Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin–Löf. We extend this result and demonstrate that QM is algorithmic $$\omega$$ -random and generic, precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo–Fraenkel Solovay random on (...) infinite-dimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M, where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real $$r\in 2^{\omega }$$, and the model undergoes random forcing extensions M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent. (shrink)

Science and Convention: Essays on Henri Poincare's Philosophy of Science and The Conventionalist Tradition contains essays concerned with Henri Poincare's philosophy of science, physics in particular, and with the conventionalist tradition in philosophy that he revived and reshaped, simultaneously with, but independently of, Pierre Duhem. Separating five essays as chapters, the book discusses the main ideas of the philosophy (Essays 1 and 5), traces at least some of its historical background (Essays 1, 2, and 3), and provides some of its (...) developments (Essays 2 and 4). (shrink)

We give an almost explicit presentation of exotic functions corresponding to some exotic smooth structure on topologically trivial R4. The construction relies on the model-theoretic tools from the previous paper. We can formulate unexpected, yet direct connection between ‘‘localized’’ exotic small R4’s and some noncommutative spaces. The formalism of QM can be interpreted in terms of exotic smooth R4’s localized in spacetime. A new way of looking at the problem of decoherence is suggested. The 4-dimensional spacetime itself has built-in means (...) which may enforce a kind of decoherence. (shrink)

A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable (...) Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ℝ4. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory. (shrink)

We consider consciousness attributed to systems in space-time which can be purely physical without biological background and focus on the mathematical understanding of the phenomenon. It is shown that the set theory based on sets in the foundations of mathematics, when switched to set theory based on ZFC models, is a very promising mathematical tool in explaining the brain/mind complex and the emergence of consciousness in natural and artificial systems. We formalise consciousness-supporting systems in physical space-time, but this is localised (...) in open domains of spatial regions and the result of this process is a family of different ZFC models. Random forcing, as in set theory, corresponds precisely to the random influence on the system of external stimuli, and the principles of reflection of set theory explain the conscious internal reaction of the system. We also develop the conscious Turing machines which have their external ZFC environment and the dynamics is encoded in the random forcing changing models of ZFC in which Turing machines with oracles are formulated. The construction is applied to cooperating families of conscious agents which, due to the reflection principle, can be reduced to the implementation of certain concurrent games with different levels of self-reflection. (shrink)

The present note is intended to draw the reader's attention to the analysis of the relation between Newtonian Mechanics and Special Relativity Mechanics (henceforth to be referred to as NM and SRM), given by Philipp Frank, one of the classics of Logical Empiricism (in Frank [1938]). Frank's analysis of the relation between NM and SRM is interesting in many ways. Firstly, it shows clearly that problems of disruptive changes and of conceptual disparity were known to and discussed by Logical Empiricists (...) and that, therefore, some of them at least were not at all blinded to 'dynamic' problems of changes in actual science by their view of physical theories as interpreted axiomatic systems and did not deal with those theories as 'static, frozen in logical mold'. Secondly, it reveals some of the reasons for the Logical Empiricist rejection of what came to be known later as the incommensurability claim. (shrink)

We show that in 4-spacetime modified at very short distances due to the weakening of classical logic, the higher dimensions emerge. We analyse the case of some smooth topoi, and the case of some class of pointless topoi. The pointless topoi raise the dimensionality due to the forcing adding “string” objects and thus replacing classical points in spacetime. Turning to strings would be something fundamental and connected with set theoretical forcing. The field theory/strings dualities originate at the set theoretical level (...) of the theories. It is argued that this fundamental level can help solving some difficulties of the physical dualities. (shrink)

We build a topological model, based on intuitionistic logic, for multi-agent biological systems (such as _Physarum polycephalum_, bacterial colonies or any other swarm), reacting to external nourishment stimuli. Our construction follows the topological description of brain activity, where particles (neurons) are activated by an external environment, represented by a topological space _X_ with an open cover \(\{U_i:i\in I\}\). The brain builds the model of this external space via the nerve (trace) of a topological space _X_. Here the body of _Physarum (...) polycephalum_ or a swarm made of networks of tubular structures represents a nerve (trace) of _X_ also which means that _Physarum polycephalum_ or a swarm gains orientation in the space of external stimuli even in the absence of any neural system. The logic of living organisms is based on open subsets of _X_ and thus can be represented by Heyting algebra (i.e. intuitionistically). We also consider the generalisation of the nerve construction to a categorical context, where the category is determined by the network structures of multi-agent biological system. This model can be generalised up to simulating the behaviour of any swarm by means of intuitionistic logic. (shrink)

Recently the cosmological evolution of the universe has been considered where 3-dimensional spatial topology undergone drastic changes. The process can explain, among others, the observed smallness of the neutrino masses and the speed of inflation. However, the entire evolution is perfectly smooth from 4-dimensional point of view. Thus the raison d’être for such topology changes is the existence of certain non-standard 4-smoothness on R4 already at very early stages of the universe. We show that the existence of such smoothness can (...) be understood as a byproduct of the quantumness of the origins of the universe. Our analysis is based on certain formal aspects of the quantum mechanical lattice of projections of infinite dimensional Hilbert spaces where formalization reaches the level of models of axiomatic set theory. (shrink)

he last 2000 years have seen a change in the prices of silver, grain, land as well as a change in living costs, the salary of hired labourers and police informers. The relationships between those figures have also changed and the monetary systems are quite different nowadays. Thus, it is difficult to answer an apparently simple question: how much is Judas’s 30 pieces of silver? An equivalent of 30 pieces of silver could be 95 zlotys, but also 46 500 zlotys (...) (i.e. about $28 and $13 700 respectively). However, it is obvious that this amount of money is dramatically little in comparison with the scale of betrayal of God and no sum of money would be adequate. But again: even if Judas had lacked faith in Jesus, the sum of money was excessively small for betraying a good and innocent man by one who was chosen by that man from the others and in whom his companions placed so much confidence. Even if Judas had realized that, it was unfortunately too late (Mt. 27: 1–5). (shrink)

Quine's reasons for recommending naturalist epistemology are: (1) knowledge, Mind and meaning are part of the world they have to do with, (2) since the cartesian quest for certainty and reductionism of carnap's 'aufbau' type have failed, Rational reconstruction has no more any advantage over psychology, (3) since phenomenalist validation of science is no longer our concern, It is not circular to appeal to psychology. Against this it is argued that (a) no definite methodological policy can be based on (1) (...) alone; (b) certainty and phenomenalist reduction are not the only purposes of rational reconstruction; (c) it would be circular to appeal to psychology in developing a theory of rational method of acquiring knowledge: (d) logic and psychology of science are complementary not mutually exclusive. (shrink)

“Discrimination on the grounds of disability is seldom malicious, but stems more from a lack of understanding.” A disabled businessman explores the need for businesses to cultivate and implement greater disability awareness. After graduating in engineering, he gained his MBA from London Business School in 1992. He now works as a consultant, capitalising on his background in business management and his personal experience of disability to assist clients in developing anti‐discriminatory policies and practice.

Chciałem Go pożegnać. Kilkakrotnie próbowałem napisać choć parę słów. O Nim samym. Okazało się, że nie potrafię. Udało mi się tylko przedstawić Jego poglądy na reizm Tadeusza Kotarbińskiego, ale uczyniłem to gdzie indziej. Muszę tu poprzestać na ofiarowaniu Mu tego, co napisałem na Jego dziewięćdziesiąte urodziny, w osiemdziesięciolecie naszej przyjaźni.

Jerzy Perzanowski s ideas were based on an original blend of logic and ontology in what he called onto/logic, where the slash is meant to suggest a quotient of ontology by logic. Perzanowski began as a logician, his early works being on modal logic, then gradually shifted his interest to logical philosophy, meaning not so much philosophy of logic as philosophy informed by logic. Perzanowski was a rare breed of analytical philosopher who thought that a philosophical theory of everything (...) was worthwhile. In this systematic spirit, he began with method. He presented his method of total analysis and synthesis quite simply: reduce the object of research to its simplest possible constituents, and then combine them in some way. Better still, combine them in every possible way, thereby producing a space of possibilities analogous to (and in certain cases identical with) the logical space. Thus, analysis and synthesis differ from a trivial disassembly and reassembly.". (shrink)

If It isn’t necessary, its non-existence is.

The following topics are discussed, from psychoanalytic, philosophical, and empirical perspectives: -/- *Sociopathy *Pedantry *The nature of bureaucrats *The nature of bureaucratic institutions *Rationalization and Repression *The relationship between ignorance and mental health *The relationship sapience and mental illness *The relationship between ignorance and the ability to act *The relationship between hyper-sapience and the inability to act. *The psychological underpinnings of addiction; and finally *The differences between men and women.

Resumo Este artigo analisa, as recentes versões, da objecção à existência de um fluxo do tempo, com base na pergunta: “Com que velocidade flui o tempo?”. O autor mostra que as soluções existentes para o problema não são plausíveis e, que em vez disso, a resolução deve ser encontrada numa nova concepção de fluxo temporal, que evite tais dificuldades. A teoria metafísica proposta desenvolve as ideias de Broad e Prior sob um novo enquadramento, que invalida a objecção resultante da questão (...) sobre o ritmo da passagem do tempo e das respostas “segundo a segundo”. Neste texto também se procura mostrar, que esta teoria metafísica, aqui proposta, possui outras virtudes que testemunham a seu favor. Palavras-chave : existência dinâmica, fluxo do tempo, presentismo, velocidade de passagem do tempoThe paper analyzes some recent versions of the objection to the existence of the flow of time based on the question: “How fast does time flow?”. It shows that the replies given to the problem are implausible and instead proposes a solution for the issue in the form of a conception of the flow of time which can avoid such a difficulty. The proposed metaphysical theory develops the ideas of Broad and Prior in a radical way and invalidates the objection resulting from the question about the rate of passage and answers “second per second”. The paper also tries to show that this metaphysical theory has other virtues which testify in favor of it. Keywords : flow of time; rate of time’s passage; presentism; dynamic existence. (shrink)

A brief but rigorous description of the logical structure of mathematical truth.