# One-to-many-mappings LO25720

From: AM de Lange (amdelange@gold.up.ac.za)
Date: 11/24/00

Dear Organlearners,

In the recent topic on Roger Bacon, Andrew Campbell writes in a footnote
that the specific heat capacity of water is not as constant as had been
assumed based on initial experiments. I explained to him that when adding
heat to water, its main (but not its only) effect is to raise the
temperature of the water. Temperature T is but one of many intensive
properties of water. Other properties are the vapour pressure P, the
viscosity V, the chemical potential C, the surface tension S, the
refractive index R, etc. When each of these properties get changed, some
of the exchanged heat /_\H gets used up. In other words, ONE /_\H gets
mapped into changes of MANY intensive properties -- the Onsager
phenomenon. One or more of these many changes are large whereas the rest

All but one may be so small that we may neglect them conveniently so as to
think of a simple one-to-one-mapping and hence make use of the
mathematical equality "=" relationship. This is the case when thinking
traditionally that the heat /_\H exchange causes only one change, namely
that in temperature T. But our our requirement for convenience and
simplicity may be our very downfall. We may begin to cling to the Mental
Model that all relationships are one-to-one-mappings which could be
converted into equality "=" expressions.

One-to-many-mappings are neither convenient nor simple as the explanation
for the variation of the supposed constant heat capacity of water
indicates. We have to keep MANY things in ONE mind to be able to deal
effectively with one-to-many-mappings. This requires a learning of many
things in advance. The easiest way is to learn these many things one by
one. However, this nothing else than one-to-one-mapping in learning
applied in a sequential manner. We still have not learned by way of a
one-to-many-mapping applied in a sweeping manner. Since we have not done
this manner of learning, we have to expect difficulties in applying
one-to-many-mappings in a sweeping manner.

One of the vexing problems in management seldom recognised, is that as a
person moves upwards in the hirarchy of management, that person has to
think less in terms of one-to-one-mappings and more in terms of
one-to-many-mappings. An essential facet of leadership is thus the ability
to think in terms of one-to-many-mappings and to assist the transformation
of thinking in terms of one-to-one-mappings into one-to-many-mappings. But
leadership itself is in a curious sense a one-to-one-mapping because one
that organisation self requiring one-to-many-mappings in thinking. By
extending leadership into stewardship this curious one-to-one-mapping has
to be transformed into a one-to-many-mapping. The steward, although one
leader of one organisation, is one steward to many organisations.

For example, Mr Nelson Mandela had been one fine leader of one country,
world wide. He is not leader any more, having had to retire with grace
because of old age. However, he also acted with outstanding stewardship
with respect to many other countries. He is still doing so, despite his
age, because of the greater need for stewardship. He does not act as
Since Mr Mandela has both leadership and stewardship qualities, he is able
to distinguish between the two rather than conflating them into one. Thus,
although a leader himself, he does not let this come into conflict with
other leaders because he deals with them as a steward rather than a

Even within South Africa Mr Mandela distinguished between leadership and
stewardship and managed our internal affairs according to it. Thus within
a couple of years most of the people began to love and honour him as a
person of exceptional integrity. I am sure that he is not doing it as I am
explaining it now formally and explicately. But he is definitely thinking
implicitly and tacitly in a way which makes my explanation a possibility.
For example, Mr Mandella is a Xhosa, one of a dozen of Banthu peoples. For
the Xhosa people he is their leader, but for the other peoples like the
Suthu or Venda, he is revered as their steward (or father as they often
say it). But South Africa also has, for example, the Xhoi and San peoples
who are indigenous, but not of Banthu origin. It is an extraordinary sight
how even these peoples also embrace him, not as a Xhosa leader, but as a
steward of multicultural South Africa. The same applies to the peoples of
European and Asian decent. Unfortunately, what can be said of Mr Mandella,
can be said only of him.

I think that we have to excercise these one-to-many-mappings in our
learning in a one-to-many-mapping manner as much as possible. While
replying to Andrew's contribution on Roger Bacon, I was seeking for an
excercise requiring not too much complexity, but illustrating sufficiently
what we have to become aware of. I could not think of one, but as I
clicked on the "send" button, one emerged in my mind. I am glad that it
happened after the "send" click because Doc Holloway had written to me in
distinction, suggesting Mr Mandella as example. Now, as Doc should be
aware by now, I seldom give an answer as a one-to-one-mapping. So I
decided to answer him with ONE reply in which I map his ONE request into
MANY topics -- heat capacity, mappings, Mr Mandella, domestic-foreign
policy and now, finally, a numerical example.

Consider the sum (addition) of the three numbers 1.357, 32.4 and 0.12. It
is something which kids in primary school learn to do. It is something
which many people nowadays cannot do without using a calculator. I wonder
how many people are aware that this is a many-to-one-mapping? How on earth
am I going to create a one-to-many-mapping out of it? Most people will
think reversibly, suggesting breaking the sum up in the three numbers
1.357, 32.4 and 0.12. But there is actually a doubly infinite number of
possibilities breaking the sum up. Firstly, we can end up with any other
three numbers rather than 1.357, 32.4 and 0.12. For exmaple, we can end up
with 1.5, 32.357 and 0.02. Secondly, we can end up with any number of
numbers like the five numbers 1, 0.357, 32, 0.4 and 0.12. These doubly
infinite possibilties are definitely not what I have in mind.

The reason why we encountered this issue of infinity, is that we were
thinking merely in terms of the system SY, in this case the system
mathematics. I was fooling around as a leader because I was not also
acting as a steward. I did it by using only ONE property, name "number".
As steward I will have to think especially about the surroundings SU which
consists of many other systems. They may have other properies than merely
the property "number". Thus I have to assign at least another property
with the property "number" to indicate with this one-to-two-mapping my
readiness to work with any one-to-many-mapping. Let me assign the property
"value". Thus entities like 1.357, 32.4 and 0.12 will have the properties
"number" and "value". The sum of 1.357, 32.4 and 0.12 is based on their
property "number". But let us see what happens to the sum when based on
their property "value" too.

When a mathematican adds up 1.357, 32.4 and 0.12, he will express the sum
as 33.877. But when a banker adds them up, he will express the sum as
33.87700 or 33.87700000. However, when a scientist adds them up, he will
express the sum as 33.9. In other words, what we have here according to
the additional property value is the one-to-many-mapping
. /==> 33.877 (mathematician)
. 1.357 + 32.4 + 0.12 = ==> 33.87700 (banker)
. \==> 33.9 (scientist)

The mathematician value the economy of numerical symbols very much. When a
mathematician writes the number 33.877, he may think of it as the decimal
(real) number 33.8770000000000000........ having an infinite number of
digits, repeating themselves from the sixth digit as the digit zero (0).
Writing any number of zero digits at the end is a wasting of symbols.

The banker value the integrity of numerical symbols very much. When a
banker writes the number 33.877, he may think of it as representing
thousands of dollars. Thus he will add two zeros (00) so nobody can cheat
on even the cents. It is easy to steal considerable amounts of money by
taking a few cents on each of millions of transactions.

The scientist value the sureness of numerical symbols very much. For him
(when not going to deep into the matter) all the digits of a number is
certain, except the last number. Thus in 1.357 the 1.35 is certain whereas
the 0.007 is uncertain. In other words, 1.357 represents any one of 1.356,
1.357 or 1.358. When adding them together, the sum will be as uncertain as
the worsest uncertainty in the parts. The number 32.4 is the most
uncertain because it ends in the first digit after the decimal dot. Hence
33.877 is also uncertain in this digit, namely the digit 8. The subsequent
digits 77 have no certainty and merely helps in rounding the digit 8 off
to the digit 9 closer to it than the digit 7. Hence the sum is 33.9

Teaching students numerical calculations in sciences like phsyics and
chemistry is an eye opener to the role of Mental Models in learning. In
their first calculations they will usually give the answer as 33.877 or
33.877000..., depending on the brand of calculator they use. If it
supplies the zeros as well, most students will write the zeros too. When
it does not supply the zeroes, many students will still add them to
signify that no fraud is intended. When multiplying 1.357, 32.4 and 0.12,
they will invariably write 5.276016 to signify that the have not frauded
the number given by the calculator. Thus they think like bankers. Only few
have the flair of the mathematician and usually none the scepticism of a
scientist.

To teach them that as a scientist they should give the value as 5.3 and
not 5.276016 takes immense effort. One has to teach them about Mental
Models (MM) and the capacity to switch from one to the other MM as the
ever changing context (surroundings) requires. This ever changing
surroundings SU is nothing else than a one-to-many-mapping resulting from
LEP (Law of Entropy Production). See for example in the topic "Work and
Free Energy" how the one order relationship /_\F < W maps itself into many
different scenariois.

The numerical example above is a serious reflection on what is going on in
our primary and secondary schools and the incredible rote learning which
mathematical leaders among them would have written persistently 33.877 and
the scientific leaders among them would have written perisistently 33.9.
They are taught virtually nothing about stewardship because they are
ingorant to the fact that

1.357 + 32.4 + 0.12,
\$1.357 + \$32.4 + \$0.12 (x 1000)
1.357meter + 32.4meter + 0.12meter

have different values. Among thousands of students over many years I had
"about ten" who could give distinctive answers for the above three cases.
I write "about ten" because I have no time to document each year's results
carefully. (Sometimes the crop of one year would not be even one student
among hundreds of them.) I have little time is because I use up much time
to teach them the different answers in such a way that they understand
clearly what they are doing or have to do whenever a new unknown situation
arises.

Dear fellow learner. Think about yourself and how you would have thought
follower, as a leader or as a steward? Are you aware that choosing one
among the three, you are logically (by way of LEM) not yet performing a
one-to-many-mapping? An authentic steward is a person who is able to
function also as a follower and a leader and not only as a steward.

One last question please. How do you think of a Learning Organisation --
as a many-to-one-mapping or as a one-to-many-mapping? In other words,
thinking of one organisation and many learners, is the LO a
learners-to-organisation-mapping or is it an
organisation-to-learners-mapping? Or is the LO both kinds of mapping with
LEM again leading us on a dubious path?

With care and best wishes

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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 <Richard@Karash.com>
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