On 9/29/2011 4:03 AM, Bruno Marchal wrote:
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 the
values are confined to true or false.
Jason
--
[SPK]
Not in general, unless one is only going to allow only Boolean
logics to exist. There have been proven to exist logics that have
truth 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 its
negation. A machine is consistent if it does not assert a
proposition and its negation.
[SPK]
Is not a machine represented mathematically by some abstract
(mathematical ) structure? I am attempting to find clarity in the
ideas surrounding the notion of "machine" and how you arrive at the
idea that the abstract notion of implementation is sufficient to
derive 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) 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 which
shows that a theory is consistent if and only if there is a
mathematical structure (called model) satisfying (in a sense which
can be made precise) the proposition proved in the theory.
[SPK]
What constraints are defined on the models by the Gödel-Henkin
completeness theorem? How do we separate out effective consistent
first-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 only
logic. There are infinitely many logics, below and above classical
propositional logic. But this cannot be used to criticize the use of
classical logic in some domain.
[SPK]
OK. My thought here was to show that classical (Boolean) logic is
not unique and should not be taken as absolute. To do so would be a
mistake similar to Kant's claim that Euclidean logic was absolute.
OK, but then why to use that fact to criticize Jason's defense of
arithmetical 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 much
more rarely intuitionistic) logic at the meta-level. You will not
find a book on fuzzy logic having fuzzy theorems, for example. Non
classical logics have multiple use, which are not related with the
kind of ontic truth we are looking for when searching a TOE.
[SPK]
Of course fuzzy logic does not have fuzzy theorem, that could be
mistaking the meaning of the word "fuzzy" with the meaning of the
word "ambiguous". I have been trying to establish the validity of the
idea that it is the rules (given as axioms, etc) that are used to
define a given mathematical structure, be it a model, or an algebra,
etc. But I think that one must be careful that the logical structure
that one uses of a means to define ontic truths is not assumed to be
absolute unless very strong reasons can be proven to exist for such
assumptions.
Usually non classical logic have epistemic or pragmatic classical
interpretations, or even classical formulation, like the classical
modal logic S4 which can emulate intuitionistic logic, or the
Brouwersche modal logic B, which can emulate weak quantum logic.
This corresponds to the fact that intuitionist logic might modelize
constructive provability, and quantum logic modelizes observability,
and not the usual notion of classical truth (as used almost
everywhere in mathematics).
[SPK]
I use the orthocomplete lattices as a representation of quantum
logic. 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 others
on this. I am not sure of the definition of "weak quantum logic" as
you 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, like
in the paper of Goldblatt that I have often refer to you. Basically it
is quantum logic without the orthomodularity axiom. It does not
distinguish 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 considering
only finite emulations of a quantum logic (such as how a classical
approximation of a QM system could be considered), how might one
apply the Tychonoff, Heine–Borel definition or Bolzano–Weierstrass
criterion of compactness to be sure that compactness obtain for the
models? If we use these compactness criteria, is it necessary that
the collection of open sets that is used in complete in an absolute
sense? Could it be that we have a way to recover the appearence of
the 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 to
parametrize 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 at
least 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 from
some S, containing /x/ contains at least one point of /S/ different
from /x/ itself. The idea is that S and x cannot be an infinite
distance (or infinite disjoint sequence of open sets) apart.
It seems to me that this would limit the implied omniscience of
the compactness criteria (via the usual axiom of choice) and it seems
more 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 doubt
about the universal truth of elementary statements in well defined
domain, like arithmetic, would lead to complete relativism, given
that you can always build some ad hoc logic/theory proving the
negation of any statement, and this would make the notion of truth
problematic. The contrary is true.
[SPK]
Relativism of that kind would be that last conclusion that I
would desire! OTOH, we do need a clear notion of contextuality as
illustrated by the way that words are defined in relation to other
words in a dictionary.
I am problem driven. I start from the problem, and use the available
math.
[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 an
interpretation of it in the classical framework.
{SPK]
This seems to be an unnecessary prejudice! Why is the classical
framework 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 it
needs classical logic on the natural numbers. Intuitionist logic can
also be used, but then the math are much more complex, and eventually
we need a non trivial use of the double negation topology. It is more
easy 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 that
QM is derivative of classical logic somehow in spite of my repeated
statements to the work of others that show that this is simply 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 many times,
but you keep coming back on that confusion. QL cannot be faithfully
extended in Boolean logic, but this does not mean that you cannot
represent 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 a
topological space, provided a classical sense for intuitionist
logic, like a lattice of linear subspaces can provide a classical
interpretation of quantum logic (indeed quantum logic is born from
such structures). It might be that nature observables obeys quantum
logic, but quantum physicists talk and reason in classical logic,
and use classical mathematical tools to describe the non classical
behavior of matter.
[SPK]
I agree but will point out that the use of classical logic could
be merely a habit and convenience.
Classical logic allows non constructive reasoning which are obligatory
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 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 as
fundamental, but this reasoning is build on the intuition that a 3p
"public" notion of communication can only be defined in Boolean
logical terms; in other words, we observe a classical reality because
that is the manner that maximally consistent collections of open sets
can bisimulate each other. Bisimulation is communication between and
within logical systems. If bisimulation cannot occur between a pair
of logics then there is no interactions between the topological
spaces dual to those logics. This gives us a way to think of seperate
physical worlds. But this reasoning requires that we treat logics and
topological spaces on an equal ontological footing. Logic cannot be
taken as the unique ontological aspect of existence.
It follows from the step 8 of UDA that if we are machine, classical
arithmetic is a theory of everything. Non classical logics are
recovered in the machine's epistemologies. S4grz1 is intuitionist and
the 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
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