Re: Universal grammar


Hans Aberg wrote:

All one can say is that the logical foundations of mathematics have
not yet been cleared up.

Mathematical logic is a techno-logic, concerned with building,
constructing, maintaining: www.seshat.ch/home/equal.htm

And here you are with the universal grammar of Rupert
Ruhstaller, based on functors and arguments, visualized
in budding circles: www.seshat.ch/home/grammar.htm

This grammar will hardly help you, so let me introduce my
grammar of equations from 1974/75. Every basic sentence
can be given as an overlapping of three sets, one of them
belonging to the word be, or it can be given as an equation,
or it can be given as an object:

overlapping sets: ball be round
sentence: The ball is round.
equation: the ball - is - a round one
object: the round ball

overlapping sets: ball be roll
sentence: The ball rolls.
equation: the ball - is - a rolling one
object: the rolling ball

You recognize an object you could touch and hold: ball.
The object is for real: be. You notice a smooth movement
along a slight slope, at the same time a revolving movement
of the object: roll. The ball, the being, and the rolling occur
at once, at the same place, in the same object. The sets
overlap, the equation is formed, the object named: the ball
is rolling, the rolling ball - also the being of the rolling ball,
the rolling of the being ball, whatever comes first to your
attention.

Most sentences are conglomerates of explicit and implicit
equations. Take a sentence from a poem, novel, scientific
text, any other text, and transform it into equations. A good
exercise. Will show you whether you understand the sentence
or not, and if not, where you fail. Here you are with two lines
from Shakespeare's poem Venus and Adonis, explained in
equations:

When he beheld his shadow in the brook
The fishes spread on it their golden gills

He - is - Adonis, or perhaps Shakespeare himself
Shakespeare or Adonis - was - making an observation
what? - was - observed or beheld by him
the beheld - was - his own shadow in the brook
fishes - were - swimming through the shadow
the fishes in the shadow - were - hardly visible
visible - were - their gills
the gills - were - reflecting a light and thus shining
golden, although the fishes themselves were hardly
visible in the shadow cast by the poet or his alter
ego in the poem

An elementary school teacher was pleased with my method
of dealing with sentences. The son of a boss of mine had
a problem grasping the concept of mathematical equations
while being good at language. I told him: forget about numbers
and give me a sentence. He invented a sentence and I turned
it into a series of linguistic equations. Then we did many more
sentences together, for nearly two hours. He got the knack
of it, and in the end he lost at least a part of his fear of
mathematical equations.

Now my proposition for you, Hans. When you drive a car
you follow a street, you hardly cruise across meadows and
bushes. Prepare also a "street" for your program: by turning
a mathematical text into basic equations. The first sentence
in chapter 1 of Andrew Wiles' proof of Fermat's Last Theorem
goes:

Let p be an odd prime.

In the case of poetry (Shakespeare above) one has to add
a lot of words. In the case of mathematics one can leave out
all the formulations that make a text nice and easy to read.
In this case we don't have to care about the "let be." We
can just say that p is an odd prime. The adjective odd is
an abbreviation of a sentence, as above, where the round
ball and the rolling ball are sentences or equations turned
into objects. So we got three basic equations:

p - is - a number

p - is - odd

p - is - prime

Transform the whole proof into basic equations (with the help
of a team of mathematicians), then run your program. Don't
worry about the big number of equations, a computer can
"handle" them. All that matters are that all mathematical
statements are correctly transformed into basic equations.

I hope this was more than a triviality.

Regards Franz Gnaedinger

.

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