9,702 research outputs found
How Far Can We Go Through Social System?
The paper elaborates an endeavor on applying the algorithmic information-theoretic computational complexity to meta-social-sciences. It is motivated by the effort on seeking the impact of the well-known incompleteness theorem to the scientific methodology approaching social phenomena. The paper uses the binary string as the model of social phenomena to gain understanding on some problems faced in the philosophy of social sciences or some traps in sociological theories. The paper ends on showing the great opportunity in recent social researches and some boundaries that limit them.meta-sociology, algorithmic information theory, incompleteness theorem, sociological theory, sociological methods
From Heisenberg to Goedel via Chaitin
In 1927 Heisenberg discovered that the ``more precisely the position is
determined, the less precisely the momentum is known in this instant, and vice
versa''. Four years later G\"odel showed that a finitely specified, consistent
formal system which is large enough to include arithmetic is incomplete. As
both results express some kind of impossibility it is natural to ask whether
there is any relation between them, and, indeed, this question has been
repeatedly asked for a long time. The main interest seems to have been in
possible implications of incompleteness to physics. In this note we will take
interest in the {\it converse} implication and will offer a positive answer to
the question: Does uncertainty imply incompleteness? We will show that
algorithmic randomness is equivalent to a ``formal uncertainty principle''
which implies Chaitin's information-theoretic incompleteness. We also show that
the derived uncertainty relation, for many computers, is physical. In fact, the
formal uncertainty principle applies to {\it all} systems governed by the wave
equation, not just quantum waves. This fact supports the conjecture that
uncertainty implies randomness not only in mathematics, but also in physics.Comment: Small change
How Far Can We Go Through Social System?
The paper elaborates an endeavor on applying the algorithmic information-theoretic computational complexity to meta-social-sciences. It is motivated by the effort on seeking the impact of the well-known incompleteness theorem to the scientific methodology approaching social phenomena. The paper uses the binary string as the model of social phenomena to gain understanding on some problems faced in the philosophy of social sciences or some traps in sociological theories. The paper ends on showing the great opportunity in recent social researches and some boundaries that limit them
Is Complexity a Source of Incompleteness?
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems
of a finitely-specified theory cannot be significantly more complex than the
theory itself}, for an appropriate measure of complexity. We show that the
measure is invariant under the change of the G\"odel numbering. For this
measure, the theorems of a finitely-specified, sound, consistent theory strong
enough to formalize arithmetic which is arithmetically sound (like
Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded
complexity, hence every sentence of the theory which is significantly more
complex than the theory is unprovable. Previous results showing that
incompleteness is not accidental, but ubiquitous are here reinforced in
probabilistic terms: the probability that a true sentence of length is
provable in the theory tends to zero when tends to infinity, while the
probability that a sentence of length is true is strictly positive.Comment: 15 pages, improved versio
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