On 10/29/2012 1:08 PM, Bruno Marchal wrote:
On 29 Oct 2012, at 14:36, Stephen P. King wrote:[Bruno Marchal wrote:] So numbers are universal and can be treated mathematically as always.I agree, but the concept of numbers has no meaning prior to the existence of objects that can be counted. To think otherwise is equivalent to claiming that unspecified statements are true or false even in the absence of the possibility of discovering the fact.
I think you confuse numbers, and the concept of numbers.
No, I do not. My claim is that Numbers are objects in the mind of conscious beings. If there does not exist worlds where entities to whom numbers are concepts then there is no such thing as a concept of numbers in such worlds. My argument is that concepts of truth and provability of theorems apply only to the concepts of numbers and their constructions, not to numbers themselves.
And then your argument is not valid, as with numbers, the miracle is that we can specify the concept of numbers, as this result in defining some arithmetical sigma_1 complete theory in terms of 0, s(0), ... and the laws of addition and multiplication, that everybody understands (unless philosophers?).
I am a philosopher! My argument rests only on the fact that the 'miracle' is exactly as you state it here: we exist and have a concept of numbers and can ascertain the truth of arithmetic statements. My claim is that truth valuations supervene on the ability of consciousness to form concepts of numbers. I question the entire idea of numbers existing as separate Platonic entities. In the absence of consciousness, there is no such thing as a concept!
BrunoPS BTW, from a computer scientist perspective, your use of NP never succeed to make sense. I don't dare to ask you to elaborate, as I am afraid you might aggravate your case. The NP question is fundamental and has many interesting feature, but it concerns a local tractability issue, and is a priori, unless justification, not relevant for the arithmetical body issue, nor number's theology (including physics) issue, etc.
It is the argument is sound and is the same kind of argument as what Kripke used to discuss the idea of possible worlds. <http://www.philosophy-index.com/kripke/> In http://en.wikipedia.org/wiki/Possible_world we read:
"There is a close relation between propositions and possible worlds. We note that every proposition is either true or false at any given possible world; then the modal status of a proposition is understood in terms of the worlds in which it is true and worlds in which it is false."
Solutions to equations or computations are not available until after they are actually solved. My solution to this is to not go so far as you do in Step 8. Let me try to be more explicit:
From your paper http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf :
"Instead of linking [the pain I feel] at space-time (x,t) to [a machine state] at space-time (x,t), we are obliged to associate [the pain I feel at space-time (x,t)] to a type or a sheaf of computations (existing forever in the arithmetical Platonia which is accepted as existing
independently of our selves with arithmetical realism). "I am pointing out that the idea of computations "existing independently of our selves" is wrong in that it conflates *the meaning and truth valuation of numbers* with *t**he existence of numbers as Platonic objects*. It is absurd to refer to the claim that the truth of "17 is prime" depends on any one person or entity, but the claim that the truth of "17 is prime" is knowable by any person is not absurd. If we stipulate that the content of knowledge exists somehow prior to that which knowledge supervenes upon, we are being absurd. The content of knowledge and the ability of knowledge occur simultaneously or not at all. Absent the "concept" of numbers there is no such thing as valuations of numbers because the notion of Platonic objects considers objects as existing independently as some singular "perfect" version that is then plurally projected somehow into the physical realm, as we see in the Allegory of the Cave. This is a one-to-many mapping, not a one-to-one mapping. How exactly is a "type" or "sheaf" a singular and "perfect" version of each and every computation and yet be something that has individuated valuations? Individual valuations of computations are only those that occur as physical instantiations of computations and thus they do not "exist" in Platonia. The Many exist in the physical worlds, no? I propose a rephrasing of your statement above: We identify the 1p qualia to a sheaf of computations (as bisimilar Boolean Algebras) that is dual to physical machine states at diffeomorphically equivalent space-time coordinates (x, y, z, t). This is a restatement of the Stone duality into COMP-like terms. ;-) (The idea of diffeomorphic equivalence is discussed in detail here: http://plato.stanford.edu/entries/spacetime-holearg/Leibniz_Equivalence.html )
When you say: <<Yes, this is the Pre-Established Harmony, but as I have argued before this concept is deeply flawed because it tries to claim that the solution to NP-Hard problem (of choosing the best possible world) is somehow accessible (for the creation of the monads by God) prior to the availability of resources with which to actually perform the computation of the solution. One cannot know the content of a solution before one computes it, even if one is omniscient!>> I don't find any sense.
How is this so difficult for you to comprehend? The Platonic Realm is defined as timeless, everything in it just 'exists', no? Therefore any argument that shows that "if A does not exist then neither does B if B requires A to exist" is true in Platonia as well, (we stipulate the existence of Platonia as defined <http://www.wku.edu/%7Ejan.garrett/302/platintr.htm#truebeing> for the sake of this statement). If a solution to a computation cannot exist until the computation is run then if the resources required to run the computation do not exist then there does not exist a solution to the computation! I propose that we can easily resolve this conundrum by stating Computational universality as: "/A computation is universal if and only if it is independent of any particular physical implementation/." This allows for the existence of physical implementations, even those that are themselves defined by correlations between sheaves for computations. This sets up a relation between computations - as abstract or immaterial objects - and physical systems that seems consistent with "COMP minus Step 8". We can recover the picture of step 8,
bijectionin a way that is truly neutral ontologically, by changing its single directed arrow to a pair of oppositely directed arrows, but this one that occurs only in the ultimate sense of the elaboration of all possible physical worlds consistent with Pratt's idea.
This idea, BTW, is consistent with the concept of Indra's Net, as an inversion of the idea that every Jewel reflects all others: Every jewel is a physical world that is defined by all computations of it. Note also that this naturally includes self-computation as jewels also reflect themselves. ;-)
I hope you don't mind my frankness. I wouldn't say this if I did not respect some intuition of yours. But math and formalism can't be a pretext for not doing the elementary reasoning in the philosophy of mind. If you use math, you have to be clearer on the link with philosophy or theology. To be understandable by others.
I am trying to be clear. I will correct and rephrase my verbiage until you understand it. I reject the idea of an entity, 'God', whose total purpose is to "observe" the Reality of the Universe! If we accept the idea that numbers exist in our complete absence, then it follows that an entity like us cannot exist just to observe the existence of numbers (or anything else). Why postulate the existence of a special entity that does what we collectively are already doing? It is our collective consciousness that Constitutes the Platonic Realm, IMHO. A theory that there is some independently existing realm is a gross violation of Occam.
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