> On 24 Sep 2020, at 13:20, Philip Thrift <cloudver...@gmail.com> wrote: > > It is interesting approach to fundamentally replace the underlaying language > (logic) of physics. > > > Topos Quantum Theory > Christopher J. Isham > > > https://fqxi.org/grants/large/awardees/view/__details/2006/isham > > > One important feature of topos theory is that a proposition such as "the > physical quantity A has a value in a certain range" need not be simply true > or false: rather, there are more possibilities that are given by the > intrinsic logic that is possessed by a topos.
That is all nice as long as you don’t confuse the language with what it is supposed to talk about. Using a non boolean topos is risky in that regards, but does make sense for the universal machine’s theory of qualia, including the quanta. This is not a problem as we have independent reason to put the physical reality in a first person (plural) realm, and the fundamental reality remains an independent Boolean structure (be it arithmetic, or a boolean topos with a "natural number object”). Bruno > > Attempt to integrate this into a programming language: > > Catlab.jl - https://github.com/AlgebraicJulia/Catlab.jl > > @philipthrift > On Thursday, September 24, 2020 at 4:05:43 AM UTC-5 Bruno Marchal wrote: >> On 22 Sep 2020, at 19:49, Lawrence Crowell <goldenfield...@gmail.com >> <applewebdata://00393538-0B03-42C2-B036-0C7E8EB047F0>> wrote: >> >> I downloaded Doering’s paper. In scanning this I see a mention of Chris >> Isham, who started this idea of Topos as a category system of physics. >> > > Which is nice. Isham is also quite open for the “many-world”. > > >> I think in a way this might be a way of looking at dualities, where if they >> have the same category or categorical topology of sheaves then these are >> dualities. While this can be elegant mathematics, such as how Grothendieke >> formulated algebraic geometry as cohomologies as topoi, this may come after >> the fact. I think honestly that physical ideas are a better way to blaze >> this trail. >> >> > > > Or philosophy of mind, also. > > > >> The complex coupling constant τ = θ/2π + 4πi/g^2 is a case where there is a >> duality between the generator of a group, generally thought of as a Lie >> algebra, and an observable which technically is in a Jordan algebra. This >> coupling constant with some element H defines g = exp(-τH), where θ is the >> vacuum angle and g the standard coupling constant. >> > > And tau is 2*pi, I guess. > > > >> This angle defines the constant wave function along orbits of gauge >> transformations. For A → UAU^{-1} - (dU)U^{-1} <>and a wave function for a >> field φ. >> >> ψ(A, φ) → ψ(UAU^{-1} - (dU)U^{-1}, Uφ <>) ≃ e^{iθ} ψ(A, φ). >> >> The angle θ is a winding number for the gauge orbits π_3(G) for the group. >> These orbits are defined for small gauge transformations by U = e^{iα} >> >> UAU^{-1} - (dU)U^{-1} ≃ <> A + i([α, A] + dα) >> >> Uφ ≃ φ + iαφ >> >> That defines ψ(A, φ) → (1 + idθ) ψ(A, φ), where dθ is an orbit map. The >> angle is a winding number. >> >> For this orbit space for the operator e^{iτH} we then have the associated >> real valued -4π/g^2. The winding number, say at a nexus of a Penrose >> diagram, is then associated with a dual form. This duality is equivalent to >> the Euclideanization of time t → it so -t/ħ = -1/kT. This is a duality (Lie >> algebraic generator) ↔ (Jordan observable). This then can in principle be >> formulated according to a topoi. >> >> > > Quite interesting. Still very mysterious, and probably related to the > material modes ([]p & <>t) in arithmetic. This should be related to knot > theory, and the quantum invariant of knots and braids, for the > arithmetical-quantum origin of space, but this led me to the > self-distributive algebra until I get stuck in complex questions of set > theory and the very large cardinals, like the cardinal of Woodin and Laver… > To be continued … in the next millenium, or the one after … (we are so slow…). > > Bruno > > > > >> LC >> >> >> On Monday, September 21, 2020 at 6:44:11 AM UTC-5 cloud...@gmail.com >> <http://gmail.com/> wrote: >> >> Some Remarks on the Logic of Quantum Gravity >> Andreas Doering >> >> https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity >> >> <https://www.academia.edu/5456772/Some_Remarks_on_the_Logic_of_Quantum_Gravity> >> >> We discuss some conceptual issues that any approach to quantum gravity has >> to confront. In particular, it is argued that one has to find a theory that >> can be interpreted in a realist manner, because theories with an >> instrumentalist interpretation are problematic for several well-known >> reasons. Since the Hilbert space formalism almost inevitably forces an >> instrumentalist interpretation on us, we suggest that a theory of quantum >> gravity should not be based on the Hilbert space formalism. We briefly >> sketch the topos approach, which makes use of the internal logic of a topos >> associated with a quantum system and comes with a natural (neo-)realist >> interpretation. Finally, we make some remarks on the relation between system >> logic and metalogic. >> >> ... >> >> Finally, when we commit ourselves to describing the whole universe using >> structures in a topos, and if we use the internal logic of the topos to >> assign truth values to propositions etc., we do not have to do all our >> proofs and mathematical arguments internally in the topos, i.e., >> constructively. Doing physics necessarily means to separate oneself from the >> system to be described, even if this system is the whole universe. Since we >> have to ‘step out’ of the system, we have to argue using the (typically >> Boolean) metalogic in which we define the mathematical structures, e.g. >> topoi and state objects, that we use in the mathematical description of the >> system at hand. It is this Boolean metalogic in which we do physics. >> >> >> @philipthrift >> > >> -- >> 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 >> email to everything-li...@googlegroups.com >> <applewebdata://00393538-0B03-42C2-B036-0C7E8EB047F0>. > >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/everything-list/17b4c67f-cc4f-4a02-a95f-e7b6d1415a76n%40googlegroups.com >> >> <https://groups.google.com/d/msgid/everything-list/17b4c67f-cc4f-4a02-a95f-e7b6d1415a76n%40googlegroups.com?utm_medium=email&utm_source=footer>. > > > -- > 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 > email to everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/4a66c130-ef6c-48f6-9c83-437764bc8cf6n%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/4a66c130-ef6c-48f6-9c83-437764bc8cf6n%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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