46 Lessons in
Early Geometry, part 10/10 / provisional version in freestyle English / a
corrected version will follow in March, April or May (hopefully) / Franz
Gnaedinger / February 2003 / www.seshat.ch

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geometry 4 / early geometry 5 / early geometry 6 / early geometry 7
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Lesson 44

Dialogue with Ahmes, scribe of the Rhind
Mathematical Papyrus, on the nature of numbers, part 1 (mistakes and bad style
are mainly owed to the insufficient bablefish component of my astrologer's
crystal ball, which, however, allows me to conversate with Ahmes over several
thousand miles and years. I guess you would gladly endure the worst English if
you could have a chat with, say, Archimedes)

Me: Good morning, professor Ahmes. Today I
would like to ask you a principal question. What are numbers? What is your idea
of numbers? What is their nature? How do they work? What are they good for?

Ahmes: Well, you announced that question last
week, and so I prepared myself to provide a short and nevertheless complete
answer. Numbers are basically things of the mind, I believe, and the very
instrument by which Osiris overcame Seth.

Me: I can imagine what you mean by the first part
of your answer, while the second one leaves me rather perplexed I must confess.

Ahmes: What is our greatest achievement? What
do you admire most?

Me: I love Egyptian painting and sculpture. The
most beautiful, charming, lovely and sensual lips ever carved in stone are
found in your works. But since you ask for the greatest achievement I should
perhaps mention the pyramids.

Ahmes: And if I tell you that we should be
admired for a greater achievement?

Me: I can't imagine anything greater and more
admirable than the pyramids of Sneferu and the

Ahmes: They are most impressive, and yet what I
have in mind is a hundred times more admirable.

Me: Tell me, professor.

Ahmes: We succeeded in turning a desert valley
into a blooming garden. We spent millennia in achieving this, and we are
spending further millennia in maintaining our paradise. Without our being here,
high floods would sweep away the fields, whereas low ones would make most plants
wither, and within a few years the chamsin would cover with ever deeper layers
of sand all our houses and even most of the temples. You may know that we
worship Osiris in the

(To be continued)

Lesson 45

Professor Ahmes? Hello? Professor Ahmes?

No response.

Well then, let me reconsider what he told me last
time.

Turning a desert valley into a blooming garden
and maintaining it a paradise was the challenge that spurred geometrical and
mathematical skills. Rushdi Said will probably agree on that. He says that
before around 7,500 years ago nobody lived in the Egyptian part of the Nile
Valley, which was a dangerous place to be, ruled by an 'angry river' (Rushdi
Said, *The River Nile*, Pergamon Press
Oxford 1993). Taming that river and turning a 600 km long desert valley into a
place to be surely was a great challenge.

The Greeks faced a completely different but no
smaller challenge. Connecting the many distant shores of their vast archipelago
required excellent ships, daring men, highly developed navigating skills, and may
well have given rise to Greek reasoning including their fabulous achievements
in geometry ... Like Isis, Penelope may symbolize the land, and like Osiris,
Odysseus may symbolize the water, while the name Homer may go back on Greek
homaereo = I unite. It was Homer's wish to unite

Enough chatting for today.

Professor Ahmes? Hello? Professor Ahmes?

Still no response. I will try again another
day, hoping to ask him what exactly he means by saying that numbers are
basically things of the mind.

Lesson 46

Me: Good morning, professor Ahmes. You told me
that numbers are basically things of the mind. Which sounded fairly good to me.
However, the more I think about your words the less I understand what you mean.

Ahmes: By handling numbers we participate in
the creative power of the gods.

Me: You always succeed in puzzling me.

Ahmes: Build a wall of bricks or blocks. It may
be a fine wall and stay firmly for years
or millennia. But even the best built and strongest wall has its weak points
and will sooner or later decay, crumble away, come tumbling down. Add numbers
instead and you will obtain a perfect wall, so to speak:

1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1
plus 1 makes 8

Regard the ones as blocks, and the sum as the
resulting wall. The numbers fit perfectly well, and the sum will last forever.

Me: I can see a block, while I can't see a
number.

Ahmes: Numbers have a double nature. They are
ideal things in the realm of building and constructing, but as such invisible.
If you wish to make them visible, they turn from ideal objects into patterns.

Me: Can you explain this?

Ahmes: You may know that we teach our aspiring
scribes from the tender age of five years on. We let them play for example with
beans. I give each pupil nine beans and ask them to lay out as many patterns as
they can imagine. That is a good way of introducing children into the realm of
numbers.

Me: Playing with beans?

Ahmes: Ready for a game? Take twelve beans and
call them one heap, or one line, or simply 1. Six beans are half a heap or
simply '2. Four beans are '3. Three beans are '4. Two beans are '6. One bean is
a twelfth of a line or simply '12. Now consider the following posibilities of
dividing a line:

oooooooooooo
1

oooooo
oooooo '2 '2

oooooo
oo oooo '2 '6 '3

oooooo
oo o ooo '2
'6 '12 '4

A few transformations and we gain a fascinating
pattern:

one
'1

one
'1x2 '2

one
'1x2 '2x3 '3

one
'1x2 '2x3 '3x4 '4

one
'1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7x8 '8x9 ...

We easily recognize mathematically gifted
pupils. They love to play with patterns that emerge at every level and are the
true guide in mathematics.

Me: Thank you, professor Ahmes.

PS

Numbers are regarded as kinds of atoms,
however, they can’t really be separated from patterns. Consider the following
stairway decimals:

1.1

1.01

1.001

1.0001

1.00001

1.000001

1.000...1

0.91

0.9914

0.999141

0.99991414

0.9999914142

0.9999999999999...1414213562373...

1.3

1.013

1.00413

1.0001413

1.000051413

1.00000951413

1.00000000...362951413

All above numbers equal 1, nevertheless they
are associated with all other numbers.

You can even calculate with these funny
numbers:

Square root of 1.000...6
equals 1.000...3

Cube root of 1.000...6
equals 1.000...2

1.000...2
times 1.000...3 equals
1.000...5

1.000...4
minus 1.000...2

--------------------------- = 2

0.999...2
minus 0.999...1

One might build ever longer numbers:

1.000...2.222...1.111...2.222...

1.0000000...1.4142135...3.1415926...

1.000...1
times 1.000...1 equals
1.000...2.000...1

1.000...1
times 2.000...1 equals
2.000...3.000...1

1.000...1
times 3.000...1 equals
3.000...4.000...1

1.1
divided by 1.1 equals 1

1.01 1.001 1.00899...

1.001 1.00001 1.0009899...

1.0001 1.0000001 1.0000999...

1.00001 1.000000001 1.00000999...

........................................................

1.000...1 1.000...0.000...1 1.000...0.999...

As long as we carry out a finite number of
steps, the infinitesimals have no influence on the value of the reals:

1 squared = 1
squared = 1 squared = 1 squared = 1 ...

This changes when we carry out an infinite
number of steps:

1.000...1
squared equals 1.000...2.000...

1.000...2
squared equals 1.000...4.000...

1.000...4
squared equals 1.000...8.000...

............................................

Go on like this forever and you obtain more
than 1. Where in this process do we pass from 1 to the next real? May it be
that the infinitesimals are hindering us from actually well-ordering the reals?
Leibniz regarded the numbers as monads that contain all other numbers. Can it
be that he played with the same stairway numbers?

"ALL IS EQUAL, ALL
UNEQUAL"

a paper for the Brno Conference 2003, Chairman
Alan Rogerson, *The Decidable and
Undecidable in Mathematics Education, a tribute to Kurt Goedel*.

Mathematical logic

As a teenager I read the first sixty pages of a
book on quantum mechanics without understaning anything, but I was fascinated
by a footnote which said that the basic equation of mathematics a = a has not
yet been thoroughly studied. Finally something I could understand. A couple of
years later I found the following explanation. Mathematics is based on the
formulas

a =
a a unequal to
b

which represent the logic of building and
constructing. Consider these examples:

b =
b = b = b = b = ...

A wall can easily be built and will stand
firmly if the bricks (b) are of the same material, size and consistency, and
have in common all other properties; in short: if they are equal.

b =
b

The bricks should neither soak up moisture nor
crumble in the rain nor crack in the summer heat; their properties must remain
stable, they must stay as they are.

2 +
1 + 3 =
6 = 2 + 1 + 3

A closed door (1) should become a part of the
wall (6); one might then wish to open the door (2+ 1+3).

0.999...
= 1

A door (0.999...) must fit into its frame (1);
otherwise it will be too tight (“klemmen” in German) or there will be a draft.

9 = 2 + 3 + 4
= 9

In order to clean or repair a machine (9) one
dismantles it into single components (2, 3, 4); then one reassembles the parts,
in order to return the machine to its former functioning state (9).

Sooner or later, a mathematical discovery finds
its way into technology. The imaginary number i (square root of minus one) was first
regarded as a strange number, yet without that funny little number no radio,
television set or computer would work today.

The logic of nature, life and
art

The logic of nature, life and art is based on
another formula:

"All is equal, all unequal ..."

(Johann Wolfgang von Goethe, *Wilhelm Meisters Wanderjahre*, *Aus Makariens Archiv*)

a =
a

An apple is an apple; yet one fruit may be red
and sweet, another green and sour, and another yellow and juicy ...

*A rose is a rose is a rose* (Gertrude Stein)

You may imagine a red rose, a white rose, a
sweet-smelling yellow tea rose, a budding rose, a flowering rose, or you may
think of a girl named Rose, Rosy, Rosemary.

A
snowflake is a snowflake

Every snowflake forms a hexagon, yet seen under
a microscope each has its own unique pattern.

A
mouse is not an elephant

Mice and elephants belong to the animal
kingdom, and are both mammals. They have a common mouse-like ancestor with a
kind of proboscis, while the hyrax, the elephant's nearest living relative,
resembles a large mouse.

We
are all equal and all different

A fair and reasonable human society is based on
the fundamental equality of all humans, while leaving room for our
individualities.

p =
p = p = p = p ...

Physicists search for elementary particles that
fulfill the above equation. They explore and expand the realm of technology,
with no end.

In his Diary of the Italian Journey, Goethe
speaks of an ever turning key. This key might well have been the formula *All is equal, all unequal,* which Goethe
successfully applied to the morphology of plants and animals, and also to works
of art. Here is a wonderful quote from a later essay on the fine arts, in the
original German:

"Alles, was uns daher als Zierde ansprechen
soll, muss gegliedert sein, und zwar im höhern Sinne, dass es aus Teilen
bestehe, die sich wechselsweise aufeinander beziehen. Hiezu wird erfordert,
dass es eine Mitte habe, ein Oben un Unten, ein Hüben und Drüben, woraus zuerst
Symmetrie entsteht, welche, wenn sie dem Verstande völlig fasslich bleibt, die
Zierde auf der geringsten Stufe genannt werden kann. Je mannigfaltiger dann
aber die Glieder werden, und je mehr jene anfängliche Symmetrie, verflochten,
versteckt, in Gegensätzen abgewechselt, als ein offenbares Geheimnis vor unsern
Augen steht, desto angenehmer wird die Zierde sein, und ganz vollkommen, wenn
wir an jene ersten Grundlagen dabei nicht mehr denken, sondern als von einem
Willkürlichen und Zufälligen überrascht werden."

A key episode from my school
days

Let me recall a key episode from my school
days. A teacher told us that we are forbidden to divide a number by zero. I
replied that I could carry out such a calculation by choosing ever smaller
divisors, until finally, I obtained an infinitely large number; and that when I
would then multiply this number by zero I would obtain 1 and any other number.
My teacher took hold of his iron key and knocked me on the head. When I
repeated my argument in the next lesson he knocked me on the head again. Thus I
came to learn that even mathematics, the kingdom of pure logic, has its
forbidden zones, where logic fails and iron keys are required.

In the light of the above insights I shall put
it like this: the division of a number by zero is a case where we leave the
realm of mathematical logic in favor of the realm of art, life and nature,
where all is equal and all unequal. Mathematical logic is a mental tool of
building and constructing. If we wish to have this tool ready to hand we are
not allowed to divide a number by zero, or to carry out similar operations; but
we are always allowed to investigate the reverse side of the mirror and to
explore the other realm of logic.

Language

Once I was invited to give private lessons to a
boy who had difficulties understanding the concept of equations. I told him:
let us put away the schoolbooks and instead have a look at language. I asked
him to give me an example of a sentence, and he did so. I then showed him that
he had just formulated a verbal equation. As I don't recall his example, I
shall invent a new one:

The
football game will take place tomorrow.

This sentence contains an equation:

the
football game - is - a tomorrow's event

We considered other sentences, and they always contained
one or several equations.

A
car passes by / a car - is - something that passes by

We see a movement, we recognize a car, and the
car and the act of moving belong together, allowing us to formulate an
equation.

The
flowers are blooming

We see a happy sway of colors, we identify it
with plants, and we formulate an equation.

Language is the means of winning the help and
care of others on whom we depend in one way or another. By speaking we are
building a world into which we invite our listener, and we shape and color it
in such a way as to please him and to win his help and support. In order to do
so we require a mathematical logic that is present in the basic structure of a
sentence. All children, even those not good at mathematics, can establish this
'verbal equation' easily and naturally. It may well be that many children who fear mathematics are
capable of building the best-formulated, most elegant and colorful sentences.

A further solace

A further solace for those less proficient at
mathematics. There are two realms of logic: a mathematical one; and another - a
treasury of mathematical and scientific laws which have not yet been
discovered. Those good at mathematics may be clever in a limited way, in the
realm of known laws; while others may be clever in handling the many still
unknown laws that nevertheless rule our lives. Dear teachers, you know well
that half of your work involves motivating your pupils, and that the best way
to this is to recognize and name the various abilities and talents of each
individual child.

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