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In the ongoing discussion about impredicativity vs predicativity, I
see Glen R. stating that he wants to understand what Robert Rosen meant by the
use of those words. I have already explained that, which is apparently not
enough information. Glen, I think it is time to tell you to RTFM, as you once
put it... (and perhaps I will make use of one of your
ubiquitous wolfish [grin]s.)
I recommend the many discussions on impredicativities in
Essays on Life, Itself. Specifically, on page 34, continuing to page
40; pages 135-140; pages 166-170; pages 292-295, to name a few. On page 160,
there is a short one which I will type into this post:
Robert Rosen wrote:
"On Entailment Structures in
General
Any system of entailments can be probed exactly as
Aristotle proposed probing causality in the material world so long ago-- namely,
by asking the (informal) question "Why?" Basically, the question "Why x?" is the
question "What entails x?"
A system that is weak in entailment is one in which very
few such why questions about members of the system possess answers
in the system-- that is, most of the things in such a system are
unentailed within the system. Accordingly, they must be separately
posited, or hypothesized, from outside. Thus a good measure of the strength of
an entailment system is precisely in how many why questions about its
members can be answered within the system, and how many are dependent on
positing or hypothesizing.
In this sense, syntactic or algorithmic systems, or
formalizable systems, are extremely weak in entailmen6t: almost everything about
them must be posited, and very little is actually entailed within such a system.
That is, almost every why question about what is in the system cannot
be answered from within the system, and thus their answers take us outside the
system. In this sense, the objective appearance of formalizable systems, their
apparent rigor, is an illusion.
In the fourteenth century, William of Occam proposed as a
principle what has become known as Occam's razor: Thou shalt not multiply
hypotheses. It is thus a characteristic of inferentially weak systems that thou
must multiply hypotheses, precisely to compensate for the inferential
weakness, the inability to answer why questions about the system within
the system.
It is a characteristic of strong systems of entailment
that they possess enough entailments to allow closed loops. In inferential
systems, such loops are often called impredicativities (Bertrand
Russell's vicious circles). From a purely syntactic point of view, these are
bad, because they create an inherently semantic element within the system;
indeed, it was the whole point of formalization, or the replacement of semantics
by syntax, to do away with them. The result was a far weaker system
inferentially, with much less entailment and much more that had to be
posited.
In a certain sense, causal entailment structures in the
material world are regarded as requiring no positing at all; every why
question about a material system possesses a material answer. Indeed, the very
concept of positing makes little sense in the context of causal entailment. Of
course, the answer to a why question about a given material system may
not be found within that system itself (unless the system has been forever
closed and isolated). However, in Life Itself, I argue that the systems
we call organisms are, in a sense, maximal in their entailment
structures; asking why about them can be answered in terms of something
else about them. They thus are inherently semantic and contain causal
counterparts of impredicative loops."
So, from this discussion, and others like it that he engages in throughout his books, it should be clear that there are groups of words/concepts that clearly go together: Predicative goes with such concepts as finite, computable, syntax, fractionable, and simple. Impredicative belongs with such concepts as infinite, non-computable, semantics, context dependence, non-fractionability, and complex. If you look in the index in any of his books, and explore the word groups, as listed, you will find many, many references to my father's usage of these terms. There are also certain names, from the history of science and
mathematics, which are associated with the predicative camp (Hilbert, Church,
Von Neumann...) On page 164, there is a discussion about Church's Thesis. It is
summed up nicely in a few paragraphs from that discussion:
Robert Rosen wrote:
"The mathematical content of Church's Thesis has to do
with the evaluation of mappings or operations-- that is, with expressions of the
form Av=w, where A is an operator of some kind, v is
a particular argument, and w is the corresponding value. Church himself
was concerned entirely with this evaluation process, and in particular with
those he deemed "effective". In very broad terms, he tried to identify this
informal notion of effectiveness with the existence of an algorithm, a
mechanical process, a computation, for the production of a value Av
from an argument v. This is how machines (the Turing machines) became
involved in the issue in the first place-- via the identification of a
program (software) for A with this evaluation procedure-- and,
in this process, via the equivocation we have mentioned on the word
machine, how the thesis came to be imported into science and
epistemology, through the idea that every model of every physical process had to
be formalizable. That is, such models are purely syntactic objects; nothing
semantic is required in them or of them.
That systems (mathematical or material) satisfying the
structures of Church's Thesis are preternaturally weak in entailment can be seen
in two distinct (but closely related) ways. The first way is to note that, in an
_expression_ of the form Av=w, the only thing one can ask why
about is the value w. The answers are "Because A" (formal plus
efficient cause; hardware plus program in a machine) and "Because v"
(material cause; input to a machine). Everything else is posited from
outside-- un-entailed. One cannot ask "Why A?" or "Why v?" and
expect answers within the system.
The second, related way is to note the difficulty of
solving inverse problems-- that is, solving the very same relation
Av=w for v, given A and w. This problem
essentially requires the production of a new operator, which we might call
[A to the power of -1, which my computer
won't allow me to notate. Sorry!-- JLR], and then evaluating
(computing) that. But how is this inverse operation itself to be entailed, let
alone evaluated? It is well known that these inverse problems cannot be solved
from within a fixed syntactic universe. They require an ineluctable
semantic aspect; indeed, it was precisely to guarantee enough
entailment to solve such inverse problems that the concept of well-posedness was
developed in the first place, through the invocation of causal entailment via
modeling relations.
It may also be noted that a description of something
through what it entails, rather than exclusively through what entails it, is the
hallmark of Aristotelian final causation. Thus, in the above equation, the
solvability of its inverse problem lets us answer the question "Why v?" by
saying "Because v entails w." This is one of the symptoms of impredicativity in
the system itself, and it is precisely what syntax alone cannot accommodate,
what in fact it was invoked to avoid."
Happy reading!
Judith
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