Re: Universal grammar

In article <1161858852.901788.39420@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Franz
Gnaedinger" <frgn@xxxxxxxxxxx> wrote:

I can't help you wit the Goethe idea, not knowing much about the context.

When I was a teenager I got a book on quantum physics.
A footnote said that the basic formula of mathematics,
q equals q, has not yet been investigated. So I began to
ponder this equation. What does it say? It speaks of a thing
or a being named q, and of another thing or being called q,
and it claims that both are identical. Are there any identical
things or beings? really identical ones? Apples vary in form,
size, color and taste. Even homozygotic twins differ, their
mothers can tell them apart ... One may read the above
equation in a second way: one thing or being q remains itself,
doesn't change in time, stays self-identical. I myself am always
changing, every day, year, decade. And what happens to an
apple when I eat it up? We may look at the basic formula of
mathematics in either way, it has no reality in our observable
world. The only objects that really satisfy the equation are
numbers and other mathematical objects. What are they good
for? They are ideals of our technical world. Let me explain this:

b = b = b = b ... --- with bricks (b, b, b, b ...) of the same
size one can build a stable wall

b = b --- if the bricks (b) keep their shape, neither soak
in the rain nor crumble in the summer heat, the wall remains
stable and solid

0.999... = 1 --- a door (0.999...) and its frame (1) must
correspond, otherwise the door is stuck, or there is a draft

9 = 1 + 3 + 5 = 9 --- one wishes to dismantle a machine (9)
into its compononents (1, 3, 5), in order to clean them, or to
repair the machine, and to reassemble them in order to get
back the well functioning machine one had before (9)

When I came this far, I read Goethe's formula "All is equal,
all unequal ..." (Wilhelm Meisters Wanderjahre, Aus Makariens
Archiv, at the end, originally meant to be in the middle) and got
excited. Goethe must have made the same discovery as I did,
long before me, and he explains his idea in many ways: he
applies it in his metamorphoses of plants and animals (one
key idea of the former has been confirmed by modern biology,
see Stephen Jay Gould, The Structure of Evolutionary Theory);
also when explaining the special symmetry of art; and in his
book on the Italian travel he speaks of an ever turning key,
which may well be the formula "All is equal, all unequal ..."

The story in working math, is that one pretty much must have a theory with
equality, and therefore there is such a concept also in formal
metamathematics. For example, two sets A, B are equal, written A = B, if
they have the same members, i.e., all x (x in A <=> x in B). Since all
modern math is thought to be expressible using some sort of set theory,
all of it has an equality, at least on the set level.

But one does not get to know an answer to the philosophical question "what
is equality, and why is it there". It is simply so that experience has it
that it is useful in theoretical modeling, and therefore it is there from
the practical point of view.

But the point is very much that there is no universal mathematical
grammar. Real working mathematics in reality passes between several formal
theories, each having its own formal language.

So we have reached an agreement. There is no universal
mathematical grammar, ...

That is, from the practical point of view, as we are dealing with humans
that need to write stuff parsable to humans. It is the experience that in
math, notions and notation must flow together. So this plus tradition
mainly dictates the notational system used.

From the formal point of view, it is believed that all modern math can be
expressed using some sort of set theory (sorry for the repetitions :-)),
so one just has to invent a notation for that, and all math becomes
expressible. But that is as useful to humans as writing computer programs
in machine code only.

...and Einstein's dream of a physics
based solely on mathematical numbers such as 1, 2, pi, e,
won't come true either:

And was there really a Newton theory that the Moon is a cheese?

...mathematics is based on the formula
a equals a, while language and physics belong to the real
world where Goethe rules: all is equal, all unequal. Note well
that his formula includes the basic equation of mathematics,
and so we can say that mathematics - the logic of building,
constructing, maintaining - is entangled with every part of the
world, while covering only half of the truth. Einstein must have
guessed this; he said mathematics is exact as long as we don't
apply it to problems of the real world, but no longer exact when
applied to a real world problem.

Most, if not all math, has its origin in modeling of the real world.
It evolves with continuos cross-feeding to the modeling of the real world.
For example, Gauss made progress in developing number theory when working
on computing planetary orbits - he was officially an astronomer, not a
mathematician. In modern days, number theory has become very useful in
cryptography, for example. Boole made his ideas in the form of logical
modeling, which later became very useful when computers as made. Cantor
developed his ideas of infinities when working on the convergence of
Fourier series. Newton and Leibnitz, who communicated their ideas of
calculus with each other, got their ideas from Archimedes, I think. And
the so called Pythagorean theorem was known in old Babylon (or
Mesopotamia), for example a 3-4-5 triangle, easy to produce with a piece
of string, was used to measure up land, there and in old Egypt. And
Archimedes realized that the Earth is round, and even measured up its
size. Columbus knew this, of course, but made reasoned the computations
were wrong error, because he then upper his chances to get money for the
trip he eventually made.

This can also be seen in
language. I remember an article in The American Scientist
on the ambiguities of English words such as same, equal,
identical. These words are fuzzy, depending on the context.
Identical in a "normal" sense means very much the same,
equal in all important features; in mathematics the same word
has a different meaning, really really identical, in every aspect.
I don't see any way to get rid of such ambiguities, ...

In math, there are no such ambiguities. For example, "identical" might
mean equal as sets. The one defines an equivalence relation, which defines
the equality used in practice.

..and assume
they belong intrinsically to language that works on many levels
at the same time. The wonder of language is that we can say
so much about the world with a limited set of some 26 letters,
a b c d e f g h i j k l m n o p q r s t u v w x y z, 10 digits, 0
1 2 3 4 5 6 7 8 9, and some extra signs.

Unicode has some 100000 plus characters in it, I think. And, in the other
direction, any positive integer can be expressed using "unary" notation
1...1. So you just need a symbol '1' and a symbol like ' ' that separates
the numbers.

This wonder is only
possible by means of ambiguity - a negative term, should
I better say polyvalency?

I think ambiguity only arises because humans are used to, and easily can
cope with such. They are not as such needed for the development of a
formal theory.

--
Hans Aberg
.

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