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James has raised some interesting issues and asked a couple of
questions. The first one is the hardest to answer so I'll go the second one
first: Yes, my father wrote extensively on the subject of Gödel's (whose name he
pronounced "Gerdel", incidentally. There should be an umlaut over the o-- ah,
the spellcheck actually puts it in! Good stuff. Anyone know the pronunciation
for sure? I've heard that name pronounced several different ways.)
"Incompleteness Theorem".
One of the best treatments of it is in Essays, on page 156. A paper
called "Syntactics and Semantics in Languages" , in which my father relates
mathematics to other languages. He speaks of the attempts to remove all
semantics, or meaning, from science as a means of achieving objectivity and how
this attempt robs science and us of any real ability to understand the natural
world. If the biological realm is "relational" then everything is connected to
everything else in some sense or other.
Robert Rosen wrote: "One area in which this strategy [removing
meaning/semantics] has been most relentlessly pursued is in a branch of language
called mathematics. The ongoing quest for axiomatization is a form of this
trend, going back to Euclid and before. In our century, in large part as a
response to the foundation crises arising in Set Theory, this quest took a
variety of forms, one of which was the formalism of David Hilbert. Hilbert was
led to deny that (pure) mathematics was about anything at all-- even about
itself. Indeed, he implicitly blamed the crises themselves entirely on
unexpunged "informal" semantic residues in mathematics, and he proposed
replacing the entire enterprise with an inherently meaningless game of pattern
generation, played with symbols on paper. Within a short time, however, Gödel
proved his celebrated Incompleteness Theorem, which in effect showed that
syntactic rules captured only an infinitesimal part of "real" mathematics-- in
effect, that Church's Thesis was false, even in this realm. Or, stated another
way, that mathematical systems that are formalizable, in the Hilbertian sense,
are infinitely rare (nongeneric) among mathematical systems in general. So, in
this realm, there is no way to reduce semantic aspects to syntactic ones in
general; the result is only a mutilation, with the creation of
artifacts."
There are many other discussions on these issues all through my
father's written work but this gives a bit of history, some context, and the
implications for science.
Judith
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