Re: Impredicativity, Rosen, life, formalism, open causality

["glen e. p. ropella" <***>]

Calvin Ostrum wrote:
> A formal system in the usual parlance is a set of abstract objects
> and a set of ways of deriving new objects from old.   A formal system
>  in this sense is an abstract object and doesn't "do" anything; a
> fortiori, it does not "self-construct".   Something in the world, the
> causal order, may be modelled by a formal system in some way or
> other, and this thing may self-construct.

Is this really true?  It seems to me that a formal system could be 
designed to "do something".  As I understand it, a formal system is 
simply an alphabet, some axioms, a grammar, and rules of inference. 
Since all the objects in a formal system are just symbols and 
ideological rules by which those symbols can be strung together, the 
formal system, itself, needs some external agent to _form_ sentences 
(that may or may not be correctly formed according to the grammar or may 
or may not be provable or refutable according to the rules of inference.

And in that sense, without the daemon out there forming new sentences, 
the formal system doesn't do anything.  But, couldn't we define a formal 
system that had some initial axioms that had some type of iterative 
definition property?  So, that all the daemon would have to do is form 
the first sentence, which would then form a new sentence, etc?  Even if 
there's no actual behavior, perhaps there could be some "maximally 
obvious next step" that could be said to be so obvious that we just 
assume it is made even if no daemon actually forms the sentence?

I'm thinking, in particular, about things like mathematical induction 
where we can take for granted that some arbitrarily large number can be 
shown to exist without manually forming all the intermediate sentences 
that show that it does exist.

Or is there something about this whole "could exist" versus "does exist" 
philosophical issue that I'm missing?  It seems to me that if we reject 
the platonic position that all the objects we discuss in a formal system 
already existed before we defined them and go with a completely 
constructivist view, where objects are generated by their definitions, 
then it might be reasonable to think that a formal system "acts" in some 
ontologically limited sense.

--
glen e. p. ropella              =><=                Hail Eris!
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