From: Paul Cockshott (clyder@GN.APC.ORG)
Date: Wed Jan 21 2004 - 17:20:13 EST
> To delve into waters about which I am still less familiar, I seem to > remember stumbling across a Chaitin paper on the web. Wasn't > there some argument to the effect that a formal system has only as > much info content as its' axioms? The world has more content than > any finite set of axioms hence formalism is inherently limited? I'm > sure I have this horribly wrong but one way or another isn't this once > more an indication of the limits of formalism not an argument for > aspiring to formalism? > Chaitins arguements stem from his algorithmic information theory which is discussed in his books 'The Limits of Mathematics' and 'The Unknowable'. His point, succinctly expressed, is that you cant get two kilos of theory out of one kilo of axioms. Ultimately, the information content of a formal theory is encoded in its axioms, since it is in principle possible for a universal turing machine to print out all possible theorems deriving from any given set of axioms, the theorems must therefore contain no more information than the axioms. However this does not imply that formalisms are invalid in studying the real world, because in any model of the real world we have both a set of dynamical laws, and a set of initial conditions. Typically the encoding of the initial conditions contains much more information than the encoding of the dynamical laws. Thus from the standpoint of Chaitin's theory, the information content of the entire simulation program is the sum of the dynamical laws plus the boundary conditions. A deterministic evolution of the system will be fully specified by the start state and thus will in a sense be encoded in the start state. But the number of possible states that the system can evolve through grows exponentially with the number of bits used to encode the starting state. One very easily gets systems whose complete evolution would not have repeated within the current life of the universe. One easily overlooks the unimaginable vastness of the finite. The apparent unpredictableness of certain formal finite systems was already evident to Babbage in the 1830's. He discusses in 'Passages from the life of a Philospher', how his difference engine could be made to produce what appear to be a regular and monotonic sequence of numbers for a very long time, but then suddenly produce a number that was apparently unexplainable. With a typical Victorian mindset he suggested that a similar process might underly 'miracles'.
This archive was generated by hypermail 2.1.5 : Sat Jan 24 2004 - 00:00:02 EST