Re: Definitions and Argument (was RE: Penrose, wave function collapse and MWI)
At 14:36 28/02/04 +, Brett Hall wrote: I think this clarifies things a little. My original way of writing what I interpreted the incompleteness theorem to be was to say that there exist (in sufficiently complex axiomatic) systems in which there are true propositions without proof. I think that this is misleading on reflection. It is more accurate to say that there exist in complex axiomatic systems (like, for example, arithmetic) propositions (or well formed formulae) that cannot be proved either true nor false, that is, which are undecidable. Such propositions do have a truth value - the law of the excluded middle still holds - so, being reasonable we have to assume that there do indeed exist statements that are true but unprovable. You should say: unprovable ... *by the system in question* Let us call S the system in question (S for some consistent System). Godel's proof entails that there are true proposition P that S cannot prove. This means also that if you add P as axiom to S, you will not get a contradiction (if not, S would prove the false statement not P), this means that S+P is (another) consistent system S'. And, obviously S' can prove P, because P is an axiom for S' so that S' proves it in one line. The other half of this way of speaking is to say that 'there also exist false statements that are unprovable' (but this, technically - is quite redundant as proving a statement is false is the same as proving as true the negation of that same statement). Does this make sense? Sure. But when you talk about a non provable proposition, you should always mention who (or which system) cannot prove it. Godel really has shown that formal provability is a relative notion (relative to the system considered). This is in total contrast with formal computability which seems to be an absolute notion not depending on any system (and that's the basic conceptual motivation for Church Thesis). As I said in another post, there is a case for *absolutely undecidable* statement. But this is trickier and is related to some formalization of informal provability. If you are interested look at the reference: REINHARDT W.N., 1985, Absolute Version of Incompleteness Theorems, Noûs, 19, pp. 317-346. REINHARDT W.N., 1986, Epistemic Theories and the Interpretation of Gödel's Incompleteness Theorems, Journal of Philosophical Logics, 15, pp. 427-474. (From Conscience and Mecanisme, where I make the case that some form of comp are *absolutely* undecidable). Bruno http://iridia.ulb.ac.be/~marchal/
Re: Tegmark is too physics-centric
At 10:33 28/02/04 +1100, Russell Standish wrote: I deliberately leave vague what is in the theory of the mind, but simply assume a small number of things about consciousness: 1) That there is a linear dimension called (psycholgical) time, in which the conscious mind find itself embedded 2) The observations are a form of a projection from the set of subsets of possibilities onto the same set. We identify a QM state with a subset of possibilities. 3) The Kolmogorov probability axioms 4) The anthropic principle 5) Sets of observers are measurable Also I assume the existance of the set of all descriptions (which I call the Schmidhuber ensemble, but perhaps more accurately should be called the Schmidhuber I ensemble to distance it from later work of his). This is roughly equivalent to your Arithmetic Realism, but probably not identical. It is the form I prefer philosophically. (I think this is the exhaustive set of assumptions - but I'm willing to have other identified) I only treat continuous time in Occams razor (hence the differential equation) however I do reference the theory of timescales which would provide a way of extending this to other types of time (discrete, rationals etc). In any case, contact with standard QM is only achieved for continuous time. The justification for assuming time is that one needs time in order to appreciate differences - and differences are the foundation of information - so in order to know anything at all, one needs to appreciate differences hence the need for a time dimension. Note - computationalism requires time in order to compute mind - therefore the assumption of time is actually a weaker assumption than computationalism. comp assumes only that the sequence 0, 1, 2, 3, 4, 5, 6, ... lives in Platonia. 3-person time apparantly does not appear. 1-person time appears through the S4Grz logic. In terms of the above assumptions, 1) is a consequence of computationalism, which I take is a basis of your theory (although I've never understood how computationalism follows from COMP). ? Wait a bit. COMP refers to computationalism. I don't understand. 2) corresponds to your 1-3 distinction. Indeed I refer to your work as justification for assuming the projection postulate. That is not clear for me. 3) Causes some people problems - however I notes that some others start from the Kolmogorov probability axioms also. No problem at all with Kolmogorov proba axioms. 4) I know the Anthropic principle causes you problems - indeed I can only remark that it is an empirical fact of our world, and leave it as a mystery to be solved later on. No problem with the so called Weak Anthropic Principle. Although obviously I prefer a Turing-Universal-Machine--thropic principle ... 5) Measurability of observers. This is the part that was buried in the derivation of linearity of QM, that caused you (and me too) some difficulty in understanding what is going on. I spoke to Stephen King on the phone yesterday, and this was one point he stumbled on also. Perhaps this is another mystery like the AP, but appears necessary to get the right answer (ie QM !) Of course a more detailed theory of the mind should give a more detailed description of physics. For example - we still don't know where 3+1 spacetime comes from, or why everything appears to be close to Newtonian dynamics. Stephen King is cooking up some more ideas in this line which seems interesting... Thanks for your clarification, Bruno http://iridia.ulb.ac.be/~marchal/
