Re: Questions about the Equivalence Principle (EP) and GR

[Brent Meeker]

On 2/27/2019 4:58 PM, agrayson2...@gmail.com wrote:
*Are you assuming uniqueness to tensors; that only tensors can produce 
covariance in 4-space? Is that established or a mathematical 
speculation? TIA, AG *


That's looking at it the wrong way around.  Anything that transforms as 
an object in space, must be representable by tensors. The informal 
definition of a tensor is something that transforms like an object, i.e. 
in three space it's something that has a location and an orientation and 
three extensions.  Something that doesn't transform as a tensor under 
coordinate system changes is something that depends on the arbitrary 
choice of coordinate system and so cannot be a fundamental physical object.


Brent

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

[agrayson2000]

On Wednesday, February 13, 2019 at 9:40:32 PM UTC-7, cdemorsella wrote:
>
> Two fascinating (and very different) approaches are presented to derive 
> Quantim Mechanics main practical tool (e.g. Born's rule). Wonder what some 
> of the physicists on here think about this research?
>
> I find the argument that no laws is the fundamental law... and that the 
> universe and its laws are emergent guided by subtle mathematical 
> statistical phenomena, at the same time both alluring and annoying it 
> is somehow unsatisfactory like being served a quite empty plate with 
> nice garnish for dinner.
>
> One example of emergence from chaotic conditions is how traffic jams (aka 
> density waves) can emerge from chaotic initial conditions, becoming self 
> re-enforcing within local domains of influence... for those unlucky to be 
> stuck in them. Density wave emergence is seen across scale, for example the 
> spiral arms of galaxies can be explained as giant gravitational pile ups 
> with some fundamentally similar parallels to say a rush hour traffic jam, 
> except on vastly different scales of course and due to other different 
> factors, in the galactic case the emergent effects of a vast number of 
> gravitational inter-actions as stars migrate through these arms on their 
> grand voyages around the galactic core.
>
> This paired with the corollary argument that any attempt to discover a 
> fundamental law seems doomed to the infinite regression of then needing to 
> explain what this foundation itself rests upon leading to the "it's 
> turtles all the way down" hall of mirrors carnival house... head-banger. 
>
> Perhaps, as Wheeler argued, the world is a self-synthesizing system, and 
> the seeming order we observe, is emergent... a law without law.
>
> Here is the link to the article:
>
>
>
> The Born Rule Has Been Derived From Simple Physical Principles | Quanta 
> Magazine 
> 
> The Born Rule Has Been Derived From Simple Physical Principles | Quanta 
> Magazine 
>
> The new work promises to give researchers a better grip on the core 
> mystery of quantum mechanics.
>
> 
>  
>
>  

> *Is there consensus that Born's rule can be, and has been derived from 
> physical principles, and/or the other postulates of QM? TIA, AG*
>

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Re: Questions about the Equivalence Principle (EP) and GR

[agrayson2000]

On Sunday, February 24, 2019 at 4:49:45 PM UTC-7, Lawrence Crowell wrote:
>
> On Sunday, February 24, 2019 at 5:31:35 PM UTC-6, agrays...@gmail.com 
> wrote:
>>
>>
>>
>> On Sunday, February 24, 2019 at 6:41:00 AM UTC-7, Lawrence Crowell wrote:
>>>
>>> On Friday, February 22, 2019 at 4:40:31 PM UTC-6, agrays...@gmail.com 
>>> wrote:



 On Friday, February 22, 2019 at 1:34:31 PM UTC-7, Brent wrote:
>
>
>
> On 2/21/2019 10:47 PM, agrays...@gmail.com wrote:
>
>
>>
> *Even if gravitons are detected, and they account for "force" 
> consistent with the other three forces, wouldn't there remain the task of 
> changing the form of gravity to make it covariant? AG*
>
>
> Gravitons, as quanta of the metric field, are already relativistic 
> particles and covariant.
>

 *I thought it's the equations of motion for the particular force, not 
 the mediating particles, that must be covariant. On a related topic for 
 this thread, where does GR depart from Mach's principle? That is, what did 
 Einstein implicitly (or explicitly) deny about Mach's principle? TIA, AG *

>
> *Would that require tensors? AG*
>
>
>>> General relativity is covariant, and curvature is expressed according to 
>>> Riemann tensors. 
>>>
>>> LC
>>>
>>
>> *Thanks, but I think you missed the thrust of my question; namely, if a 
>> theory using gravitons is independent of GR, since it would have to be 
>> covariant, could that be done without tenors, or are tensors nevertheless 
>> necessary.  AG*
>>
>
> Tensors transform homogeneously with the Lorentz group and are thus 
> covariant. Yep you need tensors. 
>
> LC 
>

*Are you assuming uniqueness to tensors; that only tensors can produce 
covariance in 4-space? Is that established or a mathematical speculation? 
TIA, AG *

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Re: Are there real numbers that cannot be defined?

[Philip Thrift]

On Wednesday, February 27, 2019 at 11:25:06 AM UTC-6, Bruno Marchal wrote:
>
>
> On 26 Feb 2019, at 19:41, Philip Thrift > 
> wrote:
>
>
>
> On Tuesday, February 26, 2019 at 5:07:38 AM UTC-6, Bruno Marchal wrote:
>>
>>
>> On 25 Feb 2019, at 20:35, Philip Thrift  wrote:
>>
>> 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.*
>>
>>
>>
>> But that argument is rather weak, as the notion of cardinality is a 
>> relative notion, depending of the model (not the theory) that we might use. 
>> There are countable models of Cantor’s theory of set and the transfinite 
>> (cf the paradox of Skolem). If you agree o identify a real number with a 
>> total computable function (from N to N), as Turing did originally, then you 
>> can prove the existence of specific non definable real number, in any rich 
>> enough extension of any essentially undecidable theory.
>>
>> It is very simple, any theory reich enough to define a universal 
>> machine/number, is automatically essentially undecidable. It is a generator 
>> of infinitely many surprises for *any* machines and super-marching, etc. We 
>> know now that we know basically nothing, with such “rich” theories. 
>> Elementary arithmetic is already essentially undecidable.
>>
>> You can change the logic, and will get quite different view on the real 
>> numbers. In brouwer’s intuitionistic logic, and in the effective topos 
>> (which generalises Kleene’s realisability notion, and is based on the 
>> category of partial combinatory algebra): we have that all real number are 
>> computable, and all functions are continuous. I am not sure if we get that 
>> all real numbers will be definable, though. They might be not-not-definable 
>> ...
>>
>>
>>
>> *” 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. *
>>
>>
>> x ≠ y ->. Ax ≠ Ay, that is Ax = Ay ->. x = y. That is the axiom of 
>> extensionality (in Combinators, lama calculus, set theories).
>>
>> I have used it in the elimination of variables in the combinations. It is 
>> a god’s gift, as it leads to simple efficacious combinators. It is, with 
>> the combinators, equivalent to [x](Ax) = A, when A has no free occurence of 
>> x. Not to be confuse with ([x]A)x = A (which is always true, and just 
>> defines what elimination of x means). 
>>
>>
>>
>> *Is there a Leibnizian structure with no definable elements? *
>>
>>
>> Yes. The classical reals, or the classical set of total computable 
>> functions.
>>
>>
>>
>> *Must indiscernible elements in a mathematical structure be automorphic 
>> images of one another? *
>>
>>
>> No. If we cannot discern them, we cannot build a morphism between them. I 
>> would say.
>>
>>
>> *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
>>
>>
>> I will take a look, but that does not make much sense, unless the logic 
>> is weakemed in some way. In some intuitionist set theory with choice, it 
>> might make sense.
>>
>> Bruno
>>
>>
>>
>>
> There is another approach vs. the JD Hamkins / set theory & mathematical 
> logic (math dept.) approach. the one of Martin Escardo / type & programming 
> language theory (computer science dept.).
>
> Martin Escardo
> http://www.cs.bham.ac.uk/~mhe/
>
> *higher-type computation = *
>
> *sets that can be exhaustively searched by an algorithm, in the sense of 
> Turing, in finite time, as those that are topologically compact*
>
>
> ?
>
> If it is an algorithm in the sense of Turing, and if the search is 
> required to be finite, then the set is finite.
>



In any case, there is programming he has written

M.H. Escardo. *Infinite sets that admit fast exhaustive search*. 

In LICS'2007, IEEE, pages 443-452, Poland, Wroclaw, July.

pdf  (paper), hs 
 (comp

Re: Modal logic, consciousness, and matter

[Bruno Marchal]

> On 26 Feb 2019, at 23:45, Philip Thrift  wrote:
> 
> 
> 
> On Tuesday, February 26, 2019 at 4:39:25 PM UTC-6, Brent wrote:
> 
> 
> On 2/26/2019 2:02 PM, Philip Thrift wrote:
>> 
>> 
>> On Tuesday, February 26, 2019 at 2:51:39 PM UTC-6, Brent wrote:
>> 
>> 
>> On 2/26/2019 11:00 AM, Philip Thrift wrote:
>>> 
>>> 
>>> On Tuesday, February 26, 2019 at 12:43:49 PM UTC-6, Brent wrote:
>>> 
>>> 
>>> Right.  Truth and existence are quite different things. 
>>> 
>>> Brent 
>>> 
>>> 
>>> 
>>> For those from the type theory, programming language theory, constructive 
>>> mathematics (whatever that clumping of schools is called):
>>> 
>>> Truth and existence are the same things.
>> 
>> So do those infer the existence of 2 and 4 from truth of 2+2=4?
>> 
>> Brent
>> 
>> Formulate arithmetic as a logic program and enter the query:
>> 
>>∃(X,Y):(X+X=Y)
>> 
>> Then via backtracking it prints out:
>> 
>> (X,Y) =
>> 
>> (0,0)
>> (1,2)
>> (2,4)
>> ...
>> 
>> 
> Formulate theology as a logic program and enter the query:
> 
>   E(x)[If P is a prefection, then Px]
> 
> Then it prints out:
> 
> x = Anslem's God.
> 
> That's the great thing about logic. You can prove anything if you just the 
> right axioms and rules of inference.
> 
> Brent
> 
> 
> 
> Automating Godel’s Ontological Proof of God’s Existence with Higher-order 
> Automated Theorem Provers
> http://page.mi.fu-berlin.de/cbenzmueller/papers/C40.pdf

This illustrates only that we can “brainwash” machine, even machine which 
reason in a valid way, by using absurd definitions or premise. That will not 
convince Brent, I’m afraid. It is logically interesting though, and not 
completely trivial, but hardly convincing as an argument of metaphysics. You 
could as well say that infinity exists because ZF say so. It is not valid 
argument. It is an argument per-authority, unless you make clear that you 
assume the axioms of ZF.

Bruno


> 
> - pt
>  
> 
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Re: Are there real numbers that cannot be defined?

[Bruno Marchal]

> On 26 Feb 2019, at 19:41, Philip Thrift  wrote:
> 
> 
> 
> On Tuesday, February 26, 2019 at 5:07:38 AM UTC-6, Bruno Marchal wrote:
> 
>> On 25 Feb 2019, at 20:35, Philip Thrift > 
>> wrote:
>> 
>> 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.
> 
> 
> But that argument is rather weak, as the notion of cardinality is a relative 
> notion, depending of the model (not the theory) that we might use. There are 
> countable models of Cantor’s theory of set and the transfinite (cf the 
> paradox of Skolem). If you agree o identify a real number with a total 
> computable function (from N to N), as Turing did originally, then you can 
> prove the existence of specific non definable real number, in any rich enough 
> extension of any essentially undecidable theory.
> 
> It is very simple, any theory reich enough to define a universal 
> machine/number, is automatically essentially undecidable. It is a generator 
> of infinitely many surprises for *any* machines and super-marching, etc. We 
> know now that we know basically nothing, with such “rich” theories. 
> Elementary arithmetic is already essentially undecidable.
> 
> You can change the logic, and will get quite different view on the real 
> numbers. In brouwer’s intuitionistic logic, and in the effective topos (which 
> generalises Kleene’s realisability notion, and is based on the category of 
> partial combinatory algebra): we have that all real number are computable, 
> and all functions are continuous. I am not sure if we get that all real 
> numbers will be definable, though. They might be not-not-definable ...
> 
> 
> 
>> ” 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.
> 
> x ≠ y ->. Ax ≠ Ay, that is Ax = Ay ->. x = y. That is the axiom of 
> extensionality (in Combinators, lama calculus, set theories).
> 
> I have used it in the elimination of variables in the combinations. It is a 
> god’s gift, as it leads to simple efficacious combinators. It is, with the 
> combinators, equivalent to [x](Ax) = A, when A has no free occurence of x. 
> Not to be confuse with ([x]A)x = A (which is always true, and just defines 
> what elimination of x means). 
> 
> 
> 
>> Is there a Leibnizian structure with no definable elements?
> 
> Yes. The classical reals, or the classical set of total computable functions.
> 
> 
> 
>> Must indiscernible elements in a mathematical structure be automorphic 
>> images of one another?
> 
> No. If we cannot discern them, we cannot build a morphism between them. I 
> would say.
> 
> 
>> 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
>>  
>> 
> I will take a look, but that does not make much sense, unless the logic is 
> weakemed in some way. In some intuitionist set theory with choice, it might 
> make sense.
> 
> Bruno
> 
> 
> 
> 
> There is another approach vs. the JD Hamkins / set theory & mathematical 
> logic (math dept.) approach. the one of Martin Escardo / type & programming 
> language theory (computer science dept.).
> 
> Martin Escardo
> http://www.cs.bham.ac.uk/~mhe/
> 
> higher-type computation = 
> 
> sets that can be exhaustively searched by an algorithm, in the sense of 
> Turing, in finite time, as those that are topologically compact

?

If it is an algorithm in the sense of Turing, and if the search is required to 
be finite, then the set is finite.





> 
> constructive mathematics, which I see as a generalization, rather than as a 
> restriction, of classical mathematics.

Hmm…. That is highly debatable, and usually, when made rigorously, it is a

Re: Modal logic, consciousness, and matter

[Bruno Marchal]

> On 26 Feb 2019, at 19:43, Brent Meeker  wrote:
> 
> 
> 
> On 2/26/2019 2:39 AM, Bruno Marchal wrote:
>>> On 26 Feb 2019, at 01:04, Brent Meeker  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.
>> 
>> Yes, but God is defined by being perfect, and existence is considered as 
>> being good, and better than non existence. So, of course, God has to exists. 
>> Then Gödel made St-Anselmes more rigorous, by making that proof in the modal 
>> logic S5, which unfortunately presuppose a metaphysics incompatible with 
>> Mechanism.
>> Such notions of God are quite away from Plato, and makes sense with 
>> physicalism, and not much sense with Mechanism, where the notion of 
>> “fundamental truth" is the closer notion to the God of Plato. There is an 
>> understanding that such a notion of “fundamental truth”, at the origin of 
>> all types of truth, is transcendant, and this fits very well with the 
>> theology (G*) of the sound machines.
> 
> Right.  Truth and existence are quite different things.


Absolutely. And there are many sort of existence, even if we postulate only one 
“universal truth”.

With mechanism, it is absolutely undecidable if there is anything more than the 
sigma_1 truth (which gives already the universal dovetailing on all 
computations, and thus of all dreaming digital machines.

Then many sort of phenomenological existence are imposed by incompleteness. The 
absolute existence can be defined by the absolute truth of existential (not 
necessarily sigma_1) formula, like Ex(x is a prime number), or Ex(x is a 
halting computation), etc.

Most interesting notion of existence, like the psychological existence (or pain 
and suffering for example), or the material existence are explained in therm of 
the modal nuances enforced by incompleteness. Physical existence becomes 
something like []<>(Ex([]<>P(x)), with the box [] being the one of Z1*, X1*, or 
S4Grz1*.

It is the advantage of the Digital Mechanist hypothesis. It might be wrong, but 
it is mathematically precise, and physically testable.

Bruno



> 
> Brent
> 
>> 
>> Only strong-atheists believe in the God of the (Roman) Christians. Educated 
>> christians usually does not, although they fake it since 529 (due to 
>> violence and authoritative argument only).
>> 
>> Of course, no argument at all can prove any existence, but all experiences 
>> makes some existence true, although not in a rationally justified way.
>> 
>> Bruno
>> 
>> 
>> 
>> 
>>> 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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