> Is mathematics language, a language, a form
> of language? This old question found an answer.
> Mathematics may be regarded as a corner in the
> triangular definition of language (as given in part 1).
> Also computer code is language, close to the
> mathematical corner, connecting people and their
> computers, machines being extensions of body
> and mind:

>   people - machines - code - machines - people

> Art is another corner of the language triangle.
> Song and dance, music, literature and painting
> - all sorts of visual and auditory and other sensual
> messages - are forms of language.

> Goethe, in his Diary of the Italian Journey, mentions
> an 'ever turning key' which I identified as his formula
> 'all is equal, all unequal' that he applied in his studies
> of the metamorphoses of plants and animals, and
> of visual art. In a later essay he wrote a wonderful
> passage on symmety and symmetry breaking:

>   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.

> I won't dare translate these lines, but render them
> in my own words, along the use I made of them in
> interpreting several paintings (turning the key, playing
> the game of equal unequal). Mathematical symmetry
> based on the equation  a = a  requires a center,
> a left and right, an above and below that mirror each
> other. This mathematical symmetry is decorative on
> the lowest level. Higher levels are obtained by varying
> the sides, and the above and below. The more variations
> come together, hiding the initial symmetry, the more
> pleasing the picture will be ... I may add the element
> of completion: all is equal, all unequal, in such a way
> that the sides and elements complete each other
> and form an entirety, at least in a great work of art.
> And this organization - the arrangement and functioning
> of the parts within the whole, on the most abstract level -
> is what we call soul in a living being, a term as real
> as information on the mathematical side.

> Allow me one more prediction. If John Archibald Wheeler
> was right and all of physics will one day be rendered in
> terms of information, the concept of the soul will emerge
> as a physical category and organization principle, useful
> in arranging information. Compiling vast masses of data
> will become a serious problem. Studying art will help
> solve that problem.

> --------------------------------------------------------------------------- ----

> > all is equal, all unequal
> > part 3
> > contrapposto

> > Our teacher of ancient Greek was a fan of Greek art
> > and placed pictures of sculptures in the showcase.
> > I remember having spent hours on a pencil drawing
> > from a beautiful marble head of a woman, and I fell
> > in love with one of the Korae of the Erechtheion on
> > the acropolis of Athens. Our teacher of Greek spoke
> > about the contrapposto: the human body appears
> > symmetrical, while variations in the pose, often
> > along a curved axis, make a sculpture more lively.
> > One day in 1965 we read a poem by Archilochos.
> > Two lines are missing, we were told; I don't remember
> > whether at the end or the beginning. I used the principle
> > of the contrapposto in order to reconstruct the missing
> > lines that should mirror the beginning or the end while
> > there should also be a difference that summarizes
> > what happens in the poem (or in a good movie, as I
> > was to find out later). My reconstruction of the missing
> > lines was convinving, although not in ancient Greek.

> > Ten years later I began reading Goethe and soon found
> > what I call his world formula: "Alles ist gleich, alles
> > ungleich (...)" in Maximen und Reflexionen, also in the
> > novel Wilhelm Meisters Wanderjahre, Aus Makariens
> > Archiv, at the end of the novel, but originally planned
> > for the middle: all is equal, all unequal ... I was delighted,
> > for this was the shortest possible way of rendering the
> > contrapposto, most generally, and meeting my study of
> > the basic mathematical equation .

> > In around 1980, when physicists hoped to complete the
> > 'zoo' of elementary particles, I predicted that this won't
> > happen, because the really elementary particles obey
> > the formula  e = e = e = e = e = e = e = ... that names
> > only the equal while leaving out the unequal. Meanwhile
> > I dare say that cosmology, based on an atomic world,
> > unavoidably encounters 'spooky' phenomena such as
> > entanglement and non-locality, greetings from the
> > other side of logic, as it were. We may well count with
> > further cosmological revolutions, and the last word
> > about the fate of the universe has not yet been said,
> > will never be said, actually.

> > Galilei called mathematics the language of nature.
> > God may be able to understand all of the world in
> > mathematical terms. Our mathematical knowledge
> > is and will be limited. For us, there is always a beyond
> > of mathematics, an unequal to the equal, and an equal
> > to the unequal.

> > ---------------------------------------------------------------------------

> > > all is equal, all unequal
> > > part 2
> > > mathematics

> > > On my 21st birthday, in 1970, I got a book on the
> > > philosophy of quantum mechanics. It contained
> > > an intriguing footnote saying that the basic equation
> > > of mathematics  p = p  had not yet been examined.
> > > In 1974/75 I found an answer. Mathematics ponders
> > > the properties and relations of ideal objects that
> > > fully satisfy the equation  a = a  meaning that an
> > > ideal object 'a' is perfectly identical with another
> > > or any other ideal object 'a', and that an ideal object
> > > 'a' remains unchanged forever. Not so in the real
> > > world. An apple is an apple, yet one apple may be
> > > read and sweet, another green and sour, and what
> > > if the apple is eaten? Mathematical equations can be
> > > interpreted in a technical context:

> > >   b = b = b = b = b = b = ...

> > > If the bricks 'b' have the same consistency, form and
> > > size, a wall is easily built

> > >   b = b

> > > and if each brick remains the same, keeps its consistency,
> > > form and size, does neither soak in the rain nor crack in
> > > the summer heat, the wall will stand.

> > >   9 = 3 + 2 + 4 = 9

> > > A machine (9) can be dismantled (3, 2, 4) and reassembled
> > > (9), either for the purpose of repairing, or cleaning.

> > >   1 = 0.999...

> > > A door (0.999...) and the door frame (1) must match,
> > > or else the door is jammed, or there is a draught.

> > >  3 + 0.999... + 6 = 10 = 3 + 0.999... + 6

> > > If you close the door (0.999...) it becomes part of the wall (10),
> > > but then you can open the door again (3 + 0.999... + 6).

> > > I invented numerous technical situations to a variety
> > > of simple equations, and over the years I saw my belief
> > > confirmed: mathematics may be regarded as the logic
> > > of building and maintaining. It is not a lucky coincidence
> > > that mathematical discoveries lead to technical inventions,
> > > the other way round: mathematics, being the logic of
> > > building and maintaining, of technics, paves the way
> > > for technology.

> > > (to be continued)

> > > --------------------------------------------------------------------------- ­---