Mathematical Foundations of Consciousness

We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical primitives. The Anti-foundation Axiom plays a significant role in our development, since among other of its features, its replacement for the Axiom of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic interpretations. These interpretations also depend on such allied notions for sets as pictures, graphs, decorations, labelings and various mappings that we use. A syntax and semantics of operators acting on sets is developed. Such features enable construction of a theory of non-well-founded sets that we use to frame mathematical foundations of consciousness. To do this we introduce a supplementary axiomatic system that characterizes experience and consciousness as primitives. The new axioms proceed through characterization of so- called consciousness operators. The Russell operator plays a central role and is shown to be one example of a consciousness operator. Neural networks supply striking examples of non-well-founded graphs the decorations of which generate associated sets, each with a Platonic aspect. Employing our foundations, we show how the supervening of consciousness on its neural correlates in the brain enables the framing of a theory of consciousness by applying appropriate consciousness operators to the generated sets in question

Should we discount future generations’ welfare? A survey on the “pure” discount rate debate.

In A Mathematical Theory of Saving (1928), Frank Ramsey not only laid the foundations of the fruitful optimal growth literature, but also launched a major moral debate: should we discount future generations’ well-being? While Ramsey regarded such “pure” discounting as “ethically indefensible”, several philosophers and economists have developed arguments justifying the “pure” discounting practice since the early 1960s. This essay consists of a survey of those arguments. After a brief examination of the – often implicit – treatment of future generations’ welfare by utilitarian thinkers before Ramsey’s view was expressed, later arguments of various kinds are analysed. It is argued that, under the assumption of perfect certainty regarding future human life, the “pure” discounting practice seems ethically untenable. However, once we account for the uncertainty regarding future generations’ existence, “pure” discounting seems more acceptable, even if strong criticisms still remain, especially regarding the adequateness of the expected utility theory in such a normative context. those limits would be faced by any other consequences-based ethical theory in front of Different Number Choices.

Doing and Showing

The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure

Linear superposition as a core theorem of quantum empiricism

Clarifying the nature of the quantum state $|\Psi\rangle$ is at the root of the problems with insight into (counterintuitive) quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure-quantum superposition-is based on a set-theoretic creation of ensemble formations and invokes the following three principia: $(\textsf{I})$ quantum statics, $(\textsf{II})$ doctrine of a number in the physical theory, and $(\textsf{III})$ mathematization of matching the two observations with each other; quantum invariance. All of the constructs rest upon a formalization of the minimal experimental entity: observed micro-event, detector click. This is sufficient for producing the $\mathbb C$-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the concept of time. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.Comment: No figures. 64 pages; 68 pages(+4), overall substantial improvements; 70 pages(+2), further improvement

Measurement and self-adjoint operators

The approximations of classical mechanics resulting from quantum mechanics are richer than a correspondence of classical dynamical variables with self-adjoint Hilbert space operators. Assertion that classical dynamic variables correspond to self-adjoint Hilbert space operators is disputable and sets unnatural limits on quantum mechanics. Well known examples of classical dynamical variables not associated with self-adjoint Hilbert space operators are discussed as a motivation for the realizations of quantum field theory that lack Hermitian field operators but exhibit interaction.Comment: 17 page

Pierre Duhem’s philosophy and history of science

LEITE (Fábio Rodrigo) – STOFFEL (Jean-François), Introduction (pp. 3-6). BARRA (Eduardo Salles de O.) – SANTOS (Ricardo Batista dos), Duhem’s analysis of Newtonian method and the logical priority of physics over metaphysics (pp. 7-19). BORDONI (Stefano), The French roots of Duhem’s early historiography and epistemology (pp. 20-35). CHIAPPIN (José R. N.) – LARANJEIRAS (Cássio Costa), Duhem’s critical analysis of mecha­ni­cism and his defense of a formal conception of theoretical phy­sics (pp. 36-53). GUEGUEN (Marie) – PSILLOS (Stathis), Anti-­scepticism and epistemic humility in Pierre Duhem’s philosophy of science (pp. 54-72). LISTON (Michael), Duhem : images of science, historical continuity, and the first crisis in physics (pp. 73-84). MAIOCCHI (Roberto), Duhem in pre-war Italian philos­ophy : the reasons of an absence (pp. 85-92). HERNÁNDEZ MÁRQUEZ (Víctor Manuel), Was Pierre Duhem an «esprit de finesse» ? (pp. 93-107). NEEDHAM (Paul), Was Duhem justified in not distinguishing between physical and chemical atomism ? (pp. 108-111). OLGUIN (Roberto Estrada), «Bon sens» and «noûs» (pp. 112-126). OLIVEIRA (Amelia J.), Duhem’s legacy for the change in the historiography of science : An analysis based on Kuhn’s writings (pp. 127-139). PRÍNCIPE (João), Poincaré and Duhem : Resonances in their first epistemological reflec­tions (pp. 140-156). MONDRAGON (Damián Islas), Book review of «Pierre Duhem : entre física y metafísica» (pp. 157-159). STOFFEL (Jean-François), Book review of P. Duhem : «La théorie physique : son objet, sa structure» / edit. by S. Roux (pp. 160-162). STOFFEL (Jean-François), Book review of St. Bordoni : «When historiography met epistemology» (pp. 163-165)