> On 6 Mar 2019, at 11:47, Lawrence Crowell <[email protected]>
> wrote:
>
> On Monday, March 4, 2019 at 6:24:35 AM UTC-6, Bruno Marchal wrote:
>
>> On 3 Mar 2019, at 20:49, Lawrence Crowell <[email protected]
>> <javascript:>> wrote:
>>
>> On Sunday, March 3, 2019 at 7:58:01 AM UTC-6, Philip Thrift wrote:
>>
>>
>> On Sunday, March 3, 2019 at 7:32:00 AM UTC-6, Lawrence Crowell wrote:
>>
>> Bringing Gödel into physics is treading on a mine field as it is. Believe
>> me, most physicists react in horror at the mere suggestion of this. I have
>> this suspicion however that quantum measurement is a a sort of Gödel
>> self-reference with quantum information or qubits. This may, at least within
>> how we describe quantum mechanics if it should turn out to be not how the
>> quantum world actually is, be one reason why we have this growing pantheon
>> of quantum interpretations and no apparent way to decide which is
>> definitively correct.
>>
>>
>>
>> I still think it's Darwin, not Gödel, that has anything to do with
>> "quantum measurement".
>>
>> But physicists recoil in horror from that.
>>
>> - pt
>>
>> Darwinian logic did put down the Aristotelian-Cartesian hierarchical
>> structure with respect to biology.
>
> OK. Darwin use both mechanism (quasi-explicitly), and is understood usually
> in the materialist frame, but Darwin just do not address that question.
>
>
>
>> Aristotle and Plato are the two most known Hellenic philosophers because
>> their systems of thought were wrapped into the New Testament Bible. Plato
>> had this idea of there being a hierarchy of being, which was taken up by St
>> Paul, carried further by Augustine, Aquinas and eventually encoded by
>> Descartes. Descartes had this hierarchy of structure over function, design
>> over material form etc, which was carried into science during the 17th and
>> 18th century. In some ways Newtonian mechanics was seen as a confirmation of
>> Descartes' metaphysics.
>
> That is true. Today we know that Newtonian Mechanics is highly not
> computable. But Newton saw that, and indeed, distrusted his Mechanics, and
> saw it as an approximation.
>
>
>
> I would say classical mechanics is NP computable.
In classical mechanics, the three body problem is Turing universal, I think. No
doubt for for the many body problem as the billiard board computer illustrates.
Any theorem complete for arbitrary finite Newtonian mechanical system will be
Turing complete, and thus essentially undecidable (in the sense of Tarski: it
means that all its effective consistent extensions are undecidable as well).
Turing universal = partial computable (not total computable).
> The problems of chaos are similar to to NP problems in that for a Turing
> machine that computes P these problems are exponential in space and time.
> Chaos is of that nature, but it is convergent. One can compute for some
> finite time the evolution of complex systems.
?
We cannot predict in advance if a machine will stop. The extensionnal equality
of machines, or combinators, is unsolvable.
P NP are complexity classes included in the total computable. Once Turing
universal, the behaviour can be non computable, not even in exponential or
super-exponential time. In no time at all.
>
>
>> Darwin struck a fatal blow to this with respect to biology.
>
> He struck the wrong view on Descartes and Mechanism, but his own Mechanism is
> a foreseen of digital mechanism, and its confirmation by molecular genetics,
> and the genetical code.
>
>
>
>>
>> Darwin did away with Aristotle and Descartes with biology. Gödel had an
>> impact on Plato, though it is not clear to me how. Gödel saw himself as a
>> Platonist and that his incompleteness theorem demonstrated how mathematical
>> truth is independent of knowing it. I tend to see this in terms of Turing
>> machines, which would say that certain problems are not computable and as
>> such no information can be derived.
>
> … can be derived mechanically. But the truth can be guessed and experience by
> non algorithmic, mechanical, means, even by a machine. Gödel’s theorem is
> already proved by machine, which can even prove their own Gödel’s theorem,
> and enforces them to be mystical, that is, to believe that there is something
> more than their own consciousness.
>
>
>
> Gödel’s theorem is proven in a computable manner, and so is an algorithm of
> sorts.
Like all theorems in mathematics and physics. A theorem has to be easy to
check. That is not possible in full second order logic (Analysis), but Analysis
can be done in effective part of second order logic, or inside first order
theories, like ZF.
What is true iw what you say, is that the (sound, or consistent) machine can
prove its own Gödel theorem. It can prove that if it is (3p) consistent, then
it can (3p)-prove its consistency. The (1p) related person ([]p & p) find on
the contrary its own consistency as the most obvious and indubitable thing.
> My point though is that physical systems
Of course that term is ambiguous in the setting of the mind-body research, as
it will admit different definitions in the immaterialist and materialist
metaphysics/theology.
My main goal was to illustrate that With the Indexical Digital Mechanist
assumption, we can transform the mind-body problem into a mathematical theory,
testable/comprable empirically. Unfortunately it is a non Aristotelian theory.
Eventually matter and the quantum is explained in terms of consciousness
differentiation. The notion of physical system cannot be taken as primitive. It
has to be derived, in a explicit very precise way from the two axioms Kxy = x
and Sxyz = xy(zy), or from RA, or from the variety of solutions of a degree
four polynomial Diophantine equation, yet, moralised from the interior by
entities “believing” in the induction axioms (corresponding to their primitive
terms). A physical system becomes an appearance stable in some type of
histories (defined by variant of provability imposed by incompleteness).
> are in a sense mechanical and as such involve a mathematical system that
> describes how the state of a system, or equivalently a'la Born rule how an
> observable changes, evolves in time. So mechanism is an algorithmic process
> as we describe it.
I use mechanism in the sense that if little daemon substitute each piece of my
brain, at some resolution level, by functional digital equivalent, then my
consciousness would not notice the difference. It is an invariance of the first
person principle, for a substation made at some level. It does not matter if
the level is high (neuronal level) or low (quantum field). Mechanism is the
assumption of the existence of that level, and the main consequence is that all
continuations of “our computation” at that level are realised in all models of
any Turing universal theory, and that physics must be extracted for the
“surviving” view (with “<>t”, i.e. consistency explicit in the mode of
provability ([]p & <>p). Physics is a sort of abstraction of the cul-de-sac
realities.
>
>
>> Whether there is a self-referential truth that is not enumerated is less
>> important. The real number line has a continuum of elements and there is not
>> enough information, even if that is infinite, to encode it all. We might say
>> in some sense that these numbers exist as if being in Plato's cave we can
>> imagine the existence of things by looking at shadows.
>
>
> Yes. For a set-theoretical realist, there are aleph_0 computable functions,
> and thus 2^aleph_0 non computable functions.
>
> Now, in many toposes, all functions are computable, and all real-functions
> are continuous. That is the case for the effective topos of Highland, based
> on Kleene’s notion of realisability. Mechanism ask for arithmetical realism,
> just to define what is a machine, but it does not asks for set-theoretical
> realism, or analytical realism.
>
>
> There are plenty of functions that are not continuous. There is the
> Weierstrass theorem that functions are integrable if they are continuous
> almost everywhere.
Assuming the third excluded principle. Intuitionist does not buy it. You can’t
prove their existence in constructive or intuitionist theories. You can add
them, but not in the Hyland topos, nor in Brouwer’s own conception.
>
> I think in some ways there is a loss of clarity on what we are meaning by
> computable.
The classical notion of computable is *terribly* clear.
A function F from N to N is computable if there is a Turing machine M which on
n gives F(n); for each n.
An equivalent definition:
A function F from N to N is computable if there is a combinator B such that Bn
= F(n), where n is the Barendrech numeral (or any other numeral system defined
with the (SK) combinator.
A computation is either a sequence of Instantaneous state of Turing machine (as
I defined once here),
Or
A sequence of combinators, such that they follow from one application of the
axioms in the right direction.
It is equivalent.
The" problem”, which is also the solution of the problem, is that we cannot
enumerate algorithmically the computable function from N to N.
What we can do, is the enumeration of the functions from subset of N to N,
including all total one (whose domain is the subset N itself).
So we can’t have any complete theory deciding if the domain of a function is N
or not, as that could be used to build such an emeuration of all and only all
total computable function. The price of universality/liberty is the abandon of
total security and total control!
Now, the function on The notion of reals does not admit a standard notion of
computability, but there are good approximations, and may alternative
refinement, depending on many axioms you could add to some intuitionist logic.
It is a jungle. Yet a fertile domain in mathematics, with many ideas born
there, finding application elsewhere. In the theology of the machine, they are
born in the ([]p & p) domain, and the total self-referential correctness makes
this probably not useful to clarify what is a real number, and what are
analytical functions.
> A function is generally computable, but not all of its domain or range are
> computable. Even the function y = x tells us to simply map an 2^{aleph_0} set
> of points from x to y.
Oh! Computationalism, Mechanism is concerned only with function from N to N (or
equivalent). Then we build from the hiierarchies and structures which are there
(although known only by logicians apparently).
> Not all the numbers are computable
All natural numbers are trivially computable.
For the real numbers, there are as many notion of reals numbers than there are
variant of intuitionist and constructive logics. Even just between S4Ggrz and
S5, there is a continuum of modal axiomatisations of so called intermediate
logics, but with the topazes, they are everywhere almost all rich mathematical
object.
> as explicit alpha numeric expressions.
Accepting A v ~A, the set of non nameable real numbers is of measure one, and
the computable, definable real numbers are of measure 0, with the usual
measure, or the cantorial one. But there are many other measure, and
topologies. The constructive, computable real numbers, is a domain as vats as
human psychology, or better, universal machine psychology. With mechanism, a
real number is a dream by a number. The ontology contains only K, S, KK, KS, …
KKK, K(KK), … All objects are finite, the infinite, be them god or universe, or
N, or aleph_1 will be ideas by numbers. They obey complex mathematics.
I guess you assume some physical universe, or some analytical structure, but
with mechanism, such existence is absolutely undecidable. No universal machine
can prove the necessity of more than just the sigma_1 truth.
> Functions that are not computable would to my mind be the result of
> differential or integral equations.
With The corrected version (by Turing himself in some footnote) of Turing's
notion of computable real numbers and the analysis on them, Pour El has written
a book showing some non computable functions have a non computable derivative,
or of limit of computable reals having a non computable limit.
But I don’t take that theory of reals too much seriously, as reals are only
(quite) useful “fiction".
>
>
>
>>
>> Kant proposed a metaphysics that is somewhat parallel to quantum mechanics.
>> The noumena of what "actually is," that Bohm insisted we could come to
>> really know, is the unknown of QM. We probably can't know if the quanta is
>> epistemic or ontic.
>
> If you *assume* Mechanism (the indexical weak version I present) then we
> already know that the quanta are epistemic.
>
>
>
>
>
>> The phenomena are the measurements and predictions. This has a certain
>> Platonic character to it, but within the physical domain. The noumena are
>> similar to Plato's pure forms and the phenomena are similar to the physical
>> forms. The employment of Gödel with physics might be compared to shifting
>> from Plato to Kant. However, if quantum interpretations are Gödel
>> self-referential physical axioms
>
>
> The whole of the appearance of the physical reality is brought by
> incompleteness, for anyone rational and saying “yes” to the doctor. To assume
> a physical reality is like assuming that a car is driven by invisible horse.
> It add nothing to the thermodynamics, and add only new question, like what
> are the invisible horse made of, and in what sense thermodynamics is not
> enough. It is like adding a new god to prevent a simple explanation.
>
> I have this idea the arrow of time we perceive is due to entanglement changes
> with quantum decoherence. This is then a manifestation of wave function
> change with measurement, observation or some general encoding of quantum
> information by quantum information we generally call decoherence. So in that
> sense some manifestation of physical reality may indeed be due to axiomatic
> incompleteness of QM.
With mechanism, any Turing universal should do. The physical reality arise from
the axiomatic incompleteness of *any* Turing universal theory, which,
importantly give rise to all Partial Computable functions, that many machines ,
most of them; whose behaviour escapes all theories we can do. Partial
computable means computable on its domain, and only God knows what they do
outside of the domain, from simple detachable loop to infinite nightmares ...
>
>
> Note that for me, Aristotle has just not understood Plato, and his philosophy
> is at the antipode of Plato. I like very much Aristotle, because he has
> always keep the scientific attitude, and has proposed refutable theories. It
> is known by everybody hat his physics was false (F = mv), and it is ignored
> by many that his theology is refuted too, at least in the cartesian
> (mechanist) frame.
>
>
>
> Aristotle has his place, but honestly I think Leucippius, Democritus, Thales
> and others had more of a scientific mind. Aristotle was very interested in
> hierarchical categories of objects and the world. He reflects a time when the
> Hellenic world was falling under despotism, as he was the mentor of Phillip
> II's son later deemed Alexander the Great.
OK.
>
> As for more below, I illustrated how a universal machine is not possible.
That sound like telling me I will show you that a prime number is not possible.
> Turing machines and related systems are iterated and involve a discrete set
> of maps according to time. An arrow of time or a timelike Killing vector
> can't be universally extended from a local region onto a global spacetime
> manifold.
Hmm… still dreaming that there is a physical universe, it seems. I hope you are
aware that you make some strong metaphysical/theological/ontological assumption
here.
But even if that is the case, I have very few doubt that the physical reality
is Turing universal. Already QM can simulate the Newtonian three bodies, which
is Turing universal, but, in fact, with QM, even the zero body problem is
Turing universal.
All physical computer illustrate that the physical reality (which with
mechanism is an emerging pattern from a sum on all (relative) computations) is
Turing universal. It is abit of a surprise, because we could have expected much
more randomness for the first person view, but the math illustrates how the
consciousness filtering makes the distinction stable enough to assure the
existence of relatively stable computers. Of course brains and even bacteria
illustrates already the Turing universality.
I recall that a number/machine u is universal means that phi_u(x, y) =
phi_x(y). We say that u mimics or emulate, or simulate exactly the machine x on
the data y. A universal machine is a machine, thus a FINITE entity which can
imitate exactly (emulate) all other machines. There is no analog notion for the
notion of computability on the reals, except natural extension based on some
representation of the real in term of (total) computable functions, but that is
again a “number’s imagination” product.
So, this very simple theory (Kxy = x, Sxyz = xz(yz)), solves the mind body
problem up to the day we find a discrepancy with nature. In that case we can
conclude that either Mechanism is false, or we belong to a malevolent
computations, relying on a normal world in which we are intentionally failed
(which I find not really plausible, but we need to add to be logically exact).
Bruno
>
> LC
>
>
>> of an auxiliary nature we are not left with any knowledge of what might be
>> called a noumena. What we face is either the fact that decoherence and the
>> outcome of a quantum event occurs for no reason at all, where quantum
>> interpretations are fantasies of sorts,
>
> I agree with you that events without a cause is nonsense.
>
>
>
>
>> or that quantum mechanics is embedded in some physical axiomatic system of
>> greater power.
>
> Not just embedded. Physics occupies a very important place in the theology of
> the universal machine, and is 100% deducible, making Mechanism empirically
> testable, and indeed confirmed by all data. Unless we add spurious axioms,
> like the wave reduction.
>
>
>
>>
>> A Darwinian viewpoint on physics is worth keeping in the back of one's head.
>> The superstring paradigm might be the endpoint of the idea there is some
>> axiomatic scheme behind physical existence.
>
>
> There is a reality before man made an axiomatic. In this case it is the
> sigma_1 arithmetical truth. Reality and truth precedes language, thought and
> axiomatic, unless you look at the numbers has being code of axiomatics and of
> machines.
>
>
>
>
>> It would be similar to Darwin, in that Darwin ended the hierarchical order
>> of life with lower animals "down there" and us humans at the top "up there"
>> as the pinnacle of creation by rules given by what we call God. Smolin and
>> other have proposed Darwinian ideas, but so far nothing has come of them.
>> Maybe physicists should keep working.
>
>
> With mechanism, there is an analogy with Darwin, given that it extends
> evolution up to the physical laws. But the evolution is made in the space of
> computations, which is in arithmetic. Quanta is brought by the consciousness
> which differentiate on all computations. That explains the weirdness of the
> quantum reality. More difficult to explain is particles, space and time, but
> there are reasons to expect an explanation for them too.
>
>
>
>>
>> The Darwinian order to the world relies upon an open world system. A closed
>> world is one where entropy does reach maximum and at equilibrium there is
>> just death. The creationists love this argument, which is true in a closed
>> world, but not open. We may then compare it to the structure of cream in
>> coffee, which has become a bit of a design game these days. If you pour
>> cream in black coffee it makes all sorts of swirls and structures, but as
>> you keep swirling the coffee that becomes a brown mix and equilibrium is
>> struck. So the analogue for an open system would be where both the coffee
>> and cream are constantly replenished so there is always black coffee and
>> there is always primarily cream whirls. It is an open system. It is not one
>> that reaches maximum entropy. Well we know the universe started at a very
>> low entropy. We know of no particular reason why there could not have been
>> lots of black holes generated so the initial entropy would be much higher.
>> So how the universe was generated, say how the inflationary spacetime with a
>> high energy false vacuum generated the low energy of the true vacuum in our
>> observable pocket, may have been an aspect of an open system.
>
> Of course the arithmetical reality is quite open in that sense. Open problem
> for the physical reality, but it would be astonishing it could be close. The
> physical reality is the border of the non physical reality seen from the
> first person view of the creature run (in infinitely many versions) in
> arithmetic.
>
>
>
>>
>> The quantum states of the universe may in fact never reach equilibrium.
>> Suppose we have a black hole of mass M and temperature T = 1/8πM in a
>> background with the same temperature. If the black hole absorbs a photon
>> from the background or emits a photon to the background as Hawking radiation
>> we then have M → M ± δM and reciprocally the temperature of the black hole
>> decreases or increases. So the black hole will by stochastic events drift
>> away from this equality situation. Equilibrium is not defined. This suggests
>> there is no general meaning to equilibrium in quantum gravitation. So as a
>> result a Darwinian concept for cosmology might be possible.
>
> With mechanism, it has been proved that this is necessary.
>
> Bruno
>
>
>>
>> LC
>>
>>
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