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 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 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?