*All IS EQUAL, ALL UNEQUAL … */
© 1975-2003 by Franz Gnaedinger, Zurich, fg(a)seshat.ch, fgn(bluemail.ch /
www.seshat.ch

*ALL IS EQUAL, ALL
UNEQUAL … (new version)*

a
paper for the *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
understanding 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, have the same 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, 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

Each
snowflake forms a hexagon, yet seen under a microscope every flake has its own
unique pattern.

A mouse is not an elephant

Mice
and elephants belong to the animal kingdom, and are 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 = p = ...

Physicists
search for elementary particles that fulfill the above equation. They explore
and expand the realm of technology, without 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, given in the original German: *Alles, was uns daher als Zierde ansprechen soll, muss
gegliedert sein, und zwar im höheren Sinne, dass es aus Teilen bestehe, die
sich wechselsweise aufeinander beziehen. Hiezu wird erfordert, dass es eine
Mitte habe, ein Oben und 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 this
calculation by choosing ever smaller divisors, until, finally, I obtained an
infinitely large number, and that when I would then multiply this number with
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 arguments 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 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’ quite 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 who are 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 do this is to recognize and
name the various abilities and talents of each individual child.

*ALL IS EQUAL, ALL UNEQUAL … (old version)*

Mathematical logic is based
on the well-known formulas

a = a

and

a unequal b

Equations can be regarded
as technical situations:

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

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

b = b

The bricks should neither
soak nor crumble in the rain, nor crackle in the summer heat; their properties
must be stable, they must remain 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), or else it will be too tight, or there will be a draft.

9 = 2 + 3 + 4 = 9

In order to clean or repair
a machine (9) one wishes to dismantle it into single components (2, 3, 4); then
one will want to reassemble the parts, in order to return it to its former
well-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 run today.

Looked at the other way
around, an achievement such as building a pyramid would not have been possible
without mathematical knowledge.

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
while one fruit may be red and sweet, another may be 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

Each snowflake forms a
hexagon, yet seen under a microscope every flake shows a particular pattern.

A mouse is not an elephant

Mice and elephants belong
to the animal kingdom, and are 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 = p = ...

Physicists search for
elementary particles that fulfill the above equation. They explore and expand
the realm of technology, without end.

Culture as a task:
integrating man-made things into our lives and natural surroundings, utilizing
them in the best possible way, minimizing their risks, preventing their abuse,
developing suitable forms of life.

Art: human measure in a
technical world.

May I add my definition of language? We depend on others and their
manifold abilities, and we convey our needs and wishes we are unable to satisfy
alone by means of language: facial expressions, signs, body language, sounds
and words. Using words we describe and explain our situation in such a way as
to gain the understanding, esteem, sympathy and help of our listeners. The more
manmade things we use the more complex and specialized our lives become, and
the more words we require. Yet every means of conveying needs and wishes in
order to obtain help from others may be regarded as language.

While the mathematical formulas
are the core of technology, a work of art conveys a more human measure.

While the natural sciences
rely on mathematics and often generalize a successful insight, the humanities
rely on art and keep alive our sense of the manifold aspects of life and
nature.

Mechanical devices and
machines have been very successful, and the mechanical paradigm has been
generalized: the cosmos regarded as a clockwork mechanism, animals seen as mere
automata, incapable of feeling.

While the natural sciences
are prone to generalize a successful insight and ignore other, no less
important aspects of life and nature, the humanities keep alive a sense of
complexity.

Some female figurines from
predynastic Egypt anticipate the Greek goddess Gaia. The idea of a biosphere, a
kind of living earth, came to the fore again in the work of James Lovelock and
Lynn Margulis.

The ancients regarded the
human being as a tiny cosmos in a living universe represented by various
deities. Are modern views so very different from ancient? Fractal geometry
allows the same form to appear on every scale, while we consider the principle
of life to be present in the universe or in matter (Carl Sagan).

Matter and energy are
equivalents (Albert Einstein). May the same hold for energy and information?
When we carry out a task in a clever way we require less energy. Might
'intelligent' matter from a new physics eventually circumvent the second law of
thermo- dynamics? (an old fantasy of mine)

Silly
questions of mine, formulated in my own personal freestyle English (on which I
hold a copyright, so please no one dare copy my mistakes ;-) On which level
does life emerge? on the level of molecules? or may there be a deeper
organization of living matter? Is an electron in a stone really the same as an
electron in a living cell? If there should be a subatomic organization of life,
how deep would it reach? where does life begin? A philosophical principle of
mine: no statement is absolutely true, and none absolutely wrong. Spirit:
innermost order of matter?