Re: Modal logic, consciousness, and matter

2019-02-25 Thread Philip Thrift


On Monday, February 25, 2019 at 6:04:28 PM UTC-6, Brent wrote:
>
>
>
> On 2/25/2019 8:55 AM, Bruno Marchal wrote: 
> > Fictionalism does not apply to the arithmetical reality, nor to 
> > physics, but to the naïve idea of a “physical universe” as being the 
> > fundamental reality. The theology of the universal machine is a priori 
> > quite non Aristotelian: there is no Creator, and there is no Creation. 
> > Just a universal dreamer which lost itself in an infinitely surprising 
> > structure and wake up from time to time, or from numbers to numbers. 
>
> There is according to St Anselm, who also thought that definitions bring 
> things into existence. 
>
> Brent 
>


Anselm would work for DC Films or Marvel Studios today.

- pt


> > 
> > I need no more than a partial applicative algebra, and each choice of 
> > the phi_i makes N into one, simply by defining an operation “*” in N 
> > such that n * m = phi_n(m). There exist numbers k and s such that 
> > 
> > ((k * n) * m) = n 
> > (((s * n) * m) * r) = (n * r) * (m * r), 
> > 
> > for all m, n, r in N. 
> > 
> > And, the key point, the operation “*” can be defined in the 
> > arithmetical language, and those statements are, for each n, m, r, 
> > provable in RA. I have shown that the converse is true. It is a very 
> > elegant Turing complete theory. With Indexical Digital Mechanism, it 
> > is absolutely undecidable if the Universe is bigger than the sigma_1 
> > reality. (But here I do a blasphemy: that can only be entirely 
> > justified by G* *only*!, It is where I have to insist that this is 
> > presented as a consequence of YD + CT (“yes doctor” + Church-Turing 
> > thesis). 
> > 
> > Such theories are essentially undecidable. It means that not only they 
> > are arithmetically incomplete, but all their effective consistent 
> > extensions are too. They are creative, you cannot capture the semantic 
> > in the way it could become complete, even in some imaginary domain 
> > concevable by the machine/theory/number. The universal machine are 
> > never entirely satisfied and a computation is always an escape 
> > forward, but their self-reflection create a mess, and illusions. 
> > 
> > The sigma_1 arithmetical reality, as seen by the universal numbers 
> > which lives there, in the first person undetermined sense, is 
> > something *very big*. It generates infinitely many surprises. There 
> > are consistent histories. 
> > 
>
>

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Re: Modal logic, consciousness, and matter

2019-02-25 Thread Brent Meeker




On 2/25/2019 8:55 AM, Bruno Marchal wrote:
Fictionalism does not apply to the arithmetical reality, nor to 
physics, but to the naïve idea of a “physical universe” as being the 
fundamental reality. The theology of the universal machine is a priori 
quite non Aristotelian: there is no Creator, and there is no Creation. 
Just a universal dreamer which lost itself in an infinitely surprising 
structure and wake up from time to time, or from numbers to numbers.


There is according to St Anselm, who also thought that definitions bring 
things into existence.


Brent



I need no more than a partial applicative algebra, and each choice of 
the phi_i makes N into one, simply by defining an operation “*” in N 
such that n * m = phi_n(m). There exist numbers k and s such that


((k * n) * m) = n
(((s * n) * m) * r) = (n * r) * (m * r),

for all m, n, r in N.

And, the key point, the operation “*” can be defined in the 
arithmetical language, and those statements are, for each n, m, r, 
provable in RA. I have shown that the converse is true. It is a very 
elegant Turing complete theory. With Indexical Digital Mechanism, it 
is absolutely undecidable if the Universe is bigger than the sigma_1 
reality. (But here I do a blasphemy: that can only be entirely 
justified by G* *only*!, It is where I have to insist that this is 
presented as a consequence of YD + CT (“yes doctor” + Church-Turing 
thesis).


Such theories are essentially undecidable. It means that not only they 
are arithmetically incomplete, but all their effective consistent 
extensions are too. They are creative, you cannot capture the semantic 
in the way it could become complete, even in some imaginary domain 
concevable by the machine/theory/number. The universal machine are 
never entirely satisfied and a computation is always an escape 
forward, but their self-reflection create a mess, and illusions.


The sigma_1 arithmetical reality, as seen by the universal numbers 
which lives there, in the first person undetermined sense, is 
something *very big*. It generates infinitely many surprises. There 
are consistent histories.




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Are there real numbers that cannot be defined?

2019-02-25 Thread Philip Thrift
via
https://twitter.com/JDHamkins/status/1100090709527408640

Joel David Hamkins   @JDHamkins

*Must there be numbers we cannot describe or define? Definability in 
mathematics and the Math Tea argument*
Pure Mathematics Research Seminar at the University of East Anglia in 
Norwich on Monday, 25 February, 2019.


Abstract:

*An old argument, heard perhaps at a good math tea, proceeds: “there must 
be some real numbers that we can neither describe nor define, since there 
are uncountably many real numbers, but only countably many definitions.” 
Does it withstand scrutiny? In this talk, I will discuss the phenomenon of 
pointwise definable structures in mathematics, structures in which every 
object has a property that only it exhibits. A mathematical structure is 
Leibnizian, in contrast, if any pair of distinct objects in it exhibit 
different properties. Is there a Leibnizian structure with no definable 
elements? Must indiscernible elements in a mathematical structure be 
automorphic images of one another? We shall discuss many elementary yet 
interesting examples, eventually working up to the proof that every 
countable model of set theory has a pointwise definable extension, in which 
every mathematical object is definable.*

http://jdh.hamkins.org/must-there-be-number-we-cannot-define-norwich-february-2019/

Lecture notes:
http://jdh.hamkins.org/wp-content/uploads/2019/02/Must-every-number-be-definable_-Norwich-Feb-2019.pdf


- pt

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Re: Modal logic, consciousness, and matter

2019-02-25 Thread Bruno Marchal

> On 25 Feb 2019, at 11:52, Philip Thrift  wrote:
> 
> 
> 
> On Monday, February 25, 2019 at 3:34:15 AM UTC-6, Bruno Marchal wrote:
> 
>> On 22 Feb 2019, at 18:44, Philip Thrift > 
>> wrote:
>> 
>> Some accept the possibility that there can be something that is immaterial.
> 
> Yes. We call them “mathematician”.
> 
> Bruno
> 
> 
> 
> This recent thesis I came across
> 
> Application and ontology in mathematics: a defence [defense] of fictionalism
> http://etheses.whiterose.ac.uk/18636/
> 
> leads to
> 
> pure mathematicians may be immaterialists,
> but applied mathematicians are materialists.
> 
> 
> 
> Abstract
> 
> The aim of this thesis is to defend fictionalism as a response to the 
> mathematical placement problem. As we will see, against the backdrop of 
> philosophical naturalism, it is difficult to see how to fit mathematical 
> objects into our best total scientific theory.
> 
Which “best total scientific theory”? Theology is still taboo, and there is no 
coherent physical theory of the universe, nor clear argument why that would 
exist.

Assuming mechanism, assuming more than the natural numbers, or than the 
combinators, or the programs, is just speculation on undecidable metaphysical 
ontologies. No doubt is put on physics as an art to put relevant order on the 
observable and predictable, but it can’t be the fundamental science. It has to 
be derived from the theology, that is from G* (actually from qZ1*, the 
observable-mode of self-reference, motivated through thought experience and/or 
Theaetetus/Parmenides/Moderatus/Plotinus. (It is an old alternative viewing of 
reality).



> On the other hand, the indispensability argument seems to suggest that 
> science itself mandates ontological commitment to mathematical entities.
> 

Not really. It needs we agree on some basic starting simple relation.

As I have demonstrate recently here, the relation 

Kxy = x
Sxyz = xz(yz)

Are enough. But classical logic + Robinson arithmetic is enough.

No need of ontological commitment other that not denying what we found almost 
obvious in primary school. You can remain formal, but it is simpler to do a bit 
of math and get the intuition that indeed it kick back and, well, 2+2 is not 
equal to 5.


> My goal is to undermine the indispensability argument by presenting an 
> account of applied mathematics as a kind of revolutionary prop-oriented 
> make-believe, the content of which is given by a mapping account of 
> mathematical applications. This kind of fictionalism faces a number of 
> challenges from various quarters. To begin with, we will have to face the 
> challenge of a different kind of indispensability argument, one that draws 
> ontological conclusions from the role of mathematical objects in scientific 
> explanations. We will then examine one recent theory of mathematical 
> scientific representation, and discover that the resulting position is 
> Platonistic. At this point we will introduce our fictionalist account, and 
> see that it defuses the Platonist consequences of mathematical 
> representation. The closing chapters of the thesis then take a 
> metaphilosophical turn. The legitmacy of a fictionalist response to the 
> mathematical placement problem is open to challenge from a metaphilosophical 
> perspective in two different ways: on the one hand, some modern pragmatists 
> have argued that this kind of metaphysics relies on questionable assumptions 
> about how langauge works. On the other, some modern philosophers have 
> developed forms of metaontological anti-realism that they believe undermine 
> the legitimacy of philosophical work in metaphysics. In the final two 
> chapters I defend the fictionalist account developed here against these 
> sceptical claims. I conclude that the fictionalist account of applied 
> mathematics offered here is our best hope for coping with the mathematical 
> placement problem. 
> 
> 


It illustrates the kind of difficulties you can meet when you take for granted 
the idea that the fundamental reality is physical. 

There is only a placement problem for mathematics because people commit 
themselves into some *place* which does not seem to be an hypothesis making 
thing simpler.

Fictionalism does not apply to the arithmetical reality, nor to physics, but to 
the naïve idea of a “physical universe” as being the fundamental reality. The 
theology of the universal machine is a priori quite non Aristotelian: there is 
no Creator, and there is no Creation. Just a universal dreamer which lost 
itself in an infinitely surprising structure and wake up from time to time, or 
from numbers to numbers.

I need no more than a partial applicative algebra, and each choice of the phi_i 
makes N into one, simply by defining an operation “*” in N such that n * m = 
phi_n(m). There exist numbers k and s such that

((k * n) * m) = n
(((s * n) * m) * r) = (n * r) * (m * r),

for all m, n, r in N.

And, the key point, the operation “*” can be defined in the 

Re: Recommend this article, Even just for the Wheeler quote near the end

2019-02-25 Thread Bruno Marchal

> On 25 Feb 2019, at 12:39, Lawrence Crowell  
> wrote:
> 
> On Monday, February 25, 2019 at 2:44:14 AM UTC-6, Bruno Marchal wrote:
> 
>> On 24 Feb 2019, at 15:24, Lawrence Crowell > > wrote:
>> 
>> On Friday, February 22, 2019 at 3:18:01 PM UTC-6, Brent wrote:
>> 
>> 
>> On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
>>> This sounds almost tautological. I have not read Masanes' paper, but he 
>>> seems to be saying the Born rule is a matter of pure logic. In some ways 
>>> that is what Born said.
>>> 
>>> The Born rule is not hard to understand. If you have a state space with 
>>> vectors |u_i> then a quantum state can be written as sum_ic_i|u_i>. For an 
>>> observable O with eigenvectors o_i the expectation values for that 
>>> observable is
>>> 
>>>  sum_{ij} = sum_{ij} = sum_ip_io_i.
>>> 
>>> So the expectations of each eigenvalue is multiple of the probability for 
>>> the system to be found in that state. It is not hard to understand, but the 
>>> problem is there is no general theorem and proof that the eigenvalues of an 
>>> operator or observable are diagonal in the probabilities.
> 
> I am not sure I understand this.
> 
> 
> 
> 
>>> In fact this has some subtle issues with degeneracies.
>> 
>> Doesn't Gleason's theorem show that there is no other consistent way to 
>> assign probabilities to subspaces of a Hilbert space?
>> 
>> Brent
>> 
>> It is close. Gleason's theorem tells us that probabilities are a consequence 
>> of certain measurements. So for a basis Q = {q_n} then in a span in Q = 
>> P{q_n}, for P a projection operator that a measure μ(Q} is given by a trace 
>> over projection operators. This is close, but it does not address the issue 
>> of eigenvalues of an operator or observable. Gleason tried to make this work 
>> for operators, but was ultimately not able to.
> 
> It should work for the projection operator, that this is the 
> yes-no-experiment, but that extends to the other measurement, by reducing (as 
> usual) the question “what is the value of A” into the (many) question “does A 
> measurement belong to this interval” … Gleason’s theorem assures that the 
> measure is unique (on the subspaces of H with dim bigger or equal to 3), so 
> the Born rule should be determined, at least in non degenerate case (but also 
> in the degenerate case when the degeneracy is due to tracing out a subsystem 
> from a bigger system. I will verify later as my mind belongs more to the 
> combinator and applicative algebra that QM for now.
> 
> 
> 
>> 
>> Many years ago I had an idea that since the trace of a density matrix may be 
>> thought of as constructed from projection operators with tr(ρ_n) = sum_n 
>> |c_n|^2P_n, that observables that commute with the density matrix might have 
>> a derived Born rule following Gleason. Further, maybe operators that do not 
>> commute then have some dual property that still upholds Born rule. I was not 
>> able to make this work.
> 
> I will think about this. Normally the measure is determine by the “right" 
> quantum logic, and the right quantum logic is determined by the any 
> “provability” box accompanied by consistency condition (like []p & p, []p & 
> <>t, …).  The main difference to be expected, is that eventually we get a 
> “quantum credibility measure”, not really a probability. It is like 
> probability, except that credibility is between 0 and infinity (not 0 and 1).
> 
> Bruno
> 
> 
> I think I ran into the issue of why Gleason's theorem does not capture the 
> Born rule. Not all operators are commutative with the density matrix. So if 
> you construct the diagonal of the density matrix, or its trace elements, with 
> projector operators and off diagonal elements with left and right acting 
> projectors (left acting hit bra vectors and right acting hit ket vectors) the 
> problem is many operators are non-commutative. In particular the usual 
> situation is for the Hamiltonian to have nontrivial commutation with the 
> density matrix.


It seems to me that Gleason theorem takes this into account. It only means that 
the probabilities does not make the same partition of the multiverse, but that 
is not a problem for someone who use physics to see if it confirms or refute 
the “observable” available to the universal numbers/machines in arithmetic.

I am not completely sure. You raise a doubt, and I’m afraid it will take some 
time I come back to Gleason theorem. But I appreciate. My conversation with 
Bruce and Brent makes me think that the notion of multiverse is far from clear. 
At least with mechanism things are crystal clear! There is only the sigma_1 
sentences, and the nuances imposed by incompleteness for the “Löbian number” 
who “lives” through them (them for the sigma_sentences, which “realises” the 
computations).

Of course I come from the other side, but if mechanism is correct, I can only 
cross physics when and where physics is correct. For now, physics is not yet a 
solved problem, as GR does not fit with QM. The very notion 

Re: Recommend this article, Even just for the Wheeler quote near the end

2019-02-25 Thread Lawrence Crowell
On Monday, February 25, 2019 at 2:44:14 AM UTC-6, Bruno Marchal wrote:
>
>
> On 24 Feb 2019, at 15:24, Lawrence Crowell  > wrote:
>
> On Friday, February 22, 2019 at 3:18:01 PM UTC-6, Brent wrote:
>>
>>
>>
>> On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
>>
>> This sounds almost tautological. I have not read Masanes' paper, but he 
>> seems to be saying the Born rule is a matter of pure logic. In some ways 
>> that is what Born said.
>>
>> The Born rule is not hard to understand. If you have a state space with 
>> vectors |u_i> then a quantum state can be written as sum_ic_i|u_i>. For an 
>> observable O with eigenvectors o_i the expectation values for that 
>> observable is
>>
>>  sum_{ij} = sum_{ij} = sum_ip_io_i.
>>
>> So the expectations of each eigenvalue is multiple of the probability for 
>> the system to be found in that state. It is not hard to understand, but the 
>> problem is there is no general theorem and proof that the eigenvalues of an 
>> operator or observable are diagonal in the probabilities. 
>>
>>
> I am not sure I understand this.
>
>
>
>
> In fact this has some subtle issues with degeneracies.
>>
>>
>> Doesn't Gleason's theorem show that there is no other consistent way to 
>> assign probabilities to subspaces of a Hilbert space?
>>
>> Brent
>>
>
> It is close. Gleason's theorem tells us that probabilities are a 
> consequence of certain measurements. So for a basis Q = {q_n} then in a 
> span in Q = P{q_n}, for P a projection operator that a measure μ(Q} is 
> given by a trace over projection operators. This is close, but it does not 
> address the issue of eigenvalues of an operator or observable. Gleason 
> tried to make this work for operators, but was ultimately not able to.
>
>
> It should work for the projection operator, that this is the 
> yes-no-experiment, but that extends to the other measurement, by reducing 
> (as usual) the question “what is the value of A” into the (many) question 
> “does A measurement belong to this interval” … Gleason’s theorem assures 
> that the measure is unique (on the subspaces of H with dim bigger or equal 
> to 3), so the Born rule should be determined, at least in non degenerate 
> case (but also in the degenerate case when the degeneracy is due to tracing 
> out a subsystem from a bigger system. I will verify later as my mind 
> belongs more to the combinator and applicative algebra that QM for now.
>
>
>
>
> Many years ago I had an idea that since the trace of a density matrix may 
> be thought of as constructed from projection operators with tr(ρ_n) = sum_n 
> |c_n|^2P_n, that observables that commute with the density matrix might 
> have a derived Born rule following Gleason. Further, maybe operators that 
> do not commute then have some dual property that still upholds Born rule. I 
> was not able to make this work.
>
>
> I will think about this. Normally the measure is determine by the “right" 
> quantum logic, and the right quantum logic is determined by the any 
> “provability” box accompanied by consistency condition (like []p & p, []p & 
> <>t, …).  The main difference to be expected, is that eventually we get a 
> “quantum credibility measure”, not really a probability. It is like 
> probability, except that credibility is between 0 and infinity (not 0 and 
> 1).
>
> Bruno
>
>
I think I ran into the issue of why Gleason's theorem does not capture the 
Born rule. Not all operators are commutative with the density matrix. So if 
you construct the diagonal of the density matrix, or its trace elements, 
with projector operators and off diagonal elements with left and right 
acting projectors (left acting hit bra vectors and right acting hit ket 
vectors) the problem is many operators are non-commutative. In particular 
the usual situation is for the Hamiltonian to have nontrivial commutation 
with the density matrix.

LC
 

>
>
> LC
>
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Re: Modal logic, consciousness, and matter

2019-02-25 Thread Philip Thrift


On Monday, February 25, 2019 at 3:34:15 AM UTC-6, Bruno Marchal wrote:
>
>
> On 22 Feb 2019, at 18:44, Philip Thrift > 
> wrote:
>
>
> Some accept the possibility that there can be something that is immaterial.
>
>
> Yes. We call them “mathematician”.
>
> Bruno
>
>
>
This recent thesis I came across

*Application and ontology in mathematics: a defence [defense] of 
fictionalism*
http://etheses.whiterose.ac.uk/18636/

leads to

pure mathematicians may be immaterialists,
but applied mathematicians are materialists.



Abstract

The aim of this thesis is to defend fictionalism as a response to the 
mathematical placement problem. As we will see, against the backdrop of 
philosophical naturalism, it is difficult to see how to fit mathematical 
objects into our best total scientific theory. On the other hand, the 
indispensability argument seems to suggest that science itself mandates 
ontological commitment to mathematical entities. My goal is to undermine 
the indispensability argument by presenting an account of applied 
mathematics as a kind of revolutionary prop-oriented make-believe, the 
content of which is given by a mapping account of mathematical 
applications. This kind of fictionalism faces a number of challenges from 
various quarters. To begin with, we will have to face the challenge of a 
different kind of indispensability argument, one that draws ontological 
conclusions from the role of mathematical objects in scientific 
explanations. We will then examine one recent theory of mathematical 
scientific representation, and discover that the resulting position is 
Platonistic. At this point we will introduce our fictionalist account, and 
see that it defuses the Platonist consequences of mathematical 
representation. The closing chapters of the thesis then take a 
metaphilosophical turn. The legitmacy of a fictionalist response to the 
mathematical placement problem is open to challenge from a 
metaphilosophical perspective in two different ways: on the one hand, some 
modern pragmatists have argued that this kind of metaphysics relies on 
questionable assumptions about how langauge works. On the other, some 
modern philosophers have developed forms of metaontological anti-realism 
that they believe undermine the legitimacy of philosophical work in 
metaphysics. In the final two chapters I defend the fictionalist account 
developed here against these sceptical claims. I conclude that the 
fictionalist account of applied mathematics offered here is our best hope 
for coping with the mathematical placement problem. 


- pt

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Re: Modal logic, consciousness, and matter

2019-02-25 Thread Bruno Marchal

> On 22 Feb 2019, at 18:44, Philip Thrift  wrote:
> 
> 
> 
> On Friday, February 22, 2019 at 3:57:49 AM UTC-6, Bruno Marchal wrote:
> 
>> On 21 Feb 2019, at 20:26, Philip Thrift > 
>> wrote:
>> 
>> 
>> 
>> On Thursday, February 21, 2019 at 8:23:15 AM UTC-6, Bruno Marchal wrote:
>> 
>>> On 18 Feb 2019, at 20:18, Philip Thrift > wrote:
>>> 
>>> On Monday, February 18, 2019 at 9:14:38 AM UTC-6, Bruno Marchal wrote:
>>> 
>>> > https://groups.google.com/d/msg/everything-list/0SIiavzPI84/jUkaOlUdAwAJ 
>>> > 
>>> This is the link to the reply in the topic "When Did Consciousness Begin?" 
>>> As I have said before, the modal logic and numerical semantics written 
>>> there is one way to approach the science of experience. But I think 
>>> ultimately this is a logical semantics (not a material semantics), but of 
>>> course belief in an actual numerical reality makes a difference.
>>> 
>>> Here is something more along those lines:
>>> 
>>> On modal logic and consciousness:
>>> 
>>> A Modal Logic for Gödelian Intuition
>>> Hasen Khudairi
>>> https://philarchive.org/archive/KHUAML 
>>> 
>>> 
>>> Towards an Axiomatic Theory of Consciousness
>>> Jim Cunningham [ https://www.doc.ic.ac.uk/~rjc/ 
>>>  ]
>>> https://www.doc.ic.ac.uk/~rjc/Cunningham.pdf 
>>> 
>>> 
>>> 
>>> However I think that there is ultimately a material semantics.
>> 
>> I can imagine a semantic for some theory of matter, but matter itself cannot 
>> be semantical. What would that mean? Even without mechanism, I have no idea 
>> what that could mean.
>> 
>> 
>> 
>> 
>> I define material semantics here:
>> 
>> Material Semantics for Unconventional Programming
>> https://codicalist.wordpress.com/2018/12/14/material-semantics-for-unconventional-programming/
>>  
>> 
>> 
>> 
>>  
>>material semantics =
>> 
>>  physical (incl. chemical+biological)
>>  +
>>  psychical (or experiential) semantics
>> 
>> 
> 
> That does not assume the existence of an ontological matter. 
> 
> 
> 
>>  
>> 
>> 
>>> 
>>> As for the ancient Greeks, forget Aristotle and look to Epicurus.
>> 
>> I assume mechanism, and I just listen to the universal machine, with machine 
>> taken in the sense of Church and Turing. The notion of matter is not assumed 
>> in such definition, and we know, basically since Gödel, that they exist in 
>> arithmetic (semantically, of course).
>> 
>> 
>> 
>>> 
>>> 
>>> 
>>> Some "offbeat" materialism I just came across that my be of interest:
>>> 
>>> Terry Eagleton 
>>> Materialism, Yale University Press
>>> excerpt 1: https://play.google.com/books/reader?id=PErJDQAAQBAJ 
>>> 
>>> excerpt 2: 
>>> http://blog.yalebooks.com/2017/02/09/material-theology-and-christian-religion/
>>>  
>>> 
>> Very nice. I would not have dare to suggest that christianism is so much 
>> materialist. I am not sure this was true during the five first hundreds 
>> years, but it is dogmatically so after 529 (closure of Plato’s academy).
>> 
>> 
>> 
>>> 
>>> 
>>> 
>>> Matter is an aggregation of cuisines whose recipes arise and combine to 
>>> form the cookbook of nature.
>> 
>> No problem with this, on the contrary. A recipe is another name for an 
>> algorithm, which is typically not made of matter, but exist in arithmetic. 
>> 
>> And for the cooking, there is no need distinguish primary matter, which 
>> cannot exist with Mechanism, and matter, which obviously exist 
>> phenomenologically. 
>> 
>> Bruno
>> 
>> 
>> 
>> 
>> Of course all algorithms (technically) are made of matter:
> 
> I disagree. An algorithm is an immaterial recipe to compute a function, or to 
> implement a process, and you can do that in any universal machinery, 
> implemented in the physical reality or not. The physical reality itself is 
> not produced by an algorithm, but emerges from the first person indeterminacy 
> on all consistent computational histories, structure by the observable mode. 
> That explains quanta and qualia, in a testable (and tested) way.
> 
> (I'don't know what this test is.)


It consists, roughly, in comparing the quantum logic extrapolated from 
observation, with the quantum logics "in the head of the machine”.  We get 
three quantum logics “in the head of the machine”: the one given (all the time 
with p for a sigma_1 (computable) sentences by []p & p, []p & <>t, and []p & 
<>t & p.
([]p is an abbreviation of Gödel’s beweisbar, <>p is ~[]~p).

Each modal nuance (with “ & <>p”, or “& p” and p sigma_1) give modal logics 
mimicking quantum logic (like S4Grz gives logics mimicking 

Re: Recommend this article, Even just for the Wheeler quote near the end

2019-02-25 Thread Bruno Marchal

> On 24 Feb 2019, at 15:24, Lawrence Crowell  
> wrote:
> 
> On Friday, February 22, 2019 at 3:18:01 PM UTC-6, Brent wrote:
> 
> 
> On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
>> This sounds almost tautological. I have not read Masanes' paper, but he 
>> seems to be saying the Born rule is a matter of pure logic. In some ways 
>> that is what Born said.
>> 
>> The Born rule is not hard to understand. If you have a state space with 
>> vectors |u_i> then a quantum state can be written as sum_ic_i|u_i>. For an 
>> observable O with eigenvectors o_i the expectation values for that 
>> observable is
>> 
>>  sum_{ij} = sum_{ij} = sum_ip_io_i.
>> 
>> So the expectations of each eigenvalue is multiple of the probability for 
>> the system to be found in that state. It is not hard to understand, but the 
>> problem is there is no general theorem and proof that the eigenvalues of an 
>> operator or observable are diagonal in the probabilities.

I am not sure I understand this.




>> In fact this has some subtle issues with degeneracies.
> 
> Doesn't Gleason's theorem show that there is no other consistent way to 
> assign probabilities to subspaces of a Hilbert space?
> 
> Brent
> 
> It is close. Gleason's theorem tells us that probabilities are a consequence 
> of certain measurements. So for a basis Q = {q_n} then in a span in Q = 
> P{q_n}, for P a projection operator that a measure μ(Q} is given by a trace 
> over projection operators. This is close, but it does not address the issue 
> of eigenvalues of an operator or observable. Gleason tried to make this work 
> for operators, but was ultimately not able to.

It should work for the projection operator, that this is the yes-no-experiment, 
but that extends to the other measurement, by reducing (as usual) the question 
“what is the value of A” into the (many) question “does A measurement belong to 
this interval” … Gleason’s theorem assures that the measure is unique (on the 
subspaces of H with dim bigger or equal to 3), so the Born rule should be 
determined, at least in non degenerate case (but also in the degenerate case 
when the degeneracy is due to tracing out a subsystem from a bigger system. I 
will verify later as my mind belongs more to the combinator and applicative 
algebra that QM for now.



> 
> Many years ago I had an idea that since the trace of a density matrix may be 
> thought of as constructed from projection operators with tr(ρ_n) = sum_n 
> |c_n|^2P_n, that observables that commute with the density matrix might have 
> a derived Born rule following Gleason. Further, maybe operators that do not 
> commute then have some dual property that still upholds Born rule. I was not 
> able to make this work.

I will think about this. Normally the measure is determine by the “right" 
quantum logic, and the right quantum logic is determined by the any 
“provability” box accompanied by consistency condition (like []p & p, []p & 
<>t, …).  The main difference to be expected, is that eventually we get a 
“quantum credibility measure”, not really a probability. It is like 
probability, except that credibility is between 0 and infinity (not 0 and 1).

Bruno


> 
> LC
> 
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