, what about Demings "profound knowledge" or
>At de Langes "creative course of time"?
Is this "creative course of time" and sometimes the synchronicity in it
not a wonderful thing to contemplate? I was preparing another contribution
to an other list when I read yours. I simply had to complete that one and
post it to that list. With Rick's permission I will now also post it to
this list.
Just a minor detail. I met Leo Minnigh on that list. One day I decided to
"spread the word" to him to join the LO list. The rest is history.
The name of the topic in that list is: The = sign means "generate"
~~~~~~~~~~~Begin Copy~~~~~~~~~~~
Leo Minnigh <L.D.Minnigh@LIBRARY.TUDELFT.NL> writes:
>PS
>While typing this answer, a mail of my friend At de Lange arived
>on the same subject. Maybe he is willing to say some very
>interesting words on the subject 'being' (in a way similar to the
>mathematical =) and becoming (similar to -->).
Greetings Leo,
Thank you very much Leo for your own creative comments on the sign
"=".
The sign "=" in mathematics is known as the identity sign. Its closest
formulation in a natural language is "is equal to". I will now use two
of them together to create the new sign "==" which will mean
"corresponds to". See if you can now understand the meaning of the
symbolic expression which I will now create:
= == is equal to
Obfuscation? No. It simply means in ordinary words:
The equality sign (=) corresponds to (==) the phrase "is equal to".
Have you noticed the path of creativity in the previous paragraph?
Most people define creativity as the ability to join two things
together so that something new emerge. (My own definition of
creativity is completely different -- an enigma to most people.) They
usually think of joining two different things. But nothing prevent us
from joining two of the same things like the "=" sign. [Whether it is
wise to connect two equal things, is a completely different question.
However, we cannot become wise before we have created!] So, following
the path of creativity (which is the creative course of time), note
that I first connected the signs "=" and "=" for the sign "==" to
emerge. Then I connected the "=" sign on the left with the phrase "is
equal to" on the right by (I hope you have noticed it) the very sign
"==" !!! Why would I do it? To give a meaning to the sign "==" !!!
Please note that I wrote: To give a meaning to the sign "==" !!!
I did not write: To give a meaning to the sign "=" !!! If I meant
this, I would have written it. I did not mean it so that I did not
write it. I wrote what I meant. It is very, very important that you
should try to understand what I meant rather than place your own
thoughts in my writing, otherwise you will not grasp the points which
I wish to make. I am fully aware that most of you would think that I
actually meant: To give a meaning to the sign "=".
For about the first forty years of my life I would have thought the
same. Actually, I was trained by 17 years of mathematical schooling to
do so -- 7 years primary school, 5 years secondary school and 5 years
university (MSc physics) training. I was trained to think
mathematically like a robot. I was never taught to think creatively in
mathematics. In fact, apart from two exceptions, in all those 17 years
I was never to taught to think creatively in any of the many subjects
which I had to study. The one exception was in my 2nd year at high
school in the subject Afrikaans (my mother tongue). The other
exception was my last year at high school in the subject English
(second language). I got the prize that year as the most creative
pupil in English -- although my teacher wanted to kick me in the pants
many a day for my poor grammer -- my Nemisis forever.
I suspect that my history fits the far majority of you.
What is mathematics? For me mathematics is a study of the form of
content. Mathematics is an eternal study because when we have created
enough form (of a certain content), we switch that very form into
content to create new forms on the older generation of forms. Let us
now study again the expression which I have created above, namely
= == is equal to
Note that I have increased the white spaces. They tell a very
important message. Let us now simply symbolise this unknown message of
the white spaces by the sign "$". Some other day we can study this
message "$". But for now we will ignore this message "$". The result
is:
=$==$is equal to
Mathematicians call any such a pattern with three members (namely "=",
"==" and "is equal to") the "associative" pattern. It has the form
a$b$c.
The associative pattern is very important in mathematics. It is needed
to formulate the law (axiom) of associativity. All mathematical
systems honour the law of associativity, even the number system. For
example, the following expression symbolises the law of associativity:
(2 + 5) + 3 = 2 + (5 + 3)
The brackets ( ) mean "first do the operation called addition in
brackets before doing the operation outside the brackets. This law
says that we are free to begin with our association. In the example it
means that on the left side we can begin with 2+5 and on the right
side we can begin with 5+3.
I do not want to go into the law of associativity itself. But I want
to ask the following questions to wet your mental appetites. Are we
really free where to begin with our connections? Engineers might say
say yes -- but chemists will definitely say no. They have learnt their
lessons. When they want to create a specific compound from distantly
related compounds so that they have to employ many succesive chemical
reactions -- they refer to all these reactions as a synthon -- they
have to begin with very definite compounds, otherwise they will never,
never, never reach their goal.
What I want to ask you, is not to confuse the "law of associativity"
with the "pattern of associativity". The law needs the pattern
otherwise we cannot formulate it, but the pattern (apparently) does
not need the law. Ok, so mathematicians need the "pattern of
associativity" to formulate the "law of associativity". But here is a
far greater surprise. Any creative person also needs the "pattern of
associativity" to come to a constructive result. On the other hand, to
produce a destructive result, a creative person can simply destroy the
"pattern of associativity" among the many possible things to do. To
say it more technically, the associative pattern is ALMOST essential
to creativity.
I wrote ALMOST because there is another thing which we should also
notice in the associative pattern
= == is equal to
The three enitities "=", "==" and "is equal to" act like units
(monads as Leibniz called them more than three hundred years ago).
Even the entity "= == is equal to" act like a monad (unit). In other
words, the expression
= == is equal to
does not only illustrate the "pattern of associativity", but also the
"pattern of monadicity" (pattern of unity). When we take these two
patterns together, we arrive at what is essential for creativity,
namely WHOLENESS.
Wholeness has two important features to it -- associativity and
monadicity. If we believe and act as if only wholeness is essential to
creativty, then we become holists embracing the philosophy of holism.
Why is wholeness essential to creativity? Study merely, as an example,
the expression
= == is equal to
We first created the sign "==" and then created the expression itself
TO CREATE A MEANING for the sign "==". Without wholeness
(associativity and monadicity) we would not have been able to create
the expression. Furthermore, we would not even have been able to
create a meaning. Why? Because again we have made use of the
associative-monadic pattern
You Internet Me
Remove anyone of the three entities and the associaitivity is broken.
Split anyone of the entities and the monadicity is broken. That is why
I have stressed in the beginning -- please note what I write and what
I mean by it. Do not split my writing from my meaning for the sake of
wholeness.
Is it possible for you to agree with me that every other expression OF
THE FORM
abcd == wxyz
tells us more about the meaning of te sign "==" as the expression
= == is equal to
did? Yes. The only difference is that the expression
= == is equal to
was the first one we have created to give a meaning to the sign "==".
Mathematicians have a very clever name for such "first created
meaningful" expressions. Guess the name -- "definition" !!!
Now for the kill.
When we write in mathematics expression such as (note the sign "=" in
each of them)
1 + 1 = 1 (boolean algebra)
1 + 1 = 2 (number algebra)
mx + c = y (linear equation, geometry)
y = f(x) (function)
A = {2, 3, 5, 7, 11, 13, 17, ....} (set theory)
we are actually producing more and more meaning for the sign "=".
But humans are creative. At a certain stage of creating more and more
meaning to the sign "=", they think stop with doing it. They have
reached equilibrium. So they make a most important mind shift. They
accept the meaning of "=" to be fixed and not growing anymore. But
they wind a new growth by saying that the sign "=" becomes their
interpreter, mediator, facilitator or "umlomo" (Zulu for mouthpiece).
In other words, rather looking at each of the the expressions
1 + 1 = 2 (number algebra)
1 + 1 = 1 (boolean algebra)
mx + c = y (linear equation, geometry)
y = f(x) (function)
A = {2, 3, 5, 7, 11, 13, 17, ....} (set theory)
as a whole, they compare the one side of the "=" sign with its other
side to increase the meaning of the other side. As an example, compare
the first two exapressions with each other. The first one (1 + 1 = 2)
we learn in primary school. But when we reach the tertiary level, some
of us are confronted with the second one (1 + 1 = 1).
Some of you might argue that since 1 + 1 = 2, it is impossible to have
1 + 1 = 1 or even 1 + 1 = 3. How is it possible that what have been
taught in primary school, can be overthrown in tertiary learning? It
does not make sense. No, it makes wonderful sense and the sense is
creativity. We have been trained to accept 1 + 1 = 2. This is true in
the "system of natural numbers". We have created that system of
"natural numbers ourselves" in our preschool years as toddlers. Our
mothers and fathers did not train us, but taught us how to do it by
their encouragement.
Then they sent us to school to be taught further. But unfortunately,
most of the teaching has been replaced by training, just as the
attention has shifted from the whole to the two sides of
1 + 1 = 2
Somebody created (we are not even told who did it as well as where and
when it was done) the "system of integers" and we are trained to
accept that
1 + 1 = 2
is true also for integers. Then we are confronted with the "system of
rational numbers" and the process repeats itself. Next follows the
"system of real numbers" and at some schools even the "system of
complex numbers". All along we are told that
1 + 1 = 2
is true for all these systems. Eventually we believe that
1 + 1 = 2
is universally true.
What a schock wait upon some of us when we learn that
1 + 1 = 1
is true in the "system of Boolean numbers". Again most of us are not
even told who did it as well as where and when it was done. Some of us
might have learnt by then how the system works, thus concluding that
the dirty deed was done by a person with the surname Boole. Some might
even learn by accident, or forced to learn by profession, that all the
computer software we use, each make use of zillions of tiny operations
which each one is a Boolean operation like
1 + 1 = 1
1 + 0 = 1
0 + 1 = 1
0 + 0 = 0
and
1 x 1 = 1
1 x 0 = 0
0 x 1 = 0
0 x 0 = 0
The first four models a very important word in our language, again an
"umlomo" like "=" or "==". It is the word OR !!! The last four models
the word AND !!!
So what had this one Boole in mind? Unfortunately, I have to hurry
myself to fulfill Leo's request. We can study Boole some other time.
Let us now think about the signs "=" and "==". Are they the same
thing? No. the sign "=" means "is equal to" while the sign "==" means
"corresponds to". There is no way how the two English phrases "is
equal to" and "corresponds to" can mean exactly the same thing. After
many centuries mathematicans are now also realising it for some thirty
years. When they mean "==", they will use words like "isomorphism" or
"adjunction" to give different flavours to "corresponds to". They use
these words in a mathematical topic which is known as Category Theory.
It seems as if we can represent all mathematics with the signs "=" and
"==". But let us look at the definition "### is a study of the form of
content". Perhaps
mathematics = ###
Perhaps it is not. I do not want to get involved in the issue what
mathematics is. I want to get involved with the study ###. Can we
represent all of our study, called ###, of the"form of content" with
these two signs?
Here is an interesting case. Make up your own minds.
During the middle of the previous century two laws were discovered
which caught the attention of scientists as probably only Newton's
laws did so two hundred years earlier. Soon afterwards they were calle
the two laws of thermodynamics. I will not refer to them as such
because for me they concersn much more than merely the discipline
thermodynamics. I will refer to them as LEC (Law of Energy
Conservation) and LEP (Law of Entropy Production). For the purposes of
this contribution you need to know nothing more about them than the
following
Law of Energy Conservation -- LEC
energy of physical universe = constant
Law of Entropy Production -- LEP
entropy of physical universe "=" increasing
Did you notice that in the formulation (action of making form) of LEC
I used the sign = while in LEP I used the sign "="? Now why will I do
such a strange thing? On the right hand side of LEC is "constant". It
refers to something which does not change. I could also have written
Law of Energy Conservation -- LEC
energy of physical universe == being
Law of Entropy Production -- LEP
entropy of physical universe == becoming
>From this we infer something very important. The sign == (corresponds
to) can be used for creating associations which involves "beings" and
"becomings". But the sign = (is equal to) is limited to associations
involving only "beings". That is why the second equality was placed in
quotation marks as "=".
Maybe you still do not believe me. So let us bring in the operation
CHANGE and apply it to each of the two formulations. It becomes for
the whole
Law of Energy Conservation -- LEC
CHANGE(energy of physical universe = constant)
Letting the CHANGE work on the two outer parts, it becomes
CHANGE(energy of physical universe) = CHANGE(constant)
But can a constant change? No -- zilch -- zero (0). Thus
CHANGE(energy of physical universe) = 0
Note that in the last formulation the = (is equal to) and the 0 (zero)
is known to all of us. Thus only the left side, namely
CHANGE(energy of physical universe)
may be unknown to many of us.
Similarly it becomes for the whole
Law of Entropy Production -- LEP
CHANGE(entropy of physical universe "=" increasing)
However, as soon as we try to let CHANGE work on the two outer parts,
the left side
CHANGE(entropy of physical universe)
gives no problem since who in whatever knows what it means. However,
the right side is the one which causes the problem because here we
have
CHANGE(increasing)
what really does
CHANGE(entropy of physical universe) "=" CHANGE(increasing)
mean? We have three unknowns here -- the left side, the "=" and the
right side. What a predicament.
However, when we study the LEP extensively, we finally come to the
conclusion that we have indeed a solution for our problem. The
formulation
CHANGE(entropy of physical universe) "=" CHANGE(increasing)
is nothing else than
CHANGE(entropy of physical universe) > 0
where the sign ">" means "is greater than".
To summarise in terms of == rather than = and "=". When we apply the
operator CHANGE on
Law of Energy Conservation -- LEC
energy of physical universe == being
Law of Entropy Production -- LEP
entropy of physical universe == becoming
the astounding result is
Law of Energy Conservation -- LEC
CHANGE(energy of physical universe) = 0
Law of Entropy Production -- LEP
CHANGE(entropy of physical universe) > 0
By now most of you should have developed the idea that the = (is equal
to) can easily be applied to "beings". But when we try to apply the =
(is equal to) "becomings", we run into deep difficulties. That is why
we use "=" and not = when try to create a sameness between becomings.
The = is suitable for beings, but not for becomings. So, how do we
overcome this lack of suitability? By BECOMING CREATIVE. Without
creativity there is no hope for us, not even in the most elementary of
all mathematics which invloves the sign = (is equal to).
Are you somebody who had horrible experiences in mathematics? Do you
run away from mathematics whenever it raises its ugly head? Do you
still dream nightmares involving mathematical exams? If you have
persisted up to here and your answer is yes, answer the following
question. Are answers to the previous three questions also YES for
your pre-school years? If not, then something terribly wrong has
happened in your years of schooling. In your pre-school years and up
to a certain year in your schooling you used your creativity to learn
mathematics.
Somewhere in some manner by somebody you were told than you need ###
(something else than creativity) to learn mathematics. I can give you
many substitutions for ###, for example "intelligence", "logic",
"discipline", "talent", "teacher" ........ . Contrary to your
experiences, you began to believe this person's claim. It is exactly
at that point where your "mathematical personality" began to die. For
some of you this death came quickly like a heart attack while for
others it came slowly like cancer. It is now time to revive your
"mathematical personality". Believe with all your mind and all your
heart in the following tenet:
TO LEARN IS TO CREATE
It is true even for mathematics. I believe that each of you have more
than enough creativity to do so.
Leo, I hope I have satisfied your request.
~~~~~~~~~~~End Copy~~~~~~~~~~~
Winfried, I hope I have brought a new perspective to your claim.
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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