### Re: Planck Length

```On Thursday, January 17, 2019 at 6:31:06 AM UTC-6, Bruno Marchal wrote:
>
>
> On 17 Jan 2019, at 09:22, agrays...@gmail.com  wrote:
>
>
>
> On Monday, January 7, 2019 at 9:25:16 PM UTC, John Clark wrote:
>>
>> On Mon, Jan 7, 2019 at 8:03 AM  wrote:
>>
>> *> How does one calculate Planck length using the fundamental constants
>>> G, h, and c, and having calculated it, how does one show that measuring a
>>> length that small with photons of the same approximate wave length, would
>>> result in a black hole? TIA, AG*
>>
>>
>> In any wave the speed of the wave is wavelength times frequency and
>> according to
>> Planck E= h*frequency  so E= C*h/wavelength.  Thus the smaller the
>> wavelength the greater the energy. According to Einstein energy is just
>> another form of mass (E = MC^2) so at some point the wavelength is so
>> small and the light photon is so energetic (aka massive) that the escape
>> velocity is greater than the speed of light and the object becomes a Black
>> Hole.
>>
>> Or you can look at it another way, we know from Heisenberg that to
>> determine the position of a particle more precisely with light you have to
>> use a smaller wavelength, and there is something called the  "Compton
>> wavelength" (Lc) ; to pin down the position of a particle of mass m to
>> within one Compton wavelength would require light of enough energy to
>> create another particle of that mass. The formula for the Compton
>> Wavelength is Lc= h/(2PI*M*c).
>>
>> Schwarzschild told us that the radius of a Black Hole (Rs), that is to
>> say where the escape velocity is the speed of light  is:  Rs= GM/c^2. At
>> some mass Lc will equal Rs and that mass is the Planck mass, and that Black
>> Hole will have the radius of the Planck Length, 1.6*10^-35 meters.
>>
>> Then if you do a little algebra:
>> GM/c^2 = h/(2PI*M*c)
>> GM= hc/2PI*M
>> GM^2 = hc/2*PI
>> M^2 = hc/2*PI*G
>> M = (hc/2*PI*G)^1/2and that is the formula for the Planck Mass ,
>> it's .02 milligrams.
>>
>> And the Planck Length turns out to be (G*h/2*PI*c^3)^1/2 and the Planck time
>> is the time it takes light to travel the Planck length.
>>
>> The Planck Temperature Tp is sort of the counterpoint to Absolute Zero,
>> Tp is as hot as things can get because the black-body radiation given off
>> by things when they are at temperature Tp have a wavelength equal to the
>> Planck Length, the distance light can move in the Planck Time of 10^-44
>> seconds. The formula for the Planck temperature is Tp = Mp*c^2/k where Mp
>> is the Planck Mass and K is Boltzmann's constant and it works out to be
>> 1.4*10^32 degrees Kelvin.  Beyond that point both Quantum Mechanics and
>> General Relativity break down and nobody understands what if anything is
>> going on.
>>
>> The surface temperature of the sun is at 5.7 *10^3  degrees Kelvin so if
>> it were 2.46*10^28 times hotter it would be at the Planck Temperature, and
>> because radiant energy is proportional to T^4 the sun would be 3.67*10^113
>> times brighter. At that temperature to equal the sun's brightness the
>> surface area would have to be reduced by a factor of 3.67*10^113, the
>> surface area of a sphere is proportional to the radius squared, so you'd
>> have to reduce the sun's radius by (3.67*10^113)^1/2, and that is
>> 6.05*10^56.
>> The sun's radius is 6.95*10^8   meters and  6.95*10^8/ 6.05*10^56  is
>> 1.15^10^-48 meters.
>>
>> That means a sphere at the Planck Temperature with a radius 10 thousand
>> billion times SMALLER than the Planck Length would be as bright as the sun,
>> but as far as we know nothing can be that small. If the radius was 10^13
>> times longer it would be as small as things can get and the object would be
>> (10^13)^2 = 10^26 times as bright as the sun. I'm just speculating but
>> perhaps that's the luminosity of the Big Bang; I say that because that's
>> how bright things would be if the smallest thing we think can exist was as
>> hot as we think things can get.
>>
>> John K Clark
>>
>
>
> *Later I'll post some questions I have about your derivation of the Planck
> length, but for now here's a philosophical question; Is there any
> difference between the claim that space is discrete, from the claim or
> conjecture that we cannot in principle measure a length shorter than the
> Planck length? *
> *TIA, AG *
>
>
> That is a very good question. I have no answer. I don’t think physicists
> have an answer either, and I do think that this requires the solution of
> the “quantum gravity” or the “quantum space-time” problem.
> With loop-gravity theory, I would say that the continuum is eventually
> replaced by something discrete, but not so with string theory; for example.
> With Mechanism, there are argument that something must stay “continuous”,
> but it might be only the distribution of probability (the real-complex
> amplitude).
>
> Bruno
>

The Planck length is just the smallest length beyond which you can isolate
a ```

### Re: The semantic view of theories and higher-order languages

```On Fri, Jan 18, 2019 at 8:30 AM Bruno Marchal  wrote:

*>Nwe cannot assume, neither a physical universe, nor analysis or set
> theory. Since recently, I have realised that we cannot even assume the
> induction axioms,*

Induction says that things are usually pretty much the same from one moment
of time to the next and from one point in space to a nearby one, if Everett
is right (and my hunch is he is) for some universes that would be true, but
such  a chaotic universe would not have structures capable of producing
thought or consciousness. Therefore  it is not only safe for us to assume
induction we DO assume it and we could not survive in the physical world
longer than about 45 seconds without it. At this very second although I
have no detailed knowledge of the wiring involved and have not seen the
blueprints I am assuming that when I hit the key marked "I" on my keyboard
a "I" symbol will appear on my screen; I assume it will happen this time
because that's what usually happened in the past, the only time it didn't
was when my keyboard was defective a few years ago but that was quickly
replaced.

John K Clark

>

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### Re: The semantic view of theories and higher-order languages

```

On Friday, January 18, 2019 at 7:30:14 AM UTC-6, Bruno Marchal wrote:
>
>
> On 18 Jan 2019, at 09:49, Philip Thrift >
> wrote:
>
>
> *The semantic view of theories and higher-order languages*
> Laurenz Hudetz
>
> "every family of set-theoretic structures has an associated language of
> higher-order logic and an up to signature isomorphism unique
> model-theoretic counterpart"
>
>
> *Several philosophers of science construe models of scientific theories as
> set-theoretic structures. Some of them moreover claim that models should
> not be construed as structures in the sense of model theory because the
> latter are language-dependent. I argue that if we are ready to construe
> models as set-theoretic structures (strict semantic view), we could equally
> well construe them as model-theoretic structures of higher-order logic
> (liberal semantic view). I show that every family of set-theoretic
> structures has an associated language of higher-order logic and an up to
> signature isomorphism unique model-theoretic counterpart, which is able to
> serve the same purposes. This allows to carry over every syntactic
> criterion of equivalence for theories in the sense of the liberal semantic
> view to theories in the sense of the strict semantic view. Taken together,
> these results suggest that the recent dispute about the semantic view and
> its relation to the syntactic view can be resolved.*
>
>
> It cannot do that in the Mechanist Frame, where we cannot assume, neither
> a physical universe, nor analysis or set theory. Since recently, I have
> realised that we cannot even assume the induction axioms, but we can use it
> in the definition of the Löbian entities, whose existence is then a
> consequence of the theories without induction. Of course, we need induction
> at the meta level, and even the whole of informal mathematics, like in any
> science pointing toward some reality independent of us.
>
> Bruno
>
>
>
>
The study of the dances between languages and their semantics is what both
philosophy and science are all about - from the view at least of
'language-oriented' philosophers.

- pt

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### Re: Coherent states of a superposition

```

On Friday, January 18, 2019 at 12:09:58 PM UTC, Bruno Marchal wrote:
>
>
> On 17 Jan 2019, at 14:48, agrays...@gmail.com  wrote:
>
>
>
> On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal wrote:
>>
>>
>> On 17 Jan 2019, at 09:33, agrays...@gmail.com wrote:
>>
>>
>>
>> On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote:
>>>
>>>
>>>
>>> On 1/16/2019 7:25 PM, agrays...@gmail.com wrote:
>>>
>>>
>>>
>>> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote:

On 1/13/2019 9:51 PM, agrays...@gmail.com wrote:

This means, to me, that the arbitrary phase angles have absolutely no
effect on the resultant interference pattern which is observed. But isn't
this what the phase angles are supposed to effect? AG

The screen pattern is determined by *relative phase angles for the
different paths that reach the same point on the screen*.  The
relative angles only depend on different path lengths, so the overall
phase
angle is irrelevant.

Brent

>>>
>>>
>>> *Sure, except there areTWO forms of phase interference in Wave
>>> Mechanics; the one you refer to above, and another discussed in the
>>> Stackexchange links I previously posted. In the latter case, the wf is
>>> expressed as a superposition, say of two states, where we consider two
>>> cases; a multiplicative complex phase shift is included prior to the sum,
>>> and different complex phase shifts multiplying each component, all of the
>>> form e^i (theta). Easy to show that interference exists in the latter case,
>>> but not the former. Now suppose we take the inner product of the wf with
>>> the ith eigenstate of the superposition, in order to calculate the
>>> probability of measuring the eigenvalue of the ith eigenstate, applying one
>>> of the postulates of QM, keeping in mind that each eigenstate is multiplied
>>> by a DIFFERENT complex phase shift.  If we further assume the eigenstates
>>> are mutually orthogonal, the probability of measuring each eigenvalue does
>>> NOT depend on the different phase shifts. What happened to the interference
>>> demonstrated by the Stackexchange links? TIA, AG *
>>>
>>> Your measurement projected it out. It's like measuring which slit the
>>> photon goes through...it eliminates the interference.
>>>
>>> Brent
>>>
>>
>> *That's what I suspected; that going to an orthogonal basis, I departed
>> from the examples in Stackexchange where an arbitrary superposition is used
>> in the analysis of interference. Nevertheless, isn't it possible to
>> transform from an arbitrary superposition to one using an orthogonal basis?
>> And aren't all bases equivalent from a linear algebra pov? If all bases are
>> equivalent, why would transforming to an orthogonal basis lose
>> interference, whereas a general superposition does not? TIA, AG*
>>
>>
>> I don’t understand this. All the bases we have used all the time are
>> supposed to be orthonormal bases. We suppose that the scalar product (e_i
>> e_j) = delta_i_j, when presenting the Born rule, and the quantum formalism.
>>
>> Bruno
>>
>
> *Generally, bases in a vector space are NOT orthonormal. *
>
>
> Right. But we can always build an orthonormal base with a decent scalar
> product, like in Hilbert space,
>
>
>
> *For example, in the vector space of vectors in the plane, any pair of
> non-parallel vectors form a basis. Same for any general superposition of
> states in QM. HOWEVER, eigenfunctions with distinct eigenvalues ARE
> orthogonal.*
>
>
> Absolutely. And when choosing a non degenerate
> observable/measuring-device, we work in the base of its eigenvectors. A
> superposition is better seen as a sum of some eigenvectors of some
> observable. That is the crazy thing in QM. The same particle can be
> superposed in the state of being here and there. Two different positions of
> one particle can be superposed.
>

*This is a common misinterpretation. Just because a wf can be expressed in
different ways (as a vector in the plane can be expressed in uncountably
many different bases), doesn't mean a particle can exist in different
positions in space at the same time. AG*

Using a non orthonormal base makes only things more complex.
>
* I posted a link to this proof a few months ago. IIRC, it was on its
>
>
> But all this makes my point. A vector by itself cannot be superposed, but
> can be seen as the superposition of two other vectors, and if those are
> orthonormal, that gives by the Born rule the probability to obtain the
> "Eigen result” corresponding to the measuring apparatus with Eigen vectors
> given by that orthonormal base.
>
> I’m still not sure about what you would be missing.
>

*You would be missing the interference! Do the math. Calculate the
probability density of a wf expressed as a superposition of orthonormal
eigenstates, where each component state has a different phase angle. All
```

### Re: Planck Length

```

On Friday, January 18, 2019 at 7:36:34 AM UTC-6, Bruno Marchal wrote:
>
>
> On 17 Jan 2019, at 21:02, Philip Thrift >
> wrote:
>
>
>
> On Thursday, January 17, 2019 at 12:45:31 PM UTC-6, Brent wrote:
>>
>>
>>
>> On 1/17/2019 12:22 AM, agrays...@gmail.com wrote:
>>
>>
>> *Later I'll post some questions I have about your derivation of the
>> Planck length, but for now here's a philosophical question; Is there any
>> difference between the claim that space is discrete, from the claim or
>> conjecture that we cannot in principle measure a length shorter than the
>> Planck length? *
>> *TIA, AG *
>>
>>
>> The theory that predicts there is a shortest measured interval assumes a
>> continuum.  There's no logical contradiction is this. But physicists tend
>> to have a positivist attitude and think that a theory that assumes things,
>> like arbitrarily short intervals, might be better expressed and simpler in
>> some way that avoids those assumptions.  This attitude does not assume the
>> mathematics itself is the reality, but only a description of reality; so
>> there can be different descriptions of the same reality.
>>
>> Brent
>>
>
>
>
> *A* theory that does this assumes a continuous mathematics.
> But that doesn't mean *every* theory has to.
>
> As Max Tegmark's little lecture to physicists says:
>
> Our challenge as physicists is to discover ... infinity-free equations.
>
>
> http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/#.XEDdLs9KiCQ
>
> Unless he is wrong in his premise, of course!
>
>
>
> That assumes non-mechanism, and thus bigger infinities. Tegmark is right:
> we cannot assume infinity at the ontological level (just the finite numbers
> 0, s(0), s(s(0)), …). But the physical reality is phenomenological, and
> requires infinite domain of indetermination, making some “observable”
> having an infinite range. The best candidate could be graham-Preskill
> frequency operator (that they use more or less rigorously to derive the
> Born rule from some “many-worlds” interpretation of QM.
>
> Bruno
>
>
>

I think it is possible some of this can be approached with what is referred
to as *higher-type computing*, where

-  *the characterization of the sets that can be exhaustively searched [1]
by an algorithm, in the sense of Turing, in finite time, as those that are
topologically compact*

- *infinite sets that can be completely inspected in finite time in an
algorithmic way, which perhaps defies intuition*

[1] Exhaustible sets in higher-type computation
https://arxiv.org/abs/0808.0441

from Martin Escardo's page
http://www.cs.bham.ac.uk/~mhe/

- pt

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### Re: Planck Length

```
> On 17 Jan 2019, at 21:02, Philip Thrift  wrote:
>
>
>
> On Thursday, January 17, 2019 at 12:45:31 PM UTC-6, Brent wrote:
>
>
> On 1/17/2019 12:22 AM, agrays...@gmail.com  wrote:
>> Later I'll post some questions I have about your derivation of the Planck
>> length, but for now here's a philosophical question; Is there any difference
>> between the claim that space is discrete, from the claim or conjecture that
>> we cannot in principle measure a length shorter than the Planck length?
>> TIA, AG
>
> The theory that predicts there is a shortest measured interval assumes a
> continuum.  There's no logical contradiction is this. But physicists tend to
> have a positivist attitude and think that a theory that assumes things, like
> arbitrarily short intervals, might be better expressed and simpler in some
> way that avoids those assumptions.  This attitude does not assume the
> mathematics itself is the reality, but only a description of reality; so
> there can be different descriptions of the same reality.
>
> Brent
>
>
>
> A theory that does this assumes a continuous mathematics.
> But that doesn't mean every theory has to.
>
> As Max Tegmark's little lecture to physicists says:
>
> Our challenge as physicists is to discover ... infinity-free equations.
>
> http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/#.XEDdLs9KiCQ
>
> Unless he is wrong in his premise, of course!

That assumes non-mechanism, and thus bigger infinities. Tegmark is right: we
cannot assume infinity at the ontological level (just the finite numbers 0,
s(0), s(s(0)), …). But the physical reality is phenomenological, and requires
infinite domain of indetermination, making some “observable” having an infinite
range. The best candidate could be graham-Preskill frequency operator (that
they use more or less rigorously to derive the Born rule from some
“many-worlds” interpretation of QM.

Bruno

>
> - pt
>
>
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### Re: The semantic view of theories and higher-order languages

```
> On 18 Jan 2019, at 09:49, Philip Thrift  wrote:
>
>
> The semantic view of theories and higher-order languages
> Laurenz Hudetz
>
> "every family of set-theoretic structures has an associated language of
> higher-order logic and an up to signature isomorphism unique model-theoretic
> counterpart"
>
> Several philosophers of science construe models of scientific theories as
> set-theoretic structures. Some of them moreover claim that models should not
> be construed as structures in the sense of model theory because the latter
> are language-dependent. I argue that if we are ready to construe models as
> set-theoretic structures (strict semantic view), we could equally well
> construe them as model-theoretic structures of higher-order logic (liberal
> semantic view). I show that every family of set-theoretic structures has an
> associated language of higher-order logic and an up to signature isomorphism
> unique model-theoretic counterpart, which is able to serve the same purposes.
> This allows to carry over every syntactic criterion of equivalence for
> theories in the sense of the liberal semantic view to theories in the sense
> of the strict semantic view. Taken together, these results suggest that the
> recent dispute about the semantic view and its relation to the syntactic view
> can be resolved.

It cannot do that in the Mechanist Frame, where we cannot assume, neither a
physical universe, nor analysis or set theory. Since recently, I have realised
that we cannot even assume the induction axioms, but we can use it in the
definition of the Löbian entities, whose existence is then a consequence of the
theories without induction. Of course, we need induction at the meta level, and
even the whole of informal mathematics, like in any science pointing toward
some reality independent of us.

Bruno

>
> - pt
>
> --
> You received this message because you are subscribed to the Google Groups
> "Everything List" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> .
> To post to this group, send email to everything-list@googlegroups.com
> .
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> .
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> .

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### Re: Coherent states of a superposition

```
> On 17 Jan 2019, at 14:48, agrayson2...@gmail.com wrote:
>
>
>
> On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal wrote:
>
>> On 17 Jan 2019, at 09:33, agrays...@gmail.com  wrote:
>>
>>
>>
>> On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote:
>>
>>
>> On 1/16/2019 7:25 PM, agrays...@gmail.com <> wrote:
>>>
>>>
>>> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote:
>>>
>>>
>>> On 1/13/2019 9:51 PM, agrays...@gmail.com <> wrote:
This means, to me, that the arbitrary phase angles have absolutely no
effect on the resultant interference pattern which is observed. But isn't
this what the phase angles are supposed to effect? AG
>>>
>>> The screen pattern is determined by relative phase angles for the different
>>> paths that reach the same point on the screen.  The relative angles only
>>> depend on different path lengths, so the overall phase angle is irrelevant.
>>>
>>> Brent
>>>
>>> Sure, except there areTWO forms of phase interference in Wave Mechanics;
>>> the one you refer to above, and another discussed in the Stackexchange
>>> links I previously posted. In the latter case, the wf is expressed as a
>>> superposition, say of two states, where we consider two cases; a
>>> multiplicative complex phase shift is included prior to the sum, and
>>> different complex phase shifts multiplying each component, all of the form
>>> e^i (theta). Easy to show that interference exists in the latter case, but
>>> not the former. Now suppose we take the inner product of the wf with the
>>> ith eigenstate of the superposition, in order to calculate the probability
>>> of measuring the eigenvalue of the ith eigenstate, applying one of the
>>> postulates of QM, keeping in mind that each eigenstate is multiplied by a
>>> DIFFERENT complex phase shift.  If we further assume the eigenstates are
>>> mutually orthogonal, the probability of measuring each eigenvalue does NOT
>>> depend on the different phase shifts. What happened to the interference
>>> demonstrated by the Stackexchange links? TIA, AG
>>>
>> Your measurement projected it out. It's like measuring which slit the photon
>> goes through...it eliminates the interference.
>>
>> Brent
>>
>> That's what I suspected; that going to an orthogonal basis, I departed from
>> the examples in Stackexchange where an arbitrary superposition is used in
>> the analysis of interference. Nevertheless, isn't it possible to transform
>> from an arbitrary superposition to one using an orthogonal basis? And aren't
>> all bases equivalent from a linear algebra pov? If all bases are equivalent,
>> why would transforming to an orthogonal basis lose interference, whereas a
>> general superposition does not? TIA, AG
>
> I don’t understand this. All the bases we have used all the time are supposed
> to be orthonormal bases. We suppose that the scalar product (e_i e_j) =
> delta_i_j, when presenting the Born rule, and the quantum formalism.
>
> Bruno
>
> Generally, bases in a vector space are NOT orthonormal.

Right. But we can always build an orthonormal base with a decent scalar
product, like in Hilbert space,

> For example, in the vector space of vectors in the plane, any pair of
> non-parallel vectors form a basis. Same for any general superposition of
> states in QM. HOWEVER, eigenfunctions with distinct eigenvalues ARE
> orthogonal.

Absolutely. And when choosing a non degenerate observable/measuring-device, we
work in the base of its eigenvectors. A superposition is better seen as a sum
of some eigenvectors of some observable. That is the crazy thing in QM. The
same particle can be superposed in the state of being here and there. Two
different positions of one particle can be superposed. Using a non orthonormal
base makes only things more complex.

> I posted a link to this proof a few months ago. IIRC, it was on its

But all this makes my point. A vector by itself cannot be superposed, but can
be seen as the superposition of two other vectors, and if those are
orthonormal, that gives by the Born rule the probability to obtain the "Eigen
result” corresponding to the measuring apparatus with Eigen vectors given by
that orthonormal base.

I’m still not sure about what you would be missing.

Bruno

>>
>> --
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>> "Everything List" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
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>> .
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>> .
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>> .
>
>
> --
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> "Everything ```

### Re: Coherent states of a superposition

```

On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote:
>
>
>
> On 1/16/2019 7:25 PM, agrays...@gmail.com  wrote:
>
>
>
> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote:
>>
>>
>>
>> On 1/13/2019 9:51 PM, agrays...@gmail.com wrote:
>>
>> This means, to me, that the arbitrary phase angles have absolutely no
>> effect on the resultant interference pattern which is observed. But isn't
>> this what the phase angles are supposed to effect? AG
>>
>>
>> The screen pattern is determined by *relative phase angles for the
>> different paths that reach the same point on the screen*.  The relative
>> angles only depend on different path lengths, so the overall phase angle is
>> irrelevant.
>>
>> Brent
>>
>
>
> *Sure, except there areTWO forms of phase interference in Wave Mechanics;
> the one you refer to above, and another discussed in the Stackexchange
> links I previously posted. In the latter case, the wf is expressed as a
> superposition, say of two states, where we consider two cases; a
> multiplicative complex phase shift is included prior to the sum, and
> different complex phase shifts multiplying each component, all of the form
> e^i (theta). Easy to show that interference exists in the latter case, but
> not the former. Now suppose we take the inner product of the wf with the
> ith eigenstate of the superposition, in order to calculate the probability
> of measuring the eigenvalue of the ith eigenstate, applying one of the
> postulates of QM, keeping in mind that each eigenstate is multiplied by a
> DIFFERENT complex phase shift.  If we further assume the eigenstates are
> mutually orthogonal, the probability of measuring each eigenvalue does NOT
> depend on the different phase shifts. What happened to the interference
> demonstrated by the Stackexchange links? TIA, AG *
>
> Your measurement projected it out. It's like measuring which slit the
> photon goes through...it eliminates the interference.
>
> Brent
>

*But if that's the case, won't the probability density of the eigenvalue
being measured (by Born's rule) be the value in the absence of
interference, which I presume is the classical value? AG *

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### The semantic view of theories and higher-order languages

```
*The semantic view of theories and higher-order languages*
Laurenz Hudetz

"every family of set-theoretic structures has an associated language of
higher-order logic and an up to signature isomorphism unique
model-theoretic counterpart"

*Several philosophers of science construe models of scientific theories as
set-theoretic structures. Some of them moreover claim that models should
not be construed as structures in the sense of model theory because the
latter are language-dependent. I argue that if we are ready to construe
models as set-theoretic structures (strict semantic view), we could equally
well construe them as model-theoretic structures of higher-order logic
(liberal semantic view). I show that every family of set-theoretic
structures has an associated language of higher-order logic and an up to
signature isomorphism unique model-theoretic counterpart, which is able to
serve the same purposes. This allows to carry over every syntactic
criterion of equivalence for theories in the sense of the liberal semantic
view to theories in the sense of the strict semantic view. Taken together,
these results suggest that the recent dispute about the semantic view and
its relation to the syntactic view can be resolved.*

- pt

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