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Re: Reality of i (was Something for Platonists)
Matt King wrote [everything-list] 6/16/03: > > Hi James, > > I don't want to get into the Platonism > discussion as I'm not of a philosophical bent, > but I would like to start discussion based on > something you wrote in one of your posts on > the subject: > > James N Rose wrote: > > > The square root of a negative number has no physical > > reality (or so it is presumed, because no abject > > examples have yet been shown/proven) but it has a most > > definite platonic ideal existence. > > The whole square root of a negative number > question boils down to the reality/unreality of > a single number, the square root of minus one, > usually called i, as every other negative > square root can be expressed as a real multiple > of this imaginary number. Now, I'd be the first > to accept that you can't have i oranges, so i > does not have the same kind of physical reality > as the natural numbers, or even the positive real > numbers. > > However, you also cannot have zero oranges, > or minus five oranges for that matter. So perhaps > it is no less physically real than the > negative numbers or zero. > > I'd also like to say that in a great deal of > physics, the imaginary number is indispensible, > at least in doing the math - could this be > sufficient evidence to declare it physically real? > Specifically, if we have used i to predict the > result of a particular experiment, and we find > that our prediction and the result match, is this > evidence for the physical reality of i? > > I'm reminded of looking out of the window > to watch the trees move, and concluding that > it is windy, even though I haven't seen or felt > the wind... > > Just a thought, > > Matt. > Matt, Thank you for your remarks, question. First, let me say I am not wholly antagonistic to platonic notions, except as the current (last 2500 year) rendering of it has, IMHO, a severe deficiency. What is absent is clear linkage between the Universal and the Real. For I do not see a coherent consistent compatible universe existing without that connection/relation. So I have spent all my cognitive life working to identify where and how that requisite is present .. and to meld the domains. Essentially, the Universal has to be isomorphic with Cantorian transfinities and Potentia -- including the emergent. It must surpass Godelian restrictions and the entire domain of information possibilities must be 'self'-accessible even if local bounds or limits are noted, used or identified. In line with your question, it is time to re-conceive 'real'. The wind and the referential cognition of wind are both real and informationally co-relevant -- and -- communicatively involved. That 'influence' might be limited to a preferential direction, is distinct from the information access. And so it is the information access among domains that I am most interested in and see as being of superior importance in any cosmology. Since Plato and Aristotle, humanity has been myopically concerned with 'association' -- restrictively understood as 'causality'. With all the attendant disconnects: form from function, materiality from process, mind from matter, relations from relations, intentionality from mechanisms from conditions from implementation necessities. We struggle to put all these back together, but like the characters in Waiting for Godot, we carry hampering concepts and weighted baggage of old ideations which are more problem than utility. For the universe and whatever aspects of it one considers (material, energectic, aetheric) to function so pervasively consistently demands not just certain absolute rules of performance, but absolute communicative relational association: communicative architecture and access and enactiveness throughout all possible timespace of the architecture, access and enactiveness. Which means -- that Platonia, that 'i' [sqrt -1], that QM and continuum, and all dimensional realms have relation (communicative access) with attendant morphings of information (codification transfers). Designating the codification transfers relations is the plateau we have reached. Relating Universal to Real. Specifying that in a certain dimensional frame of reference the [sqrt -1] -is- "real", even if it doesn't attain to qualifying as "real" in the dimensional frame of reference our perception/cognition operates in. But the overarching communicative process~architecture of Being allows the information of one frame-of-reference to transcribe/transduce to others, and so, what is 'unreal' to our tactile criteria is still accessible relevant information, just restricted to the form it can exist in our reality as. No big deal. We just have to let go of the bigotted bias that 'our' frame of reference has to be 'the' frame of reference. This led me to a strange but unavoidable conclusion re certain aspects of 'information'. Nearly without exception, there isn't a scientific mind on this plane
Re: Something for Platonists (and the platonic)
> But in fact, the only thing that privileges the set of all >computational >operations that we see in nature, is that they are instantiated by >the laws of physics. It is only through our knowledge of the physical >world >that we know of the di.erence between computable and not computable. So >it's only through our laws of physics that the nature of computation can >be understood. It can never be vice versa." methinks it be just the inverse; the "laws" of fizziks emerge from "underlying" computation, itself the result of initial rules (laws?)... CMR <--enter gratuitous quotation that implies my profundity here-->
Reality of i (was Something for Platonists)
Hi James, I don't want to get into the Platonism discussion as I'm not of a philosophical bent, but I would like to start discussion based on something you wrote in one of your posts on the subject: James N Rose wrote: > The square root of a negative number has no physical > reality (or so it is presumed, because no abject > examples have yet been shown/proven) but it has a most > definite platonic ideal existence. > The whole square root of a negative number question boils down to the reality/unreality of a single number, the square root of minus one, usually called i, as every other negative square root can be expressed as a real multiple of this imaginary number. Now, I'd be the first to accept that you can't have i oranges, so i does not have the same kind of physical reality as the natural numbers, or even the positive real numbers. However, you also cannot have zero oranges, or minus five oranges for that matter. So perhaps it is no less physically real than the negative numbers or zero. I'd also like to say that in a great deal of physics, the imaginary number is indispensible, at least in doing the math - could this be sufficient evidence to declare it physically real? Specifically, if we have used i to predict the result of a particular experiment, and we find that our prediction and the result match, is this evidence for the physical reality of i? I'm reminded of looking out of the window to watch the trees move, and concluding that it is windy, even though I haven't seen or felt the wind... Just a thought, Matt. When God plays dice with the Universe, He throws every number at once...
Re: Fw: Something for Platonists]
At 10:46 16/06/03 -0700, Hal Finney wrote: Jesse Mazer writes: > Yes, a Platonist can feel as certain of the statement "the axioms of Peano > arithmetic will never lead to a contradiction" as he is of 1+1=2, based on > the model he has of what the axioms mean in terms of arithmetic. It's hard > to see how non-Platonist could justify the same conviction, though, given > Godel's results. Since many mathematicians probably would be willing to bet > anything that the statement was true, this suggests a lot of them are at > least closet Platonists. What is the status of the possibility that a given formal system such as the one for arithmetic is inconsistent? Godel's theorem only shows that if consistent, it is incomplete, right? Are there any proofs that formal systems specifying arithmetic are consistent (and hence incomplete)? As Jesse Mazer said we all have an intuitive model of Peano Arithmetic (PA), and this should convince us of PA consistency. (We learned that model in secondary school). We can "formalize" such an argument in a set theory like ZF, that is, a model of PA can be constructed in ZF, as a first order citizen. Now this should not really convince us that PA is consistent because the ZF axioms are more demanding, and we would be entitled to ask for a proof of the consistency of ZF. By Godel second incompleteness theorem PA cannot prove the consistency of PA, ZF cannot prove the consistency of ZF. But ZF can prove the consistency of PA. Note that this latter fact *can* be proved in PA, that is: PA can prove that ZF can prove the consistency of PA, of course PA cannot prove the consistency of ZF, so this is not very useful here. A perhaps more relevant question is: does it exist a *finitary* proof of PA consistency? It all depends of course of what is meant by "finitary". If by finitary you mean "arithmetically representable", then by Godel, the answer is no. But many logicians consider that "transfinite induction" toward some reasonable ordinal can be considered as finitary. Actually Gentzen succeeds in presenting a proof of the consistency of PA through a transfinite induction up to \epsilon_0 (which is omega up to omega up to omega up to omega ...). This shows (by Godel again) that "transfinite induction up to \epsilon_0" cannot be done in PA, although it can be shown that all transfinite induction up to any \alpha little than \epsilon_0 can be done in PA. This has lend to "ordinal analysis" of formal theories where the strongness of provability of a theory is measured in term of ordinal. Remember that computability is an absolute notion (Church thesis), and formal provability is a necessary relative notion. I conjecture that the consistency of COMP should need at least a transfinite induction up to the Church-Kleene least non constructive ordinal (omega_1^CK). This should reflect the fact that the consistency of COMP is not provable by any consistent machines ... (although machines could bet on it, at their own risk and peril). Bruno
