The dichotomy between endophysical/intrinsic and exophysical/extrinsic perception concerns the question of how a model - mathematical, logical, computational - universe is perceived from inside or from outside, [71, 65, 66, 59, 60, 68, 67]. This distinction goes back in time at least to Archimedes, reported to have asked for a point outside the world from which one could move the earth. An exophysical perception is realized when the system is laid out and the experimenter peeps at the relevant features without changing the system. The information flows on a one-way road: from the system to the experimenter. An endophysical perception can be realized when the experimenter is part of the system under observation. In such a case one has a two-way informational flow; measurements and entities measured are interchangeable and any attempt to distinguish between them ends up as a convention. The general conception dominating the sciences is that the physical universe is perceivable ...

Suspicions that the world might be some sort ofa machine or algorithm existing ‘‘in the mind’ ’ ofsome symbolic number cruncher have lingered from antiquity. Although popular at times, the most radical forms of this idea never reached mainstream. Modern developments in physics and computer science have lent support to the thesis, but empirical evidence is needed before it can begin to replace our contemporary world view.

Contexts are maximal collections of co-measurable observables “bundled together ” to form a “quasi-classical mini-universe. ” Different notions of contexts are discussed for classical, quantum and generalized urn–automaton systems. PACS numbers: 02.10.-v,02.50.Cw,02.10.Ud

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.

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Proving the chaoticity of some dynamical systems is equivalent to solving the hardest problems in mathematics. Conversely, one argues that it is not unconceivable that classical physical systems may “compute the hard or even the incomputable” by measuring observables which correspond to computationally hard or even incomputable problems.

ons. They lie at the very foundations of our world conceptions. Conventions serve as a sort of "scaffolding" from which we construct our scientific worldview. Yet, they are so simple and almost self-evident that they are hardly mentioned and go unreflected. To the author, this unreflectedness and unawareness of conventionality appears to be the biggest problem related to conventions, especially if they are mistakenly considered as physical "facts" which are empirically testable. This confusion between assumption and observational, operational fact seems to be one of the biggest impediments for progressive research programs, in particular if they suggest postulates which are based on conventions different from the existing ones. In what follows we shall mainly review and discuss conventions in the two dominating theories of the 20th century: quantum mechanics and relativity theory. 2. CONVENTIONALITY OF THE CONSTANCY OF THE CHARACTERISTIC SPEED Sup

Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper-) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the self-contradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.

We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51]. That means that there does not exist any reasonable (consistent) finite formal system from which all mathematical truth is derivable. And there exists a "huge" number [11] of mathematical assertions (e.g., the continuum hypothesis, the axiom of choice) which are independent of any particular formal system. That is, they as well as their negations are compatible with the formal system. Can such formal incompleteness be translated into physics or the natural sciences in general? Is there some question about the nature of things which is provable unknowable for rational thought? Is it conceivable that the natural phenomena, even if they occur deterministically, do not allow their complete d...