3B, "The First Basic Dualism": "Science is built on dualities. Indeed,
every mode of discrimination creates one. But the most fundamental
dualism, which all others presuppose, is of course the one a
discriminator makes between self and everything else.... At any rate, =
we
know our self with ultimate certainty, even though this knowledge is
subjective ... What else is there? Whatever it is, I shall call it the
/ambience/.... this is the external world, the world of objective
reality, the world of phenomena.... Science, in fact, requires both; it
requires an external, objective world of phenomena, and an internal,
subjective world of the self, which perceives, organizes, acts and
understands."
3C, "The Second Basic Dualism": "Our second basic dualism concerns the
way we partition our ambiences, the way we manage our perceptions of =
the
external world.... It rests on a consensus /imputed/ to the ambience =
...
It is the dualism between systems and their environments.... The
partition of ambience into system and environment ... is a basic though
fateful step for science.... Systems and environments are thenceforth
perceived in entirely different ways ... system gets described by =
states
... environment is characterized rather by its effects on system.
Indeed, it is precisely at this point that, as we shall see, =
fundamental
trouble begins to creep in; already here."
3D, "Language": "An essential part of the inner world of any self is
one's language.... Language itself creates, or embodies, new dualism =
...
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 call the syntactic aspects of =
language
and its semantic aspects... Syntax involves its own inherent dualism =
...
between proposition and production rules.... The syntactical production
rules of a language are its internal vehicles for what I shall call
inferential entailment.... 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.... 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.... The 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."
Rosen concludes this section "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 ...
and (2) the use of the imperative.... 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."
3E, "On Entailment in Formal Systems". Rosen proposes: "suppose that we
step outside our formalism and contemplate one of its theorems P....
>From that perspective, we can interrogate ... we can ask: why is P true
in the system?" Rosen proposes three operations we can perform on such =
a
system: "We can change an axiom, without touching the inferential rules
... We can change an inferential rule, without changing either the
axioms or the list of which that constitutes our algorithm. Finally, we
may change the algorithm, without affecting either the axioms or the
rules themselves.... the kinds of changes we have contemplated all come
from outside the formalism ... from the standpoint of the formalism,
anything that happens outside is accordingly unentailed.... This is our
first glimpse of a peculiar thing ... namely, that though formal =
systems
allow us to talk about entailment in a coherent way, from their
standpoint everything important that affects them is itself =
unentailed."
Rosen says that this discussion should remind us of Aristotle: "we have
paralleled three of his four categories of causation; specifically, if
we call theorem P an /effect/, we may identify his idea of material
cause of P with the axioms of a formalism, his idea of efficient cause
of P with its production rules, and his idea of formal cause of P with
the specification of a particular sequence or algorithm of production
rules.... The reader may not be surprised to note that we do not see a
formal analog of Aristotle's fourth causal category, which he held to =
be
the most significant; namely, the category of final cause.... In any
formalism, there is a kind of natural flow from axioms to theorems, =
very
much like the familiar unidirectional flow of time.... The three
'traditional' causal categories (formal, material, and efficient
causation) always respect this flow of 'formal time' ... Final =
causation
gives the /appearance/, at any rate, of violating this flow."
3F. "On the Comparison of Formalisms." Rosen argues that "mathematics,
in the broadest sense, is the study of formalisms and that formalisms,
in their turn, are parts of natural language." Rosen asks: "When does
one formalism subsume another, so that the second can be in some sense
generated from the first, or embedded in it? And above all, is the
machinery for dealing with such questions, i.e., with the comparison of
formalisms, itself a formalism?"
Rosen goes on to discuss coordinate systems, transformations and
modeling relations. Rosen notes: "In order to compare [two formalisms],
we need to ... express what each formalism says to itself in the
language of the other." We need a pair of dictionaries: an encoding
dictionary and a decoding dictionary. If these can establish a modeling
relation between the inferential structures of the two formalisms such
that one is a model of the other and one is a realization of the other.
Rosen's primary point here is that the encodings and decodings are
unentailed within the formalisms themselves: "The comparison of two
inferential structures ... thus inherently involves something outside
the formalisms, in effect, a /creative act/, resulting in a new kind of
formal object, namely the modeling relation itself. It involves /art/".
3G. "Entailment in the Ambience: Causality". "The fundamental question
for us, at this point, is the following: is there, in this external
world, any kind of /entailment/, analogous to the inferential =
entailment
we have seen between propositions in a language or formalism? =
Obviously,
if there is not, we can all go home; science is not only impossible but
also inconceivable."
3H. "The Modeling Relation and Natural Law". Using models, Rosen says,
"We can compare inferential entailment in a formal system with causal
entailments, relating a bundle of phenomena that we extract from our
ambience and identify as a natural system.... the causal entailments
manifested by a natural system provide the orderliness required of the
ambience. Inferential entailment in a formal system is a way of
providing the orderliness required of the self. The act of bringing the
two into correspondence ... is the articulation of the former within =
the
latter; it is in effect science itself.... It is not generally
appreciated, especially by experimentalists ... that any measurement,
however comprehensive, is an act of abstraction ... From this
standpoint, it is ironic indeed that a mere observer regards oneself as
being in direct contact with reality and that it is 'theoretical
science' alone that deals with abstractions."
3I. "Metaphor". Rosen continues: "This modeling relation between two
natural systems N1, N2 is of the most profound importance; I shall call
it analogy.... This is another way of seeing, what I alluded to =
earlier,
that reduction to a common set of material constituents is not the only
way, nor even a very good way, of comparing natural systems.... As we
have seen, the modeling relation is intimately tied up with the notion
of prediction.... insofar as the entailment structure itself is =
captured
in a congruent model, we can actually, in a sense, pull the future of
our natural system into the present.... A large part of the cost =
imposed
by Natural Law, in return for the benefit of prediction, lies in =
finding
the right encodings. But to what extent do we really need these
encodings? Perhaps we can presume a little on Natural Law and get away
without them.... This is the essence of /metaphor/: decoding without
encoding ... Perhaps the most important for our purposes in the machine
metaphor of Descartes ... It asserts that things about machines can be
decoded into predictions about organisms ... Another one of enormous
current importance ... is what may be called the open system
metaphor.... [However] to proceed metaphorically in the above sense [we
must remember that] by giving up encoding, we also give up
/verifiability/ in any precise sense.... Hence the general =
indifference,
if not active hostility, manifested by empiricists to theory couched in
metaphorical terms."
(End 03/24/97 detailed extract from _Life Itself_ ch.3, originally read
between Jan 16 and 20, 1997.)=20
(to be continued)