Whether forcing does anything for physics, I don't know, but ultimately it
could be in future proof assistants (like Lean, Coq, etc.) and programming
languages as a continuation of
- monadic programming
- Every function can be computable!
<http://jdh.hamkins.org/every-function-can-be-computable/>
- A Haskell monad for infinite search in finite time
<http://math.andrej.com/2008/11/21/a-haskell-monad-for-infinite-search-in-finite-time/>
- Computing in Cantor’s Paradise
<https://jeapostrophe.github.io/home/static/toronto-2012flops.pdf>
https://codicalist.wordpress.com/2020/05/23/dialectical-programming/
@philipthrift
On Thursday, July 2, 2020 at 5:02:42 PM UTC-5 Lawrence Crowell wrote:
> On Thursday, July 2, 2020 at 6:48:08 AM UTC-5, Philip Thrift wrote:
>>
>> https://arxiv.org/abs/2007.00418
>>
>> [Submitted on 1 Jul 2020]
>> *Forcing as a computational process*
>>
>> Joel David Hamkins
>> <https://arxiv.org/search/math?searchtype=author&query=Hamkins%2C+J+D>,
>> Russell
>> Miller
>> <https://arxiv.org/search/math?searchtype=author&query=Miller%2C+R>, Kameryn
>> J. Williams
>> <https://arxiv.org/search/math?searchtype=author&query=Williams%2C+K+J>
>>
>> We investigate how set-theoretic forcing can be seen as a computational
>> process on the models of set theory. Given an oracle for information about
>> a model of set theory ⟨M,∈M⟩, we explain senses in which one may
>> compute M-generic filters G⊆ℙ∈M and the corresponding forcing
>> extensions M[G]. Specifically, from the atomic diagram one may compute G,
>> from the Δ0-diagram one may compute M[G] and its Δ0-diagram, and from the
>> elementary diagram one may compute the elementary diagram of M[G]. We also
>> examine the information necessary to make the process functorial, and
>> conclude that in the general case, no such computational process will be
>> functorial. For any such process, it will always be possible to have
>> different isomorphic presentations of a model of set theory M that lead to
>> different non-isomorphic forcing extensions M[G]. Indeed, there is no Borel
>> function providing generic filters that is functorial in this sense.
>>
>> @philipthrift
>>
>
> Forcing is a bizarre idea, where one introduces some new category or
> objects that extends a model. In some ways we do this with the extension of
> mathematics from the reals to complex numbers. The forcing of one system
> into another has some connections to the Jordan-Banach algebra as well. The
> replacement of a Poisson bracket structure {X, Y} → i/ħ[X, Y], is given by
> the replacement of generators that are Jordan with product rule A∘B = ½(AB
> + BA) to a Lie algebraic ad_A(B) = [A, B] product. For a general quantum
> system with a Lindblad type equation
>
> Idρ/dt = [H, ρ] + ½([Aρ, A^†] + [A, ρA^†]),
>
> such that the Hamiltonian is an element of the Lie group L, usually as a
> weight, and the operators A and A^† are operators so that AρA^†, AA^†ρ,
> and ρAA^† are operations on the density matrix that are Jordan. Depending
> upon how one looks at this we can think of the quantum mechanical theory on
> Fubini-Study metric on ℂP^n, say a metric based action or Lagrangian ℒ =
> g_{ab}y^ay^b to ℒ = g_{ab}y^ay^b + U(x_1, x_2, … ) with a general
> Euler-Lagrange equation
>
> F_i = ∂/∂x_i(∂ℒ/∂y_a)y^a - ∂ℒ/∂x_i.
>
> This is component version in Finsler geometry of a differential 1-form ω =
> πdq – dη. It should be apparent this is similar to the first law of
> thermodynamics. This is a “practical lesson” in what is meant by forcing.
> The imaginary valued system is extended to components that are further
> multiplied by i = √(-1). This is a sort of Wick rotation version of the
> forcing from reals to complex numbers.
>
>
> The Jordan-Banach algebra though is just a start. We have the skew
> symmetry that corresponds to the conjugate on I = √(-1). This Lie algebraic
> system is in contrast to the Jordan product and algebra. We also have in
> physics fundamental properties due to bosonic and fermionic statistics. In
> supersymmetry there are the Grassmann numbers that enter in to intertwine
> between these. Further, on a 2-dimensional space the interchange of two
> bosons or fermions is ambiguous. This is because a clockwise and
> counterclockwise rotational exchange are topologically distinct. In 3
> dimensions of space the paths can be exchanged, but no in two. This means a
> sign change is possible in either boson or fermion cases. The stretched
> horizon of a black hole is a 2-dimensional surface and an anionic system
> for quantum fields. I think this is a foundation for how there are these
> dichotomies in physics, such as quantum mechanics and macroscopic physics
> and different statistics. Instead of physics being founded on very large
> algebraic systems I think that Wigner had something right with the idea of
> small groups. The big symmetry systems come about from symmetry breaking or
> the removal of a degeneracy.
>
> The fibration and Noether’s theorem in differential forms is λ = p_idq_i –
> Hdt, where the 1-form has a horizonal p_idq_i and vertical Hdt principal
> bundles. This is a Finsler geometric way of seeing Lagrangian dynamics. The
> differential dλ = dp_i∧dq_i – dH∧dt, vanishes on the contact manifold. In a
> strict Finsler context the horizontal and vertical bundles are annulled
> separately. The q_i is the generator of p_i and dp_i = 0 means momentum is
> invariant under translation and similarly energy is conserved under time
> translations. If there is a Lagrange multiplier that connects the vertical
> and horizontal parts of the bundle then we have
>
> dp_i∧dq_i – dH∧dt = (dp_i/dt)dt∧dq_i – (∂H/dq_i)dq_i∧dt = [(dp_i/dt) +
> (∂H/dq_i)]dq_i∧dt
>
> which gives us the dynamical equation (dp_i/dt) = -∂H/dq_i.
>
> Hamkins et al paper is dense in set theoretic constructions. These things
> I have some introductory acquaintance with. I am also more interested in
> the practical transduction of such ideas. I will try to see what I can
> gather from this paper. Forcing does have some algorithmic connection and
> the above connects 1/kT ↔ it/ħ with macroscopic system with a temperature,
> or gravity in the case of spacetime physics, to time in the case of quantum
> mechanics.
>
> LC
>
>
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