Re: Universal grammar

Hi Franz,

Franz Gnaedinger wrote:
Hans Aberg 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?

Wittgenstein thought a lot about this. Not surprising as he was a
student of Russell, who was attempting to find a logical basis for
mathematics.

You might be interested in something I found the other day. It is...
Thomas Kuhn, The Structure of Scientific Revolutions, p.g. 44-45
(citing Ludwig Wittgenstein, Philosophical Investigations, trans. G. E.
M. Anscombe, pp 31-36):

<<<
What need we know, Wittgenstein asked, in order that we apply terms
like 'chair', or 'leaf', or 'game' unequivocally and without provoking
argument?

That question is very old and has generally been answered by saying
that we must know, consciously or intuitively, what a chair, or a leaf,
or game _is_. We must, that is, grasp some set of attributes that all
games and only games have in common. Wittgenstein, however, concluded
that, given the way we use language and the sort of world to which we
apply it, there need be no such set of characteristics. Though a
discussion of _some_ of the attributes shared by a _number_ of games or
chairs or leaves often helps us learn how to employ the corresponding
term, there is no set of characteristics that is simultaneously
applicable to all members of the class and to them alone. Instead,
confronted with a previously unobserved activity, we apply the term
'game' because what we are seeing bears a close "family resemblance" to
a number of the activities that we have previously learned to call by
that name. For Wittgenstein, in short, games, and chairs, and leaves
are natural families, each constituted by a network of overlapping and
crisscross resemblances. The existence of such a network sufficiently
accounts for our success in identifying the corresponding object or
activity. Only if the families we named overlapped and merged gradually
into one another--only, that is, if there were no _natural_
families--would our success identifying and naming provide evidence for
a set of common characteristics corresponding to each of the class
names we employ.
<<<

So first we have a definition of "meaning" as "some set of attributes
.... in common." Note: a _set_.

Then Kuhn says Wittgenstein makes a distinction:

"No natural families": "set of common characteristics", i.e. common to
all and sufficient to the definition (Note: this would mean a
representation for meaning could be compressed, because you only need
to keep this minimal set of attributes.)

"Natural families": "there is no set of characteristics that is
simultaneously applicable to all members of the class and to them
alone." (Note: this would mean a representation for meaning could not
be compressed, because there is no single set which captures all the
commonalities.)

(Though I think the actual word "natural families" is Kuhn's.)

When I came this far, I read Goethe's formula "All is equal,
all unequal ..."

I don't know much about Goethe either, but I do think this issue of
being simultaneously the same and different is at the heart of the
matter, and it is related to that of incompressibility.

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.

Compare this with Kuhn's "there is no set of characteristics that is
simultaneously applicable to all members of the class and to them
alone." Hans is really saying mathematical grammar is incompressible.

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

This is a very perceptive comment. You have hit the nail on the head.
We need to start thinking of ambiguity as re-use of symbols, and a
positive thing. Polyvalency might work very well.

The thing about ambiguity is that you can make your representation work
harder. It is the sets a token participates in which matter, not the
tokens themselves. By combining them in different sets we can make the
same tokens mean so much more.

Now, letters give us a lot of power to represent words. But they are
not the best example. You can expand out all the power of ambiguity for
letters, use it all up, in a sense. We don't use more of the power of
letters to recombine in different ways than we need to assign one
combination for each word.

What is special about language is that when we get to the level of
syntax, we give this power of ambiguity its head. We let words combine
in new ways. You can use up all the power of ambiguity of letters by
expanding out all their combinations and having a separate symbol for
each word, but you can't expand out all the sets of words to find an
unambiguous symbol for each. We can't even find sets for kinds of sets
(grammar.)

I think this is because words take part in what Kuhn calls "natural
families."

Because "natural families" are the only time you can't tame the power
of ambiguity to confer more meaning by forming new sets, systems with
attributes displaying "natural families", sets for which Kuhn says
"there is no set of characteristics that is simultaneously applicable
to all members of the class and to them alone", incompressible sets,
are the ones we call ambiguous.

"Ambiguous" has been seen as bad. "Incompressible" has been seen as
bad. But they are good things, things which give us more power, not
less.

According to Kuhn's interpretation of Wittgenstein, language, due to
irreducible sets of associations specifying meaning, is incompressible
("there is no set of characteristics that is simultaneously applicable
to all members of the class and to them alone".) I think he is right.

-Rob

.

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