Re: Universal grammar

In article <1161418795.730995.77650@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Franz
Gnaedinger" <frgn@xxxxxxxxxxx> wrote:

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

There is no problem dealing with that, as in metamathematics, one
introduces the notion of a theory with equality.

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.

Perhaps you might want to use ASTs (abstract syntax tress - see
Wikipedia), in order to get language independent representations. For
example
  The ball is round.
might be represented as the tree
         is
        /  \
      the   round
      /
    ball
That is, "is" acts as an operator with two arguments. The first argument
has what computer scientists call a closure:
     the
      |
     ball
made up of a (linguistic) quantifier and a noun. And similarly for the
second argument.

Most sentences are conglomerates of explicit and implicit
equations.

The generalization is to deal with logical equations. In computer science,
the first stepping stone are Prolog like programs. These
handle equations without bound variables (as quantifiers "all", "exist",
the Church "lambda" = function mapping "|->", etc.) and is doing only a
first depth search. The latter means that when attempting to prove the
statement X by searching a clause
  A :- A_0, ..., A_k
corresponding to a provability statement
  A_k, ..., A_0 |- A
(if A_k, ..., A_0 are valid object formulas, then so is A), it first uses
a unification u(X, A) as to obtain a valid substitution s, and adds
  s(A_0), ..., s(A_k)
as new conditions to prove, but the search is then only done sequentially.
This way, one will miss valid proofs if there is an infinite branch in the
proof, as not proof possibilities are searched.

So when turning a Prolog program into a proof checker, there are some
things one must do:
- Add operators that bind variables - very difficult to get it right in
practical implementation.
- Add breadth first search of the proof tree.
- Admit unification branching (get a sequence of substitutions from u(X, Y)).

    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.

In addition, math is structured into what corresponds to what computer
scientists call environments (constructs like {...}) and modules
(larger logical units).

In reality, working math is not dealing with a single metamathematical
object theory, but a sequence of them. In addition, it is easy for a human
to jump out of any limited metamathematical structure. There are a number
of interesting problems.

--
Hans Aberg
.

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