On Mon, Oct 20, 2008 at 5:29 PM, Abram Demski <[EMAIL PROTECTED]> wrote:
> Ben, > > "[my statement] seems to incorporate the assumption of a "finite > period of time" because a finite set of sentences or observations must > occur during a finite period of time." > > A finite set of observations, sure, but a finite set of statements can > include universal statements. Ok ... let me clarify what I meant re sentences I'll define what I mean by a **descriptive sentence** What I mean by a sentence is a finite string of symbols drawn from a finite alphabet. What I mean by a *descriptive sentence* is a sentence that is agreed by a certain community to denote some subset of the total set of observations (where all observations have finite precision and are drawn from a certain finite set). So, whether or not a descriptive sentence contains universal quantifiers or quantum-gravity quantifiers or psychospirituometaphysical quantifiers, or whatever, in the end there are some observation-sets it applies to, and some it does not. Then, what I claim is that any finite set of descriptive sentences corresponds to some computable model of reality. One never needs an uncomputable model of reality to justify a set of descriptive sentences. > > "Fractal image compression is computable." > > OK, yea, scratch the example. The point would possibly be valid if > fractal compression relied on a superset of the Mandelbrot set's math, > since the computability of that is still open as far as I know. > No ... because any algorithm that can be implemented on a digital computer, can obviously be described in purely computable terms, using assembly language. Regardless of what uncomputable semantics you may wish to assign to the expression of the algorithm in some higher-level language. > > "Based on a finite set of finite-precision observations, there is no > way to distinguish Wei Dai's black box from a black box with a Turing > machine inside." > > Sure, but the more observations, the longer the description length of > that turing machine, so that at some point it will exceed the > description length of the uncomputable alternative. We have to be careful with use of language here. It is not clear what you really mean by the "description length" of something uncomputable, since the essence of uncomputability is the property of **not being finitely describable**. One can create a Turing machine that proves theorems about uncomputable sets ... i.e., that carries out computations that we can choose to interpret as "manipulating uncomputable sets." Just as, one can create a Turing machine that carries out computations that we interpret as differential calculus operations, acting on infinitesimals. However, even though we call them "uncomputable", in reality these computations are computational, and their so-called "uncomputability" is actually just a mapping between one computable formal structure and another (the first formal structure being the algorithms/structures carrying out the computations ... the second formal structure being the formal theory of computability, which is itself a finite set of axioms that can be manipulated by a Turing machine ;-) ... There is no such thing as an uncomputable procedure with a short description length (where descriptions are finite concatenations of symbols from a finite vocabulary). There are however procedures with short description lengths that have interpretations in terms of uncomputability -- where these interpretations, if they're to be finitely describable, must also be computable ;-) -- Ben G ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
