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 and itsnegation. 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. Noneed of AUDA or complex math to understand the necessity of this, oncewe accept that we can survive with (physical, material) digital machines.

[SPK]

`Is the property of universality independent of whether or not a`

`machine has a set of properties? What is it that determines the`

`properties of a machine? I need to understand better your definition of`

`the word "machine".`

In first order logic we have Gödel-Henkin completeness theorem whichshows 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 are`

`exactly the functions that can be calculated using a mechanical`

`calculation device given unlimited amounts of time and storage space. "`

`(from http://en.wikipedia.org/wiki/Computable_function). "a computable`

`model is one whose underlying set is decidable and whose functions and`

`relations are uniformly computable. " (from`

`http://arxiv.org/abs/math/0602483).`

`A computable model, as I understand it, could be considered as a`

`representation of a system or structure whose properties can be`

`determined by some process that can itself be represented as a function`

`from the set of countable numbers to itself. This defintion seeks to`

`abstractly represent the way that we can determine the properties of a`

`physical system X or, equivalently, generate a finite list 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 use ofclassical logic in some domain.[SPK]OK. My thought here was to show that classical (Boolean) logic isnot unique and should not be taken as absolute. To do so would be amistake 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 structure`

`and the definiteness of its properties. It is my claim that prior to the`

`establishment of whether or not a method of determining or deciding what`

`the properties of a structure or system are, one can only consider the`

`possibility of the structure or system. For example, say some`

`proposition or sentence of a language exists. Does that existence`

`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 is true 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 the meaning`

`of the word "Unicorn" something that that arises simply from the`

`existence of sequence of symbols? is not meaning not something like a`

`map between some set of properties instantiated entity and some set of`

`instances of those properties in other entities? Consider an entity X`

`that had a set of properties x_i that could not be related to those 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 of theidea that it is the rules (given as axioms, etc) that are used todefine a given mathematical structure, be it a model, or an algebra,etc. But I think that one must be careful that the logical structurethat one uses of a means to define ontic truths is not assumed to beabsolute unless very strong reasons can be proven to exist for suchassumptions.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 modelizes observability,and not the usual notion of classical truth (as used almosteverywhere 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. Basically itis 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 that`

`the distributivity axiom would not make the same distinction either,`

`although Hilbert space is defined in terms of a linear algebra on a`

`vector 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>'s`

`abstract.`

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 classical`

`logic when nothing but modus`

`ponens rule is added to it, depending on the kind of the operation of`

`implication used. We also show that`

`in the usual approach the rule characterizes neither quantum nor`

`classical logic. The diff erence turns out`

`to stem from the chosen valuation on a model of a logic. Thus algebraic`

`mappings of axioms of standard`

`quantum logics would fail to yield an orthomodular lattice if a unary -`

`as opposed to binary - valuation`

`were used. Instead, non-orthomodular nontrivial varieties of orthologic`

`are obtained. We also discuss the`

`computational efficiency of the binary quantum logic and stress its`

`importance for quantum computation`

and related algorithms."

`How can we even consider the distinction of one form of abstract`

`structure, such as logical algebras or lattices, from another without`

`there existing a means to generate instantiations of the two? This`

`question 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 the`

`implications of the conjecture? The questions that I am asking here are`

`questions of the ability of the idea to give us an explanatory narrative`

`that we can use to reason about our world. You are, with your result,`

`proposing a result that implies an ontological theory: that Reality is,`

`at its primitive level, purely abstract. This seems to 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: " A point/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 it seemsmore consistent with the notion that an emulation does not need 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 is`

`partly dependent of the resolution of such a problem? If we are going to`

`seriously consider your form of ideal monism to be correct, as opposed`

`to some form of non-substance dualism or material monism or neutral`

`monism, do such questions not need to be looked at with seriousness? I`

`am very interested in ontological theories, thus my queries.`

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 the`

`math 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 Church thesis,and our intuition of numbers. We do use the comp hypothesis, and itneeds classical logic on the natural numbers. Intuitionist logic canalso be used, but then the math are much more complex, and eventuallywe need a non trivial use of the double negation topology. It is moreeasy to use, like usually in math, the meta-classical background.

[SPK]

`But it seems that you are assuming that our ability to have`

`intuitions of abstractions itself has a satisfactory explanation. You`

`seem to assume that the properties of, for example, memory obtain solely`

`from the existence of Arithmetic and that such existence is severable`

`from the physical instantiations of memory.`

This statement seems to reveal an explanation of why you believe thatQM is derivative of classical logic somehow in spite of my repeatedstatements to the work of others that show that this is simply notpossible 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 many times,but you keep coming back on that confusion. QL cannot be faithfullyextended in Boolean logic, but this does not mean that you cannotrepresent QL in a classical frame work (like it is done all the time;quantum mechanics is itself a classical theory).

[SPK]

`How is a representation of logic A in logic B not equivalent to an`

`embedding 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 are obligatoryin any modest theology, like the machine's theologies.Do you agree that a (mathematical) machine stop or ... do not stop, onsome input. We don't need more than that.

[SPK]

`A mathematical object, as I can understand it, is purely an`

`abstraction that supervenes upon the actions of a mind to have a`

`meaning. The particular properties of the object flow from the rules,`

`axioms, etc. that are used to define said object and do not depend on`

`anything else except the possibility of some instantiation of those`

`rules, axioms, etc. If a "machine" is a form of mathematical structure`

`then its existence is not predicated on any particular instantiation of`

`such a machine but its properties are not defined by the mere`

`possibility 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 abstract entities`

`but it can be represented by sequences of static relations. I`

`distinguish between the representation of a process and the process`

`itself. A map is not the territory.`

`OTOH, if we consider the idea that we can relate simulations of a`

`given process with the process itself, we are comparing one form of`

`process to another, not a static set of relations to a process. I do not`

`think of mathematical objects as static relations only, I see them more`

`as invariant patterns that occur in a background of eternal interactions`

`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 reality becausethat is the manner that maximally consistent collections of open setscan bisimulate each other. Bisimulation is communication between andwithin logical systems. If bisimulation cannot occur between a pairof logics then there is no interactions between the topologicalspaces dual to those logics. This gives us a way to think of seperatephysical worlds. But this reasoning requires that we treat logics andtopological spaces on an equal ontological footing. Logic cannot betaken as the unique ontological aspect of existence.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 then Arithmetic`

`is an explanation of everything? Maybe for an abstract and static`

`entity, but not for an entity that needs to explain the appearance of a`

`universe that is never only identically itself. I do not identify an`

`arbitrary collection of static relations with Change in a decidable one`

`to one and onto way.`

Onward! 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.