Re: Universal grammar


Hans Aberg wrote:

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.

Also set theory must be secured against paradoxa, which are
nothing else than real world logic leaking into mathematics.
In the real world, things are equal and different at the same time.
You are a human being, I am a human being, yet you are Hans
the computer crack, and I am Franz the computer moron. An
elephant is not a mouse, yet both are mammals, and descended
from a common ancestor that resembled a mouse and had a small
proboscis ...

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.

Mathematicians consider similar things, and assume that
there are equal things, identical things, absolutely identical
things, and things that never change, and then they build
a logic on these simplified objects. But one can also look
at the world with other eyes and notice differences and
varieties. When Cézanne painted a white wall as background
for a self portrait it wasn't just a white area, he made it bloom
in colors and shadings. White is white - not for a painter.

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.

Also I apologize for the repetition: set theory must be secured
against leakings of real world logic - a equal a unequal a - into
mathematics. One is not allowed to divide a number by zero,
or to construct certain sets.

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

Newton wrote 20,000 pages on astrology etc., but also one of
the most beautiful statements on the sciences: he felt like
a boy playing on a beach where he found some pretty pebbles,
yet there are many more pebbles ...

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.

Thanks for honoring the Babylonians. Also the Egyptians knew
the triangle 3-4-5 and used it for the first systematic calculation
of the circle: www.seshat.ch/home/rhind1.htm

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.

Those ambiguities disappear in mathematics, as a consequence
of allowing only things that are either equal or different, not equal
and different at the same time. This 1 and and this 1 are absolutely
the same number, and they are absolutely identical with any number
1 used by the Babylonians. Problems arise when mathematical logic
is applied to the real world where things are equal and different at
the
same time, and where they change in time.

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.

I think that polyvalency, a positive word instead of ambiguity,
is crucial for life and nature.

Regards Franz Gnaedinger

.

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