Linear thinking LO22863

AM de Lange (
Wed, 13 Oct 1999 13:49:04 +0200

Replying to LO22855 --

Dear Organlearners,

Winfried Dressler <> writes:

>I felt somehow personally asked and also embarrassed
>that I couldn't find out why you brought the straight, linear
>line into the picture.

Greetings Winfried,

Almost 50% of South Africans are illiterate (and almost 50% of the other
50% are functionally illiterate!). Most of the illiterate people (but few
of the functionally illiterate people) feel very embarrassed when they
cannot write or read in front of literate people. This embarrassment
usually influences their creativity very negatively. (See Digestor

However, I have been writing on literacy in a natural language. This
literacy rquires recognition that the language, primarily spoken and
heard, can also be supplemented with an ortography (token system) to make
writing and reading in it possible. The main problem of illiterate people
is to master the ortography. Sadly, their tutors know even less of the
little experience which they have in dealing with tokens. Thus they try to
master literacy by rote learning through which progress is very slow and
often terminated.

I now want to write on mathematical literacy and link it to disciplinarity
and eventually linear thinking. Let me again formulate how I perceive
mathematics after working almost half a century with it. The art of
mathematics is to let form emerge (be born) from content and when that
content is exhausted, to let the form collapse into content for a new
generation of form, always striving for consistency and coherency.

Please note that I have not yet written anything in symbols to set up a
mathematical formula. By this I want to stress a very important point.
Doing mathematics is PRIMARILY very much the same as speaking and hearing
a natural language, BUT NOT writing or reading it.

I see many fellow learners doing mathematics in many contributions on this
list by making form out of content consistently and coherently. Sometimes
they even let form collapse to content so as to reach out for new forms.

They are not aware that with these mental activities on content and its
form they are doing mathematics. They are "even less aware"
(hyperbolically speaking) that they use the ortography of the English
language to do so. It is like using an army knife intended for general
purposes as a surgical scalpel for delicate operations.

What they need to become aware of the mathematical beauty of their work,
is a Learning Organisation in which they and fellow learners, more mature
on the path of mathematics, team up to pursue the beauty and happiness of
mathematical thinking in all their thinking. In the "mathematics team" the
more mature learner will act as midwife, making the less mature aware of
the mathematical thinking they have already been doing as well as that
begging to emerge in deeds.

But what most of them, if not all, have been given as part of their
so-called comprehensive (well-rounded) education in school, is contact
with mathematics dished out as a discipline all on its own. In it is
eactly here, because of rote learning, they got embarrassed for not
knowing how to work with the symbolics (token system, "ortogrpahy") of
mathematics. Just like the case is with language illiterates, the
mathematics teacher seldom use their past tokenising experiences because
rote learning does not require it. Just like the case is with language
illiterates, this rote learning happens very slow and often terminates.
Thus they stop taking mathematics, thinking themselves to have no
mathematical talents. Just like the case is with language illiterates,
they become very embarrassed when they cannot do mathematics in fornt of
somebody capable of doing it. Just like the case is with language
illiterates, this embarrassment have a vast negative influence on their
creativity from a certain level upwards. Eventually they begin to command
others not to use mathematical symbolics so as not to feel embarrassed
themselves. Gradually everybody's mathematical throat gets strangled so
that eventually the mathematical spirit dies.

This is what subject disciplines do at school. The language "teacher"
instructs only language and the mathematics "teacher" instructs only
mathematics. The language teacher never point out the mathematical
beauties which emerge while studying language, not share in the pupil's
joy when such beauties do emerge. The mathematics teacher never point out
the language beauties which emerge while studying mathematics, nor share
in the pupil's joy when such beauties do emerge. Neither the language
teacher, nor the mathematics teacher, guide the pupils how to jump freely
and frequently from language to mathematics and vice versa, thus dancing
interdisciplinary thinking. What is worse, both the language teacher and
mathematics teacher themselves are incapable of identifying
correspondences between the ortography of the language and the symbolics
of the mathematics. What is even more worse than being incapable of
bridging form in language and mathematics, is their inability to bridge
the content in language and mathematics. Thus they cannot instruct the
dance of interdisciplinary thinking, even if they have to.

The result of this compartementalisation of language and mathematics is
linear thinking in each of them. The price to be paid for this exclusive
thinking is linear thinking.

>For me, linear algebra is a classical masterpiece of
>mathematical art. And I very well remember the Aha! when
>I realized what is meant by isomorphism between (physical)
>content and (mathematical) form.

The same for me! And once I had this "Aha", I began to read treatises on
algebra like fascinating fiction. I still remember how I could not make up
my mind which to read first, a science fiction novel by Robert Heinlein or
a treatise on algebraic (coordinate) geometry by Robert Bell. In the end I
read both of them, swinging from chapter to chapter between the one and
the other.

>But what I sense as beauty in order of physical diversity
>(the ability to aply one mathematical form), turns out to be a
>devastating limitation in case of living, conscious, self-aware

Yes, if we are not willing to let all form of matter (in even the highest
of it requiring most advanced mathematics) collapse, we cannot leap from
matter to spiritual content (in even the lowest of it requiring most
advanced phenomenology). Fear of creative collapses is definitely, as you
say it, a devastating limitation. It requires a shift of our focus from
the celebrated accomplishments of the past to the unexperienced attempts
of the future -- to let go of the fact that we have matured in something
so as to be born like a child with a new idea in which we have to grow up.

When the form of content gets exhausted, we easily dogmatise this form as
a celebrated accomplishment rather than pursuing its creative collapse so
as to reform ourselves to a higher level of spirituality. This is the
essence of reformation.

I have a great struggle in the parish to which I belong because most
fellow church members believe that any unexperienced attempts of the
future is a sure recipe for sinning. When I ask them what the clause
"reformed" in the name of our church means, they say it refers to the
break with the Roman Catholic Church. For them the "reformed" means
"exclusivity" while not realising that fixing such exclusivity results in
linear thinking. When I ask them whether the "reformed" makes us less
catholic (note, not Roman), they reply with a "of course, yes". When I
put to them the information that the word catholic comes from the Greek
words "kata" (throughout) and "holos" (whole), they begin to frown. If
not, I remind them about the apostolic confession which they utter in
every service and the very article "I believe in one, holy, general
church". This surely brings out the frowns.

Some where some reasoning just seems not to be a straight

Best wishes


At de Lange <> 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 <> Public Dialog on Learning Organizations -- <>