Talk:Telesis

"As observed in [235, 3, 365], if the riemannian geometries of the phase space used for the Wiener measure regularisation have nonconstant scalar curvature, then the weighting of the phase space paths is nonuniform, corresponding to the phase space point dependency of the zero-point energy. On the other hand, in a completely different context, Jaynes has argued [166, 171, 176] that the zero-point energy should be interpreted not as an ontic feature of the system, but as observer’s measure of uncertainty regarding his/her own prediction of the value of energy, as based on his/her own prior information. Quite independently from these considerations, Rodríguez has developed the theory of entropic priors [288] interpreting them as the «statistical representation of the vacuum of information in a given hypothesis space» [289]. Our formulation recombines the above insights, integrating them into a single statement: zero-point energy’s point-dependence is a manifestation of a dependence of nonlocal integrability of local inferential structure on the local geometry of user’s prior knowledge (or, equivalently prior ignorance.)" https://arxiv.org/pdf/1605.02063.pdf#page82

“While physicists often use this rule to explain the conservation of energy-momentum (or as Wheeler calls it, “momenergy”), it can be more generally interpreted with respect to information and constraint, or state and syntax. That is, the boundary is analogous to a constraint which separates an interior attribute satisfying the constraint from a complementary exterior attribute, thus creating an informational distinction.” - Langan 2002 http://infolab.ho.ua/Langan_CTMU_092902(1).pdf

Information heat engine: converting information to energy by feedback control https://arxiv.org/abs/1009.5287

“It means that it’s now possible to power nanomachines using information as the medium to transfer energy, even if there is no direct contact with the nanomachine. The task now will be to shrink the sensing system. A video camera is a hefty thing to cart around. It would clearly be handy to find some microscopic way of sensing the environment and using the information gathered to power a nanodevice.” https://www.technologyreview.com/2010/09/30/200198/physicists-convert-information-into-energy/

unbound Telesis is unbound energy if Telesis is energy

through the cumulative factorization of telesis, a dual generalization of energy properly defined to serve as the ultimate “stuff” of reality. (Emphasis added by user: HumbleBeauty)

Telesis is unquantified ontic potential

AryanOverlord:

Telesis is “infinite” in the sense that it is not bounded by cognition however even here to qualify as infinite there must be a dual-aspect one-to-one correspondence between a set and subset [such as information (set) and cognition (subset)…becoming self-transducing information] and so telesis is unquantified and may become self-consistently infinite, transfinite, finite etc. once bound in MU form. Telesis is bound by primary and secondary telic recursion, and cognition is the secondary form of telic feedback.

AryanOverlord (talk) 00:03, 14 December 2023 (UTC) AryanOverlord

“Feedback is a process where the observed outcomes of actions are taken as inputs for further action in ways that support the pursuit, maintenance, or disruption of particular conditions, forming a circular causal relationship.” https://en.wikipedia.org/wiki/Cybernetics

“More generally, recursion is a way of defining a function on any mathematical object which is “defined inductively” (in a way analogous to how the natural numbers are characterized by zero and successor). In place of the “initial value” and “successor step”, a general definition by recursion consists of giving one “clause” for each “constructor” of the inductively defined object.

Recursion is formalized in type theory by the notion of inductive type (and the corresponding elimination rule) and, equivalently, in category theory by the notion of initial algebra of an endofunctor. For F an endofunctor, a morphism of the form F(X)→X determines a collection of constructors and the recursion principle is the statement that there is a (unique) morphism f:A→X from the initial such structure F(A)→A. This f is the corresponding recursively defined function.

Viewed from just a slightly different angle, this state of affairs is the induction principle on non-dependent types.” https://ncatlab.org/nlab/show/recursion

“In traditional mathematics and set theory, the category Set of all sets is the archetypical topos. However, in predicative mathematics with its notion of sets, these do not form a topos since they do not satisfy the powerset axiom. In constructive mathematics, we may be weakly predicative and almost form a topos; the resulting weaker structure which Bishop sets (for example) form is a predicative topos (van den Berg). … A different approach to a notion of predicative topos is to assert, rather than a weak choice principle, the existence of some categorical version of higher inductive types. But this would also exclude some elementary toposes.” https://ncatlab.org/nlab/show/predicative+topos

“In predicative mathematics, the existence of power sets (along with other “impredicative” axioms) is not accepted. However we can still speak of a power set as a proper class, sometimes called a power class.

One can use power sets to construct function sets; the converse also works using excluded middle (or anything else that will guarantee the existence of the set of truth values). In particular, power sets exist in any theory containing excluded middle and function sets; thus predicative theories which include function sets must also be constructive.” https://ncatlab.org/nlab/show/power+set

“For example, even if one believes the principle of excluded middle to be true, the “internal” version of excluded middle in many interesting categories is still false; thus constructive mathematics can be useful in the study of such categories, even if mathematics is “globally” non-constructive.” https://ncatlab.org/nlab/show/constructive+mathematics

“The tautological example which is useful to see what the abstract definition by Lawvere axiomatizes is the following.

Let B=Set be the category of sets and for X a set let EX be the poset of subsets of X, regarded as the propositions about elements in X. Then comprehension exists and is given by sending a subset of X regarded as an object of EX (hence regarded as a proposition) to the same subset, but now regarded as a monomorphism in Set into X.

More generally, the same construction works for the posets of subobjects in any regular category. … For the subobject fibration of a regular category, this gives the usual (regular epi, mono) factorization system, while for the fibration of presheaves over Cat it gives the factorization of a functor into a final functor followed by a discrete fibration. (See also comprehensive factorization for a description of the latter as a factorization system in a 2-category.)“ https://ncatlab.org/nlab/show/axiom+of+separation

“A regular category is a finitely complete category which admits a good notion of image factorization. A primary raison d’être behind regular categories C is to have a decently behaved calculus of relations in C(see Rel).

Regular categories also provide a natural semantic environment to interpret a particularly well behaved positive fragment of first order logic having connectives ⊤, ∧, ∃; in other words, their internal logic is regular logic.” https://ncatlab.org/nlab/show/regular+category

“Rel is the category whose objects are sets and whose morphisms are (binary) relations between sets. It becomes a 2-category Rel (in fact, a 2-poset) by taking 2-morphisms to be inclusions of relations. … It is useful to be aware of the connections between the bicategory of relations and the bicategory of spans. Recall that a span from X to Y is a diagram of the form

X←S→Y

and there is an obvious category whose objects are spans from X to Y and whose morphisms are morphisms between such diagrams. The terminal span from X to Y is

X←π1X×Y→π2Y

and a relation from X to Y is just a subobject of the terminal span, in other words an isomorphism class of monos into the terminal span.

To each span S from X to Y, there is a corresponding relation from X to Y, defined by taking the image of the unique morphism of spans S→X×Y between X and Y. It may be checked that this yields a lax morphism of bicategories

Span→Rel

4. Limits and colimits

The category Rel does have products and coproducts; they coincide (by self-duality) and are just disjoint unions of sets. However, otherwise Rel has very few (co)limits; it doesn’t even have splittings of all idempotents. All symmetric idempotents have splittings, but the order-relation ≤ ⊆ {0,1}×{0,1} can’t be split. It follows that it can’t have (co)equalisers.

Since the category Rel is the category of free algebras (Kleisli category) for the powerset monad, there is, indeed, very little chance of a limit of such algebras being free again. To get decent limits, one has to move to the Eilenberg-Moore category of the powerset monad, viz., the category of complete suplattices.“ https://ncatlab.org/nlab/show/Rel

“A suplattice is a poset that has joins of arbitrary subsets (and in particular is a join-semilattice). By the adjoint functor theorem for posets, a suplattice necessarily has all meets as well and so is a complete lattice. However, a suplattice homomorphism preserves joins, but not necessarily meets. Furthermore, a large semilattice which has all small joins need not have all meets, but might still be considered a large suplattice (even though it may not even be a lattice).

Dually, an inflattice is a poset which has all meets, and an inflattice homomorphism in a monotone function that preserves all meets.

A frame (dual to a locale) is a suplattice in which finitary meets distribute over arbitrary joins. (Frame homomorphisms preserve all joins and finitary meets.)

The category SupLat of suplattices and suplattice homomorphisms admits a tensor product which represents “bilinear maps”, i.e. functions which preserve joins separately in each variable. Under this tensor product, the category of suplattices is a star-autonomous category in which the dualizing object is the suplattice dual to the object TV of truth-values. A semigroup in this monoidal category is a quantale, including frames as a special case when the quantale is idempotent and unital. Modules over them are modules over quantales (quantic modules with special case of localic modules, used in the localic analogue of the Grothendieck’s descent theory in Joyal-Tierney 84).” https://ncatlab.org/nlab/show/suplattice

“Due to the finite collapse, the matrix powers of implications are solved exactly. Thought is self-potent. The unique property of an implication is that being raised to the power of itself, the implication remains invariant:…which entails the exact equality of the powers and the field exponents, one of the critical result of this study:…Whenever the annihilation and creation processes converge to zero, a logical system must be reduced to identity.” https://www.sciencedirect.com/topics/social-sciences/quantum-theory

Quantum Theoretic Machines. What Is Thought From The Point Of View Of Physics https://vdoc.pub/documents/quantum-theoretic-machines-what-is-thought-from-the-point-of-view-of-physics-5pmdk8fcptj0

Matrix Logic https://vdoc.pub/documents/matrix-logic-theory-and-applications-6beon8chil30

“Context-dependence and descent theory … The glueing data, abstracted into the categorical structure of the topos of sheaves, provide the links which form a virtual reality from which the geometric object emerges. The original “ground level” fades from view, replaced by the abstract collection of glueing data itself as the only true reality. [. . . ] the place where things are really happening, and what we should concentrate on understanding, is the glueing data which explain how to pass between the various different points of view and how they are bound together.” https://www.chapman.edu/scst/conferences-and-events/grothendieck-files/halimi-slides.pdf

“Descent is best understood as a direct generalization of the situation for 0-stacks, i.e. ordinary sheaves, which we briefly recall in a language suitable for the following generalization.” https://ncatlab.org/nlab/show/descent

“Models As Universes … To put it another way, models of a formal theory are members of a universe of sets which in turn can be seen as being itself a model, of course not of a formal theory, but rather of the informal set theory that one presupposes when doing mathematics. Since this universe may thus be described as the intended model of some set metatheory, any model of formal set theory is by principle very akin to it in some ways and constitutes, so to speak, a background universe in its own right. Hence, even though the distinction between a model of ZFC and the universe is perfectly clear, a connection remains, which in fact can be read both ways: The “true” universe can be conceived of, by extension, as a kind of “monster model,” just as any model can be seen as a kind of universe. The second way of looking at things will be explored soon. But, as already said, the first one has naturally given rise to the following question: What is the connection between truth in the “big” model, and truth in all the “small” models?” https://doc-0s-8k-docs.googleusercontent.com/docs/securesc/l78ehpio6ig1sit6tkftvhd2d3qhs9b5/mp11s49tqap2ng08i2iimhqgsbukv7c0/1702596225000/12591206045410985563/18278498144166523601Z/1HF-vUdFNrUzEQcHoFEX4nkeboneaDZHN?e=download&uuid=7028bf45-6c74-48da-8edd-1eddccdfdcd4

“Homotopy Model Theory … Hence, whereas “Homotopy Type Theory” connects logic with homotopy theory through type theory, “Homotopy Model Theory” is the proposal to connect logic with homotopy theory through model theory. This is the task taken up here.

An intuitive motivation of such perspective is that the notion of boundary can easily be transposed to the context of first-order logic, formulae being conceived of as chains (in the sense of a formal sum of faces).” https://sites.google.com/view/brice-halimi/papers/math-logic

“In any finitely complete category, the kernel pair of the identity morphism id on an object X is the diagonal morphism (id,id) of X and has a coequalizer isomorphic to X itself.” https://ncatlab.org/nlab/show/finitely+complete+category

“A generalization of exact completion from finitely complete categories to categories with path objects, which turns a category with path objects into an exact category. … Let 𝒞 be the syntactic category of a dependent type theory. Then the category of setoids in 𝒞 is the homotopy exact completion of 𝒞.” https://ncatlab.org/nlab/show/homotopy+exact+completion

Free and bound variables https://ncatlab.org/nlab/show/variable

Free variables are not “implicitly universally quantified”! https://math.andrej.com/2012/12/25/free-variables-are-not-implicitly-universally-quantified/