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Howard, could you enlarge on this statement, please:
I am a Constructivist, so I would phrase your question conversely: when do
the limits of physical laws impinge on the capacity of formal symbols to
have meaning? That is, when do the purely syntactic expressions of formal mathematics go beyond what can be measured or encoded by nature? It makes sense to me that "purely syntactic" expressions of formal
mathematics move into realms which do not exist in nature (like "transfinites,"
or different "sizes" of infinities, for example). Since natural systems are
bound by contextual constraints, any purely syntactical mode of modeling is not
always going to be "congruent". Context is equivalent to "semantics". In
other words, there is meaning/information in the relations created by context.
So when mathematics is bound by the semantics of mathematics it's not a
problem-- it's only when science tries to say that nature is like mathematics
that it becomes a problem. ("We must never forget that number theory is
about numbers.") I don't think we are in any disagreement on
this.
My main area of concern is your phrasing; "the limits of physical
laws". Do you mean physics-based laws? Or do you mean what my father referred to
as "Natural Law" (meaning whatever consistencies are involved in
generating/constraining the universe, whether we know about these
consistencies or not)?
Another area of concern is the phrasing; "what can be measured or
encoded by nature". I doubt that even my father would say he could tell you the
limits of what nature can or cannot do. He could only generalize and
say that natural systems cannot behave in ways inconsistent with Natural
Law. Science in general is way behind; a small subset of all there is.
Science has deliberately limited itself by adopting the machine metaphor which
is the same as using a machine to model nature and forgetting that it's a model.
So now they are trying to make nature into a system that "realizes" their model
and only the evidence which seems to fit is allowed to be called
"scientific".
The fact that natural systems are capable of "measuring" and
"encoding" is not disputed, however. Even people who refuse to believe that
living systems are "anticipatory" in the Rosennean sense could not dispute that
natural systems are capable of measuring and encoding. On the other hand, just
because WE are natural systems which measure and encode, that doesn't mean
we do it well or accurately in any terms but our own. In other words, our
choices of what to measure and how to measure (in terms of what and how
other natural systems are measuring) may not reflect much useful reality
according to those natural systems. Their modes of measurement and encoding may
be beyond our ability to perceive, much less quantify or qualify. So how can we
say what can be measured or encoded by nature? To do so would require
that we measure their ability!
Judith
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