On 9/29/2011 10:36 AM, Stephen P. King wrote:

On 9/29/2011 4:03 AM, Bruno Marchal wrote:## Advertising

On 28 Sep 2011, at 16:44, Stephen P. King wrote:On 9/27/2011 10:47 AM, Bruno Marchal wrote:On 27 Sep 2011, at 13:49, Stephen P. King wrote:On 9/26/2011 7:56 PM, Jason Resch wrote:<snip>For well-defined propositions regarding the numbers I think thevalues are confined to true or false.Jason --[SPK]Not in general, unless one is only going to allow only Booleanlogics to exist. There have been proven to exist logics that havetruth values that range over any set of numbers, not just {0,1}.Recall the requirement for a mathematical structure to exist:Self-consistency.Consistency is a notion applied usually to theories, or (chatty)machines, not to mathematical structures.A theory is consistent if it does not prove some proposition andits negation. A machine is consistent if it does not assert aproposition and its negation.[SPK]Is not a machine represented mathematically by some abstract(mathematical ) structure? I am attempting to find clarity in theideas surrounding the notion of "machine" and how you arrive at theidea that the abstract notion of implementation is sufficient toderive the physical notion of implementation.This follows from the UD Argument, in the digital mechanist theory.No need of AUDA or complex math to understand the necessity of this,once we accept that we can survive with (physical, material) digitalmachines.[SPK]Is the property of universality independent of whether or not amachine has a set of properties? What is it that determines theproperties of a machine? I need to understand better your definitionof the word "machine".In first order logic we have Gödel-Henkin completeness theoremwhich shows that a theory is consistent if and only if there is amathematical structure (called model) satisfying (in a sense whichcan be made precise) the proposition proved in the theory.[SPK]What constraints are defined on the models by the Gödel-Henkincompleteness theorem? How do we separate out effective consistentfirst-order theories that do not have computable models?What do you mean by computable models?[SPK]Allow me to quote several definitions: "computable functions areexactly the functions that can be calculated using a mechanicalcalculation device given unlimited amounts of time and storage space." (from http://en.wikipedia.org/wiki/Computable_function). "acomputable model is one whose underlying set is decidable and whosefunctions and relations are uniformly computable. " (fromhttp://arxiv.org/abs/math/0602483).A computable model, as I understand it, could be considered as arepresentation of a system or structure whose properties can bedetermined by some process that can itself be represented as afunction from the set of countable numbers to itself. This defintionseeks to abstractly represent the way that we can determine theproperties of a physical system X or, equivalently, generate a finitelist of operations that will create an instance of X.Also, it is true that classical (Boolean) logic are not the onlylogic. There are infinitely many logics, below and above classicalpropositional logic. But this cannot be used to criticize the useof classical logic in some domain.[SPK]OK. My thought here was to show that classical (Boolean) logicis not unique and should not be taken as absolute. To do so would bea mistake similar to Kant's claim that Euclidean logic was absolute.OK, but then why to use that fact to criticize Jason's defense ofarithmetical truth independent of humans.[SPK]I am claiming a distinction between the existence of a structureand the definiteness of its properties. It is my claim that prior tothe establishment of whether or not a method of determining ordeciding what the properties of a structure or system are, one canonly consider the possibility of the structure or system. For example,say some proposition or sentence of a language exists. Does thatexistence determine the particulars of that proposition or sentence?If it can how so? How do can we claim to be able to decide that P_i istrue in the absence of a means to determine or decide what P_i means?How do you know the meaning of these word "Unicorn"? Is themeaning of the word "Unicorn" something that that arises simply fromthe existence of sequence of symbols? is not meaning not somethinglike a map between some set of properties instantiated entity and someset of instances of those properties in other entities? Consider anentity X that had a set of properties x_i that could not be related tothose of any other entity? Would this prevent the existence of X?The existence of X is the necessary possibility of X, []<>X.All treatises on any non classical logic used classical (or muchmore rarely intuitionistic) logic at the meta-level. You will notfind a book on fuzzy logic having fuzzy theorems, for example. Nonclassical logics have multiple use, which are not related with thekind of ontic truth we are looking for when searching a TOE.[SPK]Of course fuzzy logic does not have fuzzy theorem, that could bemistaking the meaning of the word "fuzzy" with the meaning of theword "ambiguous". I have been trying to establish the validity ofthe idea that it is the rules (given as axioms, etc) that are usedto define a given mathematical structure, be it a model, or analgebra, etc. But I think that one must be careful that the logicalstructure that one uses of a means to define ontic truths is notassumed to be absolute unless very strong reasons can be proven toexist for such assumptions.Usually non classical logic have epistemic or pragmatic classicalinterpretations, or even classical formulation, like the classicalmodal logic S4 which can emulate intuitionistic logic, or theBrouwersche modal logic B, which can emulate weak quantum logic.This corresponds to the fact that intuitionist logic might modelizeconstructive provability, and quantum logic modelizesobservability, and not the usual notion of classical truth (as usedalmost everywhere in mathematics).[SPK]I use the orthocomplete lattices as a representation of quantumlogic. My ideas are influenced by the work of Svozil<http://tph.tuwien.ac.at/%7Esvozil/publ/publ.html>, Calude<http://www.cs.auckland.ac.nz/%7Ecristian/10773_2006_9296_OnlinePDF.pdf>and von Benthem<http://staff.science.uva.nl/%7Ejohan/publications.html>, and otherson this. I am not sure of the definition of "weak quantum logic" asyou use it here.Svozil, Calude and van Benthem thought on the subject are very good.Weak quantum logic is the logic of sublattice of ortholattices, likein the paper of Goldblatt that I have often refer to you. Basicallyit is quantum logic without the orthomodularity axiom. It does notdistinguish finite dimensional pre-Hilbert space from Hilbert space,for example.[SPK]This paper http://www.jstor.org/pss/2274172 ? It seems to me thatthe distributivity axiom would not make the same distinction either,although Hilbert space is defined in terms of a linear algebra on avector space. Consider this paper<http://www.google.com/url?sa=t&rct=j&q=orthomodularity%20axiom&source=web&cd=5&sqi=2&ved=0CDgQFjAE&url=http%3A%2F%2Fm3k.grad.hr%2Fpapers-ps-pdf%2Fquantum-logic%2F1998-helv-phys-acta.pdf&ei=N3SETo2yFYGztwfLiYU0&usg=AFQjCNHal3UDb6B-MATSt1hloWFhSNVCnw&sig2=Fl7ESJLpFZ9qj8c8YU8S-w&cad=rja>'sabstract.http://www.google.com/url?sa=t&rct=j&q=orthomodularity%20axiom&source=web&cd=5&sqi=2&ved=0CDgQFjAE&url=http%3A%2F%2Fm3k.grad.hr%2Fpapers-ps-pdf%2Fquantum-logic%2F1998-helv-phys-acta.pdf&ei=N3SETo2yFYGztwfLiYU0&usg=AFQjCNHal3UDb6B-MATSt1hloWFhSNVCnw&sig2=Fl7ESJLpFZ9qj8c8YU8S-w&cad=rja"We show that binary orthologic becomes either quantum or classicallogic when nothing but modusponens rule is added to it, depending on the kind of the operation ofimplication used. We also show thatin the usual approach the rule characterizes neither quantum norclassical logic. The diff erence turns outto stem from the chosen valuation on a model of a logic. Thusalgebraic mappings of axioms of standardquantum logics would fail to yield an orthomodular lattice if a unary- as opposed to binary - valuationwere used. Instead, non-orthomodular nontrivial varieties oforthologic are obtained. We also discuss thecomputational efficiency of the binary quantum logic and stress itsimportance for quantum computationand related algorithms."How can we even consider the distinction of one form of abstractstructure, such as logical algebras or lattices, from another withoutthere existing a means to generate instantiations of the two? Thisquestion goes to the heart of my skepticism of your result.One question regarding the emulations. If one where consideringonly finite emulations of a quantum logic (such as how a classicalapproximation of a QM system could be considered), how might oneapply the Tychonoff, Heine–Borel definition or Bolzano–Weierstrasscriterion of compactness to be sure that compactness obtain for themodels? If we use these compactness criteria, is it necessary thatthe collection of open sets that is used in complete in an absolutesense? Could it be that we have a way to recover the appearence ofthe axiom of choice or the ultrafilter lemma?Hard and premature questions.[SPK]But do we not decide whether or not to pursue a conjecture by theimplications of the conjecture? The questions that I am asking hereare questions of the ability of the idea to give us an explanatorynarrative that we can use to reason about our world. You are, withyour result, proposing a result that implies an ontological theory:that Reality is, at its primitive level, purely abstract. This seemsto be more of an echo of the ideas of Pythagoras than those of Plato...Could it be possible to have a notion of accessibility toparametrize or weaking the word "every" as in the sentence: " Apoint /x/ in /X/ is a *limit point* of /S/ if every open set<http://en.wikipedia.org/wiki/Open_set> containing /x/ contains atleast one point of /S/ different from /x/ itself." to "A point /x/in /X/ is a *limit point* of /S/ if every open set<http://en.wikipedia.org/wiki/Open_set> , that is assessible fromsome S, containing /x/ contains at least one point of /S/ differentfrom /x/ itself. The idea is that S and x cannot be an infinitedistance (or infinite disjoint sequence of open sets) apart.It seems to me that this would limit the implied omniscience ofthe compactness criteria (via the usual axiom of choice) and itseems more consistent with the notion that an emulation does notneed to be *exact* to be informative.Perhaps. Cerrtainly open problem in comp+Theaetetus.[SPK]Does that not imply that the explanatory value of comp+Theaetus ispartly dependent of the resolution of such a problem? If we are goingto seriously consider your form of ideal monism to be correct, asopposed to some form of non-substance dualism or material monism orneutral monism, do such questions not need to be looked at withseriousness? I am very interested in ontological theories, thus myqueries.To invoke the existence of non classical logic to throw a doubtabout the universal truth of elementary statements in well defineddomain, like arithmetic, would lead to complete relativism, giventhat you can always build some ad hoc logic/theory proving thenegation of any statement, and this would make the notion of truthproblematic. The contrary is true.[SPK]Relativism of that kind would be that last conclusion that Iwould desire! OTOH, we do need a clear notion of contextuality asillustrated by the way that words are defined in relation to otherwords in a dictionary.I am problem driven. I start from the problem, and use the availablemath.[SPK]As am I. ;-) But I think that sometimes we need to look beyond themath and consider how it is that knowledge itself is possible.A non classical logic is eventually accepted when we can find aninterpretation of it in the classical framework.{SPK]This seems to be an unnecessary prejudice! Why is the classicalframework presumed to be the absolute measure of acceptability and,by implication, Reality?No. Simplicity. Together with the need of the classical Churchthesis, and our intuition of numbers. We do use the comp hypothesis,and it needs classical logic on the natural numbers. Intuitionistlogic can also be used, but then the math are much more complex, andeventually we need a non trivial use of the double negation topology.It is more easy to use, like usually in math, the meta-classicalbackground.[SPK]But it seems that you are assuming that our ability to haveintuitions of abstractions itself has a satisfactory explanation. Youseem to assume that the properties of, for example, memory obtainsolely from the existence of Arithmetic and that such existence isseverable from the physical instantiations of memory.This statement seems to reveal an explanation of why you believethat QM is derivative of classical logic somehow in spite of myrepeated statements to the work of others that show that this issimply not possible except in a crude and non-faithful manner!You repeatedly confuse the notion of embedding of a logic in another,and representing a logic in another. I have explained this manytimes, but you keep coming back on that confusion. QL cannot befaithfully extended in Boolean logic, but this does not mean that youcannot represent QL in a classical frame work (like it is done allthe time; quantum mechanics is itself a classical theory).[SPK]How is a representation of logic A in logic B not equivalent to anembedding of A in B? Maybe I am conflating a model with a representation.A non standard truth set, like the collection of open subsets of atopological space, provided a classical sense for intuitionistlogic, like a lattice of linear subspaces can provide a classicalinterpretation of quantum logic (indeed quantum logic is born fromsuch structures). It might be that nature observables obeys quantumlogic, but quantum physicists talk and reason in classical logic,and use classical mathematical tools to describe the non classicalbehavior of matter.[SPK]I agree but will point out that the use of classical logic couldbe merely a habit and convenience.Classical logic allows non constructive reasoning which areobligatory in any modest theology, like the machine's theologies.Do you agree that a (mathematical) machine stop or ... do not stop,on some input. We don't need more than that.[SPK]A mathematical object, as I can understand it, is purely anabstraction that supervenes upon the actions of a mind to have ameaning. The particular properties of the object flow from the rules,axioms, etc. that are used to define said object and do not depend onanything else except the possibility of some instantiation of thoserules, axioms, etc. If a "machine" is a form of mathematical structurethen its existence is not predicated on any particular instantiationof such a machine but its properties are not defined by the merepossibility of its existence. Additionally, the notion of "stopping"or "not stopping" has a meaning that refers to a process in some way.A process cannot be reduced to a static relation between abstractentities but it can be represented by sequences of static relations. Idistinguish between the representation of a process and the processitself. A map is not the territory.OTOH, if we consider the idea that we can relate simulations of agiven process with the process itself, we are comparing one form ofprocess to another, not a static set of relations to a process. I donot think of mathematical objects as static relations only, I see themmore as invariant patterns that occur in a background of eternalinteractions between possible aspects of Existence.I think that there may be a reason why classical logics are taken asfundamental, but this reasoning is build on the intuition that a 3p"public" notion of communication can only be defined in Booleanlogical terms; in other words, we observe a classical realitybecause that is the manner that maximally consistent collections ofopen sets can bisimulate each other. Bisimulation is communicationbetween and within logical systems. If bisimulation cannot occurbetween a pair of logics then there is no interactions between thetopological spaces dual to those logics. This gives us a way tothink of seperate physical worlds. But this reasoning requires thatwe treat logics and topological spaces on an equal ontologicalfooting. Logic cannot be taken as the unique ontological aspect ofexistence.It follows from the step 8 of UDA that if we are machine, classicalarithmetic is a theory of everything. Non classical logics arerecovered in the machine's epistemologies. S4grz1 is intuitionist andthe Z1* and X1* logics are type of quantum logics.[SPK]If we are some abstract static relational structure thenArithmetic is an explanation of everything? Maybe for an abstract andstatic entity, but not for an entity that needs to explain theappearance of a universe that is never only identically itself. I donot identify an arbitrary collection of static relations with Changein a decidable one to one and onto way.Onward! Stephen --

Dear Bruno,

`Please see the following paper for the kind of ideas that I am`

`considering:`

www.math.ru.nl/~mgehrke/Ge11.pdf "Abstract. The fact that one can associate a finite monoid with universal properties to each language recognized by an automaton is central to the solution of many practical and theoretical problems in automata theory. It is particularly useful, via the advanced theory initiated by Eilenberg and Reiterman, in separating various complexity classes and, in some

`cases it leads to decidability of such classes. In joint work with`

`Jean-´ Eric`

Pin and Serge Grigorieff we have shown that this theory may be seen as a special case of Stone duality for Boolean algebras extended to a duality between Boolean algebras with additional operations and Stone spaces equipped with Kripke style relations. This is a duality which also plays a fundamental role in other parts of the foundations of computer science, including in modal logic and in domain theory. In this talk I will give a general introduction to Stone duality and explain what this has to do with the connection between regular languages and monoids."

`I am exploring the ontological, thus philosophical, implications of`

`these ideas. My basic hypothesis is: If Abstract Objects exist then so`

`too do Topological Spaces that act as the physical worlds that implement`

`them. Reality is ontologically dual in that both abstract and concrete`

`objects exist and are related to each other such that one cannot exist`

`except if its dual also exists. Mind is an instance of an abstract`

`dynamical process, the brain is a form of implementation of mind. Logics`

`(including minds) and physical objects do not interact since they are`

`necessary and dual forms of the same neutral grundlagen, the totality of`

`existence in itself.`

Ownard! Stephen -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.