At 05:35 PM 6/10/2006, Stanley N. Salthe wrote:
>John said:
>
> > Hmm. You should read Barwise and Seligman, Information Flow: the
logic of
> >distributed Systems. Very important for understanding Quantum
Information.
> >Also, I assume that you are familiar with algorithmic complexity
theory,
> >which is certainly rigourous, Minimum Description Length (Rissanen)
and
> >Minimum Message Length (Wallace and Dowe) methods that apply
Kolomogorov
> >and Chaitin's ideas very rigourously. If you don't like the
computational
> >approaches for some reason, then you might want to look at Ingarden
et al,
> >(1997) Information Dynamics and Open Systems (Dordrecht: Kluwer).
They
> >show how probability can be derived from Boolean structures, which
are
> >based on the fundamental notion of information theory, that of making
a
> >binary distinction. So probability is based in information theory,
not the
> >other way around (there are other ways to show this, but I take the
> >Ingarden et al approach as conclusive -- Chaitin and Kolmogorov and
> >various commentators have observed the same thing). If you think
about the
> >standard foundations of probability theory, whether Bayesian
subjective
> >approaches or various objective approaches (frequency approaches fail
for
> >a number of reasons -- so they are out, but could be a counterexample
to
> >what I say next), then you will see that making distinctions and/or
the
> >idea of information present but not accessible are the grounds for
> >probability theory. Information theory is the more fundamental
notion,
> >logically, it is more general, but includes probability theory as a
> >special case. Information can be defined directly in terms of
distinctions
> >alone; probability cannot. We need to construct a measure to do that.
>
> So, I ask a follow-up question: Would the greater generality of
>information theory with respect to probability theory imply something
>concerning the even more general question of whether or not logic is more
>general than mathematics? Some seem to think that Goedel showed the
>opposite.
It depends what you mean by logic. The issue is too complicated to
get into here and now, but the simple answer is that there is no
non-arbitrary distinction between mathematics and logic. Exactly the
same reasons apply to the limits of both, and the only way to get one
more powerful than the other is to apply a double standard for proofs
and/or acceptability. So the answer, briefly, is that Goedel showed
no such thing, either way you take it, if you do not apply a double
standard for evidence.
John
----------
Professor John Collier collierj@ukzn.ac.za
Philosophy and Ethics, University of KwaZulu-Natal, Durban 4041 South Africa
T: +27 (31) 260 3248 / 260 2292 F: +27 (31) 260 3031
http://www.nu.ac.za/undphil/collier/index.html
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Received on Sat Jun 10 22:41:11 2006
