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The last of today's excerpts. From page 43 of "Life,
Itself":
Robert Rosen wrote: "An essential part of the inner world of
any self is one's language. It is a way, or reflects a way, of organizing
percepts and perhaps even of generating them.
Language itself creates, or embodies, a new dualism distinct
from (but in many ways parallel to) those we have already discussed. Indeed,
language is a unique and anomalous thing, whose acquisition, and even more,
whose correct deployment, is a kind of miracle. I cannot dwell on these matters
here but rather will concentrate on its essential role as an intermediary
between the self and its ambience, and between one aspect of part of the self
and another.
The first basic dualism inherent in language is that (1) it is
a thing in itself and (2) permits, even requires, referents external to itself.
These embody respectively what we will call the syntactic aspects of language
and its semantic aspects. Roughly speaking, syntax pertains to what language is,
as a thing in itself, while semantics pertain to extralinguistic referents.
These referents may involve the self, or the ambience, or both, or even
neither.
Let us consider syntactic aspects first. Syntax involves its
own inherent dualism, which may be roughly described as the dualism between
proposition and production rules. From a syntactical point of view, divorced
from any external referents, propositions in the language are in general not
about anything and are described entirely in terms of conventional symbol
vehicles: letters, words, sentences, and so forth. The production rules are
themselves propositions, but they do have referents, namely, other propositions
in the language. Their role is essentially a dynamic one, to enable the
construction of new propositions from given ones, or the analysis of given
propositions into simpler ones.
The syntactical production rules of a language are its internal
vehicles for what I shall call "inferential entailment". The rules thus allow us
to say, without consulting any external referent, that one proposition, or group
of propositions, implies others. More generally, inferential entailment is a
relation between propositions and means precisely that there is a string of
production rules whose successive application will take us from some of them to
the others.
Just as nobody has been able to characterize an organism in
terms of a discrete list of properties, no one has been able to characterize a
"natural language" (let us say English) in terms of a list of production rules.
Indeed, if it were possible to do this, it would be tantamount to saying that a
(natural) language can be completely characterized by syntactic properties
alone, i.e., made independent of any semantic referents whatever. There have
indeed been deadly serious attempts to do precisely this (see my remarks on
formalization below). They have all failed, often rather dramatically,
indicating (what might be obvious) that, in general, semantics cannot simply be
replaced by more syntax. Nevertheless, the attempt to do so has served to
extract various kinds of syntactical "sublanguages": these will play an
analogous rule, in the external world of the self, to the segregation of systems
in one's external world or ambience. Indeed, as we shall soon see, there is more
than just an analogy here.
We shall understand by a formalism any such "sublanguage" of a
natural language, defined by syntactic qualities alone. That is, a formalism is
a finite list of production rules, together with a generating family of
propositions on which they can act, without any specification or consideration
of extralinguistic referents. Thus, a formalism, as a fragment of natural
language, could be "about" something (i.e., endowed with extralinguistic
referents), but it need not be. A formalism, by its very nature, carries with it
no "dictionary" associating its propositions with anything outside itself. It is
propelled entirely by its own internal inferential structure, as embodied
explicitly in its production rules. These and these alone determine the
relations among the propositions of the formalism, which we have called
inferential entailment.
As we shall see, the extraction of a formalism from a natural
language has many of the properties of extracting a system from the ambience.
Therefore, I shall henceforth refer to a formalism as a "formal system"; to
distinguish formal systems from systems in the ambience or external world, I
shall call the latter "natural systems". The entire scientific enterprise, as I
shall soon argue, is an attempt to capture natural systems within formal ones,
or alternatively, to embody formal systems with external referents in such a way
as to describe natural ones. That, indeed, is what is meant by
"theory".
A prominent trend, indeed a characteristic one, of contemporary
science and mathematics is to try to dispense with extralinguistic referents
entirely and replace them with purely syntactic structures that only recognize
and manipulate the symbols of which the propositions themselves are built. This
process is, naturally enough, called "formalization". It involves the
internalization of semantic referents, in the form of additional purely symbolic
syntactic rules.
This idea of formalization, that the semantic aspects of
language can always be effectively replaced by purely syntactic ones, will turn
out to be another place where really serious trouble creeps in. Indeed, Gödel
showed in effect that it was already false for Number Theory. It will turn out
to be closely related to the reductionistic idea that there is always a "largest
model", as I shall later describe in detail (see chapter 8). For the moment,
however, I simple suggest the reader bear in mind the basic conclusion we can
distill from the discussion above: natural language is not a
formalization.
The study of formal systems is what comprises the subject of
(in the broadest sense) mathematics. Its object is the universe of formal
systems, just as real and significant a part of the self's internal world as are
the natural systems one extracts from one's ambience. Seen in another way,
mathematics is the study of inferential entailment, the art of extracting
inferents from the premises or hypotheses.
I conclude this brief consideration of language by pointing out
two aspects of natural language that will play key roles in what follows but
that never end up as part of formalisms. These are (1) the use of the
interrogative, to which I have already alluded and (2) the use of imperative.
The latter, for example, is universally presupposed, even in mathematics;
an algorithm, for example, is nothing but a string of imperatives, ordering us
to apply specific production rules to specific propositions, assuring us that if
we do so, some definite end will thereby be entailed. In the world of natural
systems, similar lists of imperatives constitute recipes, protocols, blueprints,
and the like, which govern fabrication. But, as will become apparent, the
entailment process embodied by algorithms or recipes is very different than that
governing their application. The difference, indeed, is precisely the difference
between fabrication and physiology, which I contrasted earlier (see section 1C
above). And the difference between them will provide another central feature of
our overall enterprise."
I think I'll end things there. The chapter goes on, of course, but
begins to change into a discussion of certain aspects of the points made
above...
I hope these three excerpts will be of use; particularly to
people who don't have any of the books.
Cheers,
Judith Rosen
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