In this paper we consider the possibility of a Quantum Molinism : such a view applies an analogue of the Molinistic account of free will‘s compatibility with God’s foreknowledge to God’s knowledge of (supposedly) indeterministic events at a quantum level. W e ask how (and why) a providential God could care for and know about a world with this kind of indeterminacy. We consider various formulations of such a Quantum Molinism, and after rejecting a number of options arrive at one (...) seemingly coherent formulation. (shrink)

This open access book addresses the physical phenomenon of events that seem to occur spontaneously and without any known cause. These are to be contrasted with events that happen in a determined, predictable, lawful, and causal way. All our knowledge is based on self-reflexive theorizing, as well as on operational means of empirical perception. Some of the questions that arise are the following: are these limitations reflected by our models? Under what circumstances does chance kick in? Is chance in physics (...) merely epistemic? In other words, do we simply not know enough, or use too crude levels of description for our predictions? Or are certain events "truly", that is, irreducibly, random? The book tries to answer some of these questions by introducing intrinsic, embedded observers and provable unknowns; that is, observables and procedures which are certified to be unknowable or undoable. A review of quantum mechanics is presented which is inspired by quantum logic. Postulated quantum knowables are reviewed. More exotic unknowns originate in the assumption of classical continua, and in finite automata and generalized urn models, which mimic complementarity and yet maintain value definiteness. Traditional conceptions of free will, miracles and dualistic interfaces are based on gaps in an otherwise deterministic universe. (shrink)

We study some aspects of the emergence of _lógos_ from _xáos_ on a basal model of the universe using methods and techniques from algorithmic information and Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and spurious, emerging laws surfaces. The spurious, emergent laws abound, they can be found almost everywhere. In accord with the ancient Greek theogony one could say that _lógos_, the Gods and the laws of the universe, originate from “the void,„ or from _xáos_, a picture (...) which supports the unresolvable/irreducible lawless hypothesis. The analysis presented in this paper suggests that the “laws„ discovered in science correspond merely to syntactical correlations, are local and not universal. (shrink)

The transition from unitary, reversible von Neumann-Everett quantum processes to non-unitary, irreversible processes and measurements is explored through infinite tensor products interpreted as nested, chained, or iterated Wigner’s friend scenarios. Infinite tensor products can disrupt unitary equivalence through sectorization and factorization, drawing parallels to concepts from real analysis, recursive mathematics, and statistical physics.

We construct peculiar Hilbert spaces from counterexamples to the axiom of choice. We identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with such spaces. Here a self adjoint operator is intrinsically effective if and only if the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.

The Greenberger–Horne–Zeilinger argument against noncontextual local hidden variables is recast in quantum logical terms of fundamental propositions, states and probabilities. Unlike Kochen–Specker- and Hardy-like configurations, this operator based argument proceeds within four nonintertwining contexts. The nonclassical performance of the GHZ argument is due to the choice or filtering of observables with respect to a particular state. We study the varieties of GHZ games one could play in these four contexts, depending on the chosen state of the GHZ basis.

Do the partial order and ortholattice operations of a quantum logic correspond to the logical implication and connectives of classical logic? Rephrased, How far might a classical understanding of quantum mechanics be, in principle, possible? A celebrated result of Kochen and Specker answers the above question in the negative. However, this answer is just one among various possible ones, not all negative. It is our aim to discuss the above question in terms of mappings of quantum worlds into classical ones, (...) more specifically, in terms of embeddings of quantum logics into classical logics; depending upon the type of restrictions imposed on embeddings, the question may get negative or positive answers. (shrink)

This Chapter enumerates a 200 of recursion theoretic undecidabilities not covered before.

Rather than giving a detailed account on the origin and varieties of classical determinism – which is a fascinating topic of its own a very brief sketch of some of its concepts will be given.

As stated by Milonni [370, p. xiii] and others [174, 195], “…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ldots $$\end{document} there is no vacuum in the ordinary sense of tranquil nothingness. There is instead a fluctuating quantum vacuum.” One of the observable vacuum effects is the spontaneous emission of radiation [565]: “…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ldots $$\end{document} the process of spontaneous emission of radiation is one in which “particles” are actually created. Before the (...) event, it consists of an excited atom, whereas after the event, it consists of an atom in a state of lower energy, plus a photon.”. (shrink)

Empirical evidence can solely be drawn from operational procedures accessible to embedded observers. Embeddedness means that intrinsic observers have to somehow inspect and thus interact with the object, thereby altering both the observer as well as the object inspected.

The physical theories of classical mechanics, electrodynamics and gravitation (relativity theory) have been developed alongside classical analysis. Thereby assumptions about the formal mathematical models for theoretical physics had to be made which were partly (to some degree of accuracy) corroborated empirically; and partly due to mere convenience.

The vision that self-reflexivity may, through self-intervention amounting to paradoxical situations, impose some restrictions on the performance and the capacity of physical agents to know their own states, is a challenging one. It has continued to present a source of inspiration.

According to his own narrative, and totally unaware of Saint Augustine of Hippo’s as well as Nicholas of Cusa’s (aka Nicolaus Cusanus’) notion of learned ignorance (Latin: docta ignorantia), the learned ignorance Münchhausen pulled himself (and his horse) out of a mire by his own hair (Bürger, Münchhausen. Wunderbare Reisen zu Wasser und zu Lande, 1789, [88, Chap. 4]). (This story is not contained in Raspe’s earlier collections Raspe, The Travels and the Surprising Adventures of Baron Munchausen, 1877, [427].) In (...) the following we shall be concerned with the question exactly why it is entirely hopeless to pursue the strategy suggested by the Baron Münchhausen; and why should one be concerned about this. More generally, is it (im)plausible to attempt to reach out into some external domain with purely intrinsic means; that is, by operational (from the point of view of intrinsic, embedded observers) capacities and means which cannot include any “extrinsic handle,” or Archimedean point? (shrink)

Classical physics, in particular, classical Newtonian mechanics, can be perceived as being modelled by systems of simultaneous differential equations of second order, for which the initial values of the variables and their derivatives are known. It slowly dawned on the mathematical physicists that the solutions, even if they satisfied Lipschitz continuity and thus were unique, could have a huge variety of solutions; with huge structural differences. Some of these solutions turned out to be unstable.

This chapter introduces some important epistemology. Without epistemology any inroad into the subject of (un)decidability and (in)determinism may result in confusion and incomprehensibility.

At the moment, there exists a loosely bundled canon of quantum rules subsumed under the term quantum mechanics or quantum theory.

Since theological nomenclature hardly belongs to the standard repertoire of physicists but will be used later, as some termini technici will be mentioned upfront. Thereby we will mainly follow Philipp Frank’s (informal) definitions of gaps and miracles.

In what follows the term transcendence refers to an entity or agent beyond all physical laws. (It is not used in the Kantian sense.) In contrast, immanence refers to all operational, intrinsic physical means available to embedded observers.

Induction is the inference of general rules “causing” and “generating” (in an algorithmic interpretation) physical behaviours from these very behaviours (without any extra assumptions) alone. Thus induction is “bottom up:” given the phenomena and how observers perceive them operationally, these observers somehow obtain the causes and rules which supposedly underlie these phenomena. Thereby we shall restrict ourselves to algorithmic methods of induction; others, such as intuition, guesses or oracles, or means other than intrinsic.

In what follows we shall give a very brief account of reversible evolution and, in particular, reversible computation by permutation.

Egon von Schweidler, a colleague of Exner at the University of Vienna, interpreted Rutherford’s (1902) decay law as merely probabilistically – thereby allowing deviations for small sample sizes.

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.

One of the biggest dangers in presenting quantum unknowns is sophism; a wasteful exercise in fruitless scholasticism and mysticism.

While – depending on one’s subjective optimism or pessimism often, sometimes or rarely – it is possible to predict the future, certain forecasting tasks, in particular, when it comes to self-reference, are provable unattainable, and will remain so forever.

Complementarity was first encountered in quantum mechanics. In what follows we shall present finite deterministic models featuring complementarity. The type of complementarity discussed in this chapter grew out of an attempt to understand quantum complementarity by some finite, deterministic, quasi-classical (automaton) model.

The following speculations resemble Darwin’s and also Turing’s “inversion of reason” – that is, “competence without comprehension” – forcefully put forward by the atheistic philosopher Daniel Dennett in his phrase “delere Auctorem Rerum Ut Universum Infinitum Noscas; aka DARW(=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document}UU)IN: destroy the author of things in order to know the universe”.

Penrose [40] has discussed a new point of view concerning the nature of physics that might underline conscious thought processes. He has argued that it might be the case that some physical laws are not computable, i.e. they cannot be properly simulated by computer; such laws can most probably arise on the “no-man’s-land” between classical and quantum physics. Furthermore, conscious thinking is a non-algorithmic activity. He is opposing both strong AI, and Searle’s [47] contrary viewpoint mathematical “laws”).

This chapter is for those who neither want to bother with details nor have time for the expositions and the explanatory rants of previous parts; or as a teaser to go deeper into the subject and to want to know more.

Having explored the limits and the “negative” effects of the type of self-exploration and self-examination embedded observers are bound to we shall now examine the “positive” side of self-description. In particular, we shall prove that, at least for “nontrivial” deterministic systems (in the sense of recursion theory and, by the Church-Turing thesis, capable of universal computation), it is possible to represent a complete theory or “blueprint” of itself within these very systems.

Some of the many accounts and many roots of quantum indeterminism are discussed.

In an extension of speculations that physical space–time is a fractal which might itself be embedded in a high-dimensional continuum, it is hypothesized to “compensate” for local variations of the fractal dimension by instead varying the metric in such as way that the intrinsic dimensionality remains an integer. Thereby, an extrinsic fractal continuum is intrinsically perceived as a classical continuum. Conversely, it is suggested that any variation of the metric from its Euclidean form can be “shifted” to nontrivial fractal topology. (...) Thereby “holes” or “gaps” in spacetime could give rise to curvature. (shrink)