Innopolis University

Syllabus of the Classes: Part I (November 1-2, 2019)

CLASS 9-13: Formal philosophy: The ancient age, language & realism

• The notion of formal philosophy as formalization of philosophical doctrines using the axiomatic method, as a formal tool of interdisciplinary dialogue between human and mathematical sciences – computer science and AI before all.
• It is based on the distinction between standard mathematical logic (extensional interpretation of predication as membership) and philosophical logic (intensional interpretation(s) of predication in different contexts). The philosophical logic is based on the axiomatization of modal logical calculus, of which different intensional logics are as many semantics (ontic, epistemic, deontic) of the same modal calculus.
• Exemplifying applications to the classical ontologies of the Platonic logical realism and of the Aristotelian natural realism
• Refs.: 7. 9. 13. 15. 16.

CLASS 14-15: Formal philosophy: The modern age, language & cognitivism

• The Galilei affair: apodictic vs. hypothetical method in modern Galilean science
• Descartes’ first development of analytic (algebraic) geometry and the supposed apodictic value of mathematical sciences
• This is based on self-consciousness as cognitivist foundation of self-identity of a logical tautology, extended by Newton to the self-evident character of the three laws of Newtonian mechanics (hypotheses non fingo).
• This is made explicit by Leibniz’s distinction between analytic and synthetic judgements, as well as – following Newton  – by its empiricist counterpart by Hume, Locke and Berkeley, and finally, by the Kantian theory of the synthetic a-apriori judgements about pure mathematics and physics.
• Refs.: 3. (ch. 1), and 10.

CLASS 16: Formal philosophy: The post-modern age, language & naturalism

• The birth of the hypothetical-deductive method because of the discovery of the non-Euclidean geometries and their axiomatization by Riemann’s completion of Descartes’ initial algebrization of geometry.
• The abandon of the belief in the apodictic character of mathematics determined, on the one side, the abandon of trusting the cognitivist principle of evidence in epistemology, and on the other side, the necessity of demonstrating the consistency of mathematics, and specifically of calculus, by a proper metalanguage individuated by Weierstrass and Cantor in the set theory.
• Birth of the mathematical logic by Frege’s notion of propositional function, and the discovery of the logical antinomies leading to the development of several axiomatic set theories such as ZF, all characterized by the Skolem paradox (their axioms are expressed in first-order logic, but their semantics must be necessarily of a higher order).
• All this determined in the philosophy of science the so-called “linguistic turn” by Wittengstein’s and Carnap’s logical atomism, with the birth of the neo-positivistic school and its criticism by Popper’s “evolutionary approach” on biological and then informational basis to epistemology. Even though such a linguistic turn from modern cognitivism, with its Popperian untenable irrational outcomes, is incomplete until its completion by the semiotic (algebraic) approach.
• Refs.: 3. (chs. 3-4) 10. 17.

CLASS 17: Beyond cognitivism I: the Peirce lesson of semiotics

• The algebraic origins of modern mathematical logic in Boole’s and Schröder’s work and the convergent criticism to the purely syntactic formalism of Schröder’s approach by Husserl – from the cognitivist standpoint – and by Peirce from an algebraic standpoint, vindicating the triadic character of every signifying algebraic structure underlying any predication in logic, constituting the core of his interpretation of logic as “formal semiotics”.
• This led Peirce to the development of his famous ante-predicative theory of algebraic categories (firstness-secondness-thirdness) as underlying any predicative theory of categories in logic, as well as to the development of his pioneering algebraic theory of mathematical logic that was contemporary to Frege’s mathematic logic based on his logic of classes.
• This invention by Peirce of the algebra of relations remained however not valorized till Tarski’s axiomatization of the logic of relations into a calculus of relations during the second half of XX cent., including the axiomatization also of the algebraic notion of category, underlying the actual development of the Category Theory as metalanguage of logic and mathematics in many senses wider than standard set theory.
• Refs.: 5.

Syllabus of the Classes: Part II (November 8-9, 2019)

CLASS 18: Beyond cognitivism II: the biological language of biosemiotics

• Since the pioneering N. Wiener’s extension of cybernetics as theory of communication and control from machines to biological systems the notion of “signifying signaling” among the sub-systems constituting a complex living system plays an essential role in modern theoretical and applied biology with the connected notion and measure of information. Moreover, the intrinsic triadic structure of any information flow both in artificial and biological systems opens the way to the biosemiotics approach in biology.
• The relevance of a suitable theory of information in biological sciences is today emphasized by the so-called epigenetics approach to the biological sciences, where an effective chemical signaling among the different constituents of cells, tissues, organs, …, and with their inner-outer chemical environments plays an essential role
• Therefore, such a dissipative or “open” character of any biological system, as well as the intrinsic semantic character of information exchanges in biological systems that both in philogenesis and in ontogenesis make metaphoric if not misleading the usage of communication engineering notions such as “code”, “program” and similar, as well as the usage of Shannon’s notion and measure of information that is intrinsically syntactic (it supposes a code or program by some programmer) and never semantic, as Shannon himself emphasized many times.
• Similarly, such a purely statistical and not dynamical approach to the information measures in biological systems if it is certainly congruent with the usage of the statistical “diffusive processes” to model the biological morphogenesis over the microscope plates in a biological lab, it is surely ineffective to model morphogenesis in real biological environments, where dynamic long-range correlations among resonant molecules are necessary for making the morphogenetic processes really effective, because highly selective like a purely stochastic diffusive process can never be in principle.
• All this requires an attentive development of the physical foundations of the notion of information in biological systems that can come only from quantum physics, and more precisely from quantum field theory, as fundamental theory of condensed matter physics, the biological matter included.
• Refs.: 2. 4. 11. 20. 21.

CLASS 19: Physical foundations of information I: Quantum mechanics

• We present here in a very elementary way some basic notions of the formalism of QM in the framework of statistical mechanics, and then related with the purely statistical nature of Schrödinger wave function and of its coherence/decoherence, which has not to be confused with the dynamical wave functions of oscillating and interacting physical fields, despite the fundamental formal tool of Fourier Transform applies well to both cases, often generating confusion between the statistical and the dynamic case.
• We therefore emphasize that such a formalism can study only isolated quantum systems, and then in which sense it cannot deal in principle with system phase transitions, and finally with non-equilibrium phenomena, if not in the very limited case of the near-to-equilibrium states.
• We show that such an approach is congruent with an observer-related informational approach (“information for whom?”) to QM, and then with the Shannon measure of information, and finally with the notion of Quantum Universal Turing Machine.
• Refs.: 3. (ch. 2) 6. 19.

CLASS 20-21: Physical foundations of information II: Quantum field theory

• We present here in a very elementary way some basic notions of the formalism of QFT, in a different framework of the classical interpretation of QFT as a “second quantization as to QM”, but in the framework of “many body physics”. This complementary way of approaching QFT allows to model quantum dissipative systems in far-from-equilibrium conditions. This a modeling is consistent with the Third Principle of Thermodynamics, allowing to give a thermodynamic interpretation of the Quantum Vacuum (QV) unavoidable fluctuations at the ground state |0>, i.e. at the minimum of energy, however at a temperature >0°K, because of the Third Principle. This means that in QFT the fundamental physical object is not the “particle” like in quantum and classical mechanics, but the “field”, of which particles (both bosons and fermions) are their quanta or oscillating “wave-packets”.
• We present the main notions of QFT, allowing quantum physics to deal with “open” quantum systems in far-from-equilibrium conditions, because passing through different phases, corresponding to as many “spontaneous symmetry breakings” (SSBs) of the overall quantum field dynamics at its ground state (QV condition), and corresponding to as many long-range correlations (quantum entanglement) or phase coherence domains among the quantum fields (Goldstone Theorem) at their ground state.
• The dissipative QFT is therefore the fundamental physics of condensed-matter physics giving a natural microscopic explanation of macroscopic phenomena such as the phase transitions between liquid and solid phases in solid-state physics, the ferromagnetic phase in some metals, the hot superconducting phase in some ceramic materials, or, finally the morphogenesis in living matter. All these phenomena occur in far-from-equilibrium conditions, satisfying anyway an energy balance system-thermal bath (minimum of free energy), or ground state of the balanced system, compatible anyway with several ordered states of condensed matter.
• Since the minimum free-energy function acts here as a “dynamic”, observer-independent, selection criterion among admissible states, according to the mathematical principle of the “doubling of the degrees of freedom” system/thermal-bath, the relative notion and measure of information is here semantic, just as it is semantic the “doubled qubit” of quantum computations implemented in such a quantum architecture.
• This picture is completed by the possibility of modeling – by the so-called Bogoliubov Transform, mapping a condensate of photons into another one – a quantum optical system as a “labeled state-(phase)-transition-system”, i.e., as an automaton performing quantum computations at room temperature. Indeed, it is computing dynamically, like living brains as we see, through phase coherences (functions) of electromagnetic waves, and not through coherent phases of a statistical wave functions, like in the classical QM implementations of quantum computers requiring of working at temperatures close to -273°C =0°K for not suffering decoherences.
• Refs.: 4. 6. 7. 8. 11. 12

CLASS 22: Quantum brain: QM vs. QFT modeling of the quantum brain

• Generally, the quantum approach to cognitive neurosciences has its background in the obvious evidence that it is possible to find in quantum entanglement the physical basis of consciousness. The distinction between the two main quantum approaches to cognitive neurosciences – Penrose’s and Freeman’s one – depends primarily on the ontological role that we want to attribute to consciousness.
• According to Penrose, the conscious intelligent act is related with an information gain that makes not algorithmic and then unpredictable the human intelligence act. The quantum candidate for this information gain is the Schrödinger  wave function decoherence by which the system choose one of the possible states at the macroscopic level (wave function reduction).
• Connection with Hameroff hypothesis that the quantum physics of neuron microtubules, as well as the quantum entanglement between two microtubules quantum states the candidates for implementing a neural qubit for brain quantum computations.
• The alternative quantum interpretation of brain dynamics is by Freeman and Vitiello based on QFT and the evidence that brain is a dissipative system.
• According to this interpretation, subject of the intentional acts is the person non her consciousness.
• This is consistent with the interpretation of the extended mind as including the person and its environment both physical and social.
• Refs.: 14. 6. 8. 12.

CLASS 23: Category Theory I: From Peirce to Category Theory

• We present here some basic notions of Category Theory (CT), showing, on the one side, its dependence on the algebraic and then semiotic approach to the foundations of logic by Ch. S. Peirce afterward evolved in a formal calculus of relations by the work of A. Tarski and his school, on the other side its relevance as metalanguage of the operator algebra modeling of physical and computational systems.
• From the philosophical standpoint, CT is relevant for the possibility it offers of a formal theory of the justification of predication in logic so to make of CT the best candidate as metalanguage also of the formal philosophy.

Refs.: 1. 5. 8.

CLASS 24: Category Theory II: Operator algebra in quantum physics and quantum computing

• We present in this class the relevance of CT for the development of a coalgebraic semantics of the Boolean logic in the framework of J. Rutten proposal of “Coalgebra as General Theory of Systems”. This means that programs can have their semantics in the physical states of the system in which they are implemented, as far as coalgebraic modeled.
• This has an immediate application for a computational interpretation of the coalgebraic modeling of quantum dissipative systems in QFT, developed separately by theoretical physicists and applied already with success in cognitive neuroscience.
• This picture is completed by the possibility of modeling – by the so-called Bogoliubov Transform, mapping a condensate of photons into another one – a quantum optical system as a “labeled state-(phase)-transition-system”, i.e., as an automaton performing quantum computations at room temperature. Indeed, it is computing dynamically, like living brains as we see, through phase coherences (functions) of electromagnetic waves, and not through coherent phases of a statistical wave functions, like in the classical QM implementations of quantum computers requiring of working at temperatures close to -273°C =0°K  for not suffering decoherences.
• Particularly we will show the applications of this approach for the development of a new architecture of quantum computer in a quantum optics implementation we are actually developing in Italy.

Refs.: 1. 5. 8.

CLASS 25: Conclusion: Debate on “An ontology for our information age”