George Levy wrote:
>Everytime a "measurement" is made, the set of worlds spanned >by this consciousness is defined more narrowly, but the >number in the set remains infinite.  In addition, each >simulation in the set need not belong to the same "level."
>We're faced with the strange possibility that the >consciousness spans an infinite number of simulations >distributed over widely different levels.
I agree that we are simultaneously in many simulations, and I would agree that there are uncountably many possible 'histories' or situations consistent with our consiousness and known history. But I think only a countable number of the 'classical universes' (and certainly only a small fraction) we might be in are simulations. If we look at the 'total knowledge' of any living being, including the things not consciously known but constrained to be decided by the classical history they evolved from (so, for instance, 'what killed the dinosaurs' is probably a question with a definite answer, for us, but the trajectory of a certain electron is not; the former would form part of my 'total knowledge' because if I worked out the answer by looking at enough historical evidence, I would get the same answer each time), it is always finite.
    As you suggest, any arrangement of matter consistent with this 'total knowledge' is a possibility for the universe we will find ourselves in on making further measurements, and if we live forever, and keep making new measurements, then there will be a countable number of possible universes we will encounter. But at any finite time, we will only know a finite amount, and therefore only impose a finite number of constraints on the possible universe we will find ourselves in (see my note below on living forever versus having lived forever for more on this). the relevance of this to the current issue is that the super-beings who would emulate us obey the same rules: The 'whole world' of any living being at any given time in their history, being determined only by their thoughts, and the classical histories likely to give rise to those thoughts, can be described by a finite amount of information. Therefore the total number of situations in which a conscious being is simulating our universe is countable. Also, the total number of finite chains of one being simulating another simulating a third etc. is countable (being a set of finite subsets of a countable set).
    We are left with the infinite chains of simulation. Here there are two possibilities: either the chain forms a 'loop', in whih case it has actually already been counted in the finite chains (so there are countably many of these [you may also reject them as 'unphysical']) or there is a new 'being' at every step up the chain. But I do not see how these constructions could 'really' be said to tend to any limit (which would be uncountable) as there is, unlike with real numbers, no way to say when two chains are 'getting close together': The next stage could always make them totally different, by any measure (but given the axiom of choice, these universes will be 'real', mathematically at least. But they won't be 'dense' as I'll get on to now..).
    Whether or not you accept that these limits exist, it is nevertheless the case that we are more likely, on making measurements, and reducing the number of universes we might be in to find more assortments of non-living matter than aliens, including those aliens who might 'turn out' to be controlling us. Any series of new measurements we make can be seen as adding to our 'total knowledge' as described above, a new stream of data, whih you could translate into a string of 0s and 1s (The fact that the data is genuinely new to us means that it is necessarily a discrete uniform distribution of 0s and 1s, apart from correlations you may have itroduced into the data by 'translating it' from whatever form you found it in). Then for a sufficiently long 'string' there would be a very small probability that this information would correspond to the existence of some living being, but it would be much more likely to correspond to a bunch of 'dead' particles (physically, I would see a typical measurement process as 'collecting thermal radiation from a black hole' or 'going over the cosmic horizon to see what you find' or 'absorbing a photon' [these examples are roughly in decreasing order of probability of finding 'life', but even with the last eample it is in principle possible: you could fire a super-energy gamma ray at a gold sheet and a virus might come out the other side]).For any finite 'chain' of information there is a small but finite chance of finding 'life', and even finding'life simulating us' but as you make the chain infinite (as we would have to to 'discover' an infinite chain of universes all simulating in a chain) the probability of finding forever higher and higher levels of life tends to zero. This is because at eah stage, the higher level life would need to be more complex (to simulate the extra information) so you'd likely find more 'cosmic junk' which the proposed higher level being would have to simulate as well, so he/she'd have to be even more complex. So it seems highly likely that we'd never find even one level of simulation: The stuff we'd find looking for the simulator would make the simulator ever more and more complex and hence harder to find.
 
    These rather lengthy arguments are not needed to disprove the specific world of the Matrix, though: A much more obvious inconsistency is 'why don't the robots eat what they're feeding us?' I know in sci-fi stories there's usually stuff that's not explained but you can imagine there might be some explanation for, but anyone who knows what a Carnot engine is will know that there's no possible reason to use a human as a furnace..
 
Finally, my note about living forever and having lived forever: Anyone could live forever; you just have to eat well and be made of metal, but no-one could have lived forever: Say if they try to recall their earliest memory (In the generalised 'total knowledge' sense given above)..Then the memory they recall will only ever have a finite age. At any time in the future, there will only be a certain amount of their past that is affecting them. So there will be different pasts consistent ith where they are now, and 'almost all' of them will involve a less ordered state earlier on (as there are more such states) from which their current ordered state evolves. Physically, this corresponds to the evolution of their species and situation, and the totally random state(s) found sufficiently early on correspond to the 'big bang', which I conjecture is a 'totally indeterminate' state found at the beginning of every 'universe.'

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