“Is Physics Legislated by Cosmogony?” (J. A. Wheeler)

Gianfranco Basti Faculty of Philosophy – Pontifical Lateran University – www.irafs.org IRAFS website: www.irafs.org

http://www.irafs.org - [email protected] IRAFS'18 2 http://www.irafs.org - [email protected] IRAFS'18 3 1. Introduction Theory as the proper axiomatics of formal philosophy The location of categories of Category Theory among other algebraic structures

Main -like Algebraic Structures

Closure Associativity Identity Invertibility Commutativity Semigroupoid Unneeded Needed Unneeded Unneeded Unneeded Category Unneeded Needed Needed Unneeded Unneeded Unneeded Needed Needed Needed Unneeded Needed Unneeded Unneeded Unneeded Unneeded Needed Unneeded Unneeded Needed Unneeded Loop Needed Unneeded Needed Needed Unneeded Needed Needed Unneeded Unneeded Unneeded Needed Needed Unneeded Needed Unneeded Needed Needed Needed Unneeded Unneeded Group Needed Needed Needed Needed Unneeded Needed Needed Needed Needed Needed

http://www.irafs.org - [email protected] IRAFS'18 5 Some elements of CT

▪ In CT the primitives are:

1. or arrows, f, g, – intended as a (purely relational) generalization of notions such as «function», «operator», «map», etc.

2. The identity arrow, such that, for any object A there is an identity arrow or reflexive IdA: A → A.

3. Two maps or operations from arrows to objects, dom(), codom(), assigning a domain or source and a codomain or target to each arrow.

4. The compositions of arrows, written as g; f , or f  g, in which the codomain of g is the domain of f, that is, for any three objects A, B, C in the theory, there exists a morphism composition f  g, that is, = , satisfying a transitive property among arrows. 𝑔𝑔 𝑓𝑓 ▪ These primitives must satisfy two axioms regulating compositions and𝐴𝐴 identity→𝐵𝐵→𝐶𝐶 among𝐴𝐴 → 𝐶𝐶 morphisms by which domains and codomains match appropriately:

▪ Axiom 1. (Associativity Law): = .

▪ Axiom 2. (Identity or Unity Law):ℎ ∘ f𝑔𝑔 ∘Id𝑓𝑓A = f =ℎId∘ B𝑔𝑔 ∘f 𝑓𝑓

∘ ∘

http://www.irafs.org - [email protected] IRAFS'18 6 Some elements of CT

▪ The fundamental objects of CT are arrows (morphisms), then… ▪ …any object A, B, C, characterizing a category, can be substituted by the correspondent reflexive morphism A → A constituting a relation identity IdA. Moreover, for each triple of objects, A,B,C, there exists a composition map ABC → fg → written as gf  . ▪ Therefore, a category is any structure in logic or mathematics with structure- preserving morphisms. E.g., in set-theoretic semantics, all the models of a given formal system, because sharing the same structure, constitute a category.

▪ In this way, some fundamental mathematical and logical structures are as many categories: Set (sets and functions), Grp (groups and homomorphisms), Top (topological spaces and continuous functions), Pos (partially ordered sets and monotone functions), Vect (vector spaces defined on numerical fields and linear functions), etc.

http://www.irafs.org - [email protected] IRAFS'18 7 Continuing…

▪ Another fundamental notion in CT is the notion of functor, F, that is, an operation mapping objects and arrows of a category C into another D, F: C → D, so to preserve compositions and identities. In this way, between the two categories there exists a homomorphism up to isomorphism (if the homorphism is invertible). Generally, a functor F is covariant, that is, it preserves arrow directions and composition orders, i.e.:

if fA :→→ B , then FA FB ; if fg , then Ffg (  )= FfFg ; if idA , then FidA= id FA . ▪ However, two categories can be equally homomorphic up to isomorphism if the functor G connecting them is contravariant, i.e., reversing all the arrows directions and the composition orders, i.e. G: C → Dop:

if f : A→→ B , then GB GA ; if f g , then G ( g  f )= Gg Gf ; but if idA , then GidA= id GA.

http://www.irafs.org - [email protected] IRAFS'18 8 Categorical duality

▪ Through the notion of contravariant functor, we can introduce the notion of category duality. Namely, given a category C and an endofunctor E: C → C on a category onto itself, the contravariant application of E links a category to its opposite, i.e.: Eop: C → Cop. ▪ In this way it is possible to demonstrate the dual equivalence between them, in symbols: C Cop. In CT semantics, this means that given a statement α defined on C α is true iff the statement αop defined on Cop is also true. ≃ ▪ In other terms, truth is invariant for such an exchange operation over the statements, that is, they are dually equivalent. In symbols: . ⇄ ▪ Where the equivalence symbol op 𝛼𝛼 → 𝑎𝑎

http://www.irafs.org - [email protected] IRAFS'18 9 Commutative diagrams and hom-sets

▪ Commutative diagrams are the fundamental calculus tools in CT and their uniqueness as the basis of CT universal constructions. E.g., the equations = and = corresponds to the commuting triangle and commuting square diagrams, respectively, which are the basic diagrams. I.e.: 𝑔𝑔 ∘ 𝑓𝑓 ℎ 𝑔𝑔 ∘ 𝑓𝑓 𝑘𝑘 ∘ ℎ

▪ We can define also the arrow-theoretic notion of hom-set for a category , where the prefix hom- stays for homomorphism, i.e., a structure-preserving mapping, as arrow-theoretic interpretation of a “function”. That is, for each pair of objects A, B  Ob(), we define the set: (AB , ) :=∈→{ fAr ( )| f : A B} .

http://www.irafs.org - [email protected] IRAFS'18 10 Hom-set definition of natural numbers and predicative arithmetic (Abramsky)

▪ Arrow-theoretic definition of natural number objects N in a category  with a terminal object 1 and two arrows z: 1 → N, and s: N → N, i.e., (N, z, s), without any commitment to a given set-theoretic (element-theoretic) representation of numbers (set-cardinality as an arrow-theoretic “counting”, given that we deal here with “objects without elements” exclusively as domains-codomains of arrows). ▪ For every such a triple of an object A, and arrows c: 1 → A, f: A → A, there exists a unique arrow h: N → A such that the following diagram commutes:

▪ In Set we can verify easily that = 0,1,2, … equipped with : · • ; : + 1 does indeed form a natural number object. But we are not committed to any particular set-theoretic representation of : whether as von Neumann ordinals, or Zermelo numerals or similar. Such an arrowℕ -theoretic interpretation of a natural number object (N, z, s) satisfies also 𝑧𝑧the primitive→ ℕ ∷ recursion↦ ℕ 𝑠𝑠 ℕdefinition→ ℕ ∷ 𝑛𝑛 of↦ natural𝑛𝑛 numbers without any set-theoretic impredicativity (Abramsky).

http://www.irafs.org - [email protected] IRAFS'18 11 Functions and homomorphisms (=structure preserving mappings)

▪ Injective/surjective functions in set-theory  duality monomorphism/epimorphism in CT: ∀ = ⇒= Injective if: ∀ x , x′ ∈ X(( f( x) = f( x ′′)) ⇒=( x x )) Monic if gh , ((( fg) ( fh)) gh) Surjective if: ∀∈ y Y , ∃∈ x X( f( x) = y). Epic if ∀ ghgf , ((( ) =( hf)) ⇒= gh)

Bijective relation between sets X and Z through the Scheme of the relations among different composition between an injective (X;Y) and surjective (Y;Z) homomorphisms in and in CT relations. http://www.irafs.org - [email protected] IRAFS'18 12 Two fundamental dual categories in CT: and coalgebras

▪ Coalgebras are more useful than algebras for modeling dynamic systems as labeled state(phase) transition systems (LTS), because we are not blocked in computations to polynomials by the Fundamental Theorem of Algebra. ▪ Moreover, by the principle of functorial induction by a contravariant application of the same functor F from coalgebra to algebra F* we can construct the dually equivalent algebra, given that the largest part of the modern mathematical and logical-mathematical apparatus are algebraically formulated, overall in terms of Boolean Algebras having the nice property of the inductive nature of sets on which they are defined.

▪  Universal Coalgebra as dual to Universal Algebra and as General Theory of computational and dynamic systems as STS (Rutten 2000)

http://www.irafs.org - [email protected] IRAFS'18 13 2. Formal ontology and the issue of predicate constitution In logic and in computer science

http://www.irafs.org - [email protected] IRAFS'1814 Operator Algebra from the Formal Ontology Standpoint

▪  Philosophically, it is particularly fruitful to analyze the operator algebra approach to pure and applied logic and mathematics in the framework of CT formalism from the standpoint of formal ontology, i.e., from the standpoint of the foundation of predication in formal logic, as far as non-reducible to the simple set-theoretic membership. ▪ Because of the strict relationship of formal ontology with the so-called issue of the realism of universals of the Middle-Age Tradition, this allows us immediately to compare the formal ontology with the main issues of the Western metaphysical tradition. ▪ In this framework, it is fundamental the reference to the proto-semiotic interpretation (Deely, 2001) by John Poinsot (more known as John of St. Thomas: 1589-1644) of Aquinas’ ontological theory of truth vs. the prevailing conceptualist interpretation of Aquinas given by Thomas Cajetan (1469-1534) and Francisco Suarez (1548-1617) at the beginning of Modern Age. ▪ In fact, the sign as being for something else is the third giving a signifying power to the dyadic relations cause-effect in nature and predicate-argument in logic exemplifying the triadic nature of semiosis both in Poinsot, and in Peirce.

http://www.irafs.org - [email protected] IRAFS'18 15 CT and a non-conventionalist approach to the post- modern “transcendental of sign (language)”

▪ For our aims, CT allows a formalization of the outstanding Poinsot’s proto-semiotic theory of truth based on the conformitas (homomorphism) between the subject-predicate logical relation in sentences, and the reversed causal relation genus-species (or species-individual) in nature, as previous to the related concepts (verbum mentis) in mind. That is, e.g., “The horses are mammalians iff the genus of mammalians admits the species of horses”:    n|(∀> nn m )horse∈ mammmalian←   horse ∋ mammalian Algebra(ΩΩ *) Bounded Morphism Co-Algebra( ) ▪ This homomorphism (bounded morphism) is the proper solution also of Kripke’s issue of names as rigid designators both of individuals and of natural kinds, as far as based on a causal theory of reference, unfortunately justified by Kripke (and Putnam) only on a sociological basis, and not on a naturalistic basis.

▪ On the contrary Poinsot’s theory, illustrated in his semiotics treatise De Signis the first one in the Modern Age, is based on its outstanding distinction of a third type of relations, implemented in the sensorimotor (input-output) system of humans, beside the real (causal) relations among things in nature, and the rational (logical) relations among objects in a conscious mind.

▪ Namely, the semiotic relations among signs in language (relationes secundum dici), so that the conformitas of the latter ones as to the former ones is justified before of (and independently on) any reference to the awareness of the conformitas itself in mind.

▪ In fact, the sign is the third relation (like Peirce’s “thirdness” or the “interpretant”) by which rational relations can be related to real relations, and then conscious “objects” to real “things”. In fact, ens est primum cognitum, (…) sed cognitio est effectus quidem veritatis (“being is the first known, but knowledge is a sort of effect of truth”: Aquinas, Quaestiones De Veritate, I,1).

▪ This primacy of the “post-modern transcendental of sign” as to the “modern transcendental of knowledge” can be justified if and only if we attain at the pre-logical and pre-conscious (neural) formal structures of natural languages in brains that, otherwise, for their conventional nature, justify the relativistic and hence nihilistic interpretation of post-modernity and of its “linguistic turn”.

http://www.irafs.org - [email protected] IRAFS'18 16 Poinsot’s proto-semiotic interpretation of Aquinas’ ontology of truth as “adequacy of intellect to things”

“The transcendental relation, that is nothing but a relation in language, has not the principal meaning of “relation” [i.e., it does not belong to the Aristotle’s category of dyadic “relations” (esse ad, “being-to”)], but of something “absolute” [i.e., an esse per (“being for”)] to which some relation [e.g., a dyadic (predicative) rational relation, or a dyadic (causal) real relation] can be attributed [i.e., a sign is triadic]. Indeed, if it was not implying something “absolute”, it would be not “transcendental”, that is, ranging over different predicative categories (idest vagans per diversa genera), but it would belong to only one category (Poinsot, De Signis, 578b5-579a7. In: (Deely, Tractatus de Signis. The Semiotic of John Poinsot 90)).

http://www.irafs.org - [email protected] IRAFS'18 17 A taxonomy of formal ontologies depending on truth conditions of predication (the issue about universals) ▪ Nominalism: only singular names denote something, so that the predicable universals are reduced to the predicative expressions of a given language that, by its conventional rules, determines the truth-conditions of propositions in language (Sophists, De Saussure, Neurath, …).

▪ Conceptualism: the predicable universals are expressions of mental concepts, so that the rules of thought in mind determine the truth-conditions of propositions in language (Descartes, Kant, Husserl,…).

▪ Realism: the predicable universals are expressions of properties and relations, which determine the truth-conditions of propositions in language, and existing independently of the linguistic and/or mental capacities of humans in: ▪ The logical realm, we have then the ontologies of the so-called logical realism, where the ideal logical relations determine the truth conditions of propositions (Plato, Frege, Fraenkel, …); ▪ The physical realm, we have then the ontologies of the so-called natural realism, or “naturalism”. On its turn, naturalism can be of two types: ▪ Atomistic: with only singular denotations, where the mathematical laws of mechanics with their empirical fulfilment determine the truth-conditions of propositions (Democritus, Newton, Laplace, Wittengstein’s Tractatus, Carnap, …). ▪ Relational (or Semiotic): with natural kinds (common name denotations) – the “generals” of Peirce’s semiotics –, because the real relations (causes) among things determine the linguistic relations, and then the truth-conditions of propositions in language (Aristotle, Aquinas, Poinsot, Peirce, Kripke, …). ▪  Our approach to a naturalistic formal ontology of quantum cosmology is a categorical version of the ontic structural realism (Esfeld 2011, Lyre 2004, 2011), as a particular type of relational realism, in which Kripke models have their semantics on the (co)algebraic structures of the physical reality to which they refer (=categorical duality between logical and causal necessity).

http://www.irafs.org - [email protected] IRAFS'18 18 Formal Ontologies Scheme

Nominalism

Conceptualism Logical Realism Atomistic Ontology Natural Relational

http://www.irafs.org - [email protected] IRAFS'18 19 3. A coalgebraic formal ontology of quantum physics A categorical version of the ontic structural realism in quantum phsyics

http://www.irafs.org - [email protected] IRAFS'1820 3.1. Coalgebraic relational semantics of Kripke models Kripke models in evolutionary cosmology

http://www.irafs.org - [email protected] IRAFS'1821 Physical causality in early modern conceptualist ontology

▪ Generally in modern physics the principle of causality (an effect depends on its temporally preceding cause) in physical reality is interpreted in the framework of the hypothetical-deductive method as underlying the experimental satisfaction of the measurement asserts m deduced by a theory T, including some measurement axioms, as non- logical axioms of a given theory T i.e., T  m. ▪ This implies a double supposition (Descartes, Leibniz, Kant): 1. Identifying the epistemic (logical) and the ontic (causal) necessity (cause as sufficient condition of a conditional connective «if…then») 2. Making of humans the priviliged observer of nature  foundation of truth on conscious evidence by choosing the first alternative of the epistemic conundrum: is a sentence true because it is evident, or is it evident because it is true? ▪ Clearly, this supposition depends on the modern absolute linking of causality to time, because of the absolute dependence on the initial conditions of a physical process in the Newtonian and then in the statistical mechanics, QM included.

▪ In fact, time supposes necessarily a memory and then a consciousness just like the notion of set depends o some intellectual intuition (Cantor, Goedel, Fraenkel…), in modern set-theoretic logic and mathematics .

▪  Underlying conceptualist ontology of early modern science and philosophy.

http://www.irafs.org - [email protected] IRAFS'18 22 Physical causality in relativistic and quantum physics

▪ Relativity theory  there is no privileged observer even though the relationship of causality with time remains for causal light cone of a Minkowsky space-time of special relativity, that applies also to gravitational fields of the curved space-time of general relativity, as well as on the relativistic QM and QFT. ▪ However, the temporal precedence of cause as to effect does not hold any longer in front of quantum non-locality and quantum entanglement that is indeed one of the main theoretical issues for a quantum gravity theory given the local character of a Riemannian manifold. ▪ An alternative comes from Category Theory (CT) logic that allows to formalize the Aristotelian intuition (“what is last in being is first in knowing and vice versa”), modally formalized in the Middle Age by Aquinas, as basis of the dual homomorphism (formal causality) between logical and causal entailments, where the latter (causal necessity) is the foundation of the former (logical necessity), that is, cause has an ontological not temporal precedence: ▪ Logical entailment: “It is impossible that the premise is true and the consequence false” ¬◊ ¬ (1011) ¬◊ ¬ ▪ Causal (ontic) entailment: “It is impossible that the effect exists and the cause does not exist”𝑝𝑝 ∧ 𝑞𝑞 (1101) ▪ It is evident that the logical interpretation of the cause as the sufficient condition p of a𝑞𝑞 ∧conditional𝑝𝑝 has in this way its foundation in the reversed logic of the causal entailment where p is the necessary condition  onto+logical entailment (1001) of the onto-logical biconditional of a physical law of nature obtained without supposing a logical tautology (=rational evidence) that must be fulfilled of empirical evidence like in modern conceptualist ontology, but by a categorical calculus (commuting diagram) observer-independent and time-independent.

http://www.irafs.org - [email protected] IRAFS'18 23 Why Kripke models in relational semantics of the quantum world?

▪ This approach of the ontic structural realism was introduced in the analytic philosophy debate by S. Kripke’s distinction between epistemic and ontic necessity considered by Soames, in his monumental history of analytic philosophy, as the most significant contribution to XX cent. analytic debate (Soames 2005).

• Effectively, what is relevant in Kripke semantics for an evolutionary cosmology and makes it different from Tarski semantics is that it is possible to evaluate models not with respect to one only state-of-affairs or actual world, but as to many possible worlds or world-states, of which the actual world is one of them. • In fact, it is possible to justify formally in Kripke’s relational semantics, both a second-order semantics and a first-order semantics that, differently from the former, is fully computable. • In the first case, indeed, the semantics is for all the valuations V of the propositions p over the Cartesian product of the accessibility dyadic relations R between pairs of world-states w of a world W, × , over the whole set : we are then quantifying at second-order over all the valuations V, i.e., ∀Vw. 𝑅𝑅 ⊂ 𝑊𝑊 𝑊𝑊 𝑊𝑊 • However, in Kripke semantics it is possible also to evaluate propositions p only for all the possible world-states wn³m accessible from a given world-state wm : we are then quantifying at first-order over world-states, i.e., ∀ wn³m. • This second alternative is of course fundamental for a formalized ontology of the quantum evolutionary cosmology based on QFT including also gravitational fields, where new symmetries and hence new physical laws are dynamically (causally) generated by as many unpredictable spontaneous symmetry breakings (SSBs) of the quantum vacuum (QV), and propagate themselves to subsets of future states of the universe according to a causal light cone.

http://www.irafs.org - [email protected] IRAFS'18 24 The categorical formalization of the ontic structural realism in quantum ontology: (1) The TCS contribution as to (2) the QFT contribution

«Unraveling» of a rooted- Causal light-cone tree of Kripke models using NWF-sets

• Three main steps of CT logic as to Kripke models in modal logic and in theoretical computer science (TCS) (Venema 2007): 1. CT notion of meaning function that maps a propositional formula ϕ to its extension that makes true ϕ. E.g., in set-theoretic semantics a propositional function ϕ (e.g., (p ∧ q)) is valued as true/false on the operations on corresponding sets, P, Q, that is, on its extension , i.e., ⋅ . In CT algebraic logic, the propositional variables𝜑𝜑 p can be seen as algebraic terms T  we can therefore impose an algebraic structure (frame)  on propositional formulas ϕ, , so that the truth-value of a valuation V on ϕ in 𝑃𝑃⋂𝑄𝑄 𝑝𝑝 ∧ 𝑞𝑞 → 𝑃𝑃⋂𝑄𝑄 +  can be computed as on the complex algebra (algebra+subalgebras)  , according to the unique homomorphism : 𝕊𝕊𝜑𝜑 extending V to V. That is, the validity () of a formula in the frame  corresponds to that of an equation “≈” in the complex algebra ++ 𝑉𝑉� 𝜑𝜑 𝑽𝑽� 𝕊𝕊𝝋𝝋 → 𝕊𝕊 of , and vice versa (e.g. an arithmetic equation in the case of a two-valued Boolean algebra 2 : : ) . I.e.,  �  � 𝜑𝜑    1; and        . 𝑉𝑉 𝕊𝕊 → 2𝔹𝔹𝔹𝔹

http://www.irafs.org - [email protected] IRAFS'18 25 Continuing (2)… 2. Study in TCS of the category of coalgebras Coalg: × as functorially dual to the category of algebras Alg: × , for an endofunctor Ω and its opposite Ω*, respectively. Moreover, they can be dually endowed with, respectively𝑨𝑨 → 𝑨𝑨(Venema𝑨𝑨 2007; Otto & Goranko 2007), 𝑨𝑨 𝑨𝑨 → 𝑨𝑨 ▪ A final coalgebra (=closure upper condition), if there exists in Coalg(Ω) a coalgebra such that for every coalgebra , there exists a unique homomorphism ! : , ℤ ▪ An initial algebra (=closure lower condition), if there existsℂin Alg(Ω*) an algebra , such that for every algebra thereℂ exists a unique homomorphism ! : ℂ. → ℤ 𝔸𝔸 ▪  Definition of the notion of coinduction in Coalg(Ω) as𝔹𝔹 a (upper bounded) method of set definition and proof, which𝔹𝔹 is dual as to the (standard) notion of induction𝔸𝔸 → 𝔹𝔹 in Alg(Ω*) as a (lower bounded) method of set definition and proof  new notion of finitary (upper and lower bounded) competitive computation in constructive mathematics. ▪  Given the Stone Representation Theorem for Boolean algebras stating the dual equivalence between the category of the Stone topological spaces, Stone and the category of Boolean algebras, BA:  definition of the logical dual equivalence, by application of a contravariant functor / between the category of coalgebras on Stone spaces SCoalg and the category of Boolean𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒algebras≃∗ 𝐁𝐁𝐁𝐁 ( ) ( )  dual semantic validation of Boolean formulas on their coalgebraicΩ Ω extensions. ∗ 𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 𝛀𝛀 ≃ 𝐁𝐁𝐁𝐁 𝛀𝛀

http://www.irafs.org - [email protected] IRAFS'18 26 Continuing (3)…

3. Stone coalgebras on Non-wellfounded (NWF) sets. I.e., sets represented as rooted trees, in which ZF Regularity Axiom (and then Set Total Ordering) does not hold, so that set self-inclusion and then unbounded chains of set-inclusions are allowed, but for which a Final Coalgebra Theorem holds (all NWF-sets have a unique ultimate root)  definition of Universal Coalgebra as General Theory of (Dynamic and Computing) Systems, modeled as Labeled State-Transition Systems (LTSs) defined on rooted trees of NWF-sets (Otto & Goranko 2007). ▪ I.e., the semantics of Boolean formulas (programs) is defined on the corresponding physical states of the system coalgebraically modeled  coalgebraic modeling of computations on infinite data streams, using the powerful construction of the infinite state black-box machine (Rutten 2000, Venema 2007, Basti & Vitiello 2017). ▪ Because the dyadic state-transition relation in LTSs can be immediately modeled as an accessibility relation in Kripke structures, possibility of modeling the unfolding/unraveling of rooted trees of Kripke models on a coalgebra of NWF-sets. In parenteses, Kripke model theory is demonstrably complete only in this coalgebraic fashion. ▪ Finally,in general, the direction of a formula validation relation (homomorphism) is from a formula algebraic structure  to its extension on the complex algebra +, i.e., (  ϕ) → (+  ϕ). ▪ However, if there exists in the extension a selection criterion of admissible sets, (e.g., a second-order ultrafilter Uf like in Stone Theorem, or a first-order criterion like the Vietoris construction  in the coalgebraic formalization of modal Boolean logic) the direction of the validation relation is reversed, i.e. (  ϕ) ← (Uf +  ϕ); () ( ) 𝐨𝐨𝐨𝐨 𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 ≃ 𝐌𝐌𝐌𝐌𝐌𝐌 http://www.irafs.org - [email protected] IRAFS'18 27 3.2. Coalgebraic modeling of QFT dissipative systems The QFT contribution to the formal ontology of the ontic structural realism 3.2.1. Quantum vacuum and dissipative QFT A non-perturbative interpretation of QFT The categorical formalization of the ontic structural realism in quantum ontology: (2): the QFT contribution as to the TCS contribution (1)

▪ The connection between CT coalgebraic logic just sketched and QFT has its foundation in the fact that the topologies of Stone Spaces in logic are the same of the C*-(sub-)algebras of the Hilbert spaces of the GNS-construction (Landsman 2017) in quantum physics and specifically in the coalgebraic modeling of dissipative QFT systems = semantics of the propositional formulas of the formal ontology of quantum physics, beacuse satisfying the dual equivalence between the category of coalgebras on Stone spaces SCoalg and the category of Boolean algebras ( ) ( )  dual semantic validation of Boolean formulas of the propositional calculus on their QFT coalgebraic extensions (see slides 26-27). ∗ 𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 𝛀𝛀 ≃ 𝐁𝐁𝐁𝐁 𝛀𝛀 ▪  Three steps towards the coalgebraic formalization of «dissipative QFT» (Blasone, Jizba & Vitiello, 2011; Basti, Capolupo & Vitiello, 2017):

1. Relationship between the III Principle of Thermodynamics and the notion of QV as universal energy reservoir in QFT  all quantum systems in QFT has to be interpreted as «open» systems to the unavoidable QV fluctuations in their background  necessity of a dissipative or «thermal» QFT as a unique framework including both relativistic QFT (microscopic), condensed matter physics (macroscopic), and quantum cosmology (megaloscopic).

2. The physical pillars of such a construction are:

a) The Haag Theorem;

b) The Bogoliubov Transformation;

c) The Goldstone Theorem.

http://www.irafs.org - [email protected] IRAFS'18 30 2a) The Haag Theorem

▪ Haag Theorem (1955): infinitely many degrees of freedom (phase transitions) in QFT  mismatch between the field dynamics (Heisenberg matrix equations) and its representation (the Hilbert space of physical states). ▪  The same dynamics may lead to different solutions, i.e., to infinitely many unitarily inequivalent representations (Hilbert spaces of physical states) ▪  «The choice of the representations may be considered as a boundary condition under which the Heisenberg equations have to be solved» (Blasone, Jizba & Vitiello, 2011, 73) ▪  Physical connection with the infinitely many spontaneous symmetry breakings (SSBs) compatible with the QV ground-state |0〉 of the Goldstone Theorem ruled by the Bogoliubov Transformation.

http://www.irafs.org - [email protected] IRAFS'18 31 2.b) The Bogoliubov Tranformation

▪ Bogoliubov Transformation (1958): it rules the creation/annihilation processes of bosons and fermions from the QV. ▪ I.e., given a canonical commutation relation (CCR) for a pair of creation/annihilation operators on the Hilbert space for a boson, and another pair of operators for another boson, there exists a transformation mapping the former into the latter. ▪ The same holds for a canonical anti-commutation relation (CAR) for pairs of creation/annihilation operators on the Hilbert space for fermions. ▪  Bogoliubov demonstrated that there exists an isomorphism, either of the CCR algebras, or of the CAR algebras, by which modeling different “degenerate states” of the QV at the ground state |0 , previewed by the Haag Theorem, corresponding to a process of creation/annihilation of bosons or of fermions, and/or of their condensates. ⟩

http://www.irafs.org - [email protected] IRAFS'18 32 2.c) The Goldstone Theorem

▪ Goldstone Theorem (1961): its discovery of the relationship between SSBs of the QV ground-state and the dynamic generation of long-range correlations between «material» and «interaction» force fields with their respective quanta (fermions and gauge-bosons)  constitution of physical systems as condensates of fermions and gauge-bosons, each having in a unique Nambu-Goldstone (NG) bosons condensate (=quanta of the phase-coherent modes of interaction among quantum fields: in the infinite volume limit they are not quanta of energy like gauge-bosons!) its «control-parameter», its univocal «finger-print». ▪ What is relevant is that the connection between SSBs of the QV and the formation of long-range correlations among quantum fields suggests: 1. A natural mechanism of dynamic rearrangement of the symmetries in QFT giving a precise sense to Wheeler’s statement of a causal foundation of the physical laws (symmetries) on the universe evolution. 2. A dynamic reinterpretation of wave-particle quantum duality, in terms of the duality between phase coherence of quantum fields (collective behavior of a condensate) versus their quanta (individual behavior of oscillating «particles»). 3. A dynamic rearrangement of the metrics related with the long-range correlations (entanglement) of quantum fields  macroscopic behavior of condensed matter physics has its proper explantion at the quantum microscopic level.

http://www.irafs.org - [email protected] IRAFS'18 33 QM vs QFT explanation of particle-wave duality

▪ Therefore, in QFT an uncertainty relation holds, similar to the Heisenberg uncertainty, but it is not relating two different representations (statistical blanket), particle-like, versus wave-like, as it is in the QM particle-wave duality, related with Schrödinger’s statistical wave function, namely: ∆∆xp ≥ 2 ▪ Where, x is the particle position, p is the particle momentum, and  is the Planck constant. ▪ In QFT the particle-wave duality relates dynamic entities, that is, the uncertainty on the number of the field quanta, and the uncertainty on the field phase, namely: ∆∆n ϕϕ ≥ () ▪ Where n is the number of quanta of the force field, and ϕ is the field phase. If (∆n = 0), ϕ is undefined (= no phase coherence), so that it makes sense to neglect the waveform aspect in favor of the individual, particle-like behavior. On the contrary if (∆ϕ = 0), n is undefined because an extremely high number of quanta are oscillating together according to a well-defined phase, i.e., within a given phase coherence domain. In this way, it would be nonsensical to describe the phenomenon in terms of individual particle behavior, since the collective modes of the force field prevail. ▪ Of course, in this case the probabilities of the quantum states follow a Wigner distribution, based on the notion and the measure of quasi-probability, where regions integrated under given expectation values do not represent mutually exclusive states.

http://www.irafs.org - [email protected] IRAFS'18 34 3.2.2 Non-commutative Hopf coalgebras in QFT The doubling of the algebras in QFT computations

http://www.irafs.org - [email protected] IRAFS'1835 The second step towards a dissipative QFT: its coalgebraic modeling

▪ Coalgebraic modeling of dissipative QFT. In dissipative QFT the Hopf algebra co-products × by which we calculate in a lattice of quantum numbers, e.g., the total energy of a quantum state do not commute 𝐴𝐴 → 𝐴𝐴 𝐴𝐴 between themselves since the terms represent here the values of a system state and of its correspondent thermal-bath state that cannot be treated on the same basis – differently from QM where we consider systems as closed and the terms represent, e.g., two energy values of two particles in a quantum state –  the formal tool is then the q-deformed Hopf coalgebras with non-commutative co-products, where q is a thermal parameter intrinsically related with the angle θ of the Bogoliubov Transform and its “squeezing” parameter r.

▪ Of course this doubling of the algebras extends itself also to the related doubling of Hilbert space: × with their respective C*-algebras/coalgebras, where the system state does not commute with the thermal 𝐻𝐻 → 𝐻𝐻 bath state , even though their respective , ( ) operators commute according to the Bogoliubov 𝐻𝐻� 𝑎𝑎 transform and its reversal for each , pair, consistently with the energy-arrow reversal of an energy 𝑎𝑎� 𝐴𝐴 𝜃𝜃 𝐴𝐴̃ 𝜃𝜃 balance characterizing any phase transition in QFT open systems far-from-equilibrium, each indexed by a 𝐴𝐴 𝐴𝐴̃ given value of and then of q. http://www.irafs.org - [email protected] 𝜃𝜃 IRAFS'18 36 The Quantum Vacuum “foliation” in the relativistic dissipative QFT

▪ Correspondingly, we have the “splitting” of the vacuum |0 |0 × |0 as far as annihilated by the Bogoliubov operators and : | = = | . I.e., by limiting ourselves to bosons where operators act on a hyperbolic base-function: ⟩ ≡ ⟩ ⟩ 𝑨𝑨 𝑨𝑨� 𝑨𝑨 𝟎𝟎⟩ 𝟎𝟎 𝑨𝑨� 𝟎𝟎⟩

iG kk 1 †† 0( ) e k 0 exp tanhAA 0    k  k coshk

▪ I.e., the vacuum annihilated by these operators is |0( ) ,where the subscript refers to a given unique NGB-condensate characterizing an annihilated QV state, and 0( ) 0( ) = 1, where θ effectively here denotes𝜃𝜃 ⟩𝒩𝒩 the set , for a𝒩𝒩 k-set k of quantum numbers connoting here a given bosonic condensate. 𝒩𝒩 𝜃𝜃 𝜃𝜃 𝒩𝒩 𝜃𝜃𝑘𝑘 ∀𝐤𝐤

http://www.irafs.org - [email protected] IRAFS'18 37 QV foliation in dissipative QFT ▪ Therefore, given the generator of the Bogoliubov transformations , , i.e.: , it is possible † † to demonstrate that in the infinite volume limit , infinitely many𝐴𝐴𝐤𝐤 𝐴𝐴̃ 𝐤𝐤pairs 𝐺𝐺𝐤𝐤 ≡system−𝑖𝑖 𝐴𝐴𝐤𝐤𝐴𝐴-̃thermal𝐤𝐤 − 𝐴𝐴𝐤𝐤𝐴𝐴̃ 𝐤𝐤bath for each set † † k of quantum numbers are condensed in the vacuum𝑽𝑽 → ∞|0( ) . 𝐴𝐴𝐤𝐤𝐴𝐴̃𝐤𝐤 ▪ This shows that, as far as , the phase space splits𝜃𝜃 into⟩𝒩𝒩 infinitely many inequivalent representations, as the Haag Theorem previews. 𝑉𝑉 → ∞ ▪ In thermal QFT, however, each of them is labelled by a specific θ-set: = ln , , and the Bogoliubov generator Gk maps how to pass from one system into the other. 𝜃𝜃𝐤𝐤 𝑞𝑞𝐤𝐤 ∀𝐤𝐤 ▪ This is exactly what in QFT we intend by the notion of QV-foliation, as a powerful dynamic tool used by nature of construction, of “complex systems”, in physics, and of “memories”, in biology and neurosciences, where systems are characterized by complex signaling sub-systems, generally electro-chemical, making them self-organizing systems. ▪ In fact, all the QV degenerated (doubled) states of QFT, ruled by the Bogoliubov Transform, are at the QV ground-state, so that they are very stable and then they might support a “constructive” process of complex systems / memories. ▪ Effectively one of the most successful applications of this dissipative QFT modeling is in the solution of the long- term memory or “deep learning” issue in cognitive neurosciences (Freeman, Kozma, Vitiello, Basti…: see below)

http://www.irafs.org - [email protected] IRAFS'18 38 The Doubling of the Degrees of Freedom (DDF) in relativistic QFT

▪ One can also show that the vacuum |0( ) is a generalized SU(1,1) of coherent state of condensed couples of , modes for each k. These are entangled𝜃𝜃 ⟩𝒩𝒩 modes “system-thermal bath” in the infinite volume limit. 𝐴𝐴𝐤𝐤 𝐴𝐴̃𝐤𝐤 ▪ Note that the Bogoliubov generator Gk is part of the Hamiltonian of the system i.e. of the time evolution operator. We have thus a quantum realization of the operator algebra at each time t, which can be implemented by the GNS- construction in the C*-algebra formalism and in the associated topological interpretation of QFT.

▪ Finally, because the principle of the “infinite QV-foliation” implies per se an infinite foliation also of the representational doubled Hilbert space , necessary for recovering the “closed” canonical Hamiltonian

representation of a dissipative system, the 𝑯𝑯possibility𝑯𝑯� of using the minimum free-energy function in dissipative QFT as a (thermo-)dynamic choice-criterion of physically admissible states of the doubled Hilbert space exemplifies the DDF principle in this non-perturbative approach to QFT, according to the formalism developed in (Celeghini, Rasetti, & Vitiello, 1992; Vitiello, 2007, etc.).

http://www.irafs.org - [email protected] IRAFS'18 39 The DDF principle in quantum cosmology

▪ On this regard, with respect, for instance, to famous Seyed Majid’s attempt of connecting the non-commutative Hopf coalgebras of quantum group theories (self-dual bi-coproducts) with the Riemannian manifolds of general relativity (GR), for a theory of quantum gravity, by using Mach’s principle (Majid 2000a,b; 2008), in dissipative QFT the main limit of Majid’s construction does not hold. ▪ In fact, according to Majid’s interpretation of Hopf algebras self-duality as a «maximalistic» version of Mach’s principle («it is true only what satisfies self-duality because self-representational») “all local properties of the universe should be completely determined by its global properties” (we are indeed in a quantum group theory!). This postulate is evidently untenable in GR and in a Riemannian manifold (Heller 2004).

▪ On the contrary, the category of q-deformed nonℳ-commutative Hopf coalgebras in thermal QFT displays a semigroup structure because of the non-unitary and then inequivalent character of CCRs. Therefore, the DDF principle is able to justify, differently from Majid’s theory, a local solution of the issue as it has to be in the Riemannian manifolds of GR.

▪ In fact, in dissipative QFT it is possible to suggest a thermo-field interpretation of the gravitational dynamics for non- commutative geometries, in terms of gravitation fields on a doubled Riemannian manifold × , where the two components of this doubled-dual construction have opposite directions, and where the doubled-dual Hilbert space is modelling the associated non-commutative quantum representation spaces (Sivasubramanian𝓜𝓜 ,𝓜𝓜� Srivastava, Vitiello 2014).

http://www.irafs.org - [email protected] IRAFS'18 40 Dissipative QFT in condensed matter physics

▪ Of course, dissipative QFT has its most direct application in modeling, at the fundamental level, the non-relativistic many-body physics, that is, condensed matter physics, characterized by the transition among different topological phases of matter ruled by the Bogoliubov generator Gk. ▪ In fact, it rules the transition between phases, each with its proper «order parameter» (the dynamic magnitude displaying different degrees of order at the boundary of the phase transition), and where the NG bosons condensates, acting as «a control parameter» of the different phases, acquire in literature different names, according to the different phases of matter fields they control. ▪ E.g., in solid-state physics concernig the Galileian spheric symmetry-breaking along different directions of the mechanical force field of molecule vibrations, determining the liquid (symmetry-breaking along the longitudinal direction) / solid (along the longitudinal and transversal directions) phase transition, the NG- bosons are named as phonons. ▪ In the case of the spheric symmetry-breaking of the magnetization vectors direction in the non- ferromagnetic/magnetic phase transition, NG-bosons are named as magnons, etc.

http://www.irafs.org - [email protected] IRAFS'18 41 NG bosons in living matter

▪ In the case of organic matter and in water, in which only the biological molecules are active (this is the deep reason for which >80% of our bodies is made of water, and >90% of our molecules are of water), the complex structures of the bio-molecules, and the ordered sequences of chemical reactions constituting each single biological function (their fine tuning) are ultimately derived, at the fundamental level, by the NG boson condensates named here as polarons. ▪ In fact, what characterizes both these types of molecules, of water and of organic matter, is a strong electrical dipole field. In such a way, the basic hypothesis of QFT applied to living matter is that “at the dynamic fundamental level, the living matter can be considered as a set of electrical dipoles whose rotational symmetry is broken down” (E. Del Giudice). ▪ A particular relevance for our ontological and epistemological aims has the dissipative QFT modeling of living matter in neurodynamics and cognitive neurosciences (“dissipative brains”) where the DDF principle of coalgebras acquires immediately a semantic relevance.

http://www.irafs.org - [email protected] IRAFS'18 42 3.2.3. The physical basis of intentionality in mind Intentional brain as a «dissipative brain»

http://www.irafs.org - [email protected] IRAFS'1843 The physical basis of intentionality, and the long-term memory (“deep-beliefs”) issue in brains

▪ As Perrone and myself emphasized in several papers on the physical basis of intentionality (Basti & Perrone, 1995; Basti & Perrone, 2001; Basti & Perrone, 2002; Basti, 2009), only the long-range correlations, which propagate in real-time along wide areas of the brain, and manifest themselves kinematically as “chaotic itinerancy” or instantaneous jumping among several attractors of the dynamics, can offer a valid macroscopic dynamical basis of an intentional act, always involving the simultaneous interaction among emotional, sensory and motor components, located in very far areas of the brain. ▪ Such a coordination, that constitutes also the dynamic “texture” of long-term memory phenomena, cannot be explained in terms of the usual axon-synaptic networking, too slow and then too limited in space and time, for giving a suitable explanation of this requirement. ▪ Now, a significant successes of thermal QFT is the demonstration that the chaotic trajectories in the phase space of dynamic systems, brain included, are nothing but the macroscopic trajectory in the phase space a QFT system follows for passing from one coherent phase to another one. In fact, the set |0( ) , is a , which is symplectic, and the trajectories in it are chaotic (Vitiello 2007). 𝜃𝜃 ⟩𝒩𝒩 ∀𝜃𝜃 𝐾𝐾𝑎𝑎̈ ̈ℎ𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 http://www.irafs.org - [email protected] IRAFS'18 44 The QFT dynamics underlying the chaotic dynamics in the brain as open system

▪ On the other hand, my friend Walter J. Freeman and his collaborators, during more than forty years of experimental research by the Neurophysiology Lab at the Dept. of Molecular and Cell Biology of the University of California at Berkeley, not only shared my same theoretical convictions about chaos and intentionality, but they observed, measured and modelled this type of dynamic phenomena in mammalian and human brains during intentional behaviours. ▪ The huge amount of such an experimental evidence found, during the last ten years, its proper physical-mathematical modelling in the QFT interpretation of biological condensed matter dynamics of Vitiello and his collaborators, applied to neural field many-body dynamics, so to justify the publication during the last years of several joint papers on these topics (see, for a synthesis, (Freeman & Vitiello, 2006; 2008; 2016)).

http://www.irafs.org - [email protected] IRAFS'18 45 Brain as a QFT dissipative system

▪ To sum up (Vitiello G., 2009), Freeman and his group used several advanced brain imaging techniques such as multi-electrode EEG, electro-corticograms (ECoG), and magneto-encephalogram (MEG) for studying what neurophysiologists generally consider as the background activity of the brain, often filtering it as “noise” with respect to the usual axon-synaptic activity of neurons, they are exclusively interested in.

▪ By studying these data with computational tools of signal analysis to which physicists, differently from neurophysiologists, were acquainted, they discovered the massive presence of patterns of AM/FM phase- locked oscillations. They are intermittently present in resting and/or awake subjects, as well as in the same subject actively engaged in cognitive tasks, requiring interaction with the environment.

▪ In this way, we can describe them as features of the background activity of brains, modulated in amplitude and/or in frequency by the “active engagement” of a brain with its surround. These “wave packets” extend over phase coherence domains covering much of the hemisphere in rabbits and cats (Freeman W. J., 2004a; 2004b; 2005; 2006), and regions of linear size of about 19 cm in human cortex (Freeman, Burke, Holmes, & Vanhatalo, 2003), with near zero phase-dispersion (Freeman , Ga'al, & Jornten, 2003). Synchronized oscillations of large scale neuron arrays in the β and γ ranges are observed by MEG imaging in the resting state and in the motor-task related states of the human brain (Freeman & Rogers, 2003).

http://www.irafs.org - [email protected] IRAFS'18 46 Dynamic neural fields of intentional behavior

(Left). Schematic representation of human cortex (top) and limbic system (down). (Right: left). Evidence of the intentional behavior of olfactory bulb: the same olfactory stimulus induces a modulation in amplitude (top) when the cat is hungry, and no modulation when it is full; (Right: right). Dynamic attractors (closed curves: coherent states) in the overall unstable brain field dynamics related with intentional pattern recognition. Their occurrency is of the order of ≈10-1sec.! http://www.irafs.org - [email protected] IRAFS'18 47 The quantum entanglement system-thermal bath in QFT and the doubling of degrees of freedom (DDF): the physical basis of intentional realism ▪ The “thermal bath” is not simply the “environment” for an open system but the dynamic components of the environment with which it is entangled within one phase coherence domain, so that a bi-directional energy exchange can occur only with these components, just as in a resonance phenomenon.  Principle of the “doubling” or “mirroring” (because of the inversion of the energy arrow direction (energy balance far from equilibrium) of the degrees of freedom system/thermal bath (=DDF).

We consciously see at objects in the optical part of the electromagnetic spectrum because cones and rodes of our retinas are in phase with these frequencies. Owls see at objects in the (near-)infrared part of the spectrum because their visual receptors are in phase with these frequencies.

http://www.irafs.org - [email protected] IRAFS'18 48 3.2.4. The DDF principle in quantum computing The doubled qubit in a QFT modelling of quantum computer

http://www.irafs.org - [email protected] IRAFS'1849 Neurally inspired doubled (semantic) qubit in QFT computing systems ▪ Let us consider a two level system, i.e., a qubit, described by a the orthonormal basis of two unit (pure state) vectors |0 and |1 , = , , , = 0,1 (Basti, Capolupo & Vitiello 2017). ⟩ ⟩ 𝑖𝑖 𝑗𝑗 𝛿𝛿𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗

http://www.irafs.org - [email protected] IRAFS'18 50 The doubled qubit and the minimum free-energy function as a logical evaluation function

http://www.irafs.org - [email protected] IRAFS'18 51 In fact…

http://www.irafs.org - [email protected] IRAFS'18 52 Dynamic (recursive) generation of Fibonacci’s sequences

http://www.irafs.org - [email protected] IRAFS'18 53 In conclusion…

▪ In conclusion, by recursive application of the doubled (semantic) qubit we generated a Fibonacci progression by iterative algebraic applications of the -matrices through a process that has its dynamical representation in the Hamiltonian of the system  i.e., it is a “Golden Machine” and then a Universal Computing Machine. 𝜎𝜎 ▪ On the other hand, the semantic value of the qubit in thermal QFT systems, because of the doubling of the states, allows to interpret the connected measure of the free-energy minimum as an evaluation function for the associated Boolean algebra. Indeed, the categorical dual equivalence between the category of coalgebras on Stone spaces and the category of modal Boolean algebras: () ( ) (slide 27) can be extended to the category of the q- : deformed Hopf coalgebras for the Bogoliubov endofunctor ∗ 𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 ≃ 𝐌𝐌𝐌𝐌𝐌𝐌 , having its proper matricial calculus in the associated sub-categoryℬ∗ of the composite Hilbert spaces , of the Hilbert space category, Hil. 𝒒𝒒 − 𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇 𝓑𝓑 ≃ 𝐌𝐌𝐌𝐌𝐌𝐌 𝓑𝓑 𝐇𝐇𝐇𝐇𝐇𝐇𝐴𝐴 𝐴𝐴� ▪ The theoretical connective for such a construction is the same topological properties shared by the Stone spaces on which the Boolean semantics is defined, and by the topological spaces of the C*-algebras of the composite Hilbert spaces , associated to the coalgebras of thermal QFT (Landsman 2017; Basti & Vitiello 2017).

ℋ𝐴𝐴 𝐴𝐴� ≡ ℋ𝐴𝐴⨂ℋ𝐴𝐴� http://www.irafs.org - [email protected] IRAFS'18 54 4. Ontological and epistemological consequences The formal ontology and the formal epistemology of the natural realism

http://www.irafs.org - [email protected] IRAFS'1855 From coalgebraic computational logic to coalgebraic modal logic

▪ From the standpoint of the notion of the Universal Coalgebra of a «General Theory of Dynamic and Computational Systems», both modelled as labelled state-transition systems (Rutten 2000), the coalgebraic modelling of QFT dissipative systems satisfies the computational construction of the infinite state black-box machine (Rutten 2000; Venema 2007; Basti, Capolupo & Vitiello 2017), as dynamical counterpart of the logistic Universal Quantum Turing Machine of QM (Deutch 1985). ▪ This coalgebraic computational modelling is consistent with a particular modal system, the so-called nested KD45 of the modal propositional calculus and its semantic validation on a complex coalgebraic structure + of rooted Kripke models, giving us the modal logic of the formal ontology of an evolutionary cosmology.

http://www.irafs.org - [email protected] IRAFS'18 56 Modal axioms and their validation on Kripke algebraic structures (frames)

▪ The fundamental correspondence theorem (Van Benthem 1976) between modal axioms of the propositional calculus, and first order (FO) formulas of the algebraic calculus of relations in Kripke semantics.

▪ Different systems of the modal calculus (syntax) by combining different modal axioms and the relative FO formulas  different modal semantics by adding non- logical axioms  different Kripke models of ontological, epistemological, deontic theories in formal philosophy.

▪ Main modal axioms for propositions of the modal propositional calculus: ▪ =   , where X is a set of propositions𝛼𝛼 , i.e., modal MP;

▪ 𝐊𝐊 =𝑑𝑑𝑑𝑑𝑑𝑑 𝑿𝑿 → à 𝛼𝛼;

𝑑𝑑𝑑𝑑𝑑𝑑 ▪ 𝐃𝐃 = 𝛼𝛼 → ;𝛼𝛼 Correspondence Validation of modal axioms on structures =𝑑𝑑𝑑𝑑𝑑𝑑 principle ▪ 𝐓𝐓 𝛼𝛼 → 𝛼𝛼 ; (frames) satisfying the associated FO relation ▪ 𝟒𝟒 𝑑𝑑𝑑𝑑𝑑𝑑 =𝛼𝛼 → à 𝛼𝛼 à property R 𝐄𝐄 𝐨𝐨𝐨𝐨 𝟓𝟓 𝑑𝑑𝑑𝑑𝑑𝑑 𝛼𝛼 → 𝛼𝛼

http://www.irafs.org - [email protected] IRAFS'18 57 Validation of axiomatic modal systems on structures satisfying simultaneously the associated FO relations

▪ E.g., the modal system KT5 (S5) is validated on structures (frames) of Ultra-simplified three possible-world universe possible-worlds satisfiyng forming one only equivalence class of simultaneously the reflexive and possible worlds, and then a KT5 (S5) modal Euclidean relations associated to the axioms T and 5(E), respectively, system in Kripke’s relational semantics added to axiom K, so to form one only graphical representation. equivalence class of possible worlds. ▪ E.g., the modal system KD45 (secundary S5) where an equivalence class of possible-worlds is formed by a recursive application of the serial and Euclidean relations of axioms D and 5, starting from an unaccesible world, generating symmetric, reflexive, and transitive (axiom 4) relations among the other accessed worlds.

http://www.irafs.org - [email protected] IRAFS'18 58 KT5 (S5) vs. KD45 (secundary S5) modal systems of formal ontology

▪ Generally, KT5 (or S5) is the modal system Ultra-simplified three possible-world universe giving the logical axioms (syntax) common to forming one only equivalence class of all the ontologies (semantics) of classical accessibility (causal) relations, and then a physics  all the possible states of the KT5 (S5) modal system in Kripke’s relational universe form one only equivalence class for semantics graphical representation. a given set of immutable physical laws (e.g., Laplacian physics). Simple structure of a KD45 modal system, where the possible worlds, ▪ The alternative, fitting with an evolutionary w,v,z form an equivalence class S5, cosmology modeled on a coalgebraic basis because of the unique homomorphism with a final object, is a nested KD45 system, with a final world u accessing (causally) where the different (equivalence classes of) all of them, but unaccessible to them. physical systems and the relative natural laws, emerge from the QV phase transitions (SSBs), according to a QV-foliation ruled by Passage to a the Bogoliubov generator G, as the nested KD45 endofunctor of this complex structure +of modal system of coalgebras/sub-coalgebras. rooted trees of Kripke models with an ultimate root G

http://www.irafs.org - [email protected] IRAFS'18 59 Nested KD45 system as the logical calculus underlying the formal ontology of an evolutionary cosmology and biology (see also Ehresmann 2018)

▪ From the ML standpoint, each level of emergence in the nested KD45 structure corresponds to a rooted tree of Kripke models  QV endowed with the Bogoliubov G generator = final root G of all the rooted trees of a Kripke coalgebraic complex structure +, defined on a topology of pointed NWF-sets (hom- sets) (Basti 2018). ▪ E.g., the SSB of the rotational symmetry of magnetization vectors determines the equivalence class of the physical systems satisfying the predicate/function «being a magnet» and the relative Maxwell laws, at a given step of the universe evolution, i.e., at a given step of the Kripke rooted trees unfolding, modeling the QV-foliation.

http://www.irafs.org - [email protected] IRAFS'18 60 From a conceptualist to a relational (semiotic) epistemology

▪ It is evident that the nested KD45 modal system gives us the logic of the coalgebraic Kripke semantics for the propositional logic of a naturalistic formal ontology, according to a categorical interpretation of the ontic structural realism for an evolutionary cosmology, i.e., where the necessity operator  and the related universal quantifier " are indexed at given steps n (for n³m) of the universe evolution (i.e., from a possible world to all the possible worlds causally accessed by it).

▪ That is, the nested rooted trees of Kripke models, along to different unfolding paths from a common root, can be ontically interpreted as coalgebraic structures of natural kinds (genera-species) in nature, making true the sentences referring to them in the correspondent algebraic logic (classes-subclasses: see slide 16), i.e., by a contravariant functor making bounded their homomorphism and then justifying their dual equivalence:    n|(∀≥ nn m )horse∈ mammmalian←   horse ∋ mammalian Algebra(ΩΩ *) Bounded Morphism Co-Algebra( )

http://www.irafs.org - [email protected] IRAFS'18 61 From bio-semiotics to cognitive neuroscience

▪ What is relevant from the epistemological standpoint is that this ontology is on a purely semiotic (algebraic) basis, without any reference to consciousness i.e., it concerns all communication agents (either natural (cells and organisms), or artifical (computers and automata)), and not only conscious or cognitive communication agents (animals and humans). ▪ This evidence has three fundamental consequences for the ontology of (quantum) biology and neuroscience: 1. From the standpoint of biology, this ontology is able to justify a formal biosemiotics of living systems as dissipative systems entangled with their environments, from cells to multicellular organisms. All modeled as dissipative self-organizing systems, because capable of signifying, i.e., of exchanging communication and control signals (bio-cybernetics) among themselves, and with their inner and outer environments (genetics and epigenetics), at different levels of their complex structure (see also Ehresmann 2018);

http://www.irafs.org - [email protected] IRAFS'18 62 The conscious component of our intentional acts is like an iceberg-top of unconscious processes constituting our persons or selves

2. From the standpoint of neuroscience, because it is experimentally proved that our dissipative brains entangled with their environments arrive always tenths of seconds before our self-consciousness (Libet 1983; 2007; Freeman 2000), the epistemological realism has a semiotic foundation that we cannot have in principle, if we start from consciousness (what is there, behind the Alice mirror?): consciousness as such can be only self-representational as Kant and Hegel taught us… 3. From the standpoint of anthropology, the self is not the self-consciousness, but a communication agent endowed with self-consciousness, i.e., a person, a complex communication system able of self- controlling its behaviors. The subject of our intentional cognitive and deliberative acts are neither our self-consciousness (dualism), nor our brain (monism) alone, but their psycho-physical unity. ▪ «Saying that the mind or the body alone are the subjects of our cognitive or deliberative acts is as much stupid, Aquinas said, as saying that the hammer or the chisel and not the sculptor are the authors of the statue, only because they are both necessary instruments for making a sculpture…»

▪  Personalist interpretation of neuroethics in terms of biological responsiveness to our socio-biological environment as objective foundation of our personal responsibility as conscious communication agents (Levy 2007, 2014).

http://www.irafs.org - [email protected] IRAFS'18 63 4. Conclusions: metaphysical and theological implications An overcoming of the Neo-Platonic bias in metaphysics and theology

http://www.irafs.org - [email protected] IRAFS'1864 The Aristotelian causal eduction of forms from matter vs. the Platonic methexis (participation) of forms

▪ Main thesis: the passage from a set-theoretic to an arrow-theoretic interpretation of logic and mathematics means a passage from a neo-Platonic to a neo-Aristotelian natural ontology and natural theology. ▪ Indeed, in the Aristotelian natural philosophy – against Democitus mechanism – there exists no mechanical vacuum, so that the natural forms of bodies are «educed» from matter, constituted by the chaotic motion of elements and fulfilling everything in the universe, by the action of physical causes.  Elimination of the main reason making Christian and Islamic metaphysics and theologies opposed to an evolutive cosmology and biology. ▪ “All the corporeal forms are, indeed, caused into matter not like if they would have inserted by an immaterial form, but from a matter reduced by the potency to act by some physical agent” (Aquinas, Summa Theologiae I,65,4).

▪ The definition of the potential infinity of the “primary matter” – that Aristotle denoted not as a static próte yle, but as a próte dynamis (= primary dynamism) – is indeed of “being finite and yet always different” (Phys. III, 206a, 34), i.e., without any periodicity (= Pythagorean form), like it happens in irrational numbers just discovered in Greek mathematics at that time that determined the crisis of the Pythagorean mathematics. ▪ For illustrating this notion of the always changing nature of the matter potentiality Aristotle uses the effective metaphors of the “day course” and of the “run contest in the stadium”. That is, the “being-in-potency of the primary dynamism” is like “a day is or a run contest is, because they always are becoming something different” (Phys. III, 206a, 23).

http://www.irafs.org - [email protected] IRAFS'18 65 «Cause» as a relation of «action» and «passion»

▪ “Being in potency”, and then “potentially existing” of matter therefore means “being always changing” without any periodicity or “form”. ▪ For the same reason, the matter is in potency to infinite forms just as a random infinite sequence is in potency to infinite periodicities, which can be “cut out” from it, by inducing in it some regularity. ▪ This is, ultimately, for Aristotle the effect of the acting causality by light and heat from the ordered motions of the heavenly bodies onto the earthly disordered motions of matter elements (= “migma”) and of their active- passive qualities (=elementary forces) by which they can interact to constitute bodies, but also they can transform themselves into each other ( matter dynamism is eternal not matter elements like for mechanism)  the primary dynamism is in potency also as to elements like the QV as to fermions and bosons. ▪ For this reason Aristotle defined the material forms as acts – effects of the acting causality onto the matter potency – for which he invented the term entelechy, literally “what has in itself its term”, using the slogan “etelechy chorizei”, literally “the act cuts”. Therefore, “being in act” or “actually existing” means “being unchanged in an overall becoming” or “being identical over time”, which is precisely the main character of a material substance (hypostasis) according to Aristotle.

http://www.irafs.org - [email protected] IRAFS'18 66 Natural form as entelechy of the primary dynamism of matter

▪ Another significant Aristotle’s denotation of a bodily form is as peras, literally the «house door-sill» determining the in-out boundary between the prote dynamis of matter inside and outside a body. It is therefore an entelechy: “the end (telos) of the material elements unceasing motion, and that- because-of-which (tò ou éneka) such an end is reached.” (Aristotle, Phys. II, 2, 194a, 27-29)

▪ E.g., following Aristotle’s famous example of the spherical shape (form) of a drop of water as far as composed by droplets, this form is determined for Aristotle by the regular motion of the constituting droplets on the drop surface. This motion therefore constitutes a dynamic regular boundary or “peras” between the chaotic motions of the droplets inside the drop, and the chaotic motions of the particles of air outside the drop, i.e., it is a (dynamic) form “educed” from a (dynamic) matter.

▪ Indeed, if the boundary conditions change (= outside causality, i.e., the heat quantity or temperature), this drop-form is destroyed, and other one(s) will appear. For instance, in the case of water, if we raise the temperature, we will have many singular droplets of water (vapor state) instead of few drops, but if, afterwards, we lower temperature, we have the condensation of other few drops, surely different from the original ones. The outer causality is the same, but the way it is matched by the inner causality of the material substrate motions determines a new “in-out sill” or “dynamic form”, that is, new individuals.

http://www.irafs.org - [email protected] IRAFS'18 67 Individual form as ultimate difference on which the existence of individual things and of their species depends

▪ Metaphysically, for Aquinas this notion of form as a “dynamical sill” between the causal in-out dynamic interplay, “splitting” the material substrate into the dynamically opposed pair “individual-environment”, explains how the “ultimate difference” or hecceitas of each individual as to the species to which it belongs, depends on this notion of form as “sill”, in Latin language denoted often as limen or “limit”. Indeed, such “limits” are for Aquinas “like as many individual differences for which the existence (esse simpliciter) of each individual thing is said as many times as the Dynamic (causal) explanation of the existence (of the differences are” (Aquinas, In Met., VIII, ii, forms) of an individual cat (Tom), of its species of 1694). cats (white surface), of its genus of felins (barred surface), by a layered causal in-out dynamism (see the notion of «QV foliation» in QFT). http://www.irafs.org - [email protected] IRAFS'18 68 The solution of the theological issue of the “creation from nothingness” compatible with an evolutive cosmology

▪ The composite character (relation+act+potency) of the notion of cause (it is not a predicative category like for Kant)  eduction of forms from matter by physical acting causality  physical justification of the essence (nature) of each physical entity as a dynamic synolon (“composite totality”) of matter + form. ▪  Biblical God cannot be the religious counterpart of Plato’s Demiurge who inserts from the “outside” of the material reality immaterial forms into the matter: the physical causality is sufficient for this  God creator is not the Demiurge! ▪  Aquinas’ solution of the issue by interpreting in the context of the Aristotelian theory the first two verses of the Genesis book: ▪ (1) In the beginning, God created heaven and earth. (2) Now the earth was a formless void, there was darkness over the deep, with a divine breath sweeping over the waters (Gen I). ▪ Compatibility of the “formless void” endowed with the inner dynamism of the “waving waters”, with the eternal prote dynamis of Aristotle: the formless void fulfilled with waving waters is not the mechanical vacuum of Democritus and Newton (physical counterpart of the metaphysical nothingness). ▪ The Biblical In Principle is a metaphysical not temporal and then physical notion: the creation of matter (not form) is from outside time and then including time as “measure of the material becoming”  time starts in Genesis with the progressive separations (light-darkness, lower-upper waters by the heavenly vault, sea-earth separation, etc.) from the inside the formless void, where each separation determines one of the six days of creation. ▪ Like for modern cosmology, time is inside not outside the universe  space-time can be limitless and even finite like for GR. http://www.irafs.org - [email protected] IRAFS'18 69 Creatio ex nihilo, prote dynamis, and quantum vacuum

▪ Therefore, they are completely missing the point those physicists, also eminent like S. Hawking (Hawking & Mlodinow 2010), or like L. Krauss (2012) when they pretend to explain the creatio ex nihilo of theology using the infinitely many “spontaneous symmetry breakings” of the QV. They are confuting the existence of the Neo-Platonic God-Demiurge, not of the Biblical God, of Whom the same existence of the dynamic material substrate of everything ultimately depends. ▪ The QV is not “nothingness”, just it is not “nothingness” ▪ Both the próte dynamis of Aristotle from which all the different natural things derive, as far as their natural forms are progressively educed from matter by a purely physical causality; ▪ And the “formless void” fulfilled with “waving waters”, and from which everything was created by successive separations of the Genesis Book. ▪ For this reason, Aquinas said that creation is independent of time. Therefore the universe – not in its ordered form as a cosmos, but in its material substrate – could be also eternal and nevertheless created. In fact, what is pure potency (matter) supposes necessarily what is Pure Act (God) even though from ever. ▪ “We have to say that the fact that the world has not always existed is something that cannot be proven in a demonstrative form[…]. On the other hand, the world leads to the knowledge of the creating divine power, both in the case it has not always existed, and in the case it has: everything that has not always existed was clearly caused, even though it is not so immediate that it holds also for what has always been” (Aquinas, Summa Theologiae, I, 46, 2 and 1ad 7).

http://www.stoqatpul.org - [email protected] SITA 2017 70 An arrow-theoretic notion of creation as «participation of being» (not of forms)

▪ «What is created comes to existence without any becoming or change, because every becoming and every change presuppose that something existed beforehand. Accordingly, when creating, God produces things without any change. […] If we eliminate the ‘becoming’ from an action, the relation alone will remain in place.[…] Therefore, creation in the creature is simply a certain relation to the creator, as the ultimate principle of Schematic representation of creation as «participation its being» (Aquinas, Summa of being» (3D arrows) that makes existing (a 2D Theologiae, I, 45, 3c). representation of) the curved space-time of the universe, where the existence of material things (gray circles) composing the universe are educed by physical causality (small arrows) from matter (dotted background of the space-time). http://www.irafs.org - [email protected] IRAFS'18 71 An arrow-theoretic notion of spirit in humans

▪ Of course, there exists in humans as to animals a personal relationship with the Absolute from ever and for ever, as ultimate metaphysical foundation of their existence as persons, and then of their personal responsibility in ethics (and not only of their socio- biological responsiveness as communication agents), as well as of the universal convinction of personal immortality in religions. ▪ «…Then the Lord God formed the man of dust from the ground and breathed into his nostrils the breath of life, and the man became a living creature» (Gen 2,7).

http://www.irafs.org - [email protected] IRAFS'18 72