https://arxiv.org/abs/2007.00418
Forcing as a computational process
is (type theoretic)
http://guilhem.jaber.fr/ComputationalInterpretationForcingTypeTheory.pdf
A Computational Interpretation of Forcing in Type Theory
On forcing and quantum gravity:
https://arxiv.org/search/quant-ph?searchtype=author&query=Kr%C3%B3l%2C+J
@philipthrift
> On forcing and quantum gravity:
> https://arxiv.org/search/quant-ph?searchtype=author&query=Kr%C3%B3l%2C+J
Also:
https://arxiv.org/abs/1602.02667
@philipthrift
On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.
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> On 4 Jul 2020, at 09:52, Philip Thrift <cloud...@gmail.com> wrote:
>
>
>
>
> On forcing and quantum gravity:
>
> https://arxiv.org/search/quant-ph?searchtype=author&query=Kr%C3%B3l%2C+J
I will take a closer look (in summer when I got more time). This looks more interesting … for physics, and perhaps Mechanism.
It might provides helpful tool to get the QM GR from the universal machine introspection, some day. People needs to first get well familiarised with the (mechanist) mind-body problem before (to be sure).
Bruno
On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.That is not forcing in the theoretical meaning of the papers referred to by P. Thrift.
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https://arxiv.org/abs/2007.00418[Submitted on 1 Jul 2020]Forcing as a computational process
On 4 Jul 2020, at 19:25, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote:On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.That is not forcing in the theoretical meaning of the papers referred to by P. Thrift.It is and is not. Agreed forcing is a dense set theoretic technique on how to extend a model into some other model. I think it was Hamkins who wrote on how forcing was really not that mysterious and used the example of extending integers into rationals, those into reals and those into ,,, . Set theory without some connection to actual mathematics is about as useless as tits on a boar hog.
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On 4 Jul 2020, at 19:25, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote:On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.That is not forcing in the theoretical meaning of the papers referred to by P. Thrift.It is and is not. Agreed forcing is a dense set theoretic technique on how to extend a model into some other model. I think it was Hamkins who wrote on how forcing was really not that mysterious and used the example of extending integers into rationals, those into reals and those into ,,, . Set theory without some connection to actual mathematics is about as useless as tits on a boar hog.
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I think mathematics is best when it can be used to actually calculate things, whether that be with the physical world or in expanding areas of mathematics itself. The one thing that always caused some distaste for axiomatic set theory is that it often seems so detached and remote from anything else.LC
On 7 Jul 2020, at 16:10, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Monday, July 6, 2020 at 9:29:31 AM UTC-5, Bruno Marchal wrote:On 4 Jul 2020, at 19:25, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote:On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.That is not forcing in the theoretical meaning of the papers referred to by P. Thrift.It is and is not. Agreed forcing is a dense set theoretic technique on how to extend a model into some other model. I think it was Hamkins who wrote on how forcing was really not that mysterious and used the example of extending integers into rationals, those into reals and those into ,,, . Set theory without some connection to actual mathematics is about as useless as tits on a boar hog.I think mathematics is best when it can be used to actually calculate things, whether that be with the physical world or in expanding areas of mathematics itself. The one thing that always caused some distaste for axiomatic set theory is that it often seems so detached and remote from anything else.
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On 8 Jul 2020, at 13:57, Philip Thrift <cloud...@gmail.com> wrote:On Tuesday, July 7, 2020 at 9:10:28 AM UTC-5 Lawrence Crowell wrote:I think mathematics is best when it can be used to actually calculate things, whether that be with the physical world or in expanding areas of mathematics itself. The one thing that always caused some distaste for axiomatic set theory is that it often seems so detached and remote from anything else.LCSet theory (mathematics) does seem pretty irrelevant for (computational) physics today.But forcing might be the way it could be:
Scientific proof-oriented programming (S-pop)
@philipthrift
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On 8 Jul 2020, at 13:57, Philip Thrift <cloud...@gmail.com> wrote:Scientific proof-oriented programming (S-pop)Don’t hesitate to sum up your point. Forcing is just a powerful technic to build weird model of set theory. Forcing is a particular S4-like modal logic. The relation between Smullyan-Fitting “quantisation in S4” and the material mode of the self-referential machine is a bit an open problem for me. I might say more on this in some future.Bruno
On 7 Jul 2020, at 16:10, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Monday, July 6, 2020 at 9:29:31 AM UTC-5, Bruno Marchal wrote:On 4 Jul 2020, at 19:25, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote:On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.That is not forcing in the theoretical meaning of the papers referred to by P. Thrift.It is and is not. Agreed forcing is a dense set theoretic technique on how to extend a model into some other model. I think it was Hamkins who wrote on how forcing was really not that mysterious and used the example of extending integers into rationals, those into reals and those into ,,, . Set theory without some connection to actual mathematics is about as useless as tits on a boar hog.I think mathematics is best when it can be used to actually calculate things, whether that be with the physical world or in expanding areas of mathematics itself. The one thing that always caused some distaste for axiomatic set theory is that it often seems so detached and remote from anything else.Set theory is useful to make clear the semantic of simpler system. Cantor was trying to make sense of the differential equation of heat propagation.But, unlike arithmetic, there is no real consensus of what a set should be, and there are different theories, like ZF and NF.Intuitionist set theory can have application in numerical analysis, constructive computer science, etc.Then, there is that extraordinary use of the most abstract (and theological) part of set theory (the Large large cardinal) which have been used to discover and proof the existence of some interesting order structure on the (infinite) braids, …We never know. We cannot predict if some mathematics will or will not been applied. Hardy apologised for doing useless mathematics (number theory) which today is used each time we pay something on the net…Bruno
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On 10 Jul 2020, at 20:57, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Friday, July 10, 2020 at 7:21:29 AM UTC-5, Bruno Marchal wrote:On 7 Jul 2020, at 16:10, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Monday, July 6, 2020 at 9:29:31 AM UTC-5, Bruno Marchal wrote:On 4 Jul 2020, at 19:25, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote:On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.That is not forcing in the theoretical meaning of the papers referred to by P. Thrift.It is and is not. Agreed forcing is a dense set theoretic technique on how to extend a model into some other model. I think it was Hamkins who wrote on how forcing was really not that mysterious and used the example of extending integers into rationals, those into reals and those into ,,, . Set theory without some connection to actual mathematics is about as useless as tits on a boar hog.I think mathematics is best when it can be used to actually calculate things, whether that be with the physical world or in expanding areas of mathematics itself. The one thing that always caused some distaste for axiomatic set theory is that it often seems so detached and remote from anything else.Set theory is useful to make clear the semantic of simpler system. Cantor was trying to make sense of the differential equation of heat propagation.But, unlike arithmetic, there is no real consensus of what a set should be, and there are different theories, like ZF and NF.Intuitionist set theory can have application in numerical analysis, constructive computer science, etc.Then, there is that extraordinary use of the most abstract (and theological) part of set theory (the Large large cardinal) which have been used to discover and proof the existence of some interesting order structure on the (infinite) braids, …We never know. We cannot predict if some mathematics will or will not been applied. Hardy apologised for doing useless mathematics (number theory) which today is used each time we pay something on the net…BrunoThat is a part of my understanding. We have ZF set theory, with and without the axiom of choice, and then there are other very different set theories, such as Polish set theory. Then there is the HoTT homotopy type theory. It seems the goal of set theory to reduce all of mathematics to some completely fundamental objects, whether sets or types etc, that we have a plethora of these systems and a host of complexity.
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On 10 Jul 2020, at 20:57, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Friday, July 10, 2020 at 7:21:29 AM UTC-5, Bruno Marchal wrote:On 7 Jul 2020, at 16:10, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Monday, July 6, 2020 at 9:29:31 AM UTC-5, Bruno Marchal wrote:On 4 Jul 2020, at 19:25, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote:On 3 Jul 2020, at 14:30, Lawrence Crowell <goldenfield...@gmail.com> wrote:It does not need that sort of intense structuring. The extension of real to complex numbers is a case of forcing of real numbers into pairs that obey multiplication rules for complex numbers.That is not forcing in the theoretical meaning of the papers referred to by P. Thrift.It is and is not. Agreed forcing is a dense set theoretic technique on how to extend a model into some other model. I think it was Hamkins who wrote on how forcing was really not that mysterious and used the example of extending integers into rationals, those into reals and those into ,,, . Set theory without some connection to actual mathematics is about as useless as tits on a boar hog.I think mathematics is best when it can be used to actually calculate things, whether that be with the physical world or in expanding areas of mathematics itself. The one thing that always caused some distaste for axiomatic set theory is that it often seems so detached and remote from anything else.Set theory is useful to make clear the semantic of simpler system. Cantor was trying to make sense of the differential equation of heat propagation.But, unlike arithmetic, there is no real consensus of what a set should be, and there are different theories, like ZF and NF.Intuitionist set theory can have application in numerical analysis, constructive computer science, etc.Then, there is that extraordinary use of the most abstract (and theological) part of set theory (the Large large cardinal) which have been used to discover and proof the existence of some interesting order structure on the (infinite) braids, …We never know. We cannot predict if some mathematics will or will not been applied. Hardy apologised for doing useless mathematics (number theory) which today is used each time we pay something on the net…BrunoThat is a part of my understanding. We have ZF set theory, with and without the axiom of choice, and then there are other very different set theories, such as Polish set theory. Then there is the HoTT homotopy type theory. It seems the goal of set theory to reduce all of mathematics to some completely fundamental objects, whether sets or types etc, that we have a plethora of these systems and a host of complexity.We know today that there are no foundational theory which could based all of mathematics, and it is important to continue the distinction between basic concepts like numbers and functions from their representations in set theory, or even in category theory. But will mechanism, we get a very simple ontology to explain the “physical experience”, without having the assume a physical reality. With or without Mechanism, the fact that universal machine exist makes it impossible to have any complete theory about reality, whatever that reality is. But that means that even just in arithmetic, we must expect an infinity of surprises in the exploration. With Mechanism, *we* are such surprises already.Set theory should be renamed. It is really the theory of the infinite(s). It is a sort of vertical theology: how to figure out what looks like a universe of sets. It attracts the lover of vertigo … Then, like always in mathematics, unexpected application rise up, like the apparent link between braids and some *very* large cardinal (Wooden, Laver, …).Bruno